UNIT 4
Expressions, Equations, and Relationships Contents 6.7.A 6.7.A 6.7.A
6.7.C 6.7.A 6.7.D
6.9.A 6.10.A 6.10.A
6.9.A 6.10.A 6.10.A 6.9.B
MODULE 10
Generating Equivalent Numerical Expressions
Lesson 10.1 Lesson 10.2 Lesson 10.3
Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 Prime Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 Order of Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
MODULE 11
Generating Equivalent Algebraic Expressions
Lesson 11.1 Lesson 11.2 Lesson 11.3
Modeling Equivalent Expressions . . . . . . . . . . . . . . . . . 293 Evaluating Expressions . . . . . . . . . . . . . . . . . . . . . . . . . 301 Generating Equivalent Expressions. . . . . . . . . . . . . . . . 307
MODULE 12
Equations and Relationships
Lesson 12.1 Lesson 12.2 Lesson 12.3
Writing Equations to Represent Situations . . . . . . . . . . 321 Addition and Subtraction Equations . . . . . . . . . . . . . . . 327 Multiplication and Division Equations . . . . . . . . . . . . . 335
MODULE 13
Inequalities and Relationships
Writing Inequalities . . . . . . . . . . . . . . . . . . . . . Addition and Subtraction Inequalities . . . . . . Multiplication and Division Inequalities with Positive Numbers . . . . . . . . . . . . . . . . . . . . . . Lesson 13.4 Multiplication and Division Inequalities with Rational Numbers . . . . . . . . . . . . . . . . . . . . . . Lesson 13.1 Lesson 13.2 Lesson 13.3
MODULE 14
6.11 6.6.A 6.6.B 6.6.C
. . . . . . . 349 . . . . . . . 355 . . . . . . . 361 . . . . . . . 367
Relationships in Two Variables
Graphing on the Coordinate Plane . . . . . . . . . . . Independent and Dependent Variables in Tables and Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lesson 14.3 Writing Equations from Tables . . . . . . . . . . . . . . Lesson 14.4 Representing Algebraic Relationships in Tables and Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lesson 14.1 Lesson 14.2
. . . . . 379 . . . . . 385 . . . . . 393 . . . . . 399
UNIT 4
Unit Pacing Guide 45Minute Classes Module 10 DAY 1
DAY 2
DAY 3
DAY 4
DAY 5
Lesson 10.1
Lesson 10.1
Lesson 10.2
Lesson 10.2
Lesson 10.3
DAY 6
DAY 7
Lesson 10.3
Ready to Go On? Texas Test Prep
Module 11 DAY 1
DAY 2
DAY 3
DAY 4
DAY 5
Lesson 11.1
Lesson 11.1
Lesson 11.2
Lesson 11.2
Lesson 11.3
DAY 6
DAY 7
Lesson 11.3
Ready to Go On? Texas Test Prep
Module 12 DAY 1
DAY 2
DAY 3
DAY 4
DAY 5
Lesson 12.1
Lesson 12.1
Lesson 12.2
Lesson 12.2
Lesson 12.3
DAY 6
DAY 7
Lesson 12.3
Ready to Go On? Texas Test Prep
Module 13 DAY 1
DAY 2
DAY 3
DAY 4
DAY 5
Lesson 13.1
Lesson 13.1
Lesson 13.2
Lesson 13.2
Lesson 13.3
DAY 6
DAY 7
DAY 8
DAY 9
Lesson 13.3
Lesson 13.4
Lesson 13.4
Ready to Go On? Texas Test Prep
DAY 1
DAY 2
DAY 3
DAY 4
DAY 5
Lesson 14.1
Lesson 14.1
Lesson 14.2
Lesson 14.2
Lesson 14.3
Module 14
DAY 6
DAY 7
DAY 8
DAY 9
Lesson 14.3
Lesson 14.4
Lesson 14.4
Ready to Go On? Texas Test Prep
Expressions, Equations, and Relationships
263B
Program Resources Plan
Engage and Explore
Online Teacher Edition Access a full suite of teaching resources online— plan, present, and manage classes, assignments, and activities.
RealWorld Videos Engage students with interesting and relevant applications of the mathematical content of each module.
ePlanner Easily plan your classes, create and view assignments, and access all program resources with your online, customizable planning tool.
Animated Math Online interactive simulations, tools, and games help students actively learn and practice key concepts.
Professional Development Videos Author Juli Dixon models successful teaching practices and strategies in actual classroom settings.
QR Codes Scan with your smart phone to jump directly from your print book to online videos and other resources.
Explore Activities Students interactively explore new concepts using a variety of tools and approaches.
Teacher’s Edition Support students with pointofuse Questioning Strategies, teaching tips, resources for differentiated instruction, additional activities, and more.
LESSON
7.2 Rates ?
ESSENTIAL QUESTION How do you use rates to compare quantities?
6.4.D
EXPLORE ACTIVITY
LESSON LESSON
Rates
The student is expected to: Proportionality—6.4.D Give examples of rates as the comparison by division of two quantities having different attributes, including rates as quotients.
Mathematical Processes
?
Engage
Calculating Unit Rates
6.4.D Give examples of
rates as the comparison by division of two quantities having different attributes, including rates as quotients. Math On the Spot my.hrw.com
ESSENTIAL QUESTION How do you use rates to compare quantities?
ESSENTIAL QUESTION How do you use rates to compare quantities? Sample answer: You use division to compare two quantities with different units.
Yoga Classes This month’s special:
6 classes for $90
Using Rates to Compare Prices A rate is a comparison of two quantities that have different units.
Motivate the Lesson
6 classes
Engage with the Whiteboard Have students fill in the table for each brand on the whiteboard. Ask students to write a rate for each brand, using the data from the last row of each table. Then have them compare that rate to the original rate in the problem statement. Help students to see that the two rates are equivalent.
B The cost of 2 cartons of milk is $5.50. What is the unit price?
C An airplane makes a 2,748mile flight in 6 hours. What is the airplane’s average rate of speed in miles per hour? 458 miles per hour Interactive Whiteboard Interactive example available online my.hrw.com
The unit price is $2.75 per carton of milk.
3.84
÷2
÷2
8
1.92
4
0.96 0.48 0.24
÷2
÷2
2
÷2
1
B Brand A costs $
0.24
÷5 ÷5
÷2
Ounces
Price ($)
25
4.50
5
0.90 0.18
1
The first quantity in a unit rate can be less than 1.
÷5 ÷5
÷2
The ship travels 0.4 mile per minute.
They are equivalent. per ounce.
YOUR TURN 3. There are 156 players on 13 teams. How many players are on each
Analyze Relationships Describe another method to compare the costs.
team? Personal Math Trainer Online Assessment and Intervention
EXAMPLE 1
To compare the costs, Shana must compare prices for equal amounts of juice. How can she do this?
÷50
Analyze Relationships In all of these problems, how is the unit rate related to the rate given in the original problem?
B; it costs less per ounce.
Reflect 1.
1 carton
2 cartons
Reflect
÷2
0.18
$5.50 $2.75 ________ = _______
C A cruise ship travels 20 miles in 50 minutes. 20 miles = ________ 0.4 mile __________ How far does the ship travel per minute? 50 minutes 1 minute
2.
per ounce. Brand B costs $
C Which brand is the better buy? Why?
Shana is at the grocery store comparing two brands of juice. Brand A costs $3.84 for a 16ounce bottle. Brand B costs $4.50 for a 25ounce bottle.
÷50
Brand B
Price ($)
16
$15 The unit rate _____ 1 class is the same as 15 ÷ 1 = $15 per class.
÷2
Divide the cost of each bottle by the amount of juice; 3.84 4.50 A: ____ = 0.24; B: ____ = 0.18 16 25
Explain B Michael walks 30 meters in 20 seconds. How many meters does he walk per second? 1.5 meters per second
© Houghton Mifflin Harcourt Publishing Company
EXPLORE ACTIVITY
1 class
÷6
Brand A Ounces
107 miles . The rate is _______ 2 hours
÷6
$15 $90 ________ = ______
Gerald’s yoga classes cost $15 per class.
A Complete the tables.
÷2
Chris drove 107 miles in two hours. You are comparing miles and hours.
To find the unit rate, divide both quantities in the rate by the same number so that the second quantity is 1:
107 miles . The rate is _______ 2 hours
Shana is at the grocery store comparing two brands of juice. Brand A costs $3.84 for a 16ounce bottle. Brand B costs $4.50 for a 25ounce bottle.
Explore
Connect Vocabulary
A rate is a comparison by division of two quantities that have different units.
6.4.D
$90
Use the information in the problem to write a rate: _______ 6 classes
Chris drove 107 miles in two hours. You are comparing miles and hours.
Ask: Have you ever wanted to find out which was the best buy between two products when shopping or find out how fast you are walking or running? Begin the Explore Activity to find out how to compare quantities with different units.
To compare the costs, Shana must compare prices for equal amounts of juice. How can she do this?
ADDITIONAL EXAMPLE 1 A The cost of 3 candles is $19.50. What is the unit price? $6.50 per candle
EXAMPLE 1 A Gerald pays $90 for 6 yoga classes. What is the cost per class?
6.4.D
EXPLORE ACTIVITY
6.1.G Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.
Using Rates to Compare Prices
A unit rate is a rate in which the second quantity is one unit. When the first quantity in a unit rate is an amount of money, the unit rate is sometimes called a unit price or unit cost.
12
A Complete the tables. Brand A
photograph? $
0.50
per photograph
my.hrw.com
ELL
Lesson 7.2
187
188
Unit 3
Remind students that a ratio is a comparison of two quantities expressed with the same units of measure, and a rate is a comparison of two quantities with different units of measure. A rate in which the second quantity is one unit is a unit rate.
Questioning Strategies
Mathematical Processes • How do you determine what number to divide by when finding a unit rate? Divide both quantities by the same number so that the second quantity is 1.
• How is finding a unit rate like simplifying a fraction? You find unit rates by dividing both quantities by the same number, just as you would to simplify a fraction.
YOUR TURN Avoid Common Errors
PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.G, which calls for students to “display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.” In the Explore Activity, students
Math Background The words ratio and rate come from the Latin word ratus, which means “calculation.” A unit rate is a rate expressed in its simplest form, a to b, where a may or may not be a whole number and b is 1. The terms have only 1 as a common factor. Many realworld situations involve the use of rate
Brand B
Ounces
Price ($)
÷2
16
3.84
÷2
8
1.92
4
0.96 0.48 0.24
players per team
4. A package of 36 photographs costs $18. What is the cost per
lin Harcourt Publishing Company
Texas Essential Knowledge and Skills
Proportionality—
7.2 Rates
© Houghton Mifflin Harcourt Publishing Company
7.2
Proportionality— 6.4.D Give examples of rates as the comparison by division of two quantities having different attributes, including rates as quotients.
÷2 ÷2
2 1
B Brand A costs $
0.24
Ounces ÷2 ÷2
÷5 ÷5
÷2
Price ($)
25
4.50
5
0.90 0.18
1
÷2
per ounce. Brand B costs $
0.18
per ounce.
B; it costs less per ounce.
÷5 ÷5
3 2 1
Teach
Assessment and Intervention
Math on the Spot video tutorials, featuring program authors Dr. Edward Burger and Martha SandovalMartinez, accompany every example in the textbook and give students stepbystep instructions and explanations of key math concepts.
The Personal Math Trainer provides online practice, homework, assessments, and intervention. Monitor student progress through reports and alerts. Create and customize assignments aligned to specific lessons or TEKS. • Practice – With dynamic items and assignments, students get unlimited practice on key concepts supported by guided examples, stepbystep solutions, and video tutorials. • Assessments – Choose from course assignments or customize your own based on course content, TEKS, difficulty levels, and more. • Homework – Students can complete online homework with a wide variety of problem types, including the ability to enter expressions, equations, and graphs. Let the system automatically grade homework, so you can focus where your students need help the most! • Intervention – Let the Personal Math Trainer automatically prescribe a targeted, personalized intervention path for your students.
Present engaging content on a multitude of devices, including tablets and interactive whiteboards. Continually monitor and assess student progress with integrated formative assessment.
Math Talk
Differentiated Instruction Print Resources Support all learners with Differentiated Instruction Resources, including • Leveled Practice and Problem Solving • Reteach • Reading Strategies • Success for English Learners • Challenge
Problem Solving
Calculating Unit Rates Math On the Spot my.hrw.com
You can solve rate
A unit rate is a rate in which the second quantity is one unit. When the first quantity in a unit rate is an amount of money, the unit rate is sometimes called a unit price or unit cost.
EXAMPLE 1
6.4.D
This month’s special:
6 classes for $90
To find the unit rate, divide both quantities in the rate by the same number so that the second quantity is 1:
1 class
÷2
The unit price is $2.75 per carton of milk.
112 camp _____ ers __ _____ _____ 8 campers per cabin = 14 cabins Method 2 Use equiv alent rates.
$5.50 $2.75 ________ = _______ 1 carton
2 cartons
The ship travels 0.4 mile per minute.
Divide to find the unit
rate.
÷2
Divide to find the numbe r of cabins.
16 camp ers _____ _____
112 campers _____ 2 cabins = 14 ______ cabins ×7
The camp needs 14
÷50
Animated Math
cabins.
my.hrw.com
Reflect Analyze Relationships In all of these problems, how is the unit rate related to the rate given in the original problem?
3. There are 156 players on 13 teams. How many players are on each players per team
4. A package of 36 photographs costs $18. What is the cost per
in Harcourt Publishing Company
They are equivalent.
© Houghton Mifflin Harcourt Publishing Company
Reflect
YOUR TURN
Assessment Resources Tailor assessments to meet the needs of all your classes and students, including • Leveled Module Quizzes • Leveled Unit Tests • Unit Performance Tasks • Placement, Diagnostic, and Quarterly Benchmark Tests
×7
÷50
12
my.hrw.com
ers per cabin.
C A cruise ship travels 20 miles in 50 minutes. 20 miles = ________ 0.4 mile __________ How far does the ship travel per minute? 50 minutes 1 minute
team?
Math On the Spot
There are 8 camp
÷2
Personal
rate or by using equiv alent rates.
÷2
8 camp ers _____ ____ 2 cabins = 1 cabin
$15 The unit rate _____ 1 class is the same as 15 ÷ 1 = $15 per class.
Gerald’s yoga classes cost $15 per class.
B The cost of 2 cartons of milk is $5.50. What is the unit price?
2.
Prepare students with practice similar to the Texas assessment program at every module and unit.
16 camp ers _____ _____
÷6
$90 $15 ________ = ______ ÷6
The first quantity in a unit rate can be less than 1.
Texas Test Prep
6.4.D At a summer camp , the campers are divided into grou 16 campers and ps. Each group has 2 cabins. How many cabins are needed for 112 campers? Method 1 Find the unit rate. How many campers per cabin?
$90
Use the information in the problem to write a rate: _______ 6 classes
6 classes
Raise the bar with homework and practice that incorporates higherorder thinking and mathematical processes in every lesson.
with Unit Rates
problems by using a unit
EXAMPLE 2
A Gerald pays $90 for 6 yoga classes. What is the cost per class?
Yoga Classes
Response to Intervention
5.
What If? Suppose each group has 12 campers and 3 canoe unit rate of camp ers to canoes. s. Find the 12
÷ 3 = 4; there are
4 camper per cano
4 camp ers ____ e or ____ . 1 canoe
YOUR TURN 6.
Petra jogs 3 miles in 27 minutes. At this rate, how long to jog 5 would
miles? 27 minu ____ tes____ ____ ÷3
9 minute __
it take her
Expressions, Equations, and Relationships
263D
Math Background Algebraic Expressions
TEKS 6.7.B, 6.7.C
LESSONS 11.1 to 11.3 An algebraic expression is a mathematical statement constructed from at least one variable. It may have one or more operation symbols and one or more numbers. The table shows examples and nonexamples of algebraic expressions. Algebraic Expressions Examples
x, 3y  __25 , 2xy, z 2 + 1
Nonexamples
7, 20 ÷ (4 + 1), 4x = 16
Note that an algebraic expression does not contain an equal sign. A mathematical statement that contains an equal sign is an equation. Some students may have trouble understanding that algebraic expressions can be represented in multiple equivalent forms, just like numbers and numerical expressions. Students should recognize that the following 5x . algebraic expressions are equivalent: x + 0, x, 1x, __ 5
Writing OneVariable Equations TEKS 6.7.B, 6.9.A
LESSON 12.1 Students must be able to translate English phrases and sentences into algebraic symbols. To avoid misunderstandings, there are conventions we all use when translating from words to math. The phrase “the difference of 3 and 7” translates to 3  7. An equation is a mathematical statement that two quantities are equal. An equation may involve only numbers, as in 6 + 5 = 11, or may have algebraic expressions, as in 2x = 6. When an equation contains one variable, the solution of the equation is a value of the variable that makes the equation true. For instance, x = 3 is the solution of 2x = 6 since 2(3) = 6 is a true statement. An equation such as 2x = 6 is sometimes called an open sentence. That is, it is neither true nor false until additional information (i.e., a value of x) is given. When x is replaced by
263E
a value that is a solution, the open sentence becomes a true statement. If the given value of x is not a solution, the open sentence becomes a false statement. Students should know that an equation may have no solutions, one solution, more than one solution, or infinitely many solutions. For example, the equation x + 5 = 2 + x + 3 has infinitely many solutions. Every real number is a solution of this equation. Such an equation is called an identity.
Solving OneVariable Equations TEKS 6.9.A, 6.9.B, 6.9.C, 6.10.A, 6.10.B
LESSONS 12.2 to 12.3 An equation is like a scale that is perfectly balanced. The quantities on both sides have exactly the same weight. When two quantities a and b cause a scale to balance, the same quantity c can be added to both sides of the scale while preserving the balance. Applying this idea to equations yields the Addition Property of Equality: If a = b, then a + c = b + c. Similarly, it is possible to subtract the same quantity c from both sides of the scale and preserve the balance. Applying this idea to equations gives the Subtraction Property of Equality: If a = b, then a – c = b – c. The Multiplication Property of Equality states that multiplying each side of an equation by the same nonzero number produces a new equation that has the same solutions as the original. In other words, if a = b and c ≠ 0, then ac = bc. (Strictly speaking, multiplying both sides by a constant c = 0 results in a true equation, 0 = 0, but this is not useful because we lose whatever information the original equation contained.) The Division Property of Equality states that dividing each side of an equation by the same nonzero number produces a new equation that has the same solutions as the original. That is, if a = b and c ≠ 0, then __ac = __bc .
Writing OneVariable Inequalities TEKS 6.9.A, 6.9.B, 6.9.C
LESSON 13.1 One of the essential skills of algebra is translating words into mathematics. This can be challenging for many students, especially those for whom English is a second language. Perhaps the most common error is the attempt to make a direct wordtosymbol translation that preserves the order of the words. Although this method works in many cases, it can cause problems. Consider the statement “5 less than a number n is greater than 2.” A student making the “wordorder” error might translate this incorrectly as 5  n > 2. The correct translation is n  5 > 2. Students should realize that they can check their mathematical translations in much the same way that they can check a solution. In the above example, it is helpful to choose a specific value for n, such as 20, and ask whether “5 less than 20” is represented by 5  20.
Solving OneVariable Inequalities TEKS 6.9.B, 6.10.A, 6.10.B
LESSONS 13.2 to 13.4
The Addition and Subtraction Properties of Inequality state that the same quantity may be added to or subtracted from both sides of an inequality without changing the solution set. That is, if a > b, then a + c > b + c and a  c > b  c. Multiplying or dividing both sides of an inequality by a positive number also produces an inequality with the same solution set as the original inequality. In general terms, if a > b and c > 0, then ac > bc and __ac = __bc . When multiplying or dividing both sides of an inequality by a negative number, however, the inequality symbol must be reversed. Thus, if a > b and c < 0, then ac < bc and __ac = __bc . It is often helpful for students to check the solution of an inequality by substituting specific values for the variable. To check that n ≥ 10 is the solution of 18n ≤ 180, choose a value of n that is greater than or equal to 10, such as 2. In the original inequality, this gives 18(2) ≤ 180 or 36 ≤ 180, which is a true inequality. Checking a single value can never guarantee that a solution to an inequality is correct, but it can help students catch some errors.
Solving an inequality in one variable is similar to solving an equation in one variable. The goal is to isolate the variable on one side of the inequality by writing a series of inequalities that have the same solution set.
Expressions, Equations, and Relationships
263F
UNIT 4
Expressions, Equations, and Relationships
MODULE MODULE
10 10
Generating Equivalent Numerical Expressions 6.7.A MODULE MODULE
11 11
Generating Equivalent Algebraic Expressions 6.7.A, 6.7.C, 6.7.D MODULE MODULE
12 12
Equations and Relationships 6.7.B, 6.9, 6.10
13 13 Inequalities and
MODULE MODULE
Relationships 6.9, 6.10
14 14 Relationships in Two
© Houghton Mifflin Harcourt Publishing Company • Image Credits: Andy Sotiriou/Photodisc/Getty Images
MODULE MODULE
Variables
6.6.A, 6.6.B, 6.6.C, 6.11
CAREERS IN MATH
Unit 4 Performance Task At the end of the unit, check out how botanists use math.
Botanist A botanist is a biologist who studies plants. Botanists use math to analyze data and create models of biological organisms and systems. They use these models to make predictions. They also use statistics to determine correlations. If you are interested in a career in botany, you should study these mathematical subjects: • Algebra • Trigonometry • Probability and Statistics • Calculus Research other careers that require the analysis of data and use of mathematical models.
Unit 4
263
UNIT 4
Careers in Math
Vocabulary
Botanist A botanist uses math to find correlations and to predict future results. You will learn more about using math to make predictions in the Performance Tasks at the end of the unit. For more information about careers in mathematics as well as various mathematics appreciation topics, visit the American Mathematical Society at www.ams.org
Preview
Use the puzzle to preview key vocabulary from this unit. Unscramble the circled letters within found words to answer the riddle at the bottom of the page.
T N E I C I F F E O C Z U S L
Vocabulary Preview Integrating the ELPS Use the puzzle to give students a preview of important concepts in this unit. Students may work individually, in pairs, or in groups. c.4.D Use prereading supports such as graphic organizers, illustrations, and pretaught topicrelated vocabulary to enhance comprehension of written text.
O E K O F B S D E O R H B B F
Y S O F P N O X X W U X X F W
H S P J Z U X V P S T R F Q W
S V X S O A A X O S L H U O Z
D L U F E M F L N H O O B E T
F E C P E T U D E Z O P Y K V
P B D M U T A L N W P L H Y F
P C R Y I Z D N T T R Y J K O
O E A O N O C V I Y S C F P P
T R N Q V Q P J O D S U K H U
H V I H Z C Z S Q Q R X P C H
T N U G W C H A X E S O U S U
• A number that is multiplied by a variable in an algebraic expression. (Lesson 111) • A value of the variable that makes the equation true. (Lesson 121)
J N P C I J L V W B J Q O N U
J F N I C N P K B T A C E C B
coefficient
solution
• The point where the axes intersect to form the coordinate plane. (Lesson 141) origin • The part of an expression that is added or subtracted. (Lesson 111) term • The two number lines that intersect at right angles to form a coordinate plane. (Lesson 141) axes • Tells how many times the base is used in the product. (Lesson 101) exponent
Q: Unit Resources
my.hrw.com
Before Students understand: • operations with whole numbers, decimals, and fractions • order of operations • properties of operations: inverse, identity, commutative, associative, and distributive properties • graphs in the first quadrant
Go online to access all your unit resources.
In this Unit Students will learn about: • exponents • prime factorizations • numerical and algebraic expressions • equations and inequalities • the coordinate plane
Why did the paper rip when the student tried to stretch out the horizontal axis of his graph?
A: 264
© Houghton Mifflin Harcourt Publishing Company
• The numbers in an ordered pair. (Lesson 141) coordinates
Too much X – T
E N S
I
O N!
Vocabulary Preview
After Students will learn how to: • evaluate algebraic expressions with more than one variable • write twostep equations and inequalities to represent realworld problems and write a realworld problem to represent an equation or inequality • solve twostep equations and inequalities • graph linear equations in the form y = mx + b on the coordinate plane
Expressions, Equations, and Relationships
264
Generating Equivalent Numerical Expressions How can you generate equivalent numerical expressions and use them to solve realworld problems?
© Houghton Mifflin Harcourt Publishing Company • Image Credits: Vladimir Ivanovich Danilov / Shutterstock.com
?
ESSENTIAL QUESTION
Module 10
10
LESSON 10.1
Exponents 6.7.A
You can represent realworld problems with numerical expressions and simplify the expressions by applying rules relating to exponents, prime factorization, and order of operations.
LESSON 10.2
Prime Factorization 6.7.A
LESSON 10.3
Order of Operations 6.7.A
RealWorld Video
my.hrw.com
my.hrw.com
265
MODULE
Assume that you post a video on the internet. Two of your friends view it, then two friends of each of those view it, and so on. The number of views is growing exponentially. Sometimes we say the video went viral.
my.hrw.com
Math On the Spot
Animated Math
Personal Math Trainer
Go digital with your writein student edition, accessible on any device.
Scan with your smart phone to jump directly to the online edition, video tutor, and more.
Interactively explore key concepts to see how math works.
Get immediate feedback and help as you work through practice sets.
265
DO NOT EDITChanges must be made through "File info" CorrectionKey=B
Are You Ready?
Are YOU Ready?
Assess Readiness
Complete these exercises to review skills you will need for this chapter.
Use the assessment on this page to determine if students need intensive or strategic intervention for the module’s prerequisite skills.
Whole Number Operations
2 1
Response to Intervention
1. 992 × 16
Enrichment
my.hrw.com
2. 578 × 27
3 × 270 80 × 270 (3 × 270) + (80 × 270)
3. 839 × 65
15,606
15,872
Access Are You Ready? assessment online, and receive instant scoring, feedback, and customized intervention or enrichment.
Online Assessment and Intervention
← ← ←
Find the product.
Intervention
Personal Math Trainer
270 × 83 810 + 21,600 22,410
Online Assessment and Intervention
my.hrw.com
4. 367 × 23
54,535
8,441
Use Repeated Multiplication EXAMPLE
5×5× 5× 5
↓
↓
25 × 5
Online and Print Resources
Multiply the first two factors.
↓
Multiply the result by the next factor.
↓
Multiply that result by the next factor.
125 × 5
↓
Skills Intervention worksheets
Differentiated Instruction
• Skill 34 Whole Number Operations
• Challenge worksheets
• Skill 35 Use Repeated Multiplication
Extend the Math PREAP Lesson Activities in TE
Continue until there are no more factors to multiply.
625 Find the product.
PREAP
5. 7×7×7
6. 3×3×3×3
343
7. 6×6×6×6×6
7,776
81
8. 2×2×2×2×2×2
64
© Houghton Mifflin Harcourt Publishing Company
3
270 × 83
EXAMPLE
Personal Math Trainer
Division Facts
• Skill 38 Division Facts
EXAMPLE
54 ÷ 9 =
Think:
54 ÷ 9 = 6
So, 54 ÷ 9 = 6.
9 times what number equals 54? 9 × 6 = 54
Divide. 9. 20 ÷ 4
5
266
10. 21 ÷ 7
3
11. 42 ÷ 7
6
12. 56 ÷ 8
7
Unit 4
6_MTXESE051676_U4MO10.indd 266
28/01/14 5:40 PM
PROFESSIONAL DEVELOPMENT VIDEO my.hrw.com
Author Juli Dixon models successful teaching practices as she explores equivalent numerical expressions in an actual sixthgrade classroom.
Online Teacher Edition Access a full suite of teaching resources online—plan, present, and manage classes and assignments.
Professional Development
ePlanner Easily plan your classes and access all your resources online.
my.hrw.com
Interactive Answers and Solutions Customize answer keys to print or display in the classroom. Choose to include answers only or full solutions to all lesson exercises.
Interactive Whiteboards Engage students with interactive whiteboardready lessons and activities. Personal Math Trainer: Online Assessment and Intervention Assign automatically graded homework, quizzes, tests, and intervention activities. Prepare your students with updated, TEKSaligned practice tests.
Generating Equivalent Numerical Expressions
266
Reading StartUp
Reading StartUp
Have students complete the activities on this page by working alone or with others.
Vocabulary Review Words ✔ factor (factor) factor tree (árbol de factores) ✔ integers (entero) ✔ numerical expression (expresión numérica) ✔ operations (operaciones) ✔ prime factorization (factorización prima) repeated multiplication (multiplicación repetida) simplified expression (expresión simplificada)
Visualize Vocabulary Use the ✔ words to complete the graphic. You may put more than one word in each box.
Visualize Vocabulary
Reviewing Factorization
The sequence diagram helps students review vocabulary associated with factorization and prepares them to work with exponents. After students complete the diagram, discuss the vocabulary as a class.
Factor tree
24
factor, prime factorization
integer
Understand Vocabulary Use the following explanation to help students learn the preview words. You may hear the terms power and exponent used in place of each other. However, they do not mean the same thing. An exponent is the number that is written beside and slightly above the base. It tells you how many times to use the base as a factor. A power is a number that is formed by repeated multiplication by the same factor (the base) and can be represented as the base with an exponent.
Integrating the ELPS Students can use these reading and notetaking strategies to help them organize and understand new concepts and vocabulary. c.4.D Use prereading supports such as graphic organizers, illustrations, and
pretaught topicrelated vocabulary to enhance comprehension of written text.
Additional Resources
8×3
integers, factors, operations, numerical expression
Preview Words base (base) exponent (exponente) order of operations (orden de las operaciones) power (potencia)
Understand Vocabulary Complete the sentences using the preview words.
1. A number that is formed by repeated multiplication by the same
power
factor is a
2. A rule for simplifying expressions is © Houghton Mifflin Harcourt Publishing Company
Active Reading
2×2×2×3
integers, factors, operations, numerical expression, prime factorization
3. The
base
.
order of operations
.
is a number that is multiplied. The number that
indicates how many times this number is used as a factor is the
exponent
.
Active Reading ThreePanel Flip Chart Before beginning the module, create a threepanel flip chart to help you organize what you learn. Label each flap with one of the lesson titles from this module. As you study each lesson, write important ideas like vocabulary, properties, and formulas under the appropriate flap.
Differentiated Instruction • Reading Strategies ELL Module 10
Grades 6–8 TEKS Before Students understand: • operations with whole numbers, decimals, and fractions • prime numbers • order of operations
267
Module 10
In this module Students will learn to: • generate equivalent numerical expressions using exponents • generate equivalent numerical expressions using prime factorization • simplify numerical expressions using the order of operations
After Students will connect: • order of operations and numerical expressions • numerical and algebraic expressions
267
MODULE 10
Unpacking the TEKS
Unpacking the TEKS Understanding the TEKS and the vocabulary terms in the TEKS will help you know exactly what you are expected to learn in this module.
6.7.A Generate equivalent numerical expressions using order of operations, including whole number exponents and prime factorization.
Texas Essential Knowledge and Skills Content Focal Areas
Key Vocabulary exponent (exponente) The number that indicates how many times the base is used as a factor. order of operations (orden de las operaciones) A rule for evaluating expressions: first perform the operations in parentheses, then compute powers and roots, then perform all multiplication and division from left to right, and then perform all addition and subtraction from left to right.
Expressions, equations, and relationships—6.7 The student applies mathematical process standards to develop concepts of expressions and equations.
Integrating the ELPS c.4.F Use visual and contextual support … to read gradeappropriate content area text … and develop vocabulary … to comprehend increasingly challenging language.
What It Means to You You will simplify numerical expressions using the order of operations. UNPACKING EXAMPLE 6.7.A
Ellen is playing a video game in which she captures frogs. There were 3 frogs onscreen, but the number of frogs doubled every minute when she went to get a snack. She returned after 4 minutes and captured 7 frogs. Write an expression for the number of frogs remaining. Simplify the expression. 3×2
number of frogs after 1 minute
3×2×2
number of frogs after 2 minutes
3×2×2×2
number of frogs after 3 minutes
3×2×2×2×2
number of frogs after 4 minutes
Since 3 and 2 are prime numbers, 3 × 2 × 2 × 2 × 2 is the prime factorization of the number of frogs remaining. 3 × 2 × 2 × 2 × 2 can be written with exponents as 3 × 24. The expression 3 × 24 – 7 is the number of frogs remaining after Ellen captured the 7 frogs. Use the order of operations to simplify 3 × 24 – 7.
Go online to see a complete unpacking of the .
3 × 24 – 7 = 3 × 16 – 7 = 48 – 7 = 41 41 frogs remain.
my.hrw.com
© Houghton Mifflin Harcourt Publishing Company • Image Credits: Patrik Giardino/ Photodisc/Getty Images
Use the examples on this page to help students know exactly what they are expected to learn in this module.
Visit my.hrw.com to see all the unpacked. my.hrw.com
268
Grade 6 TEKS
Lesson 10.1
Lesson 10.2
Unit 4
Lesson 10.3
6.7.A Generate equivalent numerical expressions using order of operations, including whole number exponents and prime factorization.
Generating Equivalent Numerical Expressions
268
LESSON
10.1 Exponents Texas Essential Knowledge and Skills The student is expected to: Expressions, equations, and relationships—6.7.A Generate equivalent numerical expressions using order of operations, including whole number exponents and prime factorization.
Mathematical Processes
Engage ESSENTIAL QUESTION How do you use exponents to represent numbers? Sample answer: You can use exponents to represent repeated multiplication. For example, in 5 × 5 × 5 × 5, the number 5 is multiplied 4 times, so you can represent it as 5 4.
Motivate the Lesson Ask: Have you ever heard the terms squared or cubed? Both of those expressions are used to describe exponents. Do you know what 3 squared means? Take a guess. Begin the Explore Activity to find out.
6.1.D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.
Explore EXPLORE ACTIVITY Engage with the Whiteboard Have students fill in the table on the whiteboard. Extend the table to include 8 hours, and have students fill in the table for 5, 6, 7, and 8 hours. Ask students if they could predict the total number of bacteria for hour 12 and hour 20. Discuss a rule that students could use to make those kinds of predictions.
Explain ADDITIONAL EXAMPLE 1 Use an exponent to write each expression. A 7 × 7 × 7 × 7 × 7 × 7 76 5 B __23 × __23 × __23 × __23 × __23 ( __23 )
Interactive Whiteboard Interactive example available online my.hrw.com
EXAMPLE 1 c.1.F ELL Students may know other definitions of the words raised and base. Point out that in math, raised means “to multiply by itself,” and base can mean “the foundation,” or that on which something is built. In 24, you build on the base 2 by multiplying it by itself 4 times.
Connect Vocabulary
Questioning Strategies
Mathematical Processes
4 • In B, why is the base __4 in parentheses in the power ( __4 ) ? The base is in parentheses to
5
5
show that the entire fraction is used for repeated multiplication, not just the numerator.
• Can a power have a base and an exponent that are the same number? Justify your answer. Yes; for example, 6 × 6 × 6 × 6 × 6 × 6 = 6 6.
YOUR TURN Avoid Common Errors Students may want to find the product for each expression. Review the direction line. They are not asked to simplify the expression, but to write it in exponential form.
Talk About It Check for Understanding Ask: What is the difference between the two numbers in a power? The first number is the base, which is the number that is multiplied. The second number is the exponent, which tells how many times the base is multiplied by itself.
269
Lesson 10.1
LESSON
Expressions, equations, and relationships—6.7.A Generate equivalent numerical expressions using… exponents.
10.1 Exponents
A number that is formed by repeated multiplication by the same factor is called a power. You can use an exponent and a base to write a power. For example, 73 means the product of three 7s:
Math On the Spot
73 = 7 × 7 × 7
my.hrw.com
ESSENTIAL QUESTION How do you use exponents to represent numbers?
The base is the number that is multiplied. 6.7.A
EXPLORE ACTIVITY
Power
Identifying Repeated Multiplication A realworld problem may involve repeatedly multiplying a factor by itself. A scientist observed the hourly growth of bacteria and recorded his observations in a table.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: D. Hurst/Alamy
Time (h) 0 1
6 squared, 6 to the power of 2, 6 raised to the 2nd power
73
7 cubed, 7 to the power of 3, 7 raised to the 3rd power
94
9 to the power of 4, 9 raised to 4th power
6.7.A
Use an exponent to write each expression. After 2 hours, there are 2 · 2 = ? bacteria.
2
2× 2=
3
2× 2× 2=
4
2× 2× 2× 2=
4
A 3×3×3×3×3
Math Talk
Find the base, or the number being multiplied. The base is 3.
Mathematical Processes
What is the value of a number raised to the power of 1?
8 16
When a number is raised to the power of 1, the value of the power is equal to the base.
A Complete the table. What pattern(s) do you see in the Total bacteria column?
Sample answer: Each number is 2 times the previous number. B Complete each statement. At 2 hours, the total is equal to the product of two 2s. At 3 hours, the total is equal to the product of
three
2s.
At 4 hours, the total is equal to the product of
four
2s.
Find the exponent by counting the number of 3s being multiplied. The exponent is 5. 3 × 3 × 3 × 3 × 3 = 35 5 factors of 3
B
4 _ _ × 45 × _45 × _45 5
Find the base, or the number being multiplied. The base is _45. Find the exponent by counting the number of times _45 appears in the expression. The exponent is 4.
( )
4 4 _ _ × 45 × _45 × _45 = _45 5 4 __
4 factors of
5
YOUR TURN
Reflect 1.
How to read the power
62
EXAMPLE 1
Total bacteria 1 2
The exponent tells how many times the base appears in the expression.
© Houghton Mifflin Harcourt Publishing Company
?
Using Exponents
Use exponents to write each expression.
Communicate Mathematical Ideas How is the time, in hours, related to the number of times 2 is used as a factor?
Personal Math Trainer
The number of hours is the number of times the factor
Online Assessment and Intervention
2 is repeated.
my.hrw.com
Lesson 10.1
269
270
2. 4 × 4 × 4 4. _18 × _18
43
( )
1 2 _ 8
3. 6
61
5. 5 × 5 × 5 × 5 × 5 × 5
56
Unit 4
PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.D, which calls for students to “communicate mathematical ideas, …using multiple representations, including symbols, diagrams, graphs, and language as appropriate.” Students first use tables to identify patterns involving repeated multiplication. They then use exponents to rewrite expressions that involve repeated multiplication. Finally, students find the value of expressions that are written with exponents. This process helps students understand multiple ways to represent and use exponents.
Math Background The use of the terms squared and cubed is directly related to the measurements of area and volume. The area of a square with sides 5 units long is found by multiplying, 5 × 5, or 52, or 5 squared. The volume of a cube with sides 5 units long is found by multiplying, 5 × 5 × 5, or 53, or 5 cubed.
Exponents
270
ADDITIONAL EXAMPLE 2
EXAMPLE 2
Find the value of each power.
Avoid Common Errors
A 35 243
Since powers relate to multiplication, students may confuse powers with simple multiplication. After students evaluate each power, have them compare it to a simple multiplication problem to show that the two are not equal. For example, they find that 104 = 10,000, ask them to find 10 × 4 = 40. Since the two expressions have different answers, it should be clear that 104 and 10 × 4 are not equivalent.
0
B 17 1 C (4)3 64 D 0.62 0.36 Interactive Whiteboard Interactive example available online my.hrw.com
Questioning Strategies
Mathematical Processes • If a > b, which is greater: 2a or 2b ? Explain. 2a; If the exponent a is a bigger number than the exponent b, students find then the base 2 is used as a factor more times.
• If c > d, which is greater: c3 or d3? Explain. c3; The exponent with the greater base has to be greater if the exponent is the same. For example: 53 = 5 × 5 × 5 = 125, while 43 = 4 × 4 × 4 = 64. c.4.E ELL Encourage English learners to use the active reading strategies and the illustrated, bilingual glossary as they encounter new terms and concepts.
Integrating the ELPS
YOUR TURN Engage with the Whiteboard Have students rewrite each power as repeated multiplication. Seeing the power expressed as repeated multiplication can make it easier for students to find the correct value.
Elaborate Talk About It Summarize the Lesson Ask: How can you use an exponent to represent repeated multiplication? How can you find the value of a power? You can write a repeated multiplication, such as 23 = 2 × 2 × 2. To find the value of a power, rewrite the expression without using exponents by multiplying the base the number of times shown in the exponent—for example, 54 = 5 × 5 × 5 × 5 = 625.
GUIDED PRACTICE Engage with the Whiteboard For Exercise 1, have students complete the table on the whiteboard. Discuss different methods students may have for finding the value of each power, such as using parentheses to group the repeated multiplication or multiplying the value of the previous power by 5.
Avoid Common Errors Exercises 2–5 Remind students that their answers should be expressed as a power, not as the product of a repeated multiplication. Exercises 6–20 Some students may multiply a base by its exponent instead of using the base as a factor the number of times indicated by the exponent. Remind them that 43 means that 4 is used as a factor 3 times (4 × 4 × 4). Exercise 15 If students get the answer 8, remind them that the Property of Zero as an Exponent states that the value of any nonzero number raised to the power of 0 is 1.
271
Lesson 10.1
DO NOT EDITChanges must be made through “File info” CorrectionKey=A
DO NOT EDITChanges must be made through “File info” CorrectionKey=B
Guided Practice
Finding the Value of a Power
1. Complete the table. (Explore Activity)
To find the value of a power, remember that the exponent indicates how many times to use the base as a factor.
Exponential form Math On the Spot
Property of Zero as an Exponent The value of any nonzero number raised to the power of 0 is 1.
52
5×5
25
Example: 50 = 1
53
5×5×5
125
my.hrw.com
6.7.A
Find the value of each power.
My Notes
B 0.43
625 3,125
Use an exponent to write each expression. (Example 1)
63
107
3. 10 × 10 × 10 × 10 × 10 × 10 × 10
( _34 )5
( _79 )8
5. _79 × _79 × _79 × _79 × _79 × _79 × _79 × _79
Find the value of each power. (Example 2)
Evaluate: 0.4 = 0.4 × 0.4 × 0.4 = 0.064
2
1 __ 16
12. 0.82
0.64
9.
( )
3 0 _ 5
Identify the base and the exponent. The base is _35, and the exponent is 0.
Math Talk
Mathematical Processes
Evaluate.
3
=1
Is the value of 2 the same as the value of 32? Explain.
Any number raised to the power of 0 is 1.
D 112
23 = 2 · 2 · 2 = 8 and 32 = 3 · 3 = 9, so the values are not equivalent.
Identify the base and the exponent. The base is 11, and the exponent is 2. Evaluate. 112 = (11 × 11) = 121
512
6. 83
3
The opposite of a positive number squared is a negative number.
( _14 )
15. 80 18.
? ?
( 2 )3
1 8
2,401
7. 74 10.
( _13 )
1 __ 27
3
13. 0.53
0.125
16. 121
12
19.
(  _25 )
2
4 __ 25
8. 103 11.
( _67 )
2
14. 1.12 17.
( _12 )
0
20. 92
1,000 36 __ 49
1.21 1 81
ESSENTIAL QUESTION CHECKIN
21. How do you use an exponent to represent a number such as 16?
You use an exponent to write a number that can be written as a product of equal factors. 16 = 4 × 4 (or 2 × 2 × 2 × 2), so it can be written as 42 (or 24).
© Houghton Mifflin Harcourt Publishing Company
Identify the base and the exponent. The base is 0.4, and the exponent is 3.
0
5×5×5×5×5
4. _34 × _34 × _34 × _34 × _34
Evaluate: (10)4 = 10 × (10) × (10) × (10) = 10,000
( _35 )
5×5×5×5
55
3 factors of 6
Identify the base and the exponent. The base is 10, and the exponent is 4.
5
54
2. 6 × 6 × 6
A (10)4
© Houghton Mifflin Harcourt Publishing Company
Simplified product
5
EXAMPL 2 EXAMPLE
C
Product
51
YOUR TURN Find the value of each power. 6. 34
81
7. (1)9
1
8.
( _25 )
3
8 ___ 125
Personal Math Trainer
9. 122144
Online Assessment and Intervention
my.hrw.com
Lesson 10.1
6_MTXESE051676_U4M10L1.indd 271
271
22/10/12 10:51 PM
272
Unit 4
6_MTXESE051676_U4M10L1.indd 272
28/01/14 5:51 PM
DIFFERENTIATE INSTRUCTION Kinesthetic Experience
Critical Thinking
Additional Resources
To help students remember the meaning of the base and the exponent in a power, have them use graph paper or square tiles to construct models of the squares of whole numbers 1–10. Label the models as shown below. A visual representation of a square number can help students remember that exponents represent repeated multiplication of the same factor.
Have students explore multiplication of numbers written as powers. Partners can work together to find the values of pairs of expressions such as these:
Differentiated Instruction includes: • Reading Strategies • Success for English Learners ELL • Reteach • Challenge PREAP
32 = 3 × 3
32 · 33 and 31 · 34 22 · 24 and 23 · 23 43 · 43 and 41 · 45
243 and 243; 64 and 64; 4,096 and 4,096; The values of both expressions in each pair are the same. The base is used the same number of times in each pair. The exponents in each pair have equal sums. You can add the exponents to multiply powers with the same base, for example, 2 2 · 24 = 26. Exponents
272
Personal Math Trainer Online Assessment and Intervention
Online homework assignment available
Evaluate GUIDED AND INDEPENDENT PRACTICE 6.7.A
my.hrw.com
10.1 LESSON QUIZ 6.7.A Use exponents to write each expression. 1. __37 × __37 × __37
Concepts & Skills
Practice
Explore Activity Identifying Repeated Multiplication
Exercises 1, 38–44
Example 1 Using Exponents
Exercises 2–5, 22–37
Example 2 Finding the Value of a Power
Exercises 6–20, 38–44
2. 0.9 × 0.9 × 0.9 × 0.9 Find the value of each power.
Exercise
3. 74
Depth of Knowledge (D.O.K.)
Mathematical Processes
3 4. ( __34 )
22–37
2 Skills/Concepts
1.C Select tools
Lesson Quiz available online
38–39
2 Skills/Concepts
1.A Everyday life
40
2 Skills/Concepts
1.D Multiple representations
41–42
2 Skills/Concepts
1.A Everyday life
43
3 Strategic Thinking
1.F Analyze relationships
44
2 Skills/Concepts
1.D Multiple representations
45
3 Strategic Thinking
1.G Explain and justify arguments
46–47
3 Strategic Thinking
1.F Analyze relationships
48
3 Strategic Thinking
1.G Explain and justify arguments
my.hrw.com
Answers 3 1. ( __37 )
2. 0.94 3. 2,401 27 4. __ 64
Additional Resources Differentiated Instruction includes: • Leveled Practice Worksheets
273
Lesson 10.1
Name
Class
Date
10.1 Independent Practice
Personal Math Trainer
6.7.A
my.hrw.com
Online Assessment and Intervention
Write the missing exponent.
2
22. 100 = 10
( )
1 = ___ 1 26. ____ 169 13
23. 8 = 2
2
3
24. 25 = 5
1
27. 14 = 14
28. 32 = 2
2
25. 27 = 3
5
1= 34. __ 9
(
1 _ 3
3
10
)
31. 256 =
4
4
()
64 = __ 8 29. ___ 81 9
2
35. 64 =
2
9 = 36. ___ 16
8
4
32. 16 =
33. 9 =
2
(
3 _ 4
)
Sample answer: 0.32 = 0.09; 0.3 > 0.09 44. Which power can you write to represent the volume of the cube shown? Write the power as an expression with a base and an exponent, and then find the volume of the cube.
1 in.3 ( _13 )3 = _13 × _13 × _13 = __ 27
3
FOCUS ON HIGHER ORDER THINKING
2
2
Work Area
The value of 1 raised to any power is 1. 1 multiplied by
3
2
37. 729 =
1 in. 3
45. Communicate Mathematical Ideas What is the value of 1 raised to the power of any exponent? What is the value of 0 raised to the power of any nonzero exponent? Explain.
Write the missing base. 30. 1,000 =
43. Write a power represented with a positive base and a positive exponent whose value is less than the base.
itself any number of times is 1. The value of 0 raised
3
9
to any power is 0. 0 multiplied by itself any number of times is still 0.
38. Hadley’s softball team has a phone tree in case a game is canceled. The coach calls 3 players. Then each of those players calls 3 players, and so on. How many players will be notified during the third round of calls?
46. Look for a Pattern Find the values of the powers in the following pattern: 101, 102, 103, 104… . Describe the pattern, and use it to evaluate 106 without using multiplication.
27 players
Sample answer: 10; 100; 1,000; 10,000… . Each term in
39. Tim is reading a book. On Monday he reads 3 pages. On each day after that, he reads triple the number of pages as the previous day. How many pages does he read on Thursday?
the pattern is a 1 followed by the same number of zeros as the exponent. 106 = 1,000,000
4
3 pages, or 81 pages 40. Which power can you write to represent the area of the square shown? Write the power as an expression with a base and an exponent, and then find the area of the square.
8.5 = 8.5 × 8.5 = 72.25 mm 2
47. Critical Thinking Some numbers can be written as powers of different bases. For example, 81 = 92 and 81 = 34. Write the number 64 using three different bases.
2
26, 43, and 82
41. Antonia is saving for a video game. On the first day, she saves two dollars in her piggy bank. Each day after that, she doubles the number of dollars she saved on the previous day. How many dollars does she save on the sixth day?
48. Justify Reasoning Oman said that it is impossible to raise a number to the power of 2 and get a negative value. Do you agree with Oman? Why or why not?
26 dollars, or $64
Sample answer: Agree; because the product of two numbers with the same sign is always positive, and 0
42. A certain colony of bacteria triples in length every 10 minutes. Its length is now 1 millimeter. How long will it be in 40 minutes?
raised to the power of 2 is 0.
34 mm, or 81 mm
Lesson 10.1
EXTEND THE MATH
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
8.5 mm
PREAP
Activity Every integer can be written as the sum of square numbers. For example: Sum of 2 squares: 20 = 42 + 22 Sum of 3 squares: 24 = 42 + 22 + 22 Some integers, such as 22, can be written as the sum of square numbers more than one way. Sum of 3 squares: 22 = 32 + 32 + 22 Sum of 4 squares: 22 = 42 + 22 + 12 + 12
273
Activity available online
274
Unit 4
my.hrw.com
• Can you write the number 36 as the sum of squares in more than one way? Yes; 36 = 32 + 32 + 32 + 32 = 9 + 9 + 9 + 9; 36 = 42 + 42 + 22 = 16 + 16 + 4 • Write a number on one side of an index card and on the reverse write the number as a sum of squares. Challenge a classmate to write the number. For example: Write 62 as the sum of 3 squares.
12 + 52 + 62
• Write the integers 8, 13, and 18 as the sum of 2 squares. 8 = 22 + 22; 13 = 32 + 22; 18 = 32 + 32
Exponents
274
LESSON
10.2 Prime Factorization Texas Essential Knowledge and Skills The student is expected to: Expressions, equations, and relationships—6.7.A Generate equivalent numerical expressions using order of operations, including whole number exponents and prime factorization.
Mathematical Processes
Engage
ESSENTIAL QUESTION How do you write the prime factorization of a number? Sample answer: Use a factor tree or a ladder diagram to find the prime factorization of the number, then write the prime factorization using exponents.
Explore Motivate the Lesson
6.1.E Create and use representations to organize, record, and communicate mathematical ideas.
Ask students to name two numbers that can be multiplied to get a specific product. For example, you might ask them to name two numbers that can be multiplied to get 28 (1 and 28; 2 and 14; 4 and 7). Repeat the process using 36.
Explain ADDITIONAL EXAMPLE 1 Rayshawn is designing a mural. The mural must have an area of 42 square yards. What are the possible whole number lengths and widths for the mural? The possible lengths and widths are listed: Length (yd)
42
21
14
7
Width (yd)
1
2
3
6
EXAMPLE 1 Focus on Reasoning
Mathematical Processes Point out to students that you can tell when you have found all the factors of a number when the factor pairs start to repeat.
Questioning Strategies
Mathematical Processes • For any number, which numbers are always factors? 1 and the number itself. • Is it possible for a number to have all even factors? No; 1 is a factor for all numbers.
YOUR TURN Interactive Whiteboard Interactive example available online my.hrw.com
Avoid Common Errors Remind students that when they list the factors of a number they should always begin with 1 and end with the number itself.
EXPLORE ACTIVITY 1 Animated Math Prime Factorization Students use an interactive factor tree to find prime factors of composite numbers. my.hrw.com
Connect Vocabulary
ELL
Remind students that a prime number is a number with exactly 2 factors, 1 and itself, and a composite number is a number that has more than 2 factors.
Engage with the Whiteboard Have students make alternate factor trees for 240 on the whiteboard, next to the given factor tree. Have students start with the following pairs: 8 and 30; 24 and 10; and 12 and 20. Point out to students that while the order of the factors in a factor tree may differ, the prime factors of a number are always the same.
Questioning Strategies
Mathematical Processes • When choosing the first factor pair for the branches of the factor tree for 240, does one of the factors have to be a prime number? Explain. No. A factor tree can start with any factor pair.
275
Lesson 10.2
LESSON
Expressions, equations, and relationships— 6.7.A Generate equivalent numerical expressions using prime factorization.
10.2 Prime Factorization ESSENTIAL QUESTION
The prime factorization of 12 is 2 · 3 · 2 or 22 · 3.
Finding Factors of a Number
8 · 30, 10 · 24, 12 · 20, 15 · 16
Recall that area = length · width. For Ana’s garden, 24 ft2 = length · width.
3
2 · 3
2·2·2·2·3·5 Then write the prime factorization using exponents.
24 · 3 · 5
4
6
Math Talk
8 12 24
Mathematical Processes
Give an example of a whole number that has exactly two factors? What type of number has exactly two factors?
The factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. STEP 3
2 · 6
D Write the prime factorization of 240.
4 · 6 = 6 · 4, so you only list 4 · 6.
You can also use a diagram to show the factor pairs.
2
2 · 12
C Continue adding branches until the factors at the ends of the branches are prime numbers.
List the factors of 24 in pairs. List each pair only once.
1
240
5 · 48 2 · 24
B Choose any factor pair to begin the tree. If a number in this pair is prime, circle it. If a number in the pair can be written as a product of two factors, draw additional branches and write the factors.
Ana wants to build a rectangular garden with an area of 24 square feet. What are the possible whole number lengths and widths of the garden?
© Houghton Mifflin Harcourt Publishing Company • Image Credits: Brand X Pictures/ Getty Images
1 · 240, 2 · 120, 3 · 80, 4 · 60, 5 · 48, 6 · 40,
Math On the Spot
6.7.A
24 = 1 · 24 24 = 2 · 12 24 = 3 · 8 24 = 4 · 6
my.hrw.com
Use exponents to show repeated factors.
A List the factor pairs of 240. my.hrw.com
EXAMPL 1 EXAMPLE
Animated Math
Use a factor tree to find the prime factorization of 240.
Whole numbers that are multiplied to find a product are called factors of that product. A number is divisible by its factors. For example, 4 and 2 are factors of 8 because 4 · 2 = 8, and 8 is divisible by 4 and by 2.
STEP 2
Finding the Prime Factorization of a Number The prime factorization of a number is the number written as the product of its prime factors. For example, the prime factors of 12 are 3, 2, and 2.
How do you write the prime factorization of a number?
STEP 1
6.7.A
The possible lengths and widths are: Length (ft)
24
12
8
6
Width (ft)
1
2
3
4
Reflect 5.
What If? What will the factor tree for 240 look like if you start the tree with a different factor pair? Check your prediction by creating another factor tree for 240 that starts with a different factor pair.
Sample answer: The intermediate steps on Sample answer: 13; its factors are 1 and 13; prime number
the factor tree will be different but the final prime factorization will be the same.
240
2 · 120 2 · 60 3 · 20 4 · 5 2 · 2
© Houghton Mifflin Harcourt Publishing Company
?
EXPLORE ACTIVITY 1
YOUR TURN List all the factors of each number. 1. 21 3. 42
1, 3, 7, 21 1, 2, 3, 6, 7, 14, 21, 42
Personal Math Trainer
2. 37
1, 37
4. 30
1, 2, 3, 5, 6, 10, 15, 30
Online Assessment and Intervention
my.hrw.com
Lesson 10.2
275
276
Unit 4
PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.E, which calls for students to “create and use representations to organize, record, and communicate mathematical ideas.” Students use diagrams and factor trees to organize factor pairs of a number to find prime factorizations. They also use ladder diagrams to find prime factorizations, with the “ladder” as the means of recording and communicating the prime factorization.
Math Background Ancient Greeks started to study prime numbers circa 300 B.C.E. They observed that there were an infinite number of prime numbers and that there were irregular gaps between successive prime numbers. In 1984, Samuel Yates coined the term titanic prime. He used this term to refer to any prime number with 1,000 digits or more. When he first defined a titanic prime, only 110 of them were known. Today more than 110,000 titanic primes have been identified.
Prime Factorization
276
EXPLORE ACTIVITY 2 Engage with the Whiteboard Have students complete the ladder diagram starting with different combinations of prime factors. Point out to students that while the order in which they used the prime factors may differ, the prime factors of a number are always the same.
Focus on Modeling Mathematical Processes Students who find mental math difficult may find ladder diagrams to be challenging. Model other methods for dividing by 2, such as using long division or a calculator, to show that the ladder diagram is a useful organizational tool. Emphasize that they can use a combination of division methods when using ladder diagrams. Questioning Strategies
Mathematical Processes • How do you know when 2 is a factor of a number? How do you know when 2 is not a factor of a number? Even numbers have 2 as a factor. Odd numbers do not.
• Why do you have to divide by prime numbers when using the ladder diagram? The divisors on the left show the prime factorization, so all of them must be prime numbers.
YOUR TURN Focus on Communication
Mathematical Processes Discuss with students ways to check that 2 · 3 · 3 · 3 is the prime factorization of 54. Students should understand that they can check their work two ways: by making sure that every number in the prime factorization is prime and by multiplying the expression 2 · 3 · 3 · 3 to verify that the product is 54.
Elaborate Talk About It Summarize the Lesson Ask: What is the prime factorization of a number, and how can you find the prime factorization of a number? The prime factorization of a number is an expression that shows the number as the product of its prime factors. You can use a factor tree or a ladder diagram to find the prime factorization of a number.
GUIDED PRACTICE Engage with the Whiteboard For Exercises 1–2, have students draw a diagram to list the factors of each number on the whiteboard. For Exercise 5, have students make several different factor trees on the whiteboard.
Avoid Common Errors Exercise 3 Remind students that you can tell when you have found all the factors of a number when the factor pairs start to repeat and that writing factor pairs in order makes it easier to check that all the factor pairs are listed. Exercises 4–7 Remind students that they can check their work by multiplying their answer for the prime factorization to make sure the product is the original number.
277
Lesson 10.2
DO NOT EDITChanges must be made through “File info” CorrectionKey=A
Guided Practice
6.7.A
Use a diagram to list the factor pairs of each number. (Example 1)
Using a Ladder Diagram
1. 18
A ladder diagram is another way to find the prime factorization of a number.
1
2
Use a ladder diagram to find the prime factorization of 132.
A Write 132 in the top “step” of the ladder. Choose a prime factor of 132 to write next to the step with 132. Choose 2. Divide 132 by 2 and write the quotient 66 in the next step of the ladder. B Now choose a prime factor of 66. Write the prime factor next to the step with 66. Divide 66 by that prime factor and write the quotient in the next step of the ladder.
2
4
13 26 52
1, 2, 4, 13, 26, 52
Length (ft)
72
36
24
18
12
9
Width (ft)
1
2
3
4
6
8
402
4. 402
201
E Write the prime factorization of 132 using exponents.
36
5. 36
3 · 12
· 2
4 · 3
3 · 67
22 · 3 · 11
2 · 2
2 · 3 · 67
Reflect 6. Complete a factor tree and a ladder diagram to find the prime factorization of 54.
© Houghton Mifflin Harcourt Publishing Company
1
Use a factor tree to find the prime factorization of each number. (Explore Activity 1)
to the left of the steps of the ladder.
22 · 32
Use a ladder diagram to find the prime factorization of each number. (Explore Activity 2)
54
6. 32
2 · 27
9 · 3
3 · 3
prime factorization: 2 · 33
2 32 2 16 2 8 2 4 2 2 1
7. 27
3 27 3 9 3 3 1 3 · 3 · 3 or 33
25
? ?
Communicate Mathematical Ideas If one person uses a ladder diagram and another uses a factor tree to write a prime factorization, will they get the same result? Explain.
ESSENTIAL QUESTION CHECKIN
8. Tell how you know when you have found the prime factorization of a number.
Yes; there is only one unique prime factorization for
Sample answer: when all the factors are prime and their
every integer greater than 1.
product is the original number Lesson 10.2
6_MTXESE051676_U4M10L2.indd 277
1. This is a list
2. 52
Complete the table with possible measurements of the stage.
The prime factors are 2, 2, 3, and 11. They are written
InCopy Notes
9 18
3. Karl needs to build a stage that has an area of 72 square feet. The length of the stage should be longer than the width. What are the possible whole number measurements for the length and width of the stage? (Example 1)
3 33 11 11 1
D What are the prime factors of 132? How can you tell from the ladder diagram?
7.
6
1, 2, 3, 6, 9, 18
2 132 2 66
C Keep choosing prime factors, dividing, and adding to the ladder until you get a quotient of 1.
2 54 3 27 3 9 3 3 1
3
© Houghton Mifflin Harcourt Publishing Company
EXPLORE ACTIVITY 2
DO NOT EDITChanges must be made through “File info” CorrectionKey=B
277
278
23/10/12 4:08 AM
DIFFERENTIATE INSTRUCTION InDesign Notes
Unit 4
6_MTXESE051676_U4M10L2.indd 278
InCopy Notes
1. This is a list
1. This is a list Bold, Italic, Strickthrough.
28/01/14 6:00 PM
InDesign Notes 1. This is a list
Kinesthetic Experience
Critical Thinking
Additional Resources
Students can use a method called the sieve of Eratosthenes to help identify prime numbers. Start with a 10 × 10 grid showing the numbers 1 to 100. Cross out 1, because 1 is not a prime number. Circle 2 because it is prime. Then cross out all multiples of 2, because multiples of 2 are not prime. Circle the next prime number, 3, and cross out all multiples of 3. Repeat the process until all numbers are circled or crossed out. Students can refer to this chart when deciding whether a number is prime or composite.
Discuss with students how the prime factorization of a number can be used to find all the factors of a number, by using the Associative and Commutative properties. For example, the prime factorization of 30 is 2 · 3 · 5, which can be expressed as 2 · (3 · 5), (2 · 3) · 5, or (2 · 5) · 3.
Differentiated Instruction includes: • Reading Strategies • Success for English Learners ELL • Reteach • Challenge PREAP
When you multiply the numbers inside the parentheses, the expressions simplify to 2 · (15), (6) · 5, and (10) · 3.
These numbers along with the factor pair 1 · 30, give all the factors for 30: 1, 2, 3, 5, 6, 10, 15, 30.
Prime Factorization
278
Personal Math Trainer Online Assessment and Intervention
Online homework assignment available
Evaluate GUIDED AND INDEPENDENT PRACTICE 6.7.A
my.hrw.com
10.2 LESSON QUIZ 6.7.A 1. Find all the factors of 54. 2. Find the prime factorization of 54, and then write it using exponents.
Concepts & Skills
Practice
Example 1 Finding Factors of a Number
Exercises 1–3, 9–10
Explore Activity 1 Finding the Prime Factorization of a Number
Exercises 4–5, 12–15, 17–19
Explore Activity 2 Using a Ladder Diagram
Exercises 6–7, 16
3. Find all the factors of 60. 4. Find the prime factorization of 60, and then write it using exponents. 5. Chanasia has 30 beads. She wants to put them in boxes, so that each box will contain the same whole number of beads. Use factors to list all the different ways she can put the beads into boxes.
Exercise
my.hrw.com
Answers 1. 1, 2, 3, 6, 9, 18, 27, 54 2. 2 × 3 × 3 × 3 = 2 · 3
3
3. 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 4. 2 × 2 × 3 × 5 = 2 · 3 · 5
2 Skills/Concepts
1.D Multiple representations
10
2 Skills/Concepts
1.A Everyday life
11
3 Strategic Thinking
1.G Explain and justify arguments
2 Skills/Concepts
1.C Select tools
16
3 Strategic Thinking
1.G Explain and justify arguments
17
3 Strategic Thinking
1.C Select tools
18
3 Strategic Thinking
1.G Explain and justify arguments
19
3 Strategic Thinking
1.F Analyze relationships
20
3 Strategic Thinking
1.G Explain and justify arguments
21–22
3 Strategic Thinking
1.F Analyze relationships
2
5. 1 box with 30 beads 2 boxes with 15 beads each 3 boxes with 10 beads each 5 boxes with 6 beads each 6 boxes with 5 beads each 10 boxes with 3 beads each 15 boxes with 2 beads each 30 boxes with 1 bead each
279
Lesson 10.2
Mathematical Processes
9
12–15
Lesson Quiz available online
Depth of Knowledge (D.O.K.)
Additional Resources Differentiated Instruction includes: • Leveled Practice Worksheets
DO NOT EDITChanges must be made through “File info” CorrectionKey=A
DO NOT EDITChanges must be made through “File info” CorrectionKey=B
Name
Class
Date
10.2 Independent Practice
18. In a game, you draw a card with three consecutive numbers on it. You can choose one of the numbers and find the sum of its prime factors. Then you can move that many spaces. You draw a card with the numbers 25, 26, 27. Which number should you choose if you want to move as many spaces as possible? Explain.
Personal Math Trainer
6.7.A
my.hrw.com
Online Assessment and Intervention
26; the prime factors of 25 are 5 and 5, the prime factors
9. Multiple Representations Use the grid to draw three different rectangles so that each has an area of 12 square units and they all have different widths. What are the dimensions of the rectangles?
of 26 are 2 and 13, and the prime factors of 27 are 3, 3, and 3. The sums are 10, 15, and 9. The greatest sum is 15,
1 × 12; 2 × 6; 3 × 4
so choose 26 to move 15 spaces. 19. Explain the Error When asked to write the prime factorization of the number 27, a student wrote 9 · 3. Explain the error and write the correct answer.
10. Brandon has 32 stamps. He wants to display the stamps in rows, with the same number of stamps in each row. How many different ways can he display the stamps? Explain.
9 is not a prime number; prime factorization of 27 = 33.
6 different ways; 1 row of 32 stamps; 2 rows of 16; 4 rows of 8; 32 rows of 1; 16 rows of 2; 8 rows of 4
FOCUS ON HIGHER ORDER THINKING
20. Communicate Mathematical Ideas Explain why it is possible to draw more than two different rectangles with an area of 36 square units, but it is not possible to draw more than two different rectangles with an area of 15 square units. The sides of the rectangles are whole numbers.
11. Communicate Mathematical Ideas How is finding the factors of a number different from finding the prime factorization of a number?
When you find the factors of a number, you find all
36 has five factor pairs, so five different rectangles can
factors, some of which are prime; when you find the
be drawn. 15 has only two factor pairs, so only two
prime factorization, you find only the prime factors.
14. 23
34 · 11
13. 504
23
15. 230
different rectangles can be drawn. 21. Critique Reasoning Alice wants to find all the prime factors of the number you get when you multiply 17 · 11 · 13 · 7. She thinks she has to use a calculator to perform all the multiplications and then find the prime factorization of the resulting number. Do you agree? Why or why not?
2 3 · 32 · 7 2 · 5 · 23
Disagree; the factors that are being multiplied are all prime numbers, so the prime factorization of the
16. The number 2 is chosen to begin a ladder diagram to find the prime factorization of 66. What other numbers could have been used to start the ladder diagram for 66? How does starting with a different number change the diagram?
number is 17 · 13 · 11 · 7. 22. Look for a Pattern Ryan wrote the prime factorizations shown below. If he continues this pattern, what prime factorization will he show for the number one million? What prime factorization will he show for one billion?
3 and 11 can be chosen because they are prime factors. The intermediate steps would be different, but the
10 = 5 · 2
prime factorization is the same.
100 = 52 · 22
17. Critical Thinking List five numbers that have 3, 5, and 7 as prime factors.
1,000 = 53 · 23
one million: 56 · 26; one billion: 59 · 29
Sample answer: 105, 315, 525, 735, 945 Lesson 10.2
6_MTXESE051676_U4M10L2.indd 279
InCopy Notes 1. This is a list
EXTEND THE MATH
279
280
25/10/12 11:56 AM
InDesign Notes
PREAP 1. This is a list
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
Find the prime factorization of each number. 12. 891
Work Area
Unit 4
6_MTXESE051676_U4M10L2.indd 280
InCopy Notes
1. This is a list Bold, Italic, Strickthrough. my.hrw.com Activity available online
28/01/14 6:00 PM
InDesign Notes 1. This is a list
Activity In mathematics, a perfect number is a number that is equal to the sum of all its factors (excluding the number itself ). The number 6 is an example of a perfect number. The factors of 6 are 1, 2, 3, and 6. The sum of the factors excluding 6 is 1 + 2 + 3 = 6. • Find the next largest perfect number, and show why it is perfect. 28; the factors of 28 are 1, 2, 4, 7, 14, and 28, and 1 + 2 + 4 + 7 + 14 = 28. • A student claims that 128 is a perfect number. Prove or disprove the student’s claim. False; The factors of 128 are 1, 2, 4, 8, 16, 32, 64, and 128. Their sum, excluding the number itself, is 127. • Another student says that 496 is a perfect number. Prove or disprove the student’s claim. True; The factors of 496 are 1, 2, 4, 8, 16, 31, 62, 124, 248, and 496. Their sum, excluding the number itself, is 496.
Prime Factorization
280
LESSON
10.3 Order of Operations Texas Essential Knowledge and Skills The student is expected to: Expressions, equations, and relationships—6.7.A Generate equivalent numerical expressions using order of operations, including whole number exponents and prime factorization.
Mathematical Processes
Engage
ESSENTIAL QUESTION How do you use the order of operations to simplify expressions with exponents? Sample answer: Find the value of any expressions within parentheses first. Then evaluate all powers. Then multiply or divide in order from left to right, and finally, add or subtract in order from left to right.
Motivate the Lesson
Ask: Have you ever tried to simplify an expression such as 35 + 20(122 ÷ 9)? Try it. Need help? Begin the Explore Activity to find out how to use the order of operations.
6.1.C Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.
Explore EXPLORE ACTIVITY Engage with the Whiteboard
Write the expression 2 + 3 × 4 on the whiteboard. Point out to students that this expression could have two different results (14 or 20) without guidelines to show which operation should be performed first. Show students that the correct solution is 14 based on the fact that 3 × 4 = 4 + 4 + 4. Thus the expression 2 + 3 × 4 can be written as 2 + 4 + 4 + 4, which equals 14. Now write the expressions 36  18 ÷ 6 and 7 + 24 ÷ 6 × 2 on the whiteboard and ask the students to solve them. After a few minutes, have students come up to the whiteboard and solve the equations, showing all the steps. Discuss the solutions with the class. Answers: 33 and 15
Explain ADDITIONAL EXAMPLE 1 Simplify each expression. A 30  3 × 23 6 B 128 ÷ (4 × 2)
2
32 C 40  _____ (7  5)3
EXAMPLE 1 Avoid Common Errors Students may find it easier to perform operations from left to right as they appear in an expression, rather than use the order of operations. Remind students that using the order of operations correctly ensures that everyone who simplifies the same expression will get the same answer.
2
36
Interactive Whiteboard Interactive example available online my.hrw.com
Questioning Strategies
Mathematical Processes • Can you use the order of operations with expressions that have no parentheses? Explain. Yes. The order of operations tells the order in which operations should be performed but does not require that an expression include parentheses, exponents, or all the operations. • Are the expressions 3 + 5 · 2 and 3 + (5 · 2) equivalent? Explain. Yes. In both expressions 5 · 2 should be evaluated first, and then 3 should be added to the product.
YOUR TURN Talk About It
281
Lesson 10.3
Check for Understanding Ask: In the expression 36  18 ÷ 2 + 6 × 1, which operation should you perform first? Explain. Division. This expression has no parentheses or exponents, so the first operation to perform is either multiplication or division from left to right.
LESSON
10.3 Order of Operations ?
ESSENTIAL QUESTION
EXPLORE ACTIVITY (cont’d)
Expressions, equations, and relationships— 6.7.A Generate equivalent numerical expressions using order of operations …
Reflect 1.
In C , why does it makes sense to write the values as powers? What is the pattern for the number of emails in each wave for Amy?
Sample answer: By writing the values as powers, you How do you use the order of operations to simplify expressions with exponents?
can see the exponent is equal to the wave number. The pattern would be 31, 32, 33, 34, and so on.
6.7.A
EXPLORE ACTIVITY
Exploring the Order of Operations
Simplifying Numerical Expressions
Order of Operations
A numerical expression is an expression involving numbers and operations. You can use the order of operations to simplify numerical expressions.
1. Perform operations in parentheses. Math On the Spot
2. Find the value of numbers with exponents.
my.hrw.com
EXAMPLE 1
6.7.A
3. Multiply or divide from left to right.
My Notes
4. Add or subtract from left to right.
A 5 + 18 ÷ 32 5 + 18 ÷ 32 = 5 + 18 ÷ 9
Amy 1st wave
C Amy is just one of four friends initiating the first wave of emails. Write an expression for the total number of emails sent in the 2nd wave.
=5+2
Divide.
=7
Add.
B 4 × (9 ÷ 3)2
2nd wave
4 × (9 ÷ 3)2 = 4 × 32
A Use a diagram to model the situation for Amy. Each dot represents one email. Complete the diagram to show the second wave. B Complete the table to show how many emails are sent in each wave of Amy’s diagram.
Wave
Number of emails
Power of 3
1st
3
31
2nd
9
32
4
3
Personal Math Trainer
Multiply 4 and 3/Find the value of 32
my.hrw.com
36
Multiply.
2
Online Assessment and Intervention
=
= 36
(12  8) 42 = 8 + __ 8 + _______ 2 2
D Identify the computation that should be done first to simplify the expression in C . Then simplify the expression.
9
Evaluate 32.
(12  8)2
2
The value of the expression is 4 ×
Perform operations inside parentheses.
=4×9
C 8 + _______ 2
number of people × number of emails in 2nd wave written as a power ×
Evaluate 32.
Perform operations inside parentheses.
16 = 8 + __ 2
Evaluate 42.
= 8+ 8
Divide.
= 16
Add.
© Houghton Mifflin Harcourt Publishing Company
Amy and three friends launch a new website. Each friend emails the web address to three new friends. These new friends forward the web address to three more friends. If no one receives the email more than once, how many people will receive the web address in the second wave of emails?
© Houghton Mifflin Harcourt Publishing Company
Simplify each expression.
YOUR TURN Simplify each expression using the order of operations. 2. (3  1)4 + 3
19
3.
24 ÷ (3 × 22)
2
. Lesson 10.3
281
282
Unit 4
PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.C, which calls for students to “select tools, including…paper and pencil…and techniques, including mental math…and number sense…to solve problems.” Students work with pencil and paper using the order of operations to simplify expressions. Since use of the order of operations can involve many steps, students use mental math and number sense in situations such as finding the sum of two numbers with unlike signs and raising a negative number to a positive power.
Math Background A History of Mathematical Notations by Florian Cajori describes the history of various mathematical symbols. According to Cajori, parentheses and brackets have been used as grouping symbols since the sixteenth century. A work published in 1556, General trattato di numeri e misure by Niccolò Tartaglia, is one of the first works in which parentheses are used. Brackets have been found in a manuscript edition of Algebra, by Rafael Bombelli which dates back to 1550.
Order of Operations
282
ADDITIONAL EXAMPLE 2 Simplify each expression using the order of operations. A (3)3 + 8(15  7) 32 B ____ +3×3 (4)2
11
C 18  (3) + 3 × 2 3
37 51
EXAMPLE 2 Avoid Common Errors Point out to students that, now that they are working with integers, not every minus sign will indicate subtraction. Some minus signs are used to indicate negative numbers. Remind students to read equations carefully to distinguish between the two uses of the minus sign.
Engage with the Whiteboard
Interactive Whiteboard Interactive example available online
Cover up the blue text in each part and have students circle the operation to be performed for each step on the whiteboard. Ask students to explain their choices. Then discuss the choices with the class.
my.hrw.com
Questioning Strategies
Mathematical Processes • In A, the order of operations says to perform operations in parentheses first. So why isn’t (2)2 evaluated until the second step? When you are working with negative numbers, parentheses are used to separate addition, subtraction, multiplication, and division signs from negative numbers to avoid confusion. Since there is no operation in (2)2, the first step is to subtract within the expression (3  9). (3)2
• In B, can you evaluate the expression ____ the same as (1)2? Explain. No. You cannot 3 cancel the 3s before you have evaluated (3)2.
YOUR TURN Focus on Communication It may be helpful to review the rules for adding numbers with different signs before students try to work on their own. Remind them to subtract the absolute values of the numbers and then use the sign of the number with the greater absolute value in the difference.
Elaborate Talk About It Summarize the Lesson Ask: Why is it important to use the order of operations? The order of operations is important because correctly using the order or operations ensures that everyone who simplifies the same expression will get the same answer.
GUIDED PRACTICE Engage with the Whiteboard For Exercise 1, have students complete the diagram on the whiteboard. Then have them number the branches at each level to show the numbers of each type of fish that can be formed.
Avoid Common Errors Exercises 2–3 Remind students that when they work with an expression that has both multiplication and division, they should perform the operation that occurs first in the equation from left to right. Exercises 4–5 Remind students that when an expression inside parentheses has more than one operation, they need to perform those operations according to the order of operations.
283
Lesson 10.3
DO NOT EDITChanges must be made through “File info” CorrectionKey=A
DO NOT EDITChanges must be made through “File info” CorrectionKey=B
Guided Practice
Using the Order of Operations with Integers
1. In a video game, a guppy that escapes a net turns into three goldfish. Each goldfish can turn into two betta fish. Each betta fish can turn into two angelfish. Complete the diagram and write the number of fish at each stage. Write and evaluate an expression for the number of angelfish that can be formed from one guppy. (Explore Activity)
You can use the order of operations to simplify expressions involving integers. Math On the Spot
EXAMPL 2 EXAMPLE
6.7.A
my.hrw.com
1 guppy
Simplify each expression using the order of operations.
3 goldfish
A 4(3  9) + (2)2 Perform operations inside parentheses.
B
= 4(6) + 4
Evaluate (2)2.
= 24 + 4
Multiply.
= 28
Add.
Divide.
= 18
Add.
=4+ =4+
© Houghton Mifflin Harcourt Publishing Company
2
3
)2 ÷ 3
9
=
12
angelfish
3
=2+
= 48 ÷ 3 + 1
Multiply.
= 16 + 1
=
Divide.
= 15
=
Add.
? ?
= = =
4. 2 + (24 ÷ 23) 9 = 2 + (24 ÷
Evaluate (2)3.
3. 36 ÷ 22  4 × 2 = 36 ÷
÷3
7
=
C 6 × (2)3 ÷ 3 + 1 6 × (2)3 ÷ 3 + 1 = 6 × (8) ÷ 3 + 1
×2
Complete to simplify each expression. (Examples 1 and 2)
Evaluate (3)2.
= 21 + 3
3 3 × 2 = 12 angelfish 2. 4 + (10  7)2 ÷ 3 = 4 + (
2
× 2 betta fish
2
(3)2 21 + _____ 3 (3) 21 + _____ = 21 + _93 3
3
1
3
10
8
)9
9
9 1
4
4 × 2
4 × 2 8
5. 42 × (3 × 2 + 8) = 42 × ( 6 + 8)
2
= 42 ×
9
=
9
=
16
32
×
2
ESSENTIAL QUESTION CHECKIN
6. How do you use the order of operations to simplify expressions with exponents?
Find the value of any expressions within parentheses first. Then evaluate all powers. Then multiply or divide from
YOUR TURN
left to right, and finally add or subtract from left to right.
Simplify each expression using the order of operations. 4.
7 × (4) ÷ 14  22
5.
2
© Houghton Mifflin Harcourt Publishing Company
4(3  9) + (2)2 = 4(6) + (2)2
5 (3 + 1)3  3
37
Personal Math Trainer Online Assessment and Intervention
my.hrw.com
Lesson 10.3
6_MTXESE051676_U4M10L3.indd 283
InCopy Notes 1. This is a list
283
284
10/25/12 11:12 PM
DIFFERENTIATE INSTRUCTION InDesign Notes 1. This is a list
Unit 4
6_MTXESE051676_U4M10L3.indd 284
InCopy Notes 1. This is a list Bold, Italic, Strickthrough.
1/29/14 10:37 PM
InDesign Notes 1. This is a list
Cooperative Learning
Cognitive Strategies
Additional Resources
Have students work in groups to decide which operation signs to use to make the number sentences true. They may need to use operations more than once in each number sentence.
Students may be familiar with the abbreviation PEMDAS (or Please Excuse My Dear Aunt Sally) even before being introduced to the order of operations. But its abbreviation may give the impression that multiplication is always done before division and that addition is always done before subtraction. You may wish to present the mnemonic as P E M/D A/S
Differentiated Instruction includes: • Reading Strategies • Success for English Learners ELL • Reteach • Challenge PREAP
1. Operation signs: +, , · 12
4
6
3
7 = 37
12 · 4  6 · 3 + 7 = 37
2. Operation signs: +, , ÷ 18
2
24
12
4 = 22
18 + 2  24 ÷ 12 + 4 = 22
The slashes between the M and the D, and the A and the S, can help students remember that from left to right either operation can be performed first.
Order of Operations
284
Personal Math Trainer Online Assessment and Intervention
Online homework assignment available
Evaluate GUIDED AND INDEPENDENT PRACTICE 6.7.A
my.hrw.com
10.3 LESSON QUIZ 6.7.A Simplify each expression using the order of operations. 1. 3 + (7  5)2 × 6
Concepts & Skills
Practice
Explore Activity Exploring the Order of Operations
Exercise 1
Example 1 Simplifying Numerical Expressions
Exercises 2–3, 7–12, 14–16
Example 2 Using the Order of Operations with Integers
Exercises 4–5, 13
25 2. ______ ×2 (4 + 8)
3. 8  6 ÷ 2 + 3 × 5 4. 7 × 3  15 ÷ 5
Exercise
5. 8 + 2(3 + 12 ÷ 2)2
my.hrw.com
Answers 1. 27
1.C Select tools
13
3 Strategic Thinking
1.F Analyze relationships
14
3 Strategic Thinking
1.G Explain and justify arguments
15–16
2 Skills/Concepts
1.A Everyday life
17–19
3 Strategic Thinking
1.F Analyze relationships
2. 16 3. 20
Additional Resources
4. 18
Differentiated Instruction includes: • Leveled Practice Worksheets
5. 26
285
Lesson 10.3
Mathematical Processes
2 Skills/Concepts
7–12
Lesson Quiz available online
Depth of Knowledge (D.O.K.)
Class
Date
10.3 Independent Practice
Personal Math Trainer
6.7.A
my.hrw.com
15. Ellen is playing a video game in which she captures butterflies. There are 3 butterflies onscreen, but the number of butterflies doubles every minute. After 4 minutes, she was able to capture 7 of the butterflies. a. Look for a Pattern Write an expression for the number of butterflies after 4 minutes. Use a power of 2 in your answer.
Online Assessment and Intervention
3 × 2 × 2 × 2 × 2 = 3 × 24
Simplify each expression using the order of operations. 7. 5 × 2 + 32
19
9. (11  8)2  2 × 6 2
9 11. 12 + __ 3
9
8. 15  7 × 2 + 23
3
8 + 62 12. _____ +7×2 11
39
b. Write an expression for the number of butterflies remaining after Ellen captured the 7 butterflies. Simplify the expression.
14
10. 6 + 3(13  2)  52
3 × 24  7 = 3 × 16  7 = 48  7 = 41; 41
18
butterflies remain.
13. Explain the Error Jay simplified the expression 3 × (3 + 12 ÷ 3)  4. For his first step, he added 3 + 12 to get 15. What was Jay’s error? Find the correct answer.
16. Show how to write, evaluate and simplify an expression to represent and solve this problem: Jeff and his friend each text four classmates about a concert. Each classmate then texts four students from another school about the concert. If no one receives the message more than once, how many students from the other school receive a text about the concert?
Jay worked inside the parentheses first, but he should have performed the division 12 ÷ 3 = 4 first; 25.
2 × 42 = 32; 32 students receive a text.
14. Multistep A clothing store has the sign shown in the shop window. Pani sees the sign and wants to buy 3 shirts and 2 pairs of jeans. The cost of each shirt before the discount is $12, and the cost of each pair of jeans is $19 before the discount.
SALE
Today ONLY
$3 off every purchase!
a. Write and simplify an expression to find the amount Pani pays if a $3 discount is applied to her total.
3 × 12 + 2 × 19  3; $71
17. Geometry The figure shown is a rectangle. The green shape in the figure is a square. The blue and white shapes are rectangles, and the area of the blue rectangle is 24 square inches. a. Write an expression for the area of the entire figure that includes an exponent. Then find the area.
b. Pani says she should get a $3 discount on the price of each shirt and a $3 discount on the price of each pair of jeans. Write and simplify an expression to find the amount she would pay if this is true. © Houghton Mifflin Harcourt Publishing Company
Work Area
FOCUS ON HIGHER ORDER THINKING
2 in. 6 in.
62 + 2 × 6 + 24 = 72 square inches
3 × (12  3) + 2 × (19  3); $59
b. Find the dimensions of the entire figure.
c. Analyze Relationships Why are the amounts Pani pays in a and b different?
8 in. by 9 in.
In part a, the $3 discount is applied 1 time; in b it is
18. Analyze Relationships Roberto’s teacher writes the following statement on the board: The cube of a number plus one more than the square of the number is equal to the opposite of the number. Show that the number is 1.
applied 5 times. d. If you were the shop owner, how would you change the sign? Explain.
Sample answer: If the shop owner wants to make more
(1)3 + ((1)2 + 1) = 1 + (1 + 1) = 1 + 2 = 1; 1 is
money, the sign should say “$3 off your entire purchase.”
the opposite of 1.
If customers can take the discount off every item, a lot
19. Persevere in Problem Solving Use parentheses to make this statement true: 8 × 4  2 × 3 + 8 ÷ 2 = 25
more money is discounted from each purchase.
8 × 4  (2 × 3 + 8) ÷ 2 Lesson 10.3
EXTEND THE MATH
PREAP
© Houghton Mifflin Harcourt Publishing Company • Image Credits: imagebroker/ Alamy
Name
285
286
Activity available online
Unit 4
my.hrw.com
Activity The expression (4 × 4  4) × 4 uses exactly 4 fours. When simplified, its value is 48. • Write 10 expressions that use exactly 4 fours and that equal one of the numbers 0 to 9. Use what you know about the order of operations to write the expressions. You can use addition, subtraction, multiplication, division, parentheses, and exponents in the expressions. • Justify the expressions you have written by showing how to simplify them. Sample answers: 4+444=0
(4 + 4) ÷ (4 + 4) = 1
4÷4+4÷4=2
(4 + 4 + 4) ÷ 4 = 3
4  (4  4) × 4 = 4
( __44 )
(4 + 4) 4 + _____ =6 4 (4 × 4) _____ +4=8 4
4
+4=5
4 + 4  ( __44 ) = 7
4 + 4 + ( __44 ) = 9 Order of Operations
286
MODULE QUIZ
Ready to Go On?
Ready
Assess Mastery
10.1 Exponents
Use the assessment on this page to determine if students have mastered the concepts and standards covered in this module.
Find the value of each power.
5. 3
Response to Intervention
2 1
343
1. 73
( ) 2 _ 3
3
Enrichment
10. 120
Access Ready to Go On? assessment online, and receive instant scoring, feedback, and customized intervention or enrichment. Online and Print Resources Differentiated Instruction
Differentiated Instruction
• Reteach worksheets
• Challenge worksheets
• Reading Strategies • Success for English Learners ELL
ELL
6. ( 3
)5
243
3.
( )
7. ( 2
)4
16
4.
( ) 1 _ 2
6
8. 1.42
1 __ 64
1.96
Additional Resources Assessment Resources includes: • Leveled Module Quizzes
1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120
11. 58
2 × 29
12. 212
22 × 53
13. 2,800
2 4 × 52 × 7
14. 900
22 × 3 2 × 5 2
10.3 Order of Operations Simplify each expression using the order of operations. 15. ( 21  3 ) ÷ 32
PREAP
Extend the Math PREAP Lesson activities in TE
1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96
Find the prime factorization of each number.
2
17. 17 + 15 ÷ 3  24 © Houghton Mifflin Harcourt Publishing Company
my.hrw.com
81
2. 92
8 __ 27
49 __ 81
2
Find the factors of each number. 9. 96
Online Assessment and Intervention
my.hrw.com
7 _ 9
10.2 Prime Factorization
Intervention
Personal Math Trainer
Personal Math Trainer Online Assessment and Intervention
16. 72 × ( 6 ÷ 3 )
6
98
18. ( 8 + 56 ) ÷ 4  32
19. The nature park has a pride of 7 adult lions and 4 cubs. The adults eat 6 pounds of meat each day and the cubs eat 3 pounds. Simplify 7 × 6 + 4 × 3 to find the amount of meat consumed each day by the lions.
7
54 pounds
ESSENTIAL QUESTION 20. How do you use numerical expressions to solve realworld problems?
Write an expression to model the situation. Simplify the expression using the order of operations. First perform operations in parentheses, then find the value of each power, multiply or divide from left to right, and finally add or subtract from left to right.
Module 10
Texas Essential Knowledge and Skills Lesson
Exercises
10.1
1–8
6.7.A
10.2
9–14
6.7.A
10.3
15–19
6.7.A
287
Module 10
TEKS
287
Personal Math Trainer
MODULE 10 MIXED REVIEW
Texas Test Prep
Texas Testing Tip Students can use logic to eliminate some or all of the answer choices. Item 5 Students can eliminate choices B, C, and D because they all have numbers that are not prime in the product expression. This leaves choice A as the correct answer.
Selected Response 1. Which expression has a value that is less than the base of that expression? A 2 5 B _ 6 C 32 3
( )
Item 8 Students can eliminate choices A and B because they have numbers that are not prime in the product expression.
Avoid Common Errors Item 3 Students sometimes will answer an order of operations question simply by completing the operations from left to right. Remind the students to perform operations in parentheses first. Item 7 Some students may see that 3.6 appears four times and then choose A. Remind them that multiplication is repeated addition and that exponents are needed to represent repeated multiplication.
2
2. After the game the coach bought 9 chicken meals for $5 each and 15 burger meals for $6 each. What percent of the total amount the coach spent was used for the chicken meals? 1 A 33 _% 3 B 45% 2 C 66 _% 3 D 90%
5. Which expression shows the prime factorization of 100? A 22 × 52
C
B 10 × 10
D 2 × 5 × 10
1010
A 21
C
B 23
D 27
25
7. Which expression is equivalent to 3.6 × 3.6 × 3.6 × 3.6? 34 × 64
A 3.6 × 4
C
B 36
D 3.64
3
8. Which expression gives the prime factorization of 80?
3. Which operation should you perform first when you simplify 75  ( 8 + 45 ÷ 3 ) × 7? A addition
A 24 × 10
C
B 2×5×8
D 24 × 5
23 × 5
Gridded Response 9. Alison raised 10 to the 5th power. Then she divided this value by 100. What was the quotient?
B division
multiplication
D subtraction
4. At Tanika’s school, three people are chosen in the first round. Each of those people chooses 3 people in the second round, and so on. How many people are chosen in the sixth round? A 18
.
1
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
4
4
4
4
4
4
5
5
5
5
5
5
6
6
6
6
6
6
B 216
7
7
7
7
7
7
C
8
8
8
8
8
8
9
9
9
9
9
9
243
D 729
288
Online Assessment and Intervention
6. Which number has only two factors?
D 44
C
my.hrw.com
© Houghton Mifflin Harcourt Publishing Company
Texas Test Prep
Unit 4
Texas Essential Knowledge and Skills Items
Grade 6 TEKS
Mathematical Process TEKS
1
6.7.A
6.1.F
2
6.7.A
6.1.A
3*
6.4.E, 6.7.A
6.1.C
4
6.7.A
6.1.A, 6.1.F
5
6.7.A
6.1.E
6
6.7.A
6.1.F
7
6.7.A
6.1.E
8
6.7.A
6.1.E
9
6.7.A
6.1.C
* Item integrates mixed review concepts from previous modules or a previous course.
Generating Equivalent Numerical Expressions
288
Generating Equivalent Algebraic Expressions ?
MODULE
11
LESSON 11.1
Modeling Equivalent Expressions
You can model realworld problems with variable expressions, then use algebraic rules to solve the problems.
6.7.C
LESSON 11.2
ESSENTIAL QUESTION
Evaluating Expressions
How can you generate equivalent algebraic expressions and use them to solve realworld problems?
6.7.A
LESSON 11.3
Generating Equivalent Expressions 6.7.C, 6.7.D
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Lloyd Sutton/ Alamy
RealWorld Video
my.hrw.com
my.hrw.com
289
Module 11
Carpenters use formulas to calculate a project’s materials supply. Sometimes formulas can be written in different forms. The perimeter of a rectangle can be written as P = 2(l + w) or P = 2l + 2w.
my.hrw.com
Math On the Spot
Animated Math
Personal Math Trainer
Go digital with your writein student edition, accessible on any device.
Scan with your smart phone to jump directly to the online edition, video tutor, and more.
Interactively explore key concepts to see how math works.
Get immediate feedback and help as you work through practice sets.
289
DO NOT EDITChanges must be made through "File info" CorrectionKey=B
Are You Ready?
Are YOU Ready?
Assess Readiness
Complete these exercises to review skills you will need for this chapter.
Use the assessment on this page to determine if students need intensive or strategic intervention for the module’s prerequisite skills.
Use of Parentheses
2 1
(6 + 4) × (3 + 8 + 1) = 10 × 12
Evaluate.
Intervention
Enrichment
1. 11 + (20  13)
2. (10  7)  (14  12)
3. (4 + 17)  (16  9)
4. (23  15)  (18  13)
5. 8 × (4 + 5 + 7)
6. (2 + 3) × (11  5)
my.hrw.com
Differentiated Instruction
• Skill 50 Use of Parentheses
• Challenge worksheets
• Skill 53 Words for Operations • Skill 54 Evaluate Expressions
14
30
128
Words for Operations EXAMPLE
Online and Print Resources Skills Intervention worksheets
1
3
Access Are You Ready? assessment online, and receive instant scoring, feedback, and customized intervention or enrichment.
Online Assessment and Intervention
Online Assessment and Intervention
Do the operations inside parentheses first. Multiply.
= 120
Response to Intervention
18
Personal Math Trainer
my.hrw.com
Write a numerical expression for the quotient of 20 and 5.
Think: Quotient means to divide.
20 ÷ 5
Write 20 divided by 5.
Write a numerical expression for the word expression. 7. the difference between 42 and 19
PREAP
Extend the Math PREAP Lesson Activities in TE
42  19
20 + 30
9. 30 more than 20
8. the product of 7 and 12 10. 100 decreased by 77
7 × 12
100  77
© Houghton Mifflin Harcourt Publishing Company
3
EXAMPLE
Personal Math Trainer
Evaluate Expressions EXAMPLE
Evaluate 2(5) – 32. 2(5) – 32 = 2(5)  9 = 10  9 =1
Evaluate exponents. Multiply. Subtract.
Evaluate the expression. 11. 3(8)  15
9 8
14. 4(2 + 3)  12
290
59
12. 4(12) + 11 15. 9(14  5)  42
39
13. 3(7)  4(2)
13
16. 7(8)  5(8)
16
Unit 4
6_MTXESE051676_U4MO11.indd 290
28/01/14 6:49 PM
InCopy Notes
PROFESSIONAL DEVELOPMENT VIDEO
InDesign Notes
1. This is a list Bold, Italic, Strickthrough.
1. This is a list
my.hrw.com
Author Juli Dixon models successful teaching practices as she explores equivalent algebraic expressions in an actual sixthgrade classroom.
Online Teacher Edition Access a full suite of teaching resources online—plan, present, and manage classes and assignments.
Professional Development
ePlanner Easily plan your classes and access all your resources online.
my.hrw.com
Interactive Answers and Solutions Customize answer keys to print or display in the classroom. Choose to include answers only or full solutions to all lesson exercises.
Interactive Whiteboards Engage students with interactive whiteboardready lessons and activities. Personal Math Trainer: Online Assessment and Intervention Assign automatically graded homework, quizzes, tests, and intervention activities. Prepare your students with updated, TEKSaligned practice tests.
Generating Equivalent Algebraic Expressions
290
DO NOT EDITChanges must be made through "File info" CorrectionKey=B
Reading StartUp
Reading StartUp Have students complete the activities on this page by working alone or with others.
Simplifying Expressions
operations, order of operations
Understand Vocabulary
Students can use these reading and notetaking strategies to help them organize and understand new concepts and vocabulary. c.4.D Use prereading supports such as graphic organizers, illustrations, and pretaught topicrelated vocabulary to enhance comprehension of written text.
Preview Words
base, exponent
numerical expression
×, ÷, +, 
algebraic expression (expresión algebraica) coefficient (coeficiente) constant (constante) equivalent expression (expresión equivalente) evaluating (evaluar) like terms (términos semejantes) term (término, en una expresión) variable (variable)
23 2+1+3
Understand Vocabulary Complete the sentences using the preview words.
1. An expression that contains at least one variable is an
algebraic expression
.
2. A part of an expression that is added or subtracted is a © Houghton Mifflin Harcourt Publishing Company
Integrating the ELPS
base (base) exponent (exponente) numerical expression (expresión numérica) operations (operaciones) order of operations (orden de las operaciones)
Use the review words to complete the graphic. You may put more than one word in each oval.
The graphic organizer will help students to review concepts related to simplifying expressions. If time allows, discuss as a class the mnemonic Please Excuse My Dear Aunt Sally for order of operations (Parentheses, Exponent, Multiplication, Division, Addition, Subtraction).
Active Reading
Review Words
Visualize Vocabulary
Visualize Vocabulary
Use the following explanation to help students learn the preview words. Variable is an antonym of constant. Variable means “changeable”; something that is constant does not change. In math, a variable is a letter or symbol that represents a number. A letter is used because the number is unknown, and it may vary. A constant is a numeral, not a letter. For an expression to be an algebraic expression, it must contain at least one variable.
Vocabulary
3. A
constant
term
.
is a specific number whose value does not change.
Active Reading
Additional Resources
KeyTerm Fold Before beginning the module, create a keyterm fold to help you learn the vocabulary in this module. Write the highlighted vocabulary words on one side of the flap. Write the definition for each word on the other side of the flap. Use the keyterm fold to quiz yourself on the definitions used in this module.
Differentiated Instruction • Reading Strategies ELL
Module 11
6_MTXESE051676_U4MO11.indd 291
Grades 6–8 TEKS Before Students understand: • operations with whole numbers, decimals, and fractions • order of operations • properties of operations: inverse, identity, commutative, associative, and distributive properties
291
Module 11
28/01/14 6:49 PM
InCopy Notes
InDesign Notes
1. This is a list
1. This is a list
In this module Students will learn to: • determine if two expressions are equivalent using concrete models, pictorial models, and algebraic representations • evaluate algebraic expressions for the given value of a variable • generate equivalent expressions using the properties of operations: inverse, identity, commutative, associative, and distributive properties
291
After Students will connect: • numerical and algebraic expressions • variables and symbols to translate words into math
MODULE 11
Unpacking the TEKS
Unpacking the TEKS Understanding the TEKS and the vocabulary terms in the TEKS will help you know exactly what you are expected to learn in this module.
Use the examples on this page to help students know exactly what they are expected to learn in this module.
6.7.C
What It Means to You
Determine if two expressions are equivalent using concrete models, pictorial models, and algebraic representations.
Texas Essential Knowledge and Skills
You will use models to compare expressions. UNPACKING EXAMPLE 6.7.C
On a math quiz, Tina scored 3 points more than Yolanda. Juan scored 2 points more than Yolanda and earned 2 points as extra credit. Draw models for Tina's and Juan's scores. Use your models to decide whether they made the same score.
Key Vocabulary
Content Focal Areas
equivalent expressions (expresión equivalente) Expressions that have the same value for all values of the variables.
Expressions, equations, and relationships—6.7 The student applies mathematical process standards to develop concepts of expressions and equations.
y+3 Tina
y
3 y+2+2
Integrating the ELPS Juan
y
2
2
Tina and Juan did not make the same score because the models do not show equivalent expressions.
6.7.D Generate equivalent expressions using the properties of operations: inverse, identity, commutative, associative, and distributive properties.
Go online to see a complete unpacking of the .
What It Means to You You will use the properties of operations to find an equivalent expression. UNPACKING EXAMPLE 6.7.D
William earns $13 an hour working at a movie theater. He worked h hours in concessions and three times as many hours at the ticket counter. Write and simplify an expression for the amount of money William earned.
my.hrw.com
$13 · hours at concessions + $13 · hours at ticket counter Visit my.hrw.com to see all the unpacked.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: Erik Dreyer/Getty Images
c.4.F Use visual and contextual support … to read gradeappropriate content area text … and develop vocabulary … to comprehend increasingly challenging language.
13h + 13(3h) 13h + 39h
Multiply 13 · 3h.
h(13 + 39)
Distributive Property
my.hrw.com
292
Grade 6 TEKS
Lesson 11.1
Lesson 11.2
Unit 4
Lesson 11.3
6.7.A Generate equivalent numerical expressions using order of operations, including whole number exponents and prime factorization. 6.7.C Determine if two expressions are equivalent using concrete models, pictorial models, and algebraic representations. 6.7.D Generate equivalent expressions using the properties of operations: inverse, identity, commutative, associative, and distributive properties.
Generating Equivalent Algebraic Expressions
292
LESSON
11.1 Modeling Equivalent Expressions Texas Essential Knowledge and Skills The student is expected to: Expressions, equations, and relationships—6.7.C Determine if two expressions are equivalent using concrete models, pictorial models, and algebraic expressions.
Engage
ESSENTIAL QUESTION How can you write algebraic expressions and use models to decide if expressions are equivalent? Sample answer: Write or model the variables, constants, and operations to represent each expression. Then compare the expressions or models.
Motivate the Lesson Ask: Is a quarter the same as 5 nickels? as 25 pennies? How can you describe something in different ways but not change its value? Begin the Explore Activity to find out.
Mathematical Processes 6.1.D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.
Explore EXPLORE ACTIVITY Focus on Modeling Mathematical Processes Point out to students that a balance scale can represent how two numbers or expressions compare in value. When the numbers or expressions on each side of the scale are equal in value, the scale is in balance. If the numbers or expressions are unequal, one side of the scale is higher than the other side.
Explain ADDITIONAL EXAMPLE 1 A Write each phrase as an algebraic expression. x less than 5 5  x the product of z and 8 8z B Write a phrase for each algebraic expression. __z the quotient of z and 5 5
9 + y 9 more than y
EXAMPLE 1 Engage with the Whiteboard Write a mathematical operation on the whiteboard. Next to it, have students write an expression that uses the operation. Then ask them to describe their expression with words in several different ways and to list these on the whiteboard. For example, 6x can be described as 6 times x, the product of 6 and x, and 6 multiplied by x. Discuss with students the ways to describe each operation by looking at the list provided above Example 1.
Questioning Strategies
Interactive Whiteboard Interactive example available online my.hrw.com
Mathematical Processes • How do you identify the variable in an algebraic expression? Find a letter or symbol that represents an unknown. • Does it matter which letter you choose when writing an algebraic expression? No. You can choose any letter, but mathematicians often choose the last letters of the alphabet (e.g., x, y, and z) to represent variables. • How can you find the constant in an algebraic expression? Look for a specific number whose value does not change.
Connect Vocabulary
ELL
Explain to students that in the expression 8x + 15, the number 8 is a coefficient. A coefficient is the number multiplied by the variable, x. The number 15 in this expression is a constant.
293
Lesson 11.1
11.1 ?
Modeling Equivalent Expressions
ESSENTIAL QUESTION
Expressions, equations, and relationships— 6.7.C Determine if two expressions are equivalent using concrete models, pictorial models, and algebraic representations.
Writing Algebraic Expressions An algebraic expression is an expression that contains one or more variables and may also contain operation symbols, such as + or . Math On the Spot my.hrw.com
How can you write algebraic expressions and use models to decide if expressions are equivalent?
A variable is a letter or symbol used to represent an unknown or unspecified number. The value of a variable may change. A constant is a specific number whose value does not change. constant
EXPLORE ACTIVITY
variable
6.7.C
Algebraic Expressions
Modeling Equivalent Expressions
150 + y
w+n
x
15
12  7
9 __ 16
Not Algebraic Expressions
Equivalent expressions are expressions that have the same value.
In algebraic expressions, multiplication and division are usually written without the symbols × and ÷. • Write 3 × n as 3n, 3 · n, or n · 3. • Write 3 ÷ n as _n3 . There are several different ways to describe expressions with words.
The scale shown to the right is balanced.
A Write an expression to represent the circles on the left side of the balance.
2+3
B The value of the expression on the left side is
5
.
Operation
C Write an expression to represent the circles on the right side of the balance.
1+4
D The value of the expression on the right side is
5
Words
.
Addition • • • •
added to plus sum more than
E Since the expressions have the same value, the expressions are
equivalent
subtracted from minus difference less than take away taken from
• • • •
Multiplication
Division
times multiplied by product groups of
• divided by • divided into • quotient
.
F What will happen if you remove a circle from the right side of the balance? © Houghton Mifflin Harcourt Publishing Company
Subtraction • • • • • •
EXAMPLE 1
The scale will no longer be balanced.
6.7.C
A Write each phrase as an algebraic expression.
G If you add a circle to the left side of the balance, what can you do to the right side to keep the scale in balance?
The sum of 7 and x
The operation is addition.
The algebraic expression is 7 + x.
Add a circle to the right side.
The quotient of z and 3
The operation is division.
The algebraic expression is _3z .
Reflect 1. What If? Suppose there were 2 + 5 circles on the right side of the balance and 3 on the left side of the balance. What can you do to balance the scale? Explain how the scale models equivalent expressions.
B Write a phrase for each expression. 11x
© Houghton Mifflin Harcourt Publishing Company
LESSON
The operation is multiplication.
Sample answer: Add 4 circles to the left side. 4 + 3 is equivalent
The product of 11 and x
to 2 + 5 because the value of both expressions is 7.
8y
The operation is subtraction.
y less than 8 Lesson 11.1
293
294
Unit 4
PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.D, which calls for students to “communicate mathematical ideas … using multiple representations, including symbols, diagrams…and language as appropriate.” In this lesson, students use symbols, a model of a scale, and bar models to represent equivalent expressions. They employ these multiple representations to compare algebraic expressions and solve problems in realworld situations.
Math Background François Viète (1540–1603) was a lawyer in France who devoted his spare time to mathematics. In his book In artem analyticam isagoge, he introduced the idea of using vowels for variables and using consonants for constants. This was an important step toward modern algebra. Although Viète also used + and , he had no symbol for equality. To write “equals,” he would use the Latin word aequatur. Viète is sometimes called the Father of Algebra.
Modeling Equivalent Expressions
294
YOUR TURN Avoid Common Errors ADDITIONAL EXAMPLE 2 Use a bar model to represent each expression. A 5+y
Talk About It Check for Understanding Ask: How can you tell if an expression is an algebraic expression? Look for a variable, a letter or symbol that represents an unknown. If the expression has a variable, it is an algebraic expression.
5+y
y
5
B __n4
Exercise 2 When students multiply a number by a variable, be sure that they write the number first: 4x, not x4. It’s easier to read and understand.
EXAMPLE 2
n
Questioning Strategies
Mathematical Processes • On the bar model for A, why is 7 + x written above the model, while 7 and x are written below? The total that the bar represents is 7 plus an unknown amount; the labels below show what part of the model is 7 and what part is x.
n 4
Interactive Whiteboard Interactive example available online my.hrw.com
• On the bar model for B, how do you know how many pieces to divide the bar into? The expression __z3 means “7 divided into 3 parts,” so you know you need to divide the bar into 3 equal parts.
Engage with the Whiteboard Have students change the constant in Example 2A and/or 2B. Then have students draw a model to represent the new expression. Ask volunteers to explain their models to the class, and then ask the class if the students are correct.
ADDITIONAL EXAMPLE 3 Amanda and Stuart began the week with the same amount of money. Amanda paid $7 to go to the movies. Stuart spent $4 on snacks and $3 on a pen. Write algebraic expressions and draw bar models to represent the money each has left at the end of the week. Do Amanda and Stuart have the same amount of money left? n
n 7
7 n
YOUR TURN Avoid Common Errors
Exercise 7 Some students may try to model t  2 as an addition equation. Remind students that for subtraction expressions, they must “take away” the 2 from the whole, not add to it.
EXAMPLE 3 Questioning Strategies
Mathematical Processes • How can you know which operation to use to solve the problem? Read the problem carefully. Katriana and Andrew “spent” or “took away” money from the money they started the day with. These words describe subtraction. • In Step 1, what do the labels on the model represent? The variable x represents the amount of money Katriana started with, the 5 represents the money she spent, and x  5 represents the money she has left.
Focus on Reasoning 4
3
n 4 3
The models are equivalent, so Amanda and Stuart have the same amount of money left. Interactive Whiteboard Interactive example available online my.hrw.com
295
Lesson 11.1
Mathematical Processes Have students compare and contrast the models presented in the Explore Activity and Example 3. Encourage students to debate the advantages and disadvantages of each model, supporting their arguments with examples.
YOUR TURN Avoid Common Errors If students have difficulty with drawing the models, encourage them to circle the information for Tina in one color and for Juan in a different color. Then remind them to look at the list of ways to describe math operations to determine the operation they should use in the models.
Comparing Expressions Using Models
YOUR TURN Write each phrase as an algebraic expression. 2. n times 7
7n
3. 4 minus y
12
4. 13 added to x x
+ 13
Sample answers are given.
Write a phrase for each expression. x 5. __
4y
6. 10y 10
the quotient of x and 12
Personal Math Trainer Online Assessment and Intervention
Algebraic expressions are equivalent if they are equal for all values of the variable. For example, x + 2 and x + 1 + 1 are equivalent. Math On the Spot my.hrw.com
my.hrw.com
EXAMPLE 3
6.7.C
Katriana and Andrew started the day with the same amount of money. Katriana spent 5 dollars on lunch. Andrew spent 3 dollars on lunch and 2 dollars on an afterschool snack.
multiplied by y
Do Katriana and Andrew have the same amount of money left?
EXAMPL 2 EXAMPLE
6.7.C
Math On the Spot
The variable represents the amount of money both Katriana and Andrew have at the beginning of the day.
Use a bar model to represent each expression.
A 7+x
7+x
x
7 B _3z
Divide z into 3 equal parts.
© Houghton Mifflin Harcourt Publishing Company
z
z 3
2
8. 4y
t
t 2
y
x
3
2
x 3 2
Compare the models.
STEP 3
The models are equivalent, so the expressions are equivalent. Andrew and Katriana have the same amount of money left.
YOUR TURN
Mathematical Processes
9. On a math quiz, Tina scored 3 points more than Julia. Juan scored 2 points more than Julia and earned 2 points in extra credit. Write an expression and draw a bar model to represent Tina’s score and Juan’s score. Did Tina and Juan make the same grade on the quiz? Explain.
What two phrases can you use to describe the expression _4x ? What is different about the two phrases?
Draw a bar model to represent each expression. 7. t  2
x−3−2
Math Talk
YOUR TURN
Write an algebraic expression to represent the money Andrew has left. Represent the expression with a model.
STEP 2
Sample answer: 4 divided by x; x divided into 4. The numerator, or dividend, comes first when using the term divided by. The denominator, or divisor, comes first when using the term divided into.
Combine 7 and x.
x5
5
x−5
my.hrw.com
© Houghton Mifflin Harcourt Publishing Company
Algebraic expressions can also be represented with models.
x
Write an algebraic expression to represent the money Katriana has left. Represent the expression with a model.
STEP 1
Modeling Algebraic Expressions
Juan: y + 2 + 2
Tina: y + 3
4y Personal Math Trainer
Personal Math Trainer
Online Assessment and Intervention
Online Assessment and Intervention
my.hrw.com
my.hrw.com
Lesson 11.1
295
296
y
3
y
2
2
No; the expressions are not equivalent.
Unit 4
Modeling Equivalent Expressions
296
Elaborate Talk About It Summarize the Lesson Ask: How can you find out whether algebraic expressions are equivalent? Draw models of each expression and then compare them. If the models are equivalent, the expressions are equivalent.
GUIDED PRACTICE Engage with the Whiteboard For Exercise 1, have students write expressions that will keep the scale balanced next to the scale on the whiteboard. Emphasize to students that all the expressions are equivalent. For Exercise 4, have students circle the variable and the constant in y + 12. Then have them write three different algebraic expressions that are equivalent to y + 12 on the whiteboard. Sample answer: y + 3 + 9; y + 4 + 8; y + 5 + 7 For Exercise 7, have students complete the bar models for each city on the whiteboard and then explain their reasoning.
Avoid Common Errors Exercise 2 Remind students that the order of the variable and the constant is important in subtraction expressions, y  5 is not the same as 5  y. Exercise 3 When students multiply a number by a variable, be sure that they write the number first: 4x, not x4. It’s easier to read and understand. Exercise 6 Some students may try to model m ÷ 4 as an addition equation. Remind students that for division expressions, they must “divide” the whole into parts, not add to it.
297
Lesson 11.1
Guided Practice
Name
5+ 5
2p
3. The product of 2 and p
12 added to y
p 5. __ 10
11. Write an algebraic expression with two variables and one constant.
answers are given. 10 divided into p m
t
Tucson
© Houghton Mifflin Harcourt Publishing Company
2
3
4
14. n divided by 8
n __ 8
15. p multiplied by 4
4p b + 14
Tuesday
5
3
Wednesday
8
No; the models show that the temperature in Phoenix is 1 degree less than the temperature in Tucson.
18. a take away 16 19. k less than 24
24  k
20. 3 groups of w
3w 1+q
21. the sum of 1 and q
13 __ z
22. the quotient of 13 and z
ESSENTIAL QUESTION CHECKIN
45 + c
23. c added to 45
9. How can you use expressions and models to determine if expressions are equivalent?
Write a phrase in words for each algebraic expression. Sample answers given.
Write or model the variables, constants, and operations to represent each
24. m + 83
expression. Then compare the expressions or models.
25. 42s Lesson 11.1
297
298
32. Write an expression that represents Sarah’s total pay last week. Represent her hourly wage with w.
5w + 3w
90x a  16
17. 90 times x
8. Are the expressions that represent the temperatures in the two cities equivalent? Justify your answer.
? ?
Monday
Noah
t
t 4
Sarah
Write each phrase as an algebraic expression.
16. b plus 14
t 2 3
k less than 5
Read On Bookstore Work Schedule (hours)
x
Variable(s)
the quotient of h and 12
Sarah and Noah work at Read On Bookstore and get paid the same hourly wage. The table shows their work schedule for last week.
15
Constant(s)
the product of 11 and x
12
13. Identify the parts of the algebraic expression x + 15.
7. Represent the temperature in each city with an algebraic expression and a bar model.
28. 2 + g
31. 5 − k
12. What are the variables in the expression x + 8 − y?
m 4
t minus 29 g more than 2
h 30. __
x and y
At 6 p.m., the temperature in Phoenix, AZ, t, is the same as the temperature in Tucson, AZ. By 9 p.m., the temperature in Phoenix has dropped 2 degrees and in Tucson it has dropped 4 degrees. By 11 p.m., the temperature in Phoenix has dropped another 3 degrees. (Example 3)
27. t − 29
29. 11x
Sample answer: x + 24 − y
6. Draw a bar model to represent the expression m ÷ 4. (Example 2)
Phoenix
d
Sample answer: 7 + y
Online Assessment and Intervention
9 divided by d
26. __9
10. Write an algebraic expression with the constant 7 and the variable y.
Write a phrase for each algebraic expression. (Example 1) Sample 4. y + 12
my.hrw.com
33. Write an expression that represents Noah’s total pay last week. Represent his hourly wage with w.
8w 34. Are the expressions equivalent? Did Sarah and Noah earn the same amount last week? Use models to justify your answer.
Yes, Sarah and Noah got paid the same amount last week. Check
© Houghton Mifflin Harcourt Publishing Company
7+ 3
y−3
Personal Math Trainer
6.7.C
Write each phrase as an algebraic expression. (Example 1) 2. 3 less than y
Date
11.1 Independent Practice
1. Write an expression in the right side of the scale that will keep the scale balanced. (Explore Activity)
Sample answer given.
Class
students’ models.
83 added to m 42 times s
Unit 4
DIFFERENTIATE INSTRUCTION Cognitive Strategies
Cooperative Learning
Additional Resources
Ask students for the meanings of the words variable and constant in a context such as the following: The air temperature in the desert was quite variable yesterday; it was cold overnight and warm during the day. The temperature at the equator was constant for 24 hours. Explain that the words have the same meaning in mathematics. A constant is a value that does not change, such as the number 5, and a variable is a symbol for a quantity that is not fixed, such as x.
Have students work in pairs to draw models of balance scales as shown in the Explore Activity. Instruct pairs to take turns writing simple expressions and drawing circles to represent them on the balance pans on each side of the scale. Then ask each student to write and illustrate two sets of equivalent expressions that balance when arranged on the scales. Invite pairs to explain how they chose their arrangements of circles.
Differentiated Instruction includes: • Reading Strategies • Success for English Learners ELL • Reteach • Challenge PREAP
Modeling Equivalent Expressions
298
Personal Math Trainer Online Assessment and Intervention
Evaluate GUIDED AND INDEPENDENT PRACTICE
Online homework assignment available
6.7.C
my.hrw.com
11.1 LESSON QUIZ 6.7.C 1. Write each phrase as an algebraic expression: z times 5 6 plus n 2. Write a phrase for each algebraic expression: 6 8y __ m
Concepts & Skills
Practice
Explore Activity Modeling Equivalent Expressions
Exercise 1
Example 1 Writing Algebraic Expressions
Exercises 2–5, 14–33
Example 2 Modeling Algebraic Expressions
Exercise 6
Example 3 Comparing Expressions Using Models
Exercises 7–8, 34–37
3. Use a bar model to represent 4 + m. 4. Jan and Jackie check out the same number of library books. Jan turns in 4 books after 3 weeks. Jackie returns 2 books that week and 4 books later. Write algebraic expressions and draw bar models to represent the books Jan and Jackie have left. Do they have the same number of books left? Justify your answer. Lesson Quiz available online my.hrw.com
Answers
1. 5z, 6 + n 2. 8 multiplied by y, 6 divided by m 4+m
3.
Exercise
Depth of Knowledge (D.O.K.)
10–13
2 Skills/Concepts
1.F Analyze relationships
14–33
2 Skills/Concepts
1.C Select tools
34–36
3 Strategic Thinking
1.F Analyze relationships
37–40
2 Skills/Concepts
1.A Everyday life
41
3 Strategic Thinking
1.G Explain and justify arguments
42
3 Strategic Thinking
1.G Explain and justify arguments
43
3 Strategic Thinking
1.E Create and use representations
44
3 Strategic Thinking
1.A Everyday life
45
3 Strategic Thinking
1.F Analyze relationships
Additional Resources Differentiated Instruction includes: • Leveled Practice Worksheets m
4
4. No. Sample answer: The expressions are not equivalent. If b = the number of books each checked out, Jan’s books = b  4 and Jackie’s books = b  2  4 b
b4
b24
299
Lesson 11.1
4 b
2
4
Mathematical Processes
DO NOT EDITChanges must be made through “File info” CorrectionKey=A
DO NOT EDITChanges must be made through “File info” CorrectionKey=A
35. Critique Reasoning Lisa concluded that 3 · 2 and 32 are equivalent expressions. Is Lisa correct? Explain.
39. Abby baked 48 cookies and divided them evenly into bags. Let b represent the number of bags. Write an algebraic expression to represent the number of cookies in each bag.
No; 3 · 2 = 6 and 32 = 9, 6 ≠ 9; The expressions are not
40. Eli is driving at a speed of 55 miles per hour. Let h represent the number of hours that Eli drives at this speed. Write an algebraic expression to represent the number of miles that Eli travels during this time.
equal. 36. Multiple Representations How could you represent the expressions x  5 and x  3  3 on a scale like the one you used in the Explore Activity? Would the scale balance?
41. Represent RealWorld Problems If the number of shoes in a closet is s, then how many pairs of shoes are in the closet? Explain.
x  3  3 on the right side; no; you are removing 5 from
x on the left side and you are removing 6 from x on
_s ; there are half as many pairs of shoes as there are total 2
the right.
shoes. 42. Communicate Mathematical Ideas Is 12x an algebraic expression? Explain why or why not.
37. Multistep Will, Hector, and Lydia volunteered at the animal shelter in March and April. The table shows the number of hours Will and Hector volunteered in March. Let x represent the number of hours Lydia volunteered in March.
Sample answer: 12x is an algebraic expression because it contains a variable.
March Volunteering Will
3 hours
Hector
5 hours
43. Problem Solving Write an expression that has three terms, two different variables, and one constant.
Sample answer: 2x  8y + 7.
a. Will’s volunteer hours in April were equal to his March volunteer hours plus Lydia’s March volunteer hours. Write an expression to represent Will’s volunteer hours in April.
44. Represent RealWorld Problems Describe a situation that can be modeled by the expression x − 8.
Sample answer: Sam started the day with a pack of gum. During the day he gave out 8 pieces of gum.
b. Hector’s volunteer hours in April were equal to 2 hours less than his March volunteer hours plus Lydia’s March volunteer hours. Write an expression to represent Hector’s volunteer hours in April.
52+x
45. Critique Reasoning Ricardo says that the expression y + 4 is equivalent to the expression 1y + 4. Is he correct? Explain.
c. Did Will and Hector volunteer the same number of hours in April? Explain.
Sample answer: Yes; 1y is the product of 1 and y. Since 1 times any number is equal to the number, 1 · y = y. The
Yes; The expressions are equivalent.
expression y + 4 is equivalent to 1y + 4.
38. The town of Rayburn received 6 more inches of snow than the town of Greenville. Let g represent the amount of snow in Greenville. Write an algebraic expression to represent the amount of snow in Rayburn.
g+6
Lesson 11.1
6_MTXESE051676_U4M11L1.indd 299
1. This is a list
300
299
24/12/12 12:08 PM
EXTEND THE MATH
Unit 4
6_MTXESE051676_U4M11L1.indd 300
InCopy Notes
InDesign Notes
PREAP 1. This is a list
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
3+x
InCopy Notes
55h Work Area
FOCUS ON HIGHER ORDER THINKING
Represent x  5 on the left side of the scale and
48 __ b
1. This is aonline list Bold, Italic, Strickthrough. my.hrw.com Activity available
10/25/12 3:05 PM
InDesign Notes 1. This is a list
Activity Each season the Ravens and the Hawks baseball teams play the same number of games. So far this year, the Ravens have played 3 games at home and 4 games on the road. The Hawks have played 5 games at home and 3 games on the road. Roberto drew these bar models and says that the Hawks have more games left to play in the season than the Ravens do. Is he correct? If not, what did he do incorrectly when he represented the given information? Ravens:
g
g–5–3
5
Hawks:
3
g
g–3–4
3
4
No, Roberto is not correct. He mixed up the data and mislabeled both models. 1) The tops of both bars should be labeled as g, the variable that stands for games left to play. 2) The Ravens model should have the labels g  3  4, 3, and 4. 3) The Hawks model should have the labels g  5  3, 5 and 3. Comparing the corrected models shows that the expressions are not equivalent: 7 < 8. So, the Ravens have one more game to play than the Hawks do. Modeling Equivalent Expressions
300
LESSON
11.2 Evaluating Expressions Texas Essential Knowledge and Skills The student is expected to:
Engage
ESSENTIAL QUESTION How can you use the order of operations to evaluate algebraic expressions? Sample answer: Substitute the given value for the variable in the expression and then use the order of operations to find the value of the resulting numerical expression.
Expressions, equations, and relationships—6.7.A Generate equivalent numerical expressions using order of operations, including whole number exponents and prime factorization.
Motivate the Lesson Ask: How can you evaluate an expression that includes an unknown value? Begin Example 1 to find out.
Mathematical Processes 6.1.G Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.
Explore Engage with the Whiteboard
Write the expression 2(4 + x)  5 on the whiteboard. Ask a student to evaluate the expression for x = 2. Then ask the student to explain his/her reasoning. Ask the class if the student’s work and reasoning are correct. If students have difficulty understanding which operation to perform first, review the order of operations.
ADDITIONAL EXAMPLE 1 Evaluate each expression for the given value of the variable. A b  7; b = 16 28 B __ m; m = 4
9
EXAMPLE 1
7
C 0.2t; t = 1.6
Explain Focus on Math Connections
Mathematical Processes C and D involve an expression with a coefficient, a number that is multiplied by the variable. In the expression 0.5y, the coefficient is 0.5.
0.32
D 8s: s = __12 4 Interactive Whiteboard Interactive example available online my.hrw.com
Questioning Strategies
Mathematical Processes • How might substituting a negative value into one of the expressions affect its value? Substituting a negative value could change the sign of the answer and make it larger or smaller. In A, for example, if x = 15, the answer would be 24.
YOUR TURN ADDITIONAL EXAMPLE 2 Evaluate each expression for the given value of the variable. A 6(y  6); y = 9 B 6y  6; y = 9
18 48
C n  y + x; n = 5; y = 3; x = 4 D y  2y; y = 7 2
E 4y  3; y = 7
6
35
my.hrw.com
Lesson 11.2
Check for Understanding Ask: How do you evaluate an expression for a given variable? Substitute the given value for the variable in the expression. Then perform the operations, using the order of operations to find the value of the expression.
EXAMPLE 2 Focus on Math Connections
31
Interactive Whiteboard Interactive example available online
301
Talk About It
Mathematical Processes Remind students of the correct order of operations (parentheses, exponents, multiplication/ division, addition/subtraction) and the mnemonic device PEMDAS.
Questioning Strategies
Mathematical Processes • The answers to A and B are not the same, even though the expressions are very similar. Why? The parentheses in 4(x  4) mean that you subtract first. There are no parentheses in 4x  4, so you multiply first.
11.2 ?
Evaluating Expressions
ESSENTIAL QUESTION
Expressions, equations, and relationships—6.7.A Generate equivalent numerical expressions using order of operations, including whole number exponents and prime factorization.
YOUR TURN Personal Math Trainer
Evaluate each expression for the given value of the variable.
Online Assessment and Intervention
Math On the Spot my.hrw.com
Math On the Spot my.hrw.com
A x  9; x = 15
6
Subtract.
No; in w  x + y, you subtract x from w first. Then you add y to that result. In w  y + x, you subtract y from w first. Then you add x to that result.
When x = 15, x  9 = 6. 16 B __ n ;n=8 16 __ 8
Substitute 8 for n.
2
Divide.
© Houghton Mifflin Harcourt Publishing Company
16 When n = 8, __ n = 2.
C 0.5y; y = 1.4 0.5(1.4)
Substitute 1.4 for y.
0.7
Multiply.
Math Talk
When y = 1.4, 0.5y = 0.7.
Mathematical Processes
Is w  x + y equivalent to w  y + x? Explain any difference in the order the math operations are performed.
D 6k; k = _13 6 . HINT: Think of 6 as __ 1
()
6 _13
1 for k. Substitute __ 3
2
Multiply.
m ; m = 18 3. __ 6
3
Expressions may have more than one operation or more than one variable. To evaluate these expressions, substitute the given value for each variable and then use the order of operations.
EXAMPLE 2
6.7.A
A 4(x  4); x = 7
Evaluate each expression for the given value of the variable.
Substitute 15 for x.
4.7
Evaluate each expression for the given value of the variable.
6.7.A
15  9
2. 6.5  n; n = 1.8
Using the Order of Operations
Evaluating Expressions
EXAMPL 1 EXAMPLE
32
my.hrw.com
How can you use the order of operations to evaluate algebraic expressions?
Recall that an algebraic expression contains one or more variables. You can substitute a number for that variable and then find the value of the expression. This is called evaluating the expression.
1. 4x; x = 8
When k = _13, 6k = 2.
4(7  4)
Substitute 7 for x.
4(3)
Subtract inside the parentheses.
12
Multiply.
When x = 7, 4(x  4) = 12.
B 4x  4; x = 7 4(7)  4
Substitute 7 for x.
28  4
Multiply.
24
Subtract.
When x = 7, 4x  4 = 24.
C w  x + y; w = 6, x = 5, y = 3 (6)  (5) + (3)
Substitute 6 for w, 5 for x, and 3 for y.
1+3
Subtract.
4
Add.
When w = 6, x = 5, y = 3, w  x + y = 4.
D x2  x; x = 9 (9)2  (9)
Substitute 9 for each x.
81  9
Evaluate exponents.
72
© Houghton Mifflin Harcourt Publishing Company
LESSON
Subtract.
When x = 9, x  x = 72. 2
Lesson 11.2
301
302
Unit 4
PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.G, which calls for students to “display, explain, and justify mathematical ideas … using precise mathematical language in written … communication. ” In each Example and Exercise, students use mathematical ideas and language, including the order of operations, to evaluate algebraic expressions by substituting given values for variables. Students then evaluate realworld expressions such as formulas for finding surface area and volume and converting Celsius temperatures to Fahrenheit.
Math Background We translate (or write) words into algebraic expressions by using a consistent, universally understood system. This system has evolved over thousands of years. Archaeological records indicate that Babylonian mathematicians had developed prosebased algebra by 2000 B.C. The adoption of symbols to represent operations was also part of this evolution. The symbols + and  can be traced to Johann Widman (1498); the symbol · can be traced to Gottfried Leibniz (1698); and the symbol ÷ can be traced to Johann Heinrich Rahn (1659).
Evaluating Expressions
302
YOUR TURN Avoid Common Errors Exercises 7–9 Watch for students who substitute the wrong value for the variable. Caution students to be sure that they are substituting the correct value for each variable in expressions with more than one variable.
ADDITIONAL EXAMPLE 3 The expression 3.3m gives the number of feet in m (meters). Use the expression to find the number of feet that is equivalent to 400 meters. 1,320 feet Interactive Whiteboard Interactive example available online my.hrw.com
EXAMPLE 3 Focus on Math Connections
Mathematical Processes Remind students that when there is a coefficient in front of a variable, multiplication is indicated. Therefore, when they replace the variable with a value, they need to insert parentheses. In Step 2, for example, the expression 1.8c + 32 is written as 1.8(30) + 32 when substituting 30 for c.
Questioning Strategies
Mathematical Processes • How do you find the value of the variable c? Read the text of the problem carefully. The value of the variable is given in the last sentence.
• If the expression were written as 32 + 1.8c, would you still perform the multiplication first? Explain. Addition is associative, so changing the order of the terms does not change the expression in any way. You still need to do the multiplication before the addition. c.3.B ELL Be sure English learners understand the references to Celsius and Fahrenheit in Example 3. You may want to point out that both scales measure temperature but the Celsius Scale is part of the Metric System.
Integrating the ELPS
YOUR TURN Avoid Common Errors Exercise 10 Remind students that an exponent tells how many times to use the base as a factor, so x3 means x · x · x, not 3x.
Elaborate Talk About It Summarize the Lesson Ask: How can the order of operations help you evaluate algebraic expressions? When an expression contains more than one operation, the order of operations tells which operation to perform first.
GUIDED PRACTICE Engage with the Whiteboard For Exercises 7–8, have students circle all the important information provided, including key words that indicate operations, on the whiteboard. Then have them write an expression to represent each problem. Finally, have them complete the steps to evaluate each problem.
Avoid Common Errors Exercise 2 Remind students that when there is a coefficient in front of a variable, multiplication is indicated. Therefore, when they replace the variable with a value, they need to insert parentheses. Exercises 3, 5 Remind students that the fraction bar is another way to represent division.
303
Lesson 11.2
DO NOT EDITChanges must be made through “File info” CorrectionKey=B
DO NOT EDITChanges must be made through “File info” CorrectionKey=B
Guided Practice
YOUR TURN Evaluate each expression for n = 5. 5. 4 (n  4) + 14
18
6. 6n + n2
18
8. bc + 5a
9
9. a2  (b + c)
5. _12 w + 2; w = _19
6.7.A
The expression 1.8c + 32 gives the temperature in degrees Fahrenheit for a given temperature in degrees Celsius c. Find the temperature in degrees Fahrenheit that is equivalent to 30 °C.
my.hrw.com
Find the value of c.
12(
Substitute the value into the expression.
1.8(30) + 32
Substitute 30 for c.
54 + 32
Multiply.
86
Add.
3
)+5=
36
+5=
The family spent $41
$6
Nonstudent tickets
$12
Parking
$5
41
to attend the game.
a. Write an expression that represents the perimeter of the rectangular tablecloth. Let l represent the length of the tablecloth and w represent its width. The expression would be
2w + 2l
.
b. Evaluate the expression P = 2w + 2l for l = 5 and w = 7.
YOUR TURN
2(
10. The expression 6x2 gives the surface area of a cube, and the expression x3 gives the volume of a cube, where x is the length of one side of the cube. Find the surface area and the volume of a cube with a side length of 2 m.
24
m2 ; V =
8
seconds
7
) + 2(
Stan bought
5
) = 14 +
24 feet
10
=
24
of trim to sew onto the tablecloth.
9. Essential Question Follow Up How do you know the correct order in which to evaluate algebraic expressions?
m3
11. The expression 60m gives the number of seconds in m minutes. How many seconds are there in 7 minutes?
420
Women’s Soccer Game Prices Student tickets
8. Stan wants to add trim all around the edge of a rectangular tablecloth that measures 5 feet long by 7 feet wide. The perimeter of the rectangular tablecloth is twice the length added to twice the width. How much trim does Stan need to buy? (Example 3)
86 °F is equivalent to 30 °C.
S=
50
b. Since there are three attendees, evaluate the expression 12x + 5 for x = 3.
1.8c + 32
© Houghton Mifflin Harcourt Publishing Company
6. 5(6.2 + z); z = 3.8
12x + 5 is an expression that represents the cost of one carful of nonstudent soccer fans.
c = 30 °C STEP 2
10.5
4. 9 + m; m = 1.5
1 2__ 18
a. Write an expression that represents the cost of one carful of nonstudent soccer fans. Use x as the number of people who rode in the car and attended the game.
Math On the Spot
6
2. 3a  b; a = 4, b = 6
7. The table shows the prices for games in Bella’s soccer league. Her parents and grandmother attended a soccer game. How much did they spend if they all went together in one car? (Example 3)
You can evaluate expressions to solve realworld problems.
STEP 1
2
3. _8t ; t = 4
my.hrw.com
11
Evaluating RealWorld Expressions EXAMPL 3 EXAMPLE
16
1. x  7; x = 23
Online Assessment and Intervention
Evaluate each expression for a = 3, b = 4, and c = 6. 7. ab  c
Evaluate each expression for the given value(s) of the variable(s). (Examples 1 and 2)
Personal Math Trainer
55
© Houghton Mifflin Harcourt Publishing Company
18
4. 3(n + 1)
Substitute for the variables and follow the order Personal Math Trainer
of operations that you would use for a numerical
Online Assessment and Intervention
expression.
my.hrw.com
Lesson 11.2
6_MTXESE051676_U4M11L2.indd 303
InCopy Notes 1. This is a list
303
304
28/01/14 6:58 PM
DIFFERENTIATE INSTRUCTION InDesign Notes
Unit 4
6_MTXESE051676_U4M11L2.indd 304
InCopy Notes
1. This is a list
28/01/14 6:58 PM
InDesign Notes
1. This is a list Bold, Italic, Strickthrough.
1. This is a list
Home Connection
Cooperative Learning
Additional Resources
Have students record realworld math situations they experience at home, using both words and mathematical symbols.
Have students work in groups to solve a magic square. A magic square is an array of numbers in which each row, column, and diagonal has the same sum. Ask students if the array below is a magic square if x = 4; if x = 6; or if x = 0.
Differentiated Instruction includes: • Reading Strategies • Success for English Learners ELL • Reteach • Challenge PREAP
Sample answer: Mom works out for the same length of time each day. How long does she work out in a week? 7t, where t represents the length of time she works out each day.
x+7
x
x+2
0.5x + 6
3x  5
3x
2x + 1 x+6
x+1
The array is a magic square if x = 4, but not if x = 6 or if x = 0.
Evaluating Expressions
304
Personal Math Trainer Online Assessment and Intervention
Online homework assignment available
Evaluate GUIDED AND INDEPENDENT PRACTICE 6.7.A
my.hrw.com
11.2 LESSON QUIZ 6.7.A Evaluate each expression for the given value(s) of the variable(s). 1. a + 6; a = 2
Concepts & Skills
Practice
Example 1 Evaluating Expressions
Exercises 1–6
Example 2 Using the Order of Operations
Exercises 1–6, 13
Example 3 Evaluating RealWorld Expressions
Exercises 7–8, 10–12, 14, 16
16 2. __ g ;g = 4
3. 7(m  6); m = 8 4. 7m  6; m = 8
Exercise
5. s  k + x; s = 7, k = 4, x = 6 6. The expression 4g gives the number of quarts in g gallons. How many quarts are there in 4 gallons? Lesson Quiz available online my.hrw.com
Answers 1. 8 3. 14 4. 50 5. 9 6. 16 quarts
Lesson 11.2
Mathematical Processes
2 Skills/Concepts
1.A Everyday life
13
3 Strategic Thinking
1.G Explain and justify arguments
14–16
3 Strategic Thinking
1.A Everyday life
17
3 Strategic Thinking
1.G Explain and justify arguments
18
3 Strategic Thinking
1.F Analyze relationships
10–12
Additional Resources Differentiated Instruction includes: • Leveled Practice Worksheets
2. 4
305
Depth of Knowledge (D.O.K.)
DO NOT EDITChanges must be made through “File info” CorrectionKey=B
DO NOT EDITChanges must be made through “File info” CorrectionKey=A
Class
Date
11.2 Independent Practice 6.7.A
Movie 16 Ticket Prices
x
0
1
2
3
4
5
6
$8.75
6x  x2
0
5
8
9
8
5
0
Children
$6.50
Seniors
$6.50
The value of 6x  x2 increases
8.75a + 6.5c + 6.5s or 8.75a +
6.5(c + s)
b. The Andrews family bought 2 adult tickets, 3 children’s tickets, and 1 senior ticket. Evaluate your expression in part a to find the total cost of the tickets.
8.75(2) + 6.5(3) + 6.5(1) =
© Houghton Mifflin Harcourt Publishing Company
8.75(2) + 6.5(3 + 1) = $43.50 c. The Spencer family bought 4 adult tickets and 2 children’s tickets. Did they spend the same as the Andrews family? Explain.
No; 8.75(4) + 6.5(2) = $48.00
11. The area of a triangular sail is given by the expression _21 bh, where b is the length of the base and h is the height. What is the area of a triangular sail in a model sailboat when b = 12 inches and h = 7 inches?
42
in.2
V = 5,760 ft3
13. Look for a Pattern Evaluate the expression 6x  x2 for x = 0, 1, 2, 3, 4, 5, and 6. Use your results to fill in the table and describe any pattern that you see.
Adults
a. Write an expression for the total cost of tickets.
A=
Online Assessment and Intervention
my.hrw.com
10. The table shows ticket prices at the Movie 16 theater. Let a represent the number of adult tickets, c the number of children’s tickets, and s the number of senior citizen tickets.
16. The volume of a pyramid with a square base is given by the expression _13s2h, where s is the length of a side of the base and h is the height. Find the volume of a pyramid with a square base of side length 24 feet and a height of 30 feet.
Personal Math Trainer
17. Draw Conclusions Consider the expressions 3x(x  2) + 2 and 2x2 + 3x  12. a. Evaluate each expression for x = 2 and for x = 7. Based on your results, do you know whether the two expressions are equivalent? Explain.
from 0 at x = 0 to 9 at x = 3, then
decreases back to 0 at x = 6. Also,
For x = 2, each expression has a value of 2. For x = 7,
and 6, for x = 1 and 5, and for
suggest that the expressions may be equivalent but
the values are the same for x = 0
each expression has a value of 107. These results
x = 2 and 4.
do not prove that the expressions are equivalent.
14. The kinetic energy (in joules) of a moving object can be calculated from the expression _12mv2, where m is the mass of the object in kilograms and v is its speed in meters per second. Find the kinetic energy of a 0.145kg baseball that is thrown at a speed of 40 meters per second. E=
116
b. Evaluate each expression for x = 1. Based on your results, do you know whether the two expressions are equivalent? Explain.
For x = 1, the 1st expression has a value of 1 and the 2nd expression has a value of 7. Because
the values are different, the expressions are not
joules
equivalent.
15. The area of a square is given by x2, where x is the length of one side. Mary’s original garden was in the shape of a square. She has decided to double the area of her garden. Write an expression that represents the area of Mary’s new garden. Evaluate the expression if the side length of Mary’s original garden was 8 feet.
18. Critique Reasoning Marjorie evaluated the expression 3x + 2 for x = 5 as shown: 3x + 2 = 35 + 2 = 37 What was Marjorie’s mistake? What is the correct value of 3x + 2 for x = 5?
2(x2); 2(64) = 128 square feet
3x means that 3 should be multiplied by the value
12. Ramon wants to balance his checking account. He has $2,340 in the account. He writes a check for $140. He deposits a check for $268. How much does Ramon have left in his checking account?
of x; 17
$2,468 Lesson 11.2
6_MTXESE051676_U4M11L2.indd 305
InCopy Notes 1. This is a list
Work Area
FOCUS ON HIGHER ORDER THINKING
© Houghton Mifflin Harcourt Publishing Company
Name
EXTEND THE MATH
305
306
28/01/14 6:58 PM
InDesign Notes
PREAP 1. This is a list
Unit 4
6_MTXESE051676_U4M11L2.indd 306
InCopy Notes
1. This is a list Bold, Italic, Strickthrough. my.hrw.com Activity available online
25/10/12 4:22 PM
InDesign Notes 1. This is a list
Activity Harold has a globe that has a diameter of 10 inches. He builds a globe with a radius twice as long as his original globe. Use the formulas below to find the surface area and the volume of the new globe. Use π = 3.14. Explain your process and show your work. Surface Area (SA) of a sphere = 4πr2, where r is the radius of the sphere. Volume (V) of a sphere = __43 πr3, where r is the radius of the sphere. Sample answer: First, find the radius of the new globe. The diameter of the original globe is 10 inches, so its radius is 10 ÷ 2 = 5 inches. The radius of the new globe is twice as long: 5 inches × 2 = 10 inches. Then use r = 10 inches as the value to substitute for r in each formula. SA of new globe is 4πr2 ≈ 4(3.14)(10)2 ≈ 4(3.14)(100) ≈ 4(314) ≈ 1,256 in2; V of new globe is __43 πr3 ≈ __43 (3.14)(10)3 ≈ __43 (3.14)(1,000) ≈ __43 (3,140) ≈ 4,186 __23 in3
Evaluating Expressions
306
LESSON
11.3 Generating Equivalent Expressions Texas Essential Knowledge and Skills The student is expected to: Expressions, equations, and relationships—6.7.C Determine if two expressions are equivalent using concrete models, pictorial models, and algebraic representations.
Engage
ESSENTIAL QUESTION How can you identify and write equivalent expressions? Sample answer: Substitute the same value into each expression and compare the results, or simplify each expression to see if they are equivalent.
Motivate the Lesson
Ask: Are the expressions 3x + 8 and 2x + 14 equivalent expressions for x = 6? Begin the Explore Activity to find out.
Expressions, equations, and relationships—6.7.D Generate equivalent expressions using the properties of operations: inverse, identity, commutative, associative, and distributive properties.
Explore
Mathematical Processes
EXPLORE ACTIVITY 1
6.1.D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.
Mathematical Processes Focus on Critical Thinking Mathematical Processes Point out to students that even though expressions share an element, such as 5x, and include the same operation, they are not necessarily equivalent expressions. For example, x 2 and x 3 have the same base and look similar but they are not equivalent; x 2 = x · x and x 3 = x · x · x.
Explain EXPLORE ACTIVITY 2 Focus on Modeling Mathematical Processes Point out that the number and arrangement of the algebra tiles mirrors each expression. The first model shows three groups of 1 variable plus 2 ones, and the second model shows 3 variables plus 6 ones. Discuss with students why the two algebraic expressions are equivalent. Questioning Strategies
Mathematical Processes • Earlier, you used counters to make models. How are algebra tiles similar to counters? How are they different? Both counters and algebra tiles are used the same way, one tile or counter for each number being added. The difference is that algebra tiles have an x (variable) tile used to represent an unknown quantity, while counters represent +1 or 1 only. • Which two characteristics do you look for in the models to decide whether the two expressions are equivalent? The models of both expressions should have 1) an equal number of x tiles and 2) an equal number of +1tiles or 1tiles.
YOUR TURN Avoid Common Errors Remind students to arrange the tiles in the same groups and order as shown in the expressions. Then compare the number of x tiles and the number of +1tiles or –1tiles in each model to check whether the expressions are equivalent.
307
Lesson 11.3
Generating Equivalent Expressions
LESSON
11.3 ?
EXPLORE ACTIVITY 1 (cont’d)
Expressions, equations, and relationships— 6.7.D Generate equivalent expressions using the properties of operations: inverse, identity, commutative, associative, and distributive properties. Also 6.7.C
Reflect 1.
Evaluate the expressions for a different value of x; for
ESSENTIAL QUESTION
example, when x = 1, 2x = 2 and x2 = 1.
How can you identify and write equivalent expressions?
EXPLORE ACTIVITY 1
Error Analysis Lisa evaluated the expressions 2x and x2 for x = 2 and found that both expressions were equal to 4. Lisa concluded that 2x and x2 are equivalent expressions. How could you show Lisa that she is incorrect?
6.7.C
EXPLORE ACTIVITY 2
Identifying Equivalent Expressions One way to test whether two expressions might be equivalent is to evaluate them for the same value of the variable.
Modeling Equivalent Expressions
List B
5x + 65
5x + 1
5(x + 1)
5x + 5
1 + 5x
5(13 + x)
Algebra Tiles
You can also use models to determine if two expressions are equivalent. Algebra tiles are one way to model expressions.
Match the expressions in List A with their equivalent expressions in List B. List A
6.7.C
=1 = 1 =x
Determine if the expression 3(x + 2) is equivalent to 3x + 6.
A Model each expression using algebra tiles. 3(x + 2)
3x + 6
A Evaluate each of the expressions in the lists for x = 3.
© Houghton Mifflin Harcourt Publishing Company
5(3) + 65 = 5(3 + 1) = 1 + 5(3) =
List B 5(3) + 1 =
16
20
5(3) + 5 =
20
16
5(13 + 3) =
80
3
B The model for 3(x + 2) has The model for 3x + 6 has
80
3
6
x tiles and x tiles and
6
1 tiles. 1 tiles.
C Is the expression 3(x + 2) equivalent to 3x + 6? Explain.
Yes; each expression is represented by 3 x tiles and
B Which pair(s) of expressions have the same value for x = 3?
5(x + 1) and 5x + 5; 1 + 5x and 5x + 1; 5x + 65 and
6 1 tiles.
5(13 + x)
Reflect
C How could you further test whether the expressions in each pair are equivalent?
2.
Use algebra tiles to determine if 2(x  3) is equivalent to 2x  3. Explain your answer.
© Houghton Mifflin Harcourt Publishing Company
List A
No; 2(x  3) is represented by 2 x tiles and 6 1 tiles.
Sample answer: Evaluate for several other values of x.
2x  3 is represented by 2 x tiles and 3 1 tiles.
D Do you think the expressions in each pair are equivalent? Why or why not?
Sample answer: Yes; it appears that they will always have the same value. Lesson 11.3
307
308
Unit 4
PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.D, which calls for students to “communicate mathematical ideas… using multiple representations, including symbols, diagrams…and language as appropriate.” In this lesson’s Explore Activities and Examples, students use words and operational symbols as well as algebra tiles to identify, represent, and compare algebraic expressions and to generate equivalent expressions.
Math Background Algebraic expressions are equivalent if they simplify to the same value. The Commutative, Associative, Distributive, and Identity properties give rules about how to rewrite an expression without changing its value. x + (3x + 2) + (2x + 3) x + 3x + (2 + 2x) + 3 x + 3x + (2x + 2) + 3 (x + 3x + 2x) + (2 + 3) x(1 + 3 + 2) + (2 + 3) x(6) + (5) 6x + 5
Associative Commutative Associative Distributive Addition Commutative
Generating Equivalent Expressions
308
ADDITIONAL EXAMPLE 1 Use a property to write an expression that is equivalent to n × 5. Tell which property you used. 5 × n; Commutative Property of Multiplication Interactive Whiteboard Interactive example available online my.hrw.com
EXAMPLE 1 Questioning Strategies
Mathematical Processes • How does the Commutative Property of Addition allow you to rewrite an expression without changing its value? You can change the order of the terms in an addition expression. • How do you know which properties of operations may help you identify equivalent expressions? Look at the operational symbols that appear in the given expressions. Apply properties of operations that relate to those symbols.
Engage with the Whiteboard Have students circle the operational symbol in Example 1 and write an equivalent expression on the whiteboard. Then have students draw a model of each expression, using algebra tiles to show that the expressions are equivalent.
Focus on Communication
Mathematical Processes Make sure students understand that the properties of operations are rules about how to rewrite expressions by rearranging and combining terms without changing the value of the expression.
YOUR TURN Avoid Common Errors If students have difficulty determining which property to use, remind them to begin by identifying the operation used in the given expression. Then they should look at the list of properties to see which properties apply to that operation. Point out that a given expression may have more than one equivalent expression, as more than one property can be applied.
ADDITIONAL EXAMPLE 2 Use the properties of operations to determine if the expressions are equivalent. A 6 + y; __12 (12 + y) B 2(y  3); 2y  6
not equivalent equivalent
Interactive Whiteboard Interactive example available online my.hrw.com
Animated Math Equivalent Expressions Students explore equivalent expressions using an interactive model. my.hrw.com
309
Lesson 11.3
EXAMPLE 2 Questioning Strategies
Mathematical Processes • In A, how does the Distributive Property enable you to determine that the expressions are equivalent without evaluating them? The Distributive Property states that multiplying a number by a difference, as in 3(x  2), is the same as multiplying each number in the difference and subtracting the products: (3)(x)  (3)(2) = 3x  6.
YOUR TURN Engage with the Whiteboard For Exercises 5–6, have students use algebra tiles to draw a model of each expression on the whiteboard. Then have students explain whether the expressions are equivalent or not.
DO NOT EDITChanges must be made through “File info” CorrectionKey=B
DO NOT EDITChanges must be made through “File info” CorrectionKey=A
Identifying Equivalent Expressions Using Properties
Writing Equivalent Expressions Using Properties Properties of operations can be used to identify equivalent expressions. Examples 3+4=4+3
Commutative Property of Multiplication: When multiplying, changing the order of the numbers does not change the product.
2×4=4×2
Associative Property of Addition: When adding more than two numbers, the grouping of the numbers does not change the sum.
(3 + 4) + 5 = 3 + (4 + 5)
Associative Property of Multiplication: When multiplying more than two numbers, the grouping of the numbers does not change the product.
(2 × 4) × 3 = 2 × (4 × 3)
Identity Property of Multiplication: Multiplying a number by one does not change its value.
1×7=7
Inverse Property of Addition: The sum of a number and its opposite, or additive inverse, is zero.
 3 + 3 = 0
© Houghton Mifflin Harcourt Publishing Company
EXAMPL 1 EXAMPLE
Math Talk
What property can you use to write an expression that is equivalent to 0 + c? What is the equivalent expression?
Identity Property of Addition; 0+c=c
Sample answer: Represent 3 (x  2) as three groups of x  2 with each group showing one x tile and two  1 tiles. To represent the Distributive Property, regroup the tiles to represent 3x  6 by showing a group of three x tiles and a group of six  1 tiles.
You can use the Commutative Property of Addition to write an equivalent expression: x + 3 = 3 + x
YOUR TURN For each expression, use a property to write an equivalent expression. Tell which property you used. Sample answers given.
a(bc); Associative Property of Multiplication
Personal Math Trainer
Personal Math Trainer
Online Assessment and Intervention
Online Assessment and Intervention
my.hrw.com
my.hrw.com
Lesson 11.3
6_MTXESE051676_U4M11L3.indd 309
309
310
1/29/14 10:45 PM
Distributive Property Commutative Property
YOUR TURN Use the properties of operations to determine if the expressions are equivalent. 5.
6x  8; 2(3x  5)
2(3x  5) = 6x  10;
6.
7.
2  2 + 5x; 5x
2  2 + 5x = 5x; equivalent
not equivalent
Jamal bought 2 packs of stickers and 8 individual stickers. Use x to represent the number of stickers in a pack of stickers and write an expression to represent the number of stickers Jamal bought. Is the expression equivalent to 2(4 + x)? Check your answer with algebra tile models.
Jamal bought 2x + 8 stickers. 2(4 + x) = 8 + 2x = 2x + 8; yes
Unit 4
6_MTXESE051676_U4M11L3 310
InCopy Notes
DIFFERENTIATE INSTRUCTION
Distributive Property
1 (4 + x) B 2 + x; __ 2 1x+ 2 1 (x + 4) = __ __ lk Ta th Ma 2 2 Mathematical Processes 1x = 2 + __ 2 Explain how you could use algebra tiles to 1 x. 2 + x does not equal 2 + __ represent the Distributive 2 Property in A. They are not equivalent expressions.
Mathematical Processes
6.7.D
(3 + 4)y; Distributive Property
3(x  2) = 3x  6
3(x 3  2) and 3x  6 are equivalent expressions.
my.hrw.com my.h
The operation in the expression is addition.
4. 3y + 4y =
6.7.C
A 3(x  2); 3x  6
Use a property to write an expression that is equivalent to x + 3.
3. (ab)c =
EXAMPLE 2
1. This is a list Bold, Italic, Strickthrough.
10/25/12 8:35 PM
InDesign Notes 1. This is a list
Visual Cues
Cognitive Strategies
Additional Resources
Point out to students that it is often helpful to use colored pencils to identify like terms before combining them.
A fun way for students to remember how to combine like terms is to name the variable part of like terms. For example, for the expression 2a + 5b + 4a, students can name variable a apples and variable b bananas. Thus, 2 apples + 5 bananas + 4 apples = 6 apples + 5 bananas = 6a + 5b.
Differentiated Instruction includes: • Reading Strategies • Success for English Learners ELL • Reteach • Challenge PREAP
Have students identify the like terms in the following expressions. 1. a + 2b + 2a + b + 2c 2b and b
Like terms: a and 2a,
© Houghton Mifflin Harcourt Publishing Company
9+0=9
my.hrw.com
Animated Math
6(2 + 4) = 6(2) + 6(4) 8(5  3) = 8(5)  8(3)
Identity Property of Addition: Adding zero to a number does not change its value.
Math On the Spot
my.hrw.com
Use the properties of operations to determine if the expressions are equivalent.
Properties of Operations Commutative Property of Addition: When adding, changing the order of the numbers does not change the sum.
Distributive Property: Multiplying a number by a sum or difference is the same as multiplying by each number in the sum or difference and then adding or subtracting.
Math On the Spot
2. 18 + 2d 3 + 5d + 3d 3  2d 2 Like terms: 2d 3 and 3d 3 3. 5x 3 + 3y + 7x 3  2y  4x 2 Like terms: 5x 3 and 7x 3, 3y and 2y
Generating Equivalent Expressions
310
ADDITIONAL EXAMPLE 3 Combine like terms.
EXAMPLE 3 Connect Vocabulary
A 8y2  3y2 5y2 B 4m + 3(n + 7m)
25m + 3n
C x + 8y  5y + 5x
6x + 3y
Interactive Whiteboard Interactive example available online my.hrw.com
ELL
Stress the use of correct mathematical terminology. The parts of an expression that are separated by + or  signs, such as 3x and 5x in the expression 3x + 5x, are called terms. Terms that have identical variable parts are like terms. In the expression 3x + 5x, 3x and 5x are like terms. The properties of operations allow you to rearrange and combine like terms.
Questioning Strategies
Mathematical Processes • In B, why do you have to apply the Distributive Property before adding like terms? Because of the order of operations, you need to multiply before you can add. • In C, why does y + 7y equal 8y? Because y is 1y, so 1y + 7y = (1 + 7)y = 8y.
YOUR TURN Avoid Common Errors Exercise 10 Students may neglect to add single variables. Remind them that b is 1b, so adding b to another bterm increases the coefficient by 1.
Elaborate Talk About It Summarize the Lesson Ask: How do properties of operations help you to write equivalent expressions? Properties of operations allow me to write an expression in different ways without changing its value. I can use the properties to form equivalent expressions by regrouping, reordering, and combining terms.
GUIDED PRACTICE Engage with the Whiteboard For Exercise 1, have students rewrite each expression on the whiteboard by substituting 5 for y and then simplifying the expressions. For Exercise 2, ask a student to circle the part of the model that shows that the two expressions are not equivalent on the whiteboard.
Avoid Common Errors Exercises 3–4 If students have difficulty determining which property to use, remind them to begin by identifying the operation used in the given expression. Then they should look at the list of properties to see which properties apply to that operation. Exercises 5–6 Remind students that when they apply the Distributive Property they must distribute the constant or variable that is outside the parentheses to each term that is inside the parentheses. Exercise 8 Students may neglect to add single variables. Remind them that x is 1x, so adding x to another xterm decreases the coefficient by 1.
311
Lesson 11.3
Generating Equivalent Expressions
YOUR TURN
Parts of an algebraic expression
coefficients like terms
12 + 3y2 + 4x + 2y2 + 4 12 + 3y2 + 4x + 2y2 + 4
Math On the Spot
Online Assessment and Intervention
my.hrw.com
my.hrw.com
EXAMPL 3 EXAMPLE
4 + 4y = 4(y − 1) =
2
6x and 4x are like terms.
Subtract inside the parentheses.
= 2x2
Commutative Property of Multiplication
B 3a + 2(b + 5a)
© Houghton Mifflin Harcourt Publishing Company
4y + 1 =
Distributive Property
= x2(2)
= 3a + 2b + (2 · 5)a
Distributive Property
= 3a + 2b + 10a
Associative Property of Multiplication Multiply 2 and 5.
= 3a + 10a + 2b
Commutative Property of Addition
= (3 + 10)a + 2b
Distributive Property
= 13a + 2b
Add inside the parentheses.
Math Talk
Mathematical Processes
Write 2 terms that can be combined with 7y4.
Sample answers: y4 and 6y4
y + 11x  7x + 7y = y + 7y + 11x  7x
List B
24 16 21
For each expression, use a property to write an equivalent expression. Tell which property you used. (Example 1) Sample 3. ab =
Distributive Property Commutative Property
ba
+
x4
2(x  2)
+
+
answers are given. 5(3x)  5(2)
4. 5(3x − 2) =
Distributive Prop.
Commutative Prop. of Mult.
not equivalent
6. _12(6x − 2); 3 − x
not equivalent
Combine like terms. (Example 3) 7. 32y + 12y =
= 8y + 4x
4(y + 1) = 1 + 4y =
16 24 21
2. Determine if the expressions are equivalent by comparing the models. (Explore Activity 2) not equivalent
5. _12(4 − 2x); 2  2x
y and 7y are like terms; 11x and 7x are like terms. Commutative Property
= y(1 + 7) + x(11  7)
4y − 4 =
Use the properties of operations to determine if each pair of expressions is equivalent. (Example 2)
3a + 2(b + 5a) = 13a + 2b
C y + 11x  7x + 7y
8m + 2 + 4n
2a5 + 5b
List A
A 6x2  4x2
3a + 2(b + 5a) = 3a + 2b + 2(5a)
11. 8m + 14  12 + 4n =
10. 4a5  2a5 + 4b + b =
1. Evaluate each of the expressions in the list for y = 5. Then, draw lines to match the expressions in List A with their equivalent expressions in List B. (Explore Activity 1)
Combine like terms.
6x2  4x2 = 2x2
10x2  4
Guided Practice
6.7.D
6x2  4x2 = x2 (6  4)
9. 6x2 + 4(x2  1) =
12 + 3y2 + 4x + 2y2 + 4
When an expression contains like terms, you can use properties to combine the like terms into a single term. This results in an expression that is equivalent to the original expression.
2
5y
8. 8y  3y =
? ?
44y
8. 12 + 3x − x − 12 =
2x
ESSENTIAL QUESTION CHECKIN
© Houghton Mifflin Harcourt Publishing Company
The parts of the expression that are separated by + or  signs Numbers that are multiplied by at least one variable Terms with the same variable(s) raised to the same power(s)
terms
Combine like terms.
Personal Math Trainer
9. Describe two ways to write equivalent algebraic expressions.
Use properties of operations or combine like terms
y + 11x  7x + 7y = 8y + 4x
Lesson 11.3
311
312
Unit 4
Generating Equivalent Expressions
312
Personal Math Trainer Online Assessment and Intervention
Online homework assignment available
Evaluate GUIDED AND INDEPENDENT PRACTICE 6.7.C, 6.7.D
my.hrw.com
Concepts & Skills
Practice
Explore Activity 1 Identifying Equivalent Expressions
Exercise 1
Explore Activity 2 Modeling Equivalent Expressions
Exercises 2, 14
Example 1 Writing Equivalent Expressions Using Properties
Exercises 3–4, 10–13, 27–29
Use the properties of operations to determine if the expressions are equivalent.
Example 2 Identifying Equivalent Expressions Using Properties
Exercises 5–6, 25
2. 3 + y; __13 (6 + y)
Example 3 Generating Equivalent Expressions
Exercises 7–8, 15–24, 26
11.3 LESSON QUIZ 6.7.D 1. Use one of the properties of operations to write an expression that is equivalent to 4 + m. Tell which property you used.
3. 4(y  3); 4y  12 Combine like terms. 4. 9y2  6y2
Exercise
5. 5c + 4(d+ 6c)
my.hrw.com
Answers
1. m + 4; Commutative Property of Addition 2. not equivalent 3. equivalent 4. 3y2 5. 29c + 4d 6. 5x + 6y
313
Lesson 11.3
1.C Select tools
3 Strategic Thinking
1.G Explain and justify arguments
26–27
2 Skills/Concepts
1.A Everyday life
28–29
2 Skills/Concepts
1.D Multiple representations
30
3 Strategic Thinking
1.F Analyze relationships
31
3 Strategic Thinking
1.G Explain and justify arguments
32
3 Strategic Thinking
1.F Analyze relationships
25
Lesson Quiz available online
Mathematical Processes
2 Skills/Concepts
10–24
6. x + 10y  4y + 4x
Depth of Knowledge (D.O.K.)
Additional Resources Differentiated Instruction includes: • Leveled Practice Worksheets
Class
Date
11.3 Independent Practice
Personal Math Trainer
6.7.D, 6.7.C
my.hrw.com
27. Multiple Representations Use the information in the table to write and simplify an expression to find the total weight of the medals won by the top medalwinning nations in the 2012 London Olympic Games. The three types of medals have different weights.
Online Assessment and Intervention
2012 Summer Olympics
For each expression, use a property to write an equivalent expression. Tell which property you used. Sample answers given.
dc
10. cd =
12. 4(2x − 3) =
13 + x
11. x + 13 =
Commutative Prop. of Mult.
Commutative Prop. of Addition
4(2x)  4(3)
Distributive Prop.
Silver 29 27 17
113g + 73s + 71b
Associative Prop. of Addition
Write an expression for the perimeters of each given figure. Simplify the expressions.
14. Draw algebra tile models to prove that 4 + 8x and 4(2x + 1) are equivalent.
6x + 10 mm
28.
36.4 + 4x in.
29.
10.2 in.
3x  1 mm 6 mm
4(2x + 1)
4 + 8x
x + 4 in. x + 4 in.
© Houghton Mifflin Harcourt Publishing Company
17. 6b + 7b − 10 =
13b  10
16. 32y + 5y =
37y
18. 2x + 3x + 4 =
5x + 4 6a2 + 16
19. y + 4 + 3(y + 2) =
4y + 10
20. 7a2 − a2 + 16 =
21. 3y2 + 3(4y2  2) =
15y2  6
22. z2 + z + 4z3 + 4z2 =
23. 0.5(x4  3) + 12 =
0.5x + 10.5 4
FOCUS ON HIGHER ORDER THINKING
a. Write two equivalent expressions for
4z3 + 5z2 + z
the model.
The expressions become 6  9x and 3(2  3x).
Yes; Applying the Associative Property of Addition to 3x + 12  2x
31. Communicate Mathematical Ideas Write an example of an expression that cannot be simplified, and explain how you know that it cannot be simplified.
you get 3x  2x + 12 which is equivalent to 1x + 12. Then apply
3x2  4x + 7; It does not have any like terms.
the Identity Property of Multiplication to get x + 12.
32. Problem Solving Write an expression that is equivalent to 8(2y + 4) that can be simplified.
26. William earns $13 an hour working at a movie theater. Last week he worked h hours at the concession stand and three times as many hours at the ticket counter. Write and simplify an expression for the amount of money William earned last week.
Sample answer: 2(8y + 16)
13h + 13(3h) = 13h + 39h = 52h Lesson 11.3
PREAP
4  6x; 2(2  3x)
b. What If? Suppose a third row of tiles identical to the ones above is added to the model. How does that change the two expressions?
25. Justify Reasoning Is 3x + 12  2x equivalent to x + 12? Use two properties of operations to justify your answer.
EXTEND THE MATH
Work Area
30. Problem Solving Examine the algebra tile model.
4+p
24. _14 (16 + 4p) =
x + 4 in. 10.2 in.
Combine like terms.
2x4
x + 4 in.
6 mm 3x  1 mm
15. 7x4 − 5x4 =
Bronze 29 23 19
(46 + 38 + 29)g + (29 + 27 + 17)s + (29 + 23 + 19)b;
(2 + a) + b
13. 2 + (a + b) =
Gold 46 38 29
United States China Great Britain
)PVHIUPO.JGGMJO)BSDPVSU1VCMJTIJOH$PNQBOZt*NBHF$SFEJUTª$PNTUPDL +VQJUFSJNBHFT(FUUZ*NBHFT
Name
313
Activity available online
314
Unit 4
my.hrw.com
Activity During a basketball game, Joelle scored 8 points on free throws. She also scored 2 points for each inside shot and 3 points for each outside shot she made. Joelle made n inside shots and s outside shots during the game. Write six equivalent expressions for the total number of points Joelle scored. Which properties of operations did you use to identify equivalent expressions? Sample answer: Commutative Property of Addition: 8 + 2n + 3s, 8 + 3s + 2n, 2n + 3s + 8, 2n + 8 + 3s; Distributive Property and Commutative Property of Addition: 2(4 + n) + 3s, 3s + 2(4 + n)
Generating Equivalent Expressions
314
DO NOT EDITChanges must be made through “File info” CorrectionKey=B
Module Quiz
Ready to Go On?
Ready
Assess Mastery
11.1 Modeling Equivalent Expressions
Personal Math Trainer
Write each phrase as an algebraic expression.
Use the assessment on this page to determine if students have mastered the concepts and standards covered in this module.
p __ 6
1. p divided by 6
185 + h
3. the sum of 185 and h 3
Response to Intervention
2 1
4 seasons.
4. the product of 16 and g
6. 8p; p = 9
4x
72
8. 4(d + 7); d = 2
20
7. 11 + r; r = 7 60 9. ____ m ;m=5
of a triangle with a base of 6 and a height of 8?
Differentiated Instruction
Differentiated Instruction
11.3 Generating Equivalent Expressions
• Reteach worksheets
• Challenge worksheets
11. Draw lines to match the expressions in List A with their equivalent expressions in List B.
• Success for English Learners ELL
ELL
PREAP
Extend the Math PREAP Lesson Activities in TE
Additional Resources Assessment Resources includes: • Leveled Module Quizzes
18 12
10. To find the area of a triangle, you can use the expression b × h ÷ 2, where b is the base of the triangle and h is its height. What is the area
Online and Print Resources
• Reading Strategies
16g
Evaluate each expression for the given value of the variable.
© Houghton Mifflin Harcourt Publishing Company
my.hrw.com
2. 65 less than j
11.2 Evaluating Expressions
Enrichment
Access Ready to Go On? assessment online, and receive instant scoring, feedback, and customized intervention or enrichment.
Online Assessment and Intervention
my.hrw.com
j  65
5. Let x represent the number of television show episodes that are taped in a season. Write an expression for the number of episodes taped in
Intervention
Personal Math Trainer
Online Assessment and Intervention
ESSENTIAL QUESTION
24 square units
List A 7x + 14 7 + 7x 7( x  1 )
List B 7( 1 + x ) 7x  7 7( x + 2 )
12. How can you determine if two algebraic expressions are equivalent?
Sample answer: Model the expressions with bar models or algebra tiles to determine if the expressions are equivalent; write equivalent expressions using properties of operations, order of operations, and combining like terms.
Module 11
6_MTXESE051676_U4M11RT.indd 315
Texas Essential Knowledge and Skills Lesson
Exercises
11.1
1–5
6.7.C
11.2
6–10
6.7.C
11.3
11
6.7.D
315
Module 11
TEKS
315
28/01/14 7:18 PM
DO NOT EDITChanges must be made through “File info” CorrectionKey=B
Texas Test Prep
Texas Testing Tip Students can circle or underline key words and phrases to identify important information. Item 1 Students should underline the word product. Product means the operation used is multiplication. Thus students can quickly see that choice C is the correct answer. Item 3 Students should underline the phrase divided them evenly between p pages. Since the operation used is division, students can eliminate choices A and B, and then check the order of division in the two remaining answer choices to reveal that choice D is the correct answer.
Selected Response 1. Which expression represents the product of 83 and x? A 83 + x
Item 5 Some students may forget that 7w means the product of 7 and the variable and mistakenly substitute 9 for w to make the number 79. Remind students that the variable in a term is multiplied by the coefficient. Item 6 Caution students to read the question carefully. Some students may not realize that this question is asking for how many pages Katie has left to read and may indicate that choice D is correct.
83x
A the product of r and 9 B the quotient of r and 9
9 less than r
3. Rhonda was organizing photos in a photo album. She took 60 photos and divided them evenly among p pages. Which algebraic expression represents the number of photos on each page? p
A p  60
C
B 60  p
60 D __ p
__ 60
4. Using the algebraic expression 4n + 6, what is the greatest wholenumber value of n that will give you a result less than 100? A 22
C
B 23
D 25
C
49
D 77
316
7. The expression 12(x + 4) represents the total cost of CDs Mei bought in April and May at $12 each. Which property is applied to write the equivalent expression 12x + 48? A Associative Property of Addition
Commutative Property of Multiplication
C
D Distributive Property
Gridded Response 8. When traveling in Europe, Bailey converts the temperature given in degrees Celsius to a Fahrenheit temperature by using the expression 9x ÷ 5 + 32, where x is the Celsius temperature. Find the temperature in degrees Fahrenheit when it is 15 °C.
.
5
9
0
0
0
0
0
0
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
4
4
4
4
4
4
5
5
5
5
5
5
6
6
6
6
6
6
7
7
7
7
7
7
8
8
8
8
8
8
9
9
9
9
9
9
24
5. Evaluate 7w  14 for w = 9. B 18
200
B Associative Property of Multiplication
D r more than 9
A 2
A 40
D 250
D 83  x
Avoid Common Errors
6. Katie has read 32% of a book. If she has read 80 pages, how many more pages does Katie have left to read?
C
2. Which phrase describes the algebraic expression _9r ?
C
Online Assessment and Intervention
B 170
B 83 ÷ x
C
my.hrw.com
© Houghton Mifflin Harcourt Publishing Company
Texas Test Prep
Personal Math Trainer
Module 11 MIXed ReVIeW
Unit 4
6_MTXESE051676_U4M11RT.indd 316
1/29/14 10:53 PM
Texas Essential Knowledge and Skills Items
Grade 6 TEKS
Mathematical Process TEKS
1
6.7.C
6.1.D
2
6.7.C
6.1.D
3
6.7.C
6.1.A, 6.1.D
4
6.7.A
6.1.F
5
6.7.D
6*
6.5.A
6.1.A
7
6.7.D
6.1.A
8
6.7.D
6.1.A, 6.1.D
* Item integrates mixed review concepts from previous modules or a previous course.
Generating Equivalent Algebraic Expressions
316
Equations and Relationships ?
ESSENTIAL QUESTION How can you use equations and relationships to solve realworld problems?
MODULE
You can model realworld problems with equations, then use algebraic rules to solve the equations.
12
LESSON 12.1
Writing Equations to Represent Situations 6.7.B, 6.9.A, 6.10.B
LESSON 12.2
Addition and Subtraction Equations 6.9.B, 6.9.C, 6.10.A
LESSON 12.3
Multiplication and Division Equations
© Houghton Mifflin Harcourt Publishing Company • Image Credits: JoeFox/Alamy
6.9.B, 6.9.C, 6.10.A
RealWorld Video
my.hrw.com
my.hrw.com
317
Module 12
People often attempt to break World Records. To model how many seconds faster f an athlete's time a seconds must be to match a record time r seconds, write the equation f = a  r.
my.hrw.com
Math On the Spot
Animated Math
Personal Math Trainer
Go digital with your writein student edition, accessible on any device.
Scan with your smart phone to jump directly to the online edition, video tutor, and more.
Interactively explore key concepts to see how math works.
Get immediate feedback and help as you work through practice sets.
317
Are YOU Ready?
Are You Ready? Use the assessment on this page to determine if students need intensive or strategic intervention for the module’s prerequisite skills. 3 2 1
Personal Math Trainer
Complete these exercises to review skills you will need for this chapter.
Assess Readiness
Evaluate Expressions EXAMPLE
Response to Intervention
Online Assessment and Intervention
my.hrw.com
Evaluate 8(3+2)  52 8(3+2)  52 = 8(5)  52 = 8(5)  25 = 40  25 = 15
Perform operations inside parentheses first. Evaluate exponents. Multiply. Subtract.
Evaluate the expression.
Online Assessment and Intervention
my.hrw.com
Online and Print Resources Skills Intervention worksheets
Differentiated Instruction
• Skill 54 Evaluate Expressions
• Challenge worksheets
7. 2(8 + 3) + 42
38
9. 8(2 +1)2  42
56
39
4. 6(8  3) + 3(7  4)
1
5. 10(6  5)  3(9  6)
64
2. 8(2 + 4) + 16
5
3. 3(14  7)  16
Access Are You Ready? assessment online, and receive instant scoring, feedback, and customized intervention or enrichment.
Personal Math Trainer
29
1. 4(5 + 6)  15
Enrichment
29
6. 7(4 + 5 + 2)  6(3 + 5) 8. 7(14  8)  62
6
Connect Words and Equations EXAMPLE
PREAP
• Skill 56 Connect Words and Extend the Math PREAP Equations Lesson Activities in TE
The product of a number and 4 is 32. The product of x and 4 is 32. Represent the unknown with a variable. Determine the operation. 4 × x is 32. 4 × x = 32. Determine the placement of the equal sign.
© Houghton Mifflin Harcourt Publishing Company
Intervention
Write an algebraic equation for the word sentence.
10. A number increased by 7.9 is 8.3.
x + 7.9 = 8.3
12. The quotient of a number and 8 is 4.
x÷8=4
14. The difference between 31 and a number is 7.
318
31  x = 7
11. 17 is the sum of a number and 6.
17 = x + 6
13. 81 is three times a number.
81 = 3x 15. Eight less than a number is 19.
x  8 = 19
Unit 4
PROFESSIONAL DEVELOPMENT VIDEO my.hrw.com
Author Juli Dixon models successful teaching practices as she explores equation concepts in an actual sixthgrade classroom.
Online Teacher Edition Access a full suite of teaching resources online—plan, present, and manage classes and assignments.
Professional Development
ePlanner Easily plan your classes and access all your resources online.
my.hrw.com
Interactive Answers and Solutions Customize answer keys to print or display in the classroom. Choose to include answers only or full solutions to all lesson exercises.
Interactive Whiteboards Engage students with interactive whiteboardready lessons and activities. Personal Math Trainer: Online Assessment and Intervention Assign automatically graded homework, quizzes, tests, and intervention activities. Prepare your students with updated, TEKSaligned practice tests.
Equations and Relationships
318
DO NOT EDITChanges must be made through "File info" CorrectionKey=B
Reading StartUp
Reading StartUp Have students complete the activities on this page by working alone or with others.
Visualize Vocabulary Use the ✔ words to complete the graphic.
Visualize Vocabulary The main idea web will help students review vocabulary and concepts related to algebraic expressions. Ask students to think of other algebraic expressions and their components and share them with the class, making sure to identify the variable, coefficient, terms, and constant in each expression.
4x + 5
Use the following explanations to help students learn the preview words. The words expression and equation are not synonyms. An algebraic expression is a mathematical statement that contains one or more variables. An equation is a mathematical statement stating that two expressions are equal. You evaluate an expression and solve an equation.
Students can use these reading and notetaking strategies to help them organize and understand new concepts and vocabulary. c.4.D Use prereading supports such as graphic organizers, illustrations, and pretaught topicrelated vocabulary to enhance comprehension of written text.
Review Words ✔ algebraic expression (expresión algebraica) ✔ coefficient (coeficiente) ✔ constant (constante) evaluating (evaluar) like terms (términos semejantes) ✔ term (término, en una expresión) ✔ variable (variable)
Preview Words
5
4x and 5
constant
terms
Understand Vocabulary
equation (ecuación) equivalent expression (expresión equivalente) properties of operations (propiedades de las operaciones) solution (solución)
Match the term on the left to the correct expression on the right.
© Houghton Mifflin Harcourt Publishing Company
Integrating the ELPS
x
variable
algebraic expression
Understand Vocabulary
Active Reading
4
coefficient
Vocabulary
1. algebraic expression
A. A mathematical statement that two expressions are equal.
2. equation
B. A value of the variable that makes the statement true.
3. solution
C. A mathematical statement that includes one or more variables.
Active Reading
Additional Resources
Booklet Before beginning the module, create a booklet to help you learn the concepts in this module. Write the main idea of each lesson on each page of the booklet. As you study each lesson, write important details that support the main idea, such as vocabulary and formulas. Refer to your finished booklet as you work on assignments and study for tests.
Differentiated Instruction • Reading Strategies ELL Module 12
6_MTXESE051676_U4MO12.indd 319
28/01/14 7:25 PM
Grades 6–8 TEKS Before Students understand: • operations with rational numbers • properties of operations: inverse, identity, commutative, associative, and distributive properties
319
Module 12
In this module Students will learn to: • write onevariable, onestep equations to represent constraints or conditions within problems • model and solve onevariable, onestep equations that represent problems • write corresponding realworld problems given onevariable, onestep equations
319
After Students will learn how to: • write onevariable, twostep equations to represent realworld problems • write a realworld problem to represent a onevariable, twostep equation • solve onevariable, twostep equations
MODULE 12
Unpacking the TEKS
Unpacking the TEKS Understanding the TEKS and the vocabulary terms in the TEKS will help you know exactly what you are expected to learn in this module.
6.9.A Write onevariable, onestep equations and inequalities to represent constraints or conditions within problems.
Texas Essential Knowledge and Skills Content Focal Areas Expressions, equations, and relationships—6.7 The student applies mathematical process standards to develop concepts of expressions and equations.
Expressions, equations, and relationships—6.10 The student applies mathematical process standards to use equations and inequalities to solve problems.
You will learn to write an equation or inequality to represent a situation.
Key Vocabulary
UNPACKING EXAMPLE 6.9.A
equation (ecuación) A mathematical sentence that shows that two expressions are equivalent.
The Falcons won their football game with a score of 30 to 19. Kevin scored 12 points for the Falcons. Write an equation to determine how many points Kevin’s teammates scored.
inequality (desigualdad) A mathematical sentence that shows the relationship between quantities that are not equal.
Expressions, equations, and relationships—6.9 The student applies mathematical process standards to use equations and inequalities to represent situations.
What It Means to You
6.10.B Determine if the given value(s) make(s) onevariable, onestep equations or inequalities true.
Integrating the ELPS
Kevin’s points
+
Teammates’ points
=
Total points
12
+
t
=
30
What It Means to You You can substitute a given value for the variable in an equation or inequality to check if that value makes the equation or inequality true. UNPACKING EXAMPLE 6.10.B
Melanie bought 6 tickets to a play. She paid a total of $156. Write an equation to determine whether each ticket cost $26 or $28.
c.4.F Use visual and contextual support … to read gradeappropriate
content area text … and develop vocabulary … to comprehend increasingly challenging language.
Number of tickets bought 6
· ·
Price per ticket p
=
Total cost
=
156
Substitute 26 and 28 for p to see which equation is true.
Go online to see a complete unpacking of the .
Visit my.hrw.com to see all the unpacked.
6p = 156
6p = 156
? 156 6 · 26 =
? 156 6 · 28 =
? 156 ✓ 156 =
© Houghton Mifflin Harcourt Publishing Company • Image Credits: Robert Llewellyn/Corbis Super RF/Alamy Limited
Use the examples on this page to help students know exactly what they are expected to learn in this module.
? 156 ✗ 168 =
The cost of a ticket to the play was $26. my.hrw.com
my.hrw.com
Grade 6 TEKS
320
Lesson 12.1
Lesson 12.2
Unit 4
Lesson 12.3
6.7.B Distinguish between expressions and equations verbally, numerically, and algebraically. 6.9.A Write onevariable, onestep equations and inequalities to represent constraints or conditions within problems. 6.9.B Represent solutions for onevariable, onestep equations and inequalities on number lines. 6.9.C Write corresponding realworld problems given onevariable, onestep equations or inequalities. 6.10.A Model and solve onevariable, onestep equations and inequalities that represent problems, including geometric concepts. 6.10.B Determine if the given value(s) make(s) onevariable, onestep equations or inequalities true.
Equations and Relationships
320
LESSON
12.1 Writing Equations to Represent Situations Texas Essential Knowledge and Skills The student is expected to: Expressions, equations, and relationships—6.9.A Write onevariable, onestep equations and inequalities to represent constraints or conditions within problems. Expressions, equations, and relationships—6.7.B Distinguish between expressions and equations verbally, numerically, and algebraically. Expressions, equations, and relationships—6.10.B Determine if the given value(s) make(s) onevariable, onestep equations or inequalities true.
Mathematical Processes 6.1.D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.
ADDITIONAL EXAMPLE 1 Determine whether the given value is a solution of the equation. A x + 15 = 10; x = 5 m B __ = 11; m = 33 3
C 5n = 42; x = 7
yes no
no
Interactive Whiteboard Interactive example available online my.hrw.com
ADDITIONAL EXAMPLE 2 Eli is y years old. His 9yearold cousin Jen is 4 years younger than he is. Write an equation to represent this situation. Sample answer: y  4 = 9 Interactive Whiteboard Interactive example available online my.hrw.com
321
Lesson 12.1
Engage
ESSENTIAL QUESTION How do you write equations and determine whether a number is a solution of an equation? Write a statement that links two expressions with an equals sign. Substitute a number for the variable and simplify. If the final statement is true, the number is a solution.
Motivate the Lesson Ask: Ty wants to buy a video game. He has $57, which is $38 less than he needs. Does the game cost $90 or $95? Begin Example 1 to find out how to solve this problem.
Explore Engage with the Whiteboard
Write x + 4 and x + 4 = 9 on the whiteboard. Ask students how they differ. Explain the difference between an expression and an equation. Then have students model both sides of the equation, using algebra tiles on the whiteboard. Ask them what x must represent for the two sides to be equal. Tell them that x = 5 is the solution of the equation.
Explain EXAMPLE 1 Focus on Modeling Mathematical Processes Explain to students that an equation is like a balanced scale. Just as the weights on both sides of a balanced scale are exactly the same, the expressions on both sides of an equation represent exactly the same value. Show examples of equations on a balance scale. Questioning Strategies
Mathematical Processes • In A, why is x = 6 a solution of the equation? because the resulting statement, 15 = 15, is a true statement In B, why is y = 8 not a solution of the equation? because the resulting statement, 2 = 32, is a false statement
YOUR TURN Avoid Common Errors Watch for students who substitute the given value incorrectly. Caution students to doublecheck their work for accuracy.
EXAMPLE 2 Questioning Strategies
Mathematical Processes • Could you write another equation to represent the situation? Yes. I can write the equation p = Total points  Mark’s points or p = 46  17 because subtraction is the opposite of addition.
YOUR TURN Avoid Common Errors If students have difficulty writing equations to represent the word problems, encourage them to make a model, like the one in Example 2, to organize the information.
DO NOT EDITChanges must be made through “File info” CorrectionKey=A
12.1 ?
Writing Equations to Represent Situations
ESSENTIAL QUESTION
Expressions, equations, and relationships—6.9.A Write onevariable, onestep equations … to represent constraints or conditions within problems. Also 6.7.B, 6.10.B
How do you write equations and determine whether a number is a solution of an equation?
YOUR TURN Personal Math Trainer
Determine whether the given value is a solution of the equation. 1. 11 = n + 6; n = 5
Online Assessment and Intervention
yes
5+4
5+4=9 a number plus 4 is 9.
n+4
Algebraic
Math On the Spot my.hrw.com
n+4=9
EXAMPL 1 EXAMPLE
my.hrw.com
EXAMPLE 2
An equation relates two expressions using symbols for is or equals.
© Houghton Mifflin Harcourt Publishing Company
Mark’s points
17
Math Talk
6.10.B
Mathematical Processes
Use words to describe two expressions that represent the total points scored. What does the equation say about the expressions you wrote?
A x + 9 = 15; x = 6 ? 6 + 9 = 15 Substitute 6 for x. ? 15 = 15 Add. y __ = 32; y = 8 4 8 ? ___ = 32 Substitute 8 for y. 4
=
Total points
+
p
=
46
HOME
PERIOD
GUEST
Math Talk: The sum of Mark’s points and his teammates’ points; the total points scored; the equation uses the symbol = to show that the two expressions are equal.
Write an equation to represent each situation.
Divide.
4. Marilyn has a fish tank that contains 38 fish. There are 9 goldfish and f other fish.
5. Juanita has 102 beads to make n necklaces. Each necklace will have 17 beads.
6. Craig is c years old. His 12yearold sister Becky is 3 years younger than Craig.
7. Sonia rented ice skates for h hours. The rental fee was $2 per hour and she paid a total of $8.
f + 9 = 38
y
8 is not a solution of the equation _4 = 32.
C 8x = 72; x = 9 ? 8(9) = 72 ? 72 = 72
+
Teammates’ points
YOUR TURN
6 is a solution of x + 9 = 15.
? 2 = 32
6.9.A
Mark scored 17 points for the home team in a basketball game. His teammates as a group scored p points. Write an equation to represent this situation.
Determine whether the given value is a solution of the equation.
B
You can represent some realworld situations with an equation. Making a model first can help you organize the information.
Math On the Spot
Equation
a number plus 4
Numerical
yes
Writing Equations to Represent Situations
An equation is a mathematical statement that two expressions are equal. An equation may or may not contain variables. For an equation that has a variable, a solution of the equation is a value of the variable that makes the equation true. An equation represents a relationship between two An expression represents values. a single value.
Words
no
36 3. __ x = 9; x = 4
my.hrw.com
Determining Whether Values Are Solutions
Expression
2. y  6 = 24; y = 18
Substitute 9 for x.
Personal Math Trainer
Multiply.
Online Assessment and Intervention
9 is a solution of 8x = 72.
my.hrw.com
Lesson 12.1
6_MTXESE051676_U4M12L1 321
321
25/10/12 5:34 PM
322
c  3 = 12
17n = 102
© Houghton Mifflin Harcourt Publishing Company
LESSON
DO NOT EDITChanges must be made through “File info” CorrectionKey=B
2h = 8
Unit 4
6_MTXESE051676_U4M12L1.indd 322
28/01/14 7:36 PM
PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.D, which calls for students to “Communicate mathematical ideas, …using multiple representations, including symbols…and language as appropriate.” Students begin by distinguishing equations from expressions. Then they determine whether a given number is a solution of the equation. Next, students write equations that represent realworld situations expressed in words. Finally, students use substitution to check whether a value for a variable makes an equation true.
Math Background The equals sign, which first appeared about 450 years ago, is a relatively new math concept. The term equation comes from a Latin word meaning “to set equal.” There are, essentially, four types of equations: • True equation: 2 + 3 = 5 • False equation: 3 + 4 = 8 • Conditional equation: y  6 = 7 (not true for all values) • Identity: 3n + 5n = 8n (true for all values)
Writing Equations to Represent Situations
322
ADDITIONAL EXAMPLE 3 Suki paid $132 for 6 DVDs. Write an equation to determine whether each DVD cost $17 or $22. Sample equation: 6n = $132; $22 Interactive Whiteboard Interactive example available online
EXAMPLE 3 Questioning Strategies
Mathematical Processes • Is the equation x  47 = 18 true? Explain. The equation is true only when x = 65. An equation is a balanced mathematical statement, and only the correct solution will maintain the balance. • How can you check your answer by using a different mathematical operation? Add $18 and $47 to find the original amount on the card. You should get $65.
my.hrw.com
Connect Multiple Representations Mathematical Processes Have students explain why adding $18 to $47 to check that $65 is the answer will yield the same result as subtracting $47 from $65 to see that $18 is the money left over. Students should recognize that addition and subtraction are inverse operations. Animated Math Modeling Equations Students model equations using interactive algebra tiles. my.hrw.com
Focus on Math Connections Point out that an equals sign with a question mark above (≟) is used immediately after a variable has been substituted by a number. This symbol indicates that it is not yet known whether the equation is true or false.
YOUR TURN Avoid Common Errors If students have difficulty writing equations to represent the word problems, encourage them to make a model, like the one in Example 3, to organize the information. Then remind students to check that their solution makes the original equation true.
Elaborate Talk About It Summarize the Lesson Ask: How do you know when a number is a solution to a equation? When the number is substituted for the variable, it makes the equation true. An equation is true when the values of the expressions on opposite sides of the equals sign are the same.
GUIDED PRACTICE Engage with the Whiteboard For Exercises 1–2, have student fill in the boxes and determine whether the given values are solutions to the equations. For Exercise 13, have a student circle the important information in the problem statement on the whiteboard. Then have another student complete the model for the word equation and write the equation next to the model.
Avoid Common Errors Exercises 3–12 Watch for students who substitute the given value incorrectly. Caution students to doublecheck their work for accuracy. Exercises 14–16 If students have difficulty writing equations to represent the word problems, encourage them to make a model, like the one in Example 3, to organize the information. Then remind students to check that their solution makes the original equation true.
323
Lesson 12.1
DO NOT EDITChanges must be made through “File info” CorrectionKey=A
DO NOT EDITChanges must be made through “File info” CorrectionKey=B
Guided Practice
Writing an Equation and Checking Solutions
Determine whether the given value is a solution of the equation. (Example 1) Math On the Spot my.hrw.com
EXAMPL 3 EXAMPLE
6.10.B
Sarah used a gift card to buy $47 worth of groceries. Now she has $18 left on her gift card. Write an equation to determine whether Sarah had $65 or $59 on the gift card before buying groceries. STEP 1
Amount on card STEP 2

Amount spent
=
my.hrw.com
STEP 3
=
Amount left on card
x

47
=
18
Substitute 65 and 59 for x to see which equation is true.
The amount spent and the amount left on the card are the known quantities. Substitute those values in the equation.
© Houghton Mifflin Harcourt Publishing Company
13
4 yes
7. 21 = m + 9; m = 11 9. d  4 = 19; d = 15
no no
6. 2.5n = 45; n = 18
yes
8. 21  h = 15; h = 6
yes
10. 5 + x = 47; x = 52
yes
yes
12. 5q = 31; q = 13
no no
Number of rooms
×
on each floor
=
Total number of rooms
8r = 256
14. In the school band, there are 5 trumpet players and f flute players. There are twice as many flute players as there are trumpet players. Write an equation to represent this situation. (Example 3)
Sample answer: _2f = 5
15. Pedro bought 8 tickets to a basketball game. He paid a total of $208. Write an equation to determine whether each ticket cost $26 or $28. (Example 3)
Sample equation: 8x = 208; $26
8. What expressions are represented in the equation x  47 = 18? How does the relationship represented in the equation help you determine if the equation is true?
16. The high temperature was 92°F. This was 24°F higher than the overnight low temperature. Write an equation to determine whether the low temperature was 62°F or 68°F. (Example 3)
x  47 and 18; Subtract 47 from the given number and
Sample equation: x + 24 = 92; 68°F
if the difference is 18 the equation is true.
? ?
YOUR TURN 9. On Saturday morning, Owen earned $24 raking leaves. By the end of the afternoon he had earned a total of $62. Write an equation to determine whether Owen earned $38 or $31 on Saturday afternoon.
? =4
4. 17y = 85; y = 5
no
Number
Reflect
Personal Math Trainer
Substitute the number for the variable and simplify. If the final statement is true, the number is a solution.
my.hrw.com
323
17/01/13 10:16 PM
ESSENTIAL QUESTION CHECKIN
17. Tell how you can determine whether a number is a solution of an equation.
Online Assessment and Intervention
Lesson 12.1
6_MTXESE051676_U4M12L1 323
52
? _____ = 4
of floors
x  47 = 18 x  47 = 18 ? ? 65  47 = 18 59  47 = 18 ? ? 18 = 18 12 = 18 The amount on Sarah’s gift card before she bought groceries was $65.
Sample equation: x + 24 = 62; $38
5
yes
13. Each floor of a hotel has r rooms. On 8 floors, there are a total of 256 rooms. Write an equation to represent this situation. (Example 2)
Let x be the amount on the card. Amount spent
? 23 =
n 2. __ = 4; n = 52 13
9
11. w  9 = 0; w = 9
Rewrite the equation using a variable for the unknown quantity and the given values for the known quantities.

14
3. 14 + x = 46; x = 32 Animated Math
Amount left on card
Amount on card
? 23 =
5. 25 = _5k ; k = 5
Identify the three quantities given in the problem.
no
1. 23 = x  9; x = 14
© Houghton Mifflin Harcourt Publishing Company
You can substitute a given value for the variable in an equation to check if that value makes the equation true.
324
Unit 4
6_MTXESE051676_U4M12L1.indd 324
28/01/14 7:37 PM
DIFFERENTIATE INSTRUCTION Visual Cues
Critical Thinking
Additional Resources
Display a set of pan balance scales. For each of the following equations, write the left side over the left scale and the right side over the right scale. Ask what the value of the variable must be for the scales to remain balanced.
Give students the following problem to solve.
1. x + 3 = 7 x = 4
Remind students that to compare quantities, they need to use the same units.
Differentiated Instruction includes: • Reading Strategies • Success for English Learners ELL • Reteach • Challenge PREAP
3. 4y = 28 y = 7
17 ≠ 350 ÷ 20; No, they do not have the same amount of money.
2. t  5 = 3 t = 8 w 4. __ = 6 w = 12 2
Rebecca has 17 onedollar bills. Courtney has 350 nickels. Do the two girls have the same amount of money?
Writing Equations to Represent Situations
324
Personal Math Trainer Online Assessment and Intervention
Online homework assignment available
Evaluate GUIDED AND INDEPENDENT PRACTICE 6.9.A, 6.10.B
my.hrw.com
12.1 LESSON QUIZ 6.9.A, 6.10.B Determine whether the given value is a solution of the equation. Write yes or no. 1. w – 7 = 20; w = 13
Concepts & Skills
Practice
Example 1 Determining Whether Values Are Solutions
Exercises 1–12
Example 2 Writing Equations to Represent Situations
Exercises 13, 18–21, 24–25
Example 3 Writing an Equation and Checking Solutions
Exercises 14–16, 26
2. 15t = 120; t = 8 y 3. __ = 2; y = 24 12
Exercise
4. 5 = x + 5; x = 10 5. In a choir there are 16 altos and s sopranos. There are twice as many sopranos as altos. Write an equation to represent this situation. 6. Jerome is onethird the age of his aunt, who is 51 years old. Write an equation to determine whether Jerome is 14 or 17.
Depth of Knowledge (D.O.K.) 2 Skills/Concepts
1.A Everyday life
22
3 Strategic Thinking
1.G Explain and justify arguments
23
2 Skills/Concepts
1.F Analyze relationships
24
3 Strategic Thinking
1.G Explain and justify arguments
25–26
3 Strategic Thinking
1.D Multiple representations
27–29
3 Strategic Thinking
1.G Explain and justify arguments
18–21
Lesson Quiz available online my.hrw.com
Answers 1. no 2. yes 3. no 4. yes 5. __2s = 16 6. 3x = 51; Jerome is 17 years old.
325
Lesson 12.1
Mathematical Processes
Additional Resources Differentiated Instruction includes: • Leveled Practice Worksheets
Class
Date Personal Math Trainer
12.1 Independent Practice 6.7.B, 6.9.A, 6.10.B
my.hrw.com
18. Andy is onefourth as old as his grandfather, who is 76 years old. Write an equation to determine whether Andy is 19 or 22 years old.
4n = 76; 19 years old 19. A sleeping bag weighs 8 pounds. Your backpack and sleeping bag together weigh 31 pounds. Write an equation to determine whether the backpack without the sleeping bag weighs 25 or 23 pounds.
x + 8 = 31; 23 pounds
22. Write an equation that involves multiplication, contains a variable, and has a solution of 5. Can you write another equation that has the same solution and includes the same variable and numbers but uses division? If not, explain. If possible, write the equation.
© Houghton Mifflin Harcourt Publishing Company
29 32
a. Write an equation that relates Cindy’s age to her dad’s age when Cindy is 18.
Cindy’s Age 2 years old
36 years old
10 years old
?
18 years old
b. Determine if 42 is a solution to the equation. Show your work.
No; 42  26 = 16; 16 is not equal to 18.
Sample answer: An expression
c. Explain the meaning of your answer in part b.
represents one value such as
Cindy’s father will not be 42 when she is 18.
2 + 5 and an equation represents FOCUS ON HIGHER ORDER THINKING
the relationship between two equation is a statement that the
Yes, because 4 + f = 12 means there are 8 flute players,
equivalent. Each expression is
and 8 is twice 4.
24. Explain the Error The problem states that Ursula earns $9 per hour. To write an expression that tells how much money Ursula earns for h hours, Joshua wrote _h9 . Sarah wrote 9h. Whose expression is correct and why?
Sarah is correct because the
x is the distance between
28. Problem Solving Ronald paid $162 for 6 tickets to a basketball game. During the game he noticed that his friend paid $130 for 5 tickets. The price of each ticket was $26. Was Ronald overcharged? Justify your answer.
Yes, Ronald was overcharged because 6(26) = 156 and 156 < 162. 29. Communicate Mathematical Ideas Tariq said you can write an equation by setting an expression equal to itself. Would an equation like this be true? Explain.
earnings are the number of
b. Describe what your variable represents.
Work Area
27. Critical Thinking In the school band, there are 4 trumpet players and f flute players. The total number of trumpet and flute players is 12. Are there twice as many flute players as trumpet players? Explain.
two expressions, 2 + 5 and 7, are just part of that statement.
29  13 = x; 13 + x = 29
hours times the amount per
Yes, setting an expression equal to itself forms an
hour, or 9h.
equation that will always be true regardless of the value of the variable(s) or numbers in the expression.
Artaville and Greenville Lesson 12.1
EXTEND THE MATH
Dad’s Age 28 years old
x  26 = 18
values such as 2 + 5 = 7. An
a. Write two equations that state the relationship of the distances between Greenville, Artaville, and Jonesborough.
26. Multiple Representations The table shows ages of Cindy and her dad.
23. How are expressions and equations different? Explain using a numerical example.
x  23 = 48; 71 students
Maybern
is 44  7 = x.
20 ÷ x = 4 or x = 20 ÷ 4
21. The table shows the distance between Greenville and nearby towns. The distance between Artaville and Greenville is 13 miles less than the distance between Greenville and Jonesborough. Distance between Greenville and Nearby Towns (miles)
Both equations are correct. Another correct equation
Sample answer: 4x = 20; Yes:
20. Halfway through a bus route, 23 students have been dropped off and 48 students remain on the bus. Write an equation to determine whether there are 61 or 71 students on the bus at the beginning of the route.
Jonesborough
Online Assessment and Intervention
25. Communicate Mathematical Ideas A dog weighs 44 pounds and the veterinarian thinks it needs to lose 7 pounds. Mikala wrote the equation x + 7 = 44 to represent the situation. Kirk wrote the equation 44  x = 7. Which equation is correct? Can you write another equation that represents the situation?
© Houghton Mifflin Harcourt Publishing Company
Name
PREAP
325
Activity available online
326
Unit 4
my.hrw.com
Activity Following the release of the hit movie Attack of the Giant Muffins, muffin fever spread across the country. This resulted in a contest to choose the plot of the sequel Attack of the Giant Muffins, Part 2. The table shows the number of contest entries from some towns in one state.
Town Entries Hillville .....................................40 Dos Rios ..................................30 High Corn ..............................105 Moose .......................................n HoHum ..................................25
Write an equation for each statement. 1. The sum of the number of entries from Hillville and Moose was 90. n + 40 = 90 2. The difference between the number of Moose entries and HoHum entries was 25. n  25 = 25 3. The total number of entries is 5 times the number of Moose entries. 5n = 250 Writing Equations to Represent Situations
326
LESSON
12.2 Addition and Subtraction Equations Texas Essential Knowledge and Skills The student is expected to: Expressions, equations, and relationships—6.10.A Model and solve onevariable, onestep equations and inequalities that represent problems, including geometric concepts. Expressions, equations, and relationships—6.9.B Represent solutions for onevariable, onestep equations and inequalities on number lines. Expressions, equations, and relationships—6.9.C Write corresponding realworld problems given onevariable, onestep equations or inequalities.
Mathematical Processes 6.1.E Create and use representations to organize, record, and communicate mathematical ideas.
ADDITIONAL EXAMPLE 1 Solve the equation y + 8 = 22. Graph the solution on a number line. y = 14 10
15
20
Interactive Whiteboard Interactive example available online my.hrw.com
Engage
ESSENTIAL QUESTION How do you solve equations that contain addition and subtraction? Sample answer: Apply the inverse operation—subtraction for addition equations and addition for subtraction equations—to both sides of the equation.
Motivate the Lesson Ask: Does anyone have a puppy? Have you noticed how fast they grow? Begin the Explore Activity to find out how to model one puppy’s weight gain, using an equation and algebra tiles.
Explore EXPLORE ACTIVITY Focus on Modeling Mathematical Processes Remind students that an equation is like a balanced scale. If you increase or decrease the weights by the same amount on both sides, the scale will remain in balance. Emphasize that students must remove the same number of tiles from both sides of the mat.
Explain EXAMPLE 1 Focus on Math Connections
Mathematical Processes Remind students that subtraction is the inverse, or opposite, of addition. If an equation contains addition, solve it by subtracting from both sides to “undo” the addition.
Questioning Strategies
Mathematical Processes • Would you solve 15 + a = 26 differently from a + 15 = 26? no because addition is associative and the order of the variable and the added number does not change the process of subtracting 15 from both sides
• How can you use substitution to check an answer to an addition equation? Substitute the value for the variable into the original equation and simplify. If the result is a true statement, the value is the solution. • Why is the graph only a single point? because there is only one answer for the equation
YOUR TURN Avoid Common Errors Watch for students who perform the inverse operation of subtraction only on the side with the variable. Stress that to keep the equation “balanced” the same amount must be taken away from each side.
327
Lesson 12.2
DO NOT EDITChanges must be made through “File info” CorrectionKey=A
Addition and Subtraction Equations
12.2 ?
ESSENTIAL QUESTION
Expressions, equations, and relationships—6.10.A Model and solve onevariable, onestep equations ... that represent problems, including geometric concepts. Also 6.9.B, 6.9.C
How do you solve equations that contain addition or subtraction?
EXPLORE ACTIVITY
Using Subtraction to Solve Equations Removing the same number of tiles from each side of an equation mat models subtracting the same number from both sides of an equation. Math On the Spot my.hrw.com
You can subtract the same number from both sides of an equation, and the two sides will remain equal. When an equation contains addition, solve by subtracting the same number from both sides.
6.10.A
Modeling Equations
EXAMPLE 1
A puppy weighed 6 ounces at birth. After two weeks, the puppy weighed 14 ounces. How much weight did the puppy gain?
6
+
Weight gained
=
Weight after 2 weeks
+
x
=
14
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Petra Wegner / Alamy
a + 15 = 26
Notice that the number 15 is added to a.
a + 15 = 26  15 15 ___ a = 11
Subtract 15 from both sides of the equation.
Check: a + 15 = 26
To answer this question, you can solve the equation 6 + x = 14. Algebra tiles can model some equations. An equation mat represents the two sides of an equation. To solve the equation, remove the same number of tiles from both sides of the mat until the x tile is by itself on one side.
B How many 1 tiles must you remove on the left 6 side so that the x tile is by itself? Cross out these tiles on the equation mat.
1
1
1
1
1
1
x
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Substitute 11 for a.
26 = 26
Add on the left side.
6+ x
5 6 7 8 9 10 11 12 13 14 15
14
Reflect 2.
C Whenever you remove tiles from one side of the mat, you must remove the same number of tiles from the other side of the mat. Cross out the tiles that should be removed on the right side of the mat. D How many tiles remain on the right side of the mat? This is the solution of the equation.
8
8
ounces.
variable is isolated, or alone, on one side of the equation.
Mathematical Processes
Why did you remove tiles from each side of your model?
You want to remove tiles equally from both The x tile is alone on one side of the mat. The solution is sides to keep the equation balanced. the number of small tiles on the other side. Communicate Mathematical Ideas How do you know when the model shows the final solution? How do you read the solution?
Lesson 12.2
6_MTXESE051676_U4M12L2 327
Communicate Mathematical Ideas How do you decide which number to subtract from both sides?
Subtract the number added to the variable so that the
Math Talk
YOUR TURN
Reflect 1.
? 11 + 15 = 26
Graph the solution on a number line.
A Model 6 + x = 14.
The puppy gained
6.10.A, 6.9.B
Solve the equation a + 15 = 26. Graph the solution on a number line.
Let x represent the number of ounces gained. Weight at birth
Subtraction Property of Equality
327
25/10/12 3:56 PM
Personal Math Trainer Online Assessment and Intervention
my.hrw.com
328
3. Solve the equation 5 = w + 1.5. Graph the solution on a number line. w=
5 4 3 2 1
© Houghton Mifflin Harcourt Publishing Company
LESSON
DO NOT EDITChanges must be made through “File info” CorrectionKey=B
0 1 2 3 4 5
3.5
Unit 4
6_MTXESE051676_U4M12L2.indd 328
28/01/14 7:46 PM
PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.E, which calls for students to “create and use representations to organize, record, and communicate mathematical ideas.” Students use algebra tiles and number lines to model the solutions to onestep equations. They proceed to solve algebraic equations symbolically by using inverse operations to isolate the variable. They then apply the algebraic method to solve equations representing realworld situations.
Math Background In Elements, Book I, Euclid listed five axioms that he called “common notions.” 1. Things which are equal to the same thing are also equal to one another. 2. If equals be added to equals, the wholes are equal. 3. If equals be subtracted from equals, the remainders are equal. 4. Things which coincide with one another are equal to one another. 5. The whole is greater than the part.
Addition and Subtraction Equations
328
ADDITIONAL EXAMPLE 2
Solve the equation y −13 = –12. Graph the solution on a number line. y=1 5
0
5
Interactive Whiteboard Interactive example available online my.hrw.com
EXAMPLE 2 Focus on Math Connections
Mathematical Processes Remind students that addition and subtraction are inverse operations. If an equation contains subtraction, solve it by adding to both sides to “undo” the subtraction.
Questioning Strategies
Mathematical Processes • How is solving an equation containing subtraction similar to solving an equation containing addition? Solve both types of equations by using the inverse operation to get the variable by itself on one side of the equation. • Does it make sense that the solution is greater than 18? Explain. Yes. The equation indicates that subtracting 21 from some numbers gives an answer of 18, so the solution must be greater than 18.
Engage with the Whiteboard Cover up the sentences in blue next to each step of the solution and have students write a description of what is happening in each step. Then have the students graph the solution on the number line.
Focus on Modeling
Mathematical Processes Guide students to see that it makes sense to draw only the needed portion of the number line when graphing a solution, especially when the numbers are large.
YOUR TURN Focus on Math Connections
ADDITIONAL EXAMPLE 3 Write and solve an equation to find the measure of the unknown angle. z + 48° = 90°; z = 42°
z
Mathematical Processes Remind students that to add or subtract fractions with different denominators, it is necessary to rewrite the fractions with a common denominator. A fraction greater than 1 should be written as a mixed number for ease of graphing.
EXAMPLE 3 Questioning Strategies
Mathematical Processes • How do you know that x + 60 = 180 is the correct equation to use to find the measure of x? The two angles shown in the drawing are supplementary. • Does the sketch of the unknown angle x appear to be twice 60°? Explain. Yes. If I divide x into two halves, the two angles appear to have measures that are similar to the 60° angle.
48°
Interactive Whiteboard Interactive example available online my.hrw.com
Focus on Math Connections
Mathematical Processes Have students explain what will be true about the unknown measure of any angle if its supplement is less than 90°. Elicit that it will be an obtuse angle. Then ask the same question about angles whose supplements are greater than 90°.
YOUR TURN ADDITIONAL EXAMPLE 4
Solve the equation n + 6.5 = 20. Then write a realworld problem that involves adding these two quantities. n = 13.5; Sample answer: Ivan needed 6.5 more points to take the lead with 20 points. How many points had he already scored? Interactive Whiteboard Interactive example available online my.hrw.com
329
Lesson 12.2
Focus on Math Connections
Mathematical Processes Point out to students that the angle represented here is a right angle and that a right angle measures exactly 90°. Remind them that if the sum of the measures of two angles is 90°, the two angles are complementary angles. This information should help students to write and solve an equation to find the measure of the unknown angle.
EXAMPLE 4 Connect to Daily Life
Mathematical Processes Point out to students that the equation has decimals containing hundredths and that money is commonly represented as decimals containing hundredths. Thus, a good realworld application for this equation would be a situation involving money. Encourage students to brainstorm situations from everyday life that could make sense.
DO NOT EDITChanges must be made through “File info” CorrectionKey=B
DO NOT EDITChanges must be made through “File info” CorrectionKey=B
Using Addition to Solve Equations
Solving Equations that Represent Geometric Concepts
When an equation contains subtraction, solve by adding the same number to both sides.
You can write equations to represent geometric relationships.
Addition Property of Equality
Math On the Spot
Math On the Spot
my.hrw.com
my.hrw.com
Recall that a straight line has an angle measure of 180°. Two angles whose measures have a sum of 180° are called supplementary angles. Two angles whose measures have a sum of 90° are called complementary angles.
You can add the same number to both sides of an equation, and the two sides will remain equal.
EXAMPLE 3 EXAMPL 2 EXAMPLE
Find the measure of the unknown angle.
6.10.A, 6.9.B
STEP 1
Solve the equation y  21 = 18. Graph the solution on a number line. y  21 = 18
Notice that the number 21 is subtracted from y.
y  21 = 18 + 21 +21 ___ y = 39
Add 21 to both sides of the equation. STEP 2
Subtract.
=
180°
Write a description to represent the model. Include a question for the unknown angle.
STEP 3
Write an equation.
STEP 4
Solve the equation.
x + 60 = 180
35 36 37 38 39 40 41 42 43 44 45
x+
Reflect © Houghton Mifflin Harcourt Publishing Company
60°
The sum of an unknown angle and a 60° angle is 180°. What is the measure of the unknown angle?
Substitute 39 for y.
Graph the solution on a number line.
4.
+
60°
x
60 = 180 60 60 _ _
Subtract 60 from each side.
Communicate Mathematical Ideas How do you know whether to add on both sides or subtract on both sides when solving an equation?
x
If the equation contains addition, subtract. If the
The unknown angle measures 120°.
=
120
The final answer includes units of degrees.
equation contains subtraction, add. YOUR TURN 6. Write and solve an equation to find the measure of the unknown angle.
YOUR TURN Solve the equation h  _12 = _43 .
5.
x + 65 = 90; x = 25°
2
Graph the solution on a number line. h=
1
0
1
5 _ , or 1_14 4
2 Personal Math Trainer
Personal Math Trainer
Online Assessment and Intervention
Online Assessment and Intervention
my.hrw.com
my.hrw.com
Lesson 12.2
6_MTXESE051676_U4M12L2.indd 329
329
330
28/01/14 7:46 PM
x
© Houghton Mifflin Harcourt Publishing Company
18 = 18
Write the information in the boxes. Unknown angle
Check: y  21 = 18 ? 39  21 = 18
6.10.A
65°
7. Write and solve an equation to find the measure of a complement of an angle that measures 42°.
x + 42 = 90; x = 48°
Unit 4
6_MTXESE051676_U4M12L2.indd 330
28/01/14 7:46 PM
DIFFERENTIATE INSTRUCTION Critical Thinking
Cognitive Strategies
Additional Resources
Present the magic square shown below. Explain that the sum of any three numbers added across, down, or diagonally should have the same sum.
Have students work with a partner to create “mindreading” puzzles that depend upon inverse operations to return to the original number. The following puzzle is an example. Pick a number.
Differentiated Instruction includes: • Reading Strategies • Success for English Learners ELL • Reteach • Challenge PREAP
Have students write and solve equations to fill in the table. The magic sum for this square is 465. 248
31
186
93
155
217
124
279
62
1. Add 4.
5. Subtract 4.
2. Subtract 1.
6. Add 5.
3. Add 7.
7. Subtract 7.
4. Subtract 5.
8. Add 1.
Your answer is the number you started with.
Addition and Subtraction Equations
330
Questioning Strategies
Mathematical Processes • Using number sense, what can you determine about the value of x, given that the sum of two numbers is 25? The value of x will be less than 25.
• How else could you write the equation to solve for x? Explain. Sample answer: You could write x + 21.79 = 25; addition is commutative, so x + 21.79 = 21.79 + x. • How is the question in a realworld problem related to its equation? The question asks about the value of the variable.
YOUR TURN Focus on Communication
Mathematical Processes Guide students to write problems that use sensible data, and have them explain, step by step, how to solve their equations. Challenge each student to create original word problems based on unique situations.
Elaborate Talk About It Summarize the Lesson Ask: How do you solve and check an equation containing only addition or only subtraction? You apply the inverse operation—subtraction for addition equations and addition for subtraction equations—to both sides of the equation.
GUIDED PRACTICE Engage with the Whiteboard For Exercise 1, have a student complete the verbal model on the whiteboard and then draw a model with algebra tiles below the verbal model. Ask students to provide an explanation of the solution.
Avoid Common Errors Exercises 2–6 Remind students to use the inverse operation of subtraction to undo addition or addition to undo subtraction. Exercise 7 Remind students to begin by identifying the angle they are using, as right, straight, or obtuse so that they will know what angle measure to use in their equation.
331
Lesson 12.2
DO NOT EDITChanges must be made through “File info” CorrectionKey=B
DO NOT EDITChanges must be made through “File info” CorrectionKey=B
Guided Practice
Writing RealWorld Problems for a Given Equation Math On the Spot
1. A total of 14 guests attended a birthday party. Three guests stayed after the party to help clean up. How many guests left when the party ended? (Explore Activity) a. Let x represent the number
my.hrw.com
EXAMPL 4 EXAMPLE
b.
6.9.C
+
left at end of party
Write a realworld problem for the equation 21.79 + x = 25. Then solve the equation.
x
21.79 + x = 25 STEP 1
Number that
of guests who left when the party ended. Number that
stayed to clean 3
+
c. Draw algebra tiles to model the equation.
Examine each part of the equation.
11
Total at party
=
14
+ + +
+
friends left when the party ended.
=
x is the unknown or quantity we are looking for. 21.79 is added to x. = 25 means that after adding 21.79 and x, the result is 25. STEP 2
Write a realworld situation that involves adding two quantities. Joshua wants to buy his mother flowers and a card for Mother’s Day. Joshua has $25 to spend and selects roses for $21.79. How much can he spend on a card?
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Brand X Pictures/Getty Images
STEP 3
Math Talk
Mathematical Processes
How is the question in a realworld problem related to its equation?
Solve the equation. 21.79 + x = 25 21.79 21.79 __ __ x=
3.21
The final answer includes units of money in dollars.
The question asks about the value of the variable.
Solve each equation. Graph the solution on a number line. (Examples 1 and 2) 2. 2 = x  3
x=
5 4 3 2 1
5
3. s + 12.5 = 14 5 4 3 2 1
0 1 2 3 4 5
s=
1.5
0 1 2 3 4 5
Solve each equation. (Examples 1 and 2) 4. h + 6.9 = 11.4 h=
4.5
6. n + _12 = _74
5. 82 + p = 122
40
p=
n=
5 _ 4
7. Write and solve an equation to find the measure of the unknown angle. (Example 3)
Joshua can spend $3.21 on a Mother’s Day card.
x + 45 = 180; x = 135°
Reflect 8.
+ + + + + + + + + + + + + +
45°
x
8. Write a realworld problem for the equation x  75 = 200. Then solve the equation. (Example 4)
What If? How might the realworld problem change if the equation were x  21.79 = 25 and Joshua still spent $21.79 on roses?
Check students’ answers; x = 275
21.79 is subtracted from the unknown value, and the result would remain 25. The variable would be the amount Joshua started with and
? ?
25 would the amount of money available after buying roses.
ESSENTIAL QUESTION CHECKIN
9. How do you solve equations that contain addition or subtraction?
You apply the inverse operation—subtraction for
YOUR TURN 9. Write a realworld problem for the equation x  100 = 40. Then solve the equation.
Check students’ problems; x = 140
addition equations and addition for subtraction
Personal Math Trainer
equations—to both sides of the equation.
Online Assessment and Intervention
my.hrw.com
Lesson 12.2
6_MTXESE051676_U4M12L2.indd 331
© Houghton Mifflin Harcourt Publishing Company
You can write a realworld problem for a given equation. Examine each number and mathematical operation in the equation.
331
28/01/14 8:45 PM
332
Unit 4
6_MTXESE051676_U4M12L2.indd 332
28/01/14 8:45 PM
Addition and Subtraction Equations
332
Personal Math Trainer Online Assessment and Intervention
Online homework assignment available
Evaluate GUIDED AND INDEPENDENT PRACTICE 6.9.B, 6.9.C, 6.10.A
my.hrw.com
12.2 LESSON QUIZ 6.10.A, 6.9.C Solve each equation. 1. y + 8.3 = 12.7 2. w − 14 = 23 3. 18 + x = 6 4. t – 1.9 = 15.7 5. __12 = a  __45 6. Ellie spent $88.79 at the computer store. She then had $44.50 left to buy a cool hat. How much money did she originally have? Write and solve an equation to answer the question. 7. Write a realworld problem for the equation x + 12 = 35. Then solve the equation. Lesson Quiz available online my.hrw.com
Answers 1. y = 4.4
2. w = 37 3. x = –12 4. t = 17.6 5. a = __74 6. x − $88.79 = $44.50; x = $133.29 7. See students’ answers; x = 23
333
Lesson 12.2
Concepts & Skills
Practice
Explore Activity Modeling Equations
Exercise 1
Example 1 Using Subtraction to Solve Equations
Exercises 2–6, 10, 12–14
Example 2 Using Addition to Solve Equations
Exercises 2–6, 11, 15–16
Example 3 Solving Equations that Represent Geometric Concepts
Exercise 7
Example 4 Writing RealWorld Problems for a Given Equation
Exercises 8, 17
Exercise
Depth of Knowledge (D.O.K.)
Mathematical Processes
10–16
2 Skills/Concepts
1.A Everyday life
17–18
3 Strategic Thinking
1.F Analyze relationships
19
3 Strategic Thinking
1.A Everyday life
20
3 Strategic Thinking
1.G Explain and justify arguments
21
3 Strategic Thinking
1.F Analyze relationships
Additional Resources Differentiated Instruction includes: • Leveled Practice Worksheets
Name
Class
Date
12.2 Independent Practice 6.9.B, 6.9.C, 6.10.A
my.hrw.com
Write and solve an equation to answer each question.
Handy Dandy Grocery Regular price
Sample answer: m  123.45 = 36.55; $160
Sample answer: e + 8 = 31; 23
11. My sister is 14 years old. My brother says that his age minus twelve is equal to my sister’s age. How old is my brother?
Both a and r are equal to $1.50, so the discount is the same. 20. Critical Thinking An orchestra has twice as many woodwind instruments as brass instruments. There are a total of 150 brass and woodwind instruments.
17. Represent RealWorld Problems Write a realworld situation that can be represented by 15 + c = 17.50. Then solve the equation and describe what your answer represents for the problem situation.
Sample answer: x + 8.95 = 21.35; $12.40
a. Write two different addition equations that describe this situation. Use w for woodwinds and b for brass.
w + b = 150; w = b + b = 2b
Check students’ answers. c = 2.50
13. The Acme Car Company sold 37 vehicles in June. How many compact cars were sold in June?
w = 100; b = 50
Number sold
SUV
8
Compact
?
18. Critique Reasoning Paula solved the equation 7 + x = 10 and got 17, but she is not certain if she got the correct answer. How could you explain Paula’s mistake to her?
Sample answer: x + 8 = 37;
21. Look for a Pattern Assume the following: a + 1 = 2, b + 10 = 20, c + 100 = 200, d + 1,000 = 2,000, ... a. Solve each equation for each variable.
a = 1; b = 10, c = 100, d = 1,000, ...
Paula added the 7 to 10 instead
29 compact cars
b. What pattern do you notice between the variables?
of subtracting. She should have
14. Sandra wants to buy a new MP3 player that is on sale for $95. She has saved $73. How much more money does she need?
c. What would be the value of g if the pattern continues?
g = 1,000,000
the equation and found that
Sample answer: 73 + b = 95 or
x = 3.
95  b = 73; $22
Lesson 12.2
EXTEND THE MATH
Every variable is ten times the one before it.
subtracted 7 from both sides of
© Houghton Mifflin Harcourt Publishing Company
b. How many woodwinds and how many brass instruments satisfy the given information?
Acme Car Company  June Sales Type of car
$3.99
b. Which fruit has a greater discount? Explain.
t = $773
12. Kim bought a poster that cost $8.95 and some colored pencils. The total cost was $21.35. How much did the colored pencils cost?
$2.99
5pound bag of oranges
1.49 + a = 2.99; 2.49 + r = 3.99
Sample answer: 548 = t  225;
Sample answer: 14 = b  12; b = 26
5pound bag of apples
a. Write an equation to find the discount for each situation using a for apples and r for oranges.
16. Brita withdrew $225 from her bank account. After her withdrawal, there was $548 left in Brita’s account. How much money did Brita have in her account before the withdrawal?
elephants
Work Area
19. Multistep Handy Dandy Grocery is having a sale this week. If you buy a 5pound bag of apples for the regular price, you can get another bag for $1.49. If you buy a 5pound bag of oranges at the regular price, you can get another bag for $2.49.
Online Assessment and Intervention
15. Ronald spent $123.45 on school clothes. He counted his money and discovered that he had $36.55 left. How much money did he originally have?
10. A wildlife reserve had 8 elephant calves born during the summer and now has 31 total elephants. How many elephants were in the reserve before summer began?
© Houghton Mifflin Harcourt Publishing Company
FOCUS ON HIGHER ORDER THINKING
Personal Math Trainer
333
334
Activity available online
PREAP
Unit 4
my.hrw.com
Activity Use the code shown in the table below to send messages. A
B
C
D
E
F
G
H
I
J
K
L
M
1
2
3
4
5
6
7
8
9
10
11
12
13
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
14
15
16
17
18
19
20
21
22
23
24
25
26
• Write a short secret message on a piece of paper. • Use the chart to find the number that matches each letter of your message. Write the number above each letter in your message. • To send your message in code, write a series of equations on a separate piece of paper. Use n as your variable and write the equations in the same order as the letters appear in your message. For example, for the letter G, n = 7. You might write n + 5 = 12. • Trade equations and decode classmates’ messages. Addition and Subtraction Equations
334
LESSON
12.3 Multiplication and Division Equations Texas Essential Knowledge and Skills The student is expected to: Expressions, equations, and relationships—6.10.A Model and solve onevariable, onestep equations and inequalities that represent problems, including geometric concepts.
Engage
ESSENTIAL QUESTION How do you solve equations that contain multiplication or division? Sample answer: You apply the inverse operation—division for multiplication equations and multiplication for division equations—to both sides of the equation.
Motivate the Lesson Ask: How many of you like to bake? Can you imagine writing an equation to calculate the ingredients you need? Begin the Explore Activity to see an example for a cookie recipe.
Expressions, equations, and relationships—6.9.B Represent solutions for onevariable, onestep equations and inequalities on number lines. Expressions, equations, and relationships—6.9.C Write corresponding realworld problems given onevariable, onestep equations or inequalities.
Mathematical Processes
Explore EXPLORE ACTIVITY Engage with the Whiteboard For B, have a student circle the remaining two groups of the model on the whiteboard. Then have students fill in the answer for C and the final answer. Ask how the model helped them understand the problem.
6.1.C Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.
Explain EXAMPLE 1 Focus on Math Connections
ADDITIONAL EXAMPLE 1 Solve each equation. Graph the solution on a number line.
5
0
B 27 = 9z z = 3 5
0
Questioning Strategies
Mathematical Processes • How will you know what number to divide both sides of an equation by in order to solve it? Divide both sides by the number that the variable is multiplied by.
A 6y = 24 y = 4 10
Mathematical Processes Remind students that division is the inverse, or opposite, of multiplication. To solve an equation that contains multiplication, use division to “undo” the multiplication.
5
Interactive Whiteboard Interactive example available online my.hrw.com
• Why must you divide both sides of the equation by the same number? You do so to maintain the equality, or the balance, of the equation.
Focus on Critical Thinking Mathematical Processes Challenge students to describe how solving a multiplication equation is similar to solving an addition or subtraction equation. Students should indicate that the same inverse operation must be applied to both sides of the equation, leaving the variable isolated on one side of the equation.
YOUR TURN Avoid Common Errors Watch for students who multiply both sides of the equation when they should divide. Remind students to be certain they are using the inverse operation and to check their answers.
335
Lesson 12.3
DO NOT EDITChanges must be made through “File info” CorrectionKey=A
12.3 ?
Multiplication and Division Equations
ESSENTIAL QUESTION
Expressions, equations, and relationships—6.10.A Model and solve onevariable, onestep equations . . . that represent problems, including geometric concepts. Also 6.9.B, 6.9.C
How do you solve equations that contain multiplication or division?
EXPLORE ACTIVITY
Using Division to Solve Equations Separating the tiles on both sides of an equation mat into an equal number of groups models dividing both sides of an equation by the same number. Math On the Spot my.hrw.com
You can divide both sides of an equation by the same nonzero number, and the two sides will remain equal. When an equation contains multiplication, solve by dividing both sides of the equation by the same nonzero number.
6.9.B, 6.9.C, 6.10.A
Modeling Equations
EXAMPLE 1
Deanna has a cookie recipe that requires 12 eggs to make 3 batches of cookies. How many eggs are needed per batch of cookies?
Number of eggs per batch
·
=
A 9a = 54 Total eggs
x
1 1 1 1
x
1 1 1 1
x
1 1 1 1
© Houghton Mifflin Harcourt Publishing Company
3x
4
Divide both sides of the equation by 9.
Check: 9a = 54 ? 9(6) = 54 54 = 54
12
18 = 3d
x
1 1 1 1
x
1 1 1 1
x
1 1 1 1
4
eggs are needed per batch of cookies.
Reflect 1.
Notice that 9 is multiplied by a.
9a __ __ = 54 9 9
0 1 2 3 4 5 6 7 8 9 10
Substitute 6 for a. Multiply on the left side.
B 18 = 3d
B There are 3 xtiles, so draw circles to separate the tiles into 3 equal groups. One group has been circled for you.
C How many +1tiles are in each group? This is the solution of the equation.
9a = 54 a=6
3 x 12 · = To answer this question, you can use algebra tiles to solve 3x = 12.
A Model 3x = 12.
6.10.A, 6.9.B
Solve each equation. Graph the solution on a number line.
Let x represent the number of eggs needed per batch. Number of batches
Division Property of Equality
Look for a Pattern Why does it make sense to arrange the 12 tiles in 3 rows of 4 instead of any other arrangement of 12 + 1tiles, such as 2 rows of 6?
Notice that 3 is multiplied by d.
18 3d ___ = ____ 3 3
Math Talk
Mathematical Processes
Check: 18 = 3d ? 18 = 3(6)
Why is the solution to the equation the number of tiles in each group?
18 = 18
The number of +1tiles in each group equals one xtile.
109 8 7 6 5 4 3 2 1 0
Substitute 6 for d. Multiply on the right side.
YOUR TURN Personal Math Trainer
3 rows are evenly divisible by 3.
Divide both sides of the equation by 3.
6 = d
© Houghton Mifflin Harcourt Publishing Company
LESSON
DO NOT EDITChanges must be made through “File info” CorrectionKey=B
Online Assessment and Intervention
Solve the equation 3x = 21. Graph the solution on a number line. 2. x =
7
10 9 8 7 6 5 4 3 2 1 0
my.hrw.com
Lesson 12.3
6_MTXESE051676_U4M12L3 335
335
25/10/12 8:50 PM
336
Unit 4
6_MTXESE051676_U4M12L3.indd 336
28/01/14 9:02 PM
PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.C, which calls for students to “select tools…and techniques, including mental math…and number sense to solve problems.” Students first use number sense to identify which inverse operation they should use to isolate the variable on one side of an equation. They then use pencil and paper to solve equations, to check their solutions by using substitution, and to graph the solutions on a number line. In Example 3, students use number sense to translate words to algebraic equations. Students should be encouraged to use mental math to check their answers.
Math Background Division is defined as multiplication by the reciprocal. To solve 3x = 15, for example, we could multiply both sides by __13 , which would be equivalent to dividing both sides by 3. While most students would rather divide by 3 than multiply by __13 , this alternate interpretation is useful when trying to solve equations such as __2 x = 4. Dividing by __2 is equivalent to multiplying 3 3 by __32 .
Multiplication and Division Equations
336
ADDITIONAL EXAMPLE 2 Solve each equation. Graph the solution on a number line. y A __7 = 2 y = 14
20
15
10
15
20
y
B 5 = __3 y = 15 10
Interactive Whiteboard Interactive example available online my.hrw.com
EXAMPLE 2
Focus on Math Connections
Mathematical Processes Remind students that multiplication and division are inverse operations. To solve an equation that contains division, use multiplication to “undo” the division.
Questioning Strategies
Mathematical Processes • How is solving an equation containing division similar to solving an equation containing multiplication? You solve both by using the inverse operation on both sides to get the variable by itself. • Does it make sense that the solution is greater than 15? Yes. Since some number divided by 2 is 15, the unknown number will be greater than (double) the unknown number.
Engage with the Whiteboard Cover up the sentences in blue next to each step of the solution in B and have students write a description of what is happening in each step on the whiteboard. Then have the students graph the solution on the number line.
YOUR TURN
Avoid Common Errors Watch for students who divide both sides of the equation when they should multiply. Remind students to be certain that they are using the inverse operation and to check their answers.
ADDITIONAL EXAMPLE 3 The area of Danielle’s garden is onetwelfth the area of her entire yard. The area of the garden is 10 square feet. What is the area of the yard? Write and solve an equation to solve the y problem. 10 = __ ; 120 square feet 12 Interactive Whiteboard Interactive example available online my.hrw.com
EXAMPLE 3
Questioning Strategies
Mathematical Processes • Why does it make sense to express Juanita’s total scrapbooking time as the decimal 2.5? How else might you express the time? You can compute easily with the number 2.5. Another way is to express the time as the mixed number 2__12 or as the fraction __52 . • Is Julia’s scrapbooking speed best described as a ratio, a rate, or a unit rate? Explain. It is a unit rate because it not only compares quantities in different units, but it also has a denominator of 1. • Suppose Juanita works at her usual rate for 6 hours one weekend. How many pages can she expect to complete? about 54 pages
Focus on Math Connections
Mathematical Processes In solving the problem about Juanita’s scrapbooking, students need to subtract after they solve the division equation, to compare her rate last week to her usual rate. This Example not only reinforces the usefulness of the fourstep problemsolving process but also prepares students for solving twostep equations—those containing more than one operation.
Integrating the ELPS
c.4.G ELL Encourage English learners to take notes on new terms or concepts and to write them in familiar language.
YOUR TURN
Focus on Reasoning
Mathematical Processes In this multistep problem, students first need to find the total number of cards Roberto started with. Have students describe why __5x = 9 is the correct equation for finding that number.
337
Lesson 12.3
DO NOT EDITChanges must be made through “File info” CorrectionKey=A
DO NOT EDITChanges must be made through “File info” CorrectionKey=B
Using Multiplication to Solve Equations
Using Equations to Solve Problems You can use equations to solve realworld problems. Math On the Spot
Math On the Spot
my.hrw.com
my.hrw.com
Multiplication Property of Equality
My Notes
You can multiply both sides of an equation by the same number, and the two sides will remain equal.
EXAMPLE 3
Problem Solving
Juanita is scrapbooking. She usually completes about 9 pages per hour. One night last week she completed pages 23 through 47 in 2.5 hours. Did she work at her average rate?
Solve each equation. Graph the solution on a number line.
© Houghton Mifflin Harcourt Publishing Company
Math Talk
0
10
20
30
40
50
Divide on the left side.
Notice that r is divided by the number 2. Multiply both sides of the equation by 2. 0 5 10 15 20 25 30 35 40 45 50
Check: 15 = __r 2 ? 30 15 = ___ 2 15 = 15
• Compare the number of pages Juanita can expect to complete with the number of pages she actually completed. Justify and Evaluate Solve
Both equations are solved by applying an inverse operation to both sides of the equation. The inverse operations applied are different.
Substitute 50 for x.
B 15 = __r 2 15 = __r 2 2 · 15 = 2 · __r 2 30 = r
• Solve an equation to find the number of pages Juanita can expect to complete.
How is solving a multiplication equation similar to solving a division equation? How are they different?
Multiply both sides of the equation by 5. 20 10
Formulate a Plan
Mathematical Processes
Notice that x is divided by the number 5.
Substitute 30 for r.
TICKET TICKET
Identify the important information. • Worked for 2.5 hours • Starting page: 23 Ending page: 47 • Scrapbooking rate: 9 pages per hour
6.10.A, 6.9.B
x = 10 A __ 5 x = 10 __ 5 x = 5 · 10 5 · __ 5 x = 50 x = 10 Check: __ 5 ? 50 = ___ 10 5 10 = 10
Sixth Grade MEMORIES Goodmes! ti
Analyze Information
EXAMPL 2 EXAMPLE
6.9.C 6.9.C
Let n represent the number of pages Juanita can expect to complete in 2.5 hours if she works at her average rate of 9 pages per hour. Write an equation. n =9 ___ 2.5
n = 2.5 · 9 2.5 · ___ 2.5
Write the equation. Multiply both sides by 2.5.
n = 22.5 Juanita can expect to complete 22.5 pages in 2.5 hours. Juanita completed pages 23 through 47, a total of 25 pages. Because 25 > 22.5, she worked faster than her expected rate.
Divide on the right side.
Justify and Evaluate
You used an equation to find the number of pages Juanita could expect to complete in 2.5 hours if she worked at her average rate. You found that she could complete 22.5 pages.
YOUR TURN y
Solve the equation __ = 1. Graph the solution on a number line. 9 y 3. _ = 1 9
y=
9
Online Assessment and Intervention
The answer makes sense, because Juanita completed 25 pages in 2.5 hours, which is equivalent to a rate of 10 pages in 1 hour. Since 10 > 9, you know that she worked faster than her average rate.
my.hrw.com
Lesson 12.3
6_MTXESE051676_U4M12L3.indd 337
Since 22.5 pages is less than the 25 pages Juanita completed, she worked faster than her average rate.
Personal Math Trainer
10 9 8 7 6 54 3 2 1 0
338
337
28/01/14 9:02 PM
© Houghton Mifflin Harcourt Publishing Company
When an equation contains division, solve by multiplying both sides of the equation by the same number.
Unit 4
6_MTXESE051676_U4M12L3.indd 338
18/01/13 4:59 AM
DIFFERENTIATE INSTRUCTION Number Sense
Have students use a fraction bar to indicate division of both sides of the equation. This provides an easy visual check to compare the coefficient of the variable and the number chosen for dividing both sides of the equation.
Kinesthetic Experience
Give each group of students a set of nine cards numbered 1–9. On signal, each group picks a card at random and passes it to a different group. The number on this card becomes the divisor in a division equation. On a second signal, each group passes a different card to the same group, and this card becomes the quotient. Each group solves their equation, using the following form: (x ÷ first card) = (second card). The first group to solve their equation wins.
Additional Resources
Differentiated Instruction includes: ••Reading Strategies ••Success for English Learners ELL ••Reteach ••Challenge PREAP
Multiplication and Division Equations
338
EXAMPLE 4 Focus on Reasoning
Mathematical Processes Encourage students to begin by analyzing the equation and using number sense to find situations from everyday life that could make sense. For example, if an equation contained decimals, a situation involving money would most likely be a good realworld situation.
Questioning Strategies
Mathematical Processes • Using number sense, what can you determine about the value of x given that the product of two numbers is 72? That the value of x will be less than 72.
• How is the question in a realworld problem related to its equation? The question asks about the value of the variable.
YOUR TURN Focus on Communication
Mathematical Processes Guide students to write problems that use sensible data, and have them explain step by step how to solve their equations. Challenge each student to create original word problems based on unique situations.
Elaborate Talk About It Summarize the Lesson Ask: How do you solve equations that contain multiplication or division? You apply the inverse operation—division for multiplication equations and multiplication for division equations—to both sides of the equation.
GUIDED PRACTICE Engage with the Whiteboard For Exercise 1, have students complete the verbal model on the whiteboard and then draw a model with algebra tiles below the verbal model. Ask students to explain their reasoning.
Avoid Common Errors Exercises 2–3 Remind students to use the inverse operation of division to undo multiplication or multiplication to undo division. Exercise 4 Remind students that the formula for area of a rectangle is A = l · w.
339
Lesson 12.3
DO NOT EDITChanges must be made through “File info” CorrectionKey=B
DO NOT EDITChanges must be made through “File info” CorrectionKey=B
Guided Practice YOUR TURN 4. Roberto is dividing his baseball cards equally among himself, his brother, and his 3 friends. Roberto was left with 9 cards. How many cards did Roberto give away? Write and solve an equation to solve the problem.
Sample answer: _5x = 9; x = 45; Roberto gave away
Personal Math Trainer
1. Caroline ran 15 miles in 5 days. She ran the same distance each day. Write and solve an equation to determine the number of miles she ran each day. (Explore Activity)
b.
Number of
days
Math On the Spot my.hrw.com
8x = 72 Caroline ran
Examine each part of the equation. x is the unknown or quantity we are looking for.
2. x ÷ 3 = 3
= 72 means that after multiplying 8 and x, the result is 72.
x=
Write a realworld situation that involves multiplying two quantities. A hot air balloon flew at 8 miles per hour. How many hours did it take this balloon to travel 72 miles?
© Houghton Mifflin Harcourt Publishing Company
+ + + + +
Total number of miles
=
15
+ + + + +
+ + + + +
miles each day.
Solve each equation. Graph the solution on a number line. (Examples 1 and 2)
8 is multiplied by x.
3. 4x = 32
9
x=
0 1 2 3 4 5 6 7 8 9 10
Solve the equation.
8
10 9 8 7 6 5 4 3 2 1 0
4. The area of the rectangle shown is 24 square inches. How much longer is its length than its width? (Example 3)
8x = 72 8x __ __ = 72 8 8
3
+ + + + +
=
6 in.
24 = 6w; w = 4; 6  4 = 2 inches longer
Divide both sides by 8.
? ?
x=9 The balloon traveled for 9 hours.
ESSENTIAL QUESTION CHECKIN
5. How do you solve equations that contain multiplication or division?
You apply the inverse operation—division for multiplication equations and multiplication for division YOUR TURN 5. Write a realworld problem for the equation 11x = 385. Then solve the equation.
Check students’ problems; x = 35
equations—to both sides of the equation.
Personal Math Trainer Online Assessment and Intervention
my.hrw.com
Lesson 12.3
6_MTXESE051676_U4M12L3.indd 339
w
© Houghton Mifflin Harcourt Publishing Company
6.9.C
Write a realworld problem for the equation 8x = 72. Then solve the equation.
STEP 3
x
·
c. Draw algebra tiles to model the equation.
You can write a realworld problem for a given equation.
STEP 2
· Number of miles
run each day
5
Writing RealWorld Problems
STEP 1
.
my.hrw.com
45  9 = 36 cards.
EXAMPL 4 EXAMPLE
number of miles run each day
a. Let x represent the
Online Assessment and Intervention
339
28/01/14 9:44 PM
340
Unit 4
6_MTXESE051676_U4M12L3.indd 340
1/29/14 11:00 PM
Multiplication and Division Equations
340
Personal Math Trainer Online Assessment and Intervention
Online homework assignment available
Evaluate GUIDED AND INDEPENDENT PRACTICE 6.9.B, 6.10.A
my.hrw.com
Concepts & Skills
Practice
Explore Activity Modeling Equations
Exercise 1
Example 1 Using Division to Solve Equations
Exercise 3
Example 2 Using Multiplication to Solve Equations
Exercise 2
Example 3 Using Equations to Solve Problems
Exercises 4, 6–12
5. The area of a rectangle is 48 square inches. The length is 8 inches. What is the measure of its width? Write and solve an equation.
Example 4 Writing RealWord Problems
Exercises 13–14
Lesson Quiz available online
Exercise
12.3 LESSON QUIZ 6.10.A Solve each equation. 1. __4x = 12.5 2. 8w = 120 z 3. 6 = ___ 4.5
4. 20 = 2.5m
my.hrw.com
Depth of Knowledge (D.O.K.)
Mathematical Processes
2 Skills/Concepts
1.A Everyday life
13
3 Strategic Thinking
1.C Select tools
1. x = 50
14
3 Strategic Thinking
1.F Analyze relationships
2. w = 15
15
3 Strategic Thinking
1.G Explain and justify arguments
16–17
3 Strategic Thinking
1.F Analyze relationships
Answers
3. z = 27
6–12
4. m = 8 5. Sample answer: 8x = 48 inches; x = 6 in.
Additional Resources Differentiated Instruction includes: • Leveled Practice Worksheets
341
Lesson 12.3
DO NOT EDITChanges must be made through “File info” CorrectionKey=A
DO NOT EDITChanges must be made through “File info” CorrectionKey=B
Name
Class
Date
12.3 Independent Practice 6.9.B, 6.9.C, 6.10.A
my.hrw.com
Write and solve an equation to answer each question. Sample answers are given. 6. Jorge baked cookies for his math class’s endofyear party. There are 28 people in Jorge’s math class including Jorge and his teacher. Jorge baked enough cookies for everyone to get 3 cookies each. How many cookies did Jorge bake?
14. Representing RealWorld Problems Write and solve a problem involving money that can be solved with a division equation and has a solution of 1,350.
Personal Math Trainer
Sample answer: Marcy split her income from last week
Online Assessment and Intervention
equally between paying her student loans, rent, and
11. Dharmesh has a square garden with a perimeter of 132 feet. Is the area of the garden greater than 1,000 square feet?
savings. She put $450 in savings. How much did Marcy earn last week? _3x = 450, x = 1,350; Marcy earned
S
$1,350 last week. S
c __ = 3; 84 cookies 28
FOCUS ON HIGHER ORDER THINKING
4s = 132; s = 33 ft;
15. Communicate Mathematical Ideas Explain why 7 · _7x = x. How does this relate to solving division equations?
33 × 33 = 1,089 square feet
7. Sam divided a rectangle into 8 congruent rectangles that each have the area shown. What is the area of the rectangle before Sam divided it?
7x 7 · _7x = __ = 1x = x; 7 divided by 7 is 1. 1x is the same as x. 7
Yes, the area of the garden is greater than 1,000 square feet.
When you solve division equations, you multiply both
Area = 5 cm2
sides of the equation by the number that will give you 1x, or x.
12. Ingrid walked her dog and washed her car. The time she spent walking her dog was onefourth the time it took her to wash her car. It took Ingrid 14 minutes to walk the dog. How long did it take Ingrid to wash her car?
8. Carmen participated in a readathon. Mr. Cole pledged $4.00 per book and gave Carmen $44. How many books did Carmen read?
4k = 44; 11 books
16. Critical Thinking A number tripled and tripled again is 729. What is the number?
w __ = 14; 56 minutes 4
3(3x) = 729;
13. Representing RealWorld Problems Write and solve a problem involving money that can be solved with a multiplication equation.
9. Lee drove 420 miles and used 15 gallons of gasoline. How many miles did Lee’s car travel per gallon of gasoline?
15m = 420; 28 mi/gal
10. On some days, Melvin commutes 3.5 hours per day to the city for business meetings. Last week he commuted for a total of 14 hours. How many days did he commute to the city?
9x = 729;
x = 81;
the number is 81.
17. Multistep Andre has 4 times as many model cars as Peter, and Peter has onethird as many model cars as Jade. Andre has 36 model cars. a. Write and solve an equation to find how many model cars Peter has.
Sample answer: Jacque earned
4p = 36; p = 9; Peter has 9 model cars.
$168 for babysitting over 6 weeks. If she earned the same
b. Using your answer from part a, write and solve an equation to find how many model cars Jade has.
amount each week, how much
1 _ j = 9; j = 27; Jade has 27 model cars. 3
did she earn for one week? 6x = 168; x = $28.
© Houghton Mifflin Harcourt Publishing Company
a __ = 5; 40 square centimeters 8
© Houghton Mifflin Harcourt Publishing Company
Work Area
3.5d = 14; 4 days
Lesson 12.3
6_MTXESE051676_U4M12L3.indd 341
341
26/10/12 12:03 AM
EXTEND THE MATH
PREAP
342
Unit 4
6_MTXESE051676_U4M12L3.indd 342
Activity available online
28/01/14 9:52 PM
my.hrw.com
Activity Here is one way you can name 14, using four 7s and common arithmetic operations: 7×7 ____ +7 7
Use four 7s and the signs for common operations to name each given number. 1. 2 = __________ 2. 3 = __________ 3. 4 = __________ 4. 5 = __________
7+7+7
7+7
77 Sample solutions: 1. _77 + _77 2. _______ 3. __  7 4. 7  _____ 7 7 7
Multiplication and Division Equations
342
MODULE QUIZ
Ready to Go On?
Ready
Assess Mastery
12.1 Writing Equations to Represent Situations
Use the assessment on this page to determine if students have mastered the concepts and standards covered in this module.
Determine whether the given value is a solution of the equation.
3 1
no
1. p  6 = 19; p = 13
yes
b 3. __ = 5; b = 60 12
Response to Intervention
2
Personal Math Trainer
5. 18  h = 13; h = 5
no
Online Assessment and Intervention
my.hrw.com
2. 62 + j = 74; j = 12
yes
4. 7w = 87; w = 12
no
6. 6g = 86; g = 16
no
Write an equation to represent the situation. 7. The number of eggs in the refrigerator e decreased by 5 equals 18.
Intervention
e  5 = 18
Enrichment
Access Ready to Go On? assessment online, and receive instant scoring, feedback, and customized intervention or enrichment.
Personal Math Trainer
my.hrw.com
p + 17 = 29
12.2 Addition and Subtraction Equations Solve each equation.
Online and Print Resources
9. r  38 = 9
Differentiated Instruction
Differentiated Instruction
• Reteach worksheets
• Challenge worksheets
• Reading Strategies • Success for English Learners ELL
ELL
11. n + 75 = 155
PREAP
Extend the Math PREAP Lesson Activities in TE
Additional Resources Assessment Resources includes: • Leveled Module Quizzes
r = 47 n = 80
10. h + 17 = 40
h = 23
12. q  17 = 18
q = 35
12.3 Multiplication and Division Equations Solve each equation. © Houghton Mifflin Harcourt Publishing Company
Online Assessment and Intervention
8. The number of new photos p added to the 17 old photos equals 29.
13. 8z = 112 f 15. __ = 24 28
z = 14 f = 672
d 14. __ =7 14
16. 3a = 57
d = 98 a = 19
ESSENTIAL QUESTION 17. How can you solve problems involving equations that contain addition, subtraction, multiplication, or division?
Write an equation for the situation. Then apply the inverse operation to get the variable alone on one side of the equation.
Module 12
Texas Essential Knowledge and Skills Lesson
Exercises
12.1
1–8
6.7.B, 6.9.A, 6.10.B
12.2
9–12
6.9.B, 6.9.C, 6.10.A, 6.10.B
12.3
13–16
6.9.B, 6.9.C, 6.10.A, 6.10.B
343
Module 12
TEKS
343
Personal Math Trainer
MODULE 12 MIXED REVIEW
Texas Test Prep
Texas Test Prep
Item 8 Students can write the equation 17x = 680 to represent the situation and then start substituting answers in the equation to find which answer makes the sentence true. Choices A and B result in false equations, but choice C results in 680 = 680. Thus, choice C is the correct answer.
Selected Response 1. Kate has gone up to the chalkboard to do math problems 5 more times than Andre. Kate has gone up 11 times. Which equation represents this situation? A a  11 = 5 B 5a = 11 C
Item 1 Some students will see the word times, automatically think of multiplication, and quickly pick choice B. Encourage them to read carefully to identify all key words or phrases such as more times, which indicates addition, not multiplication. Item 4 Some students may select choice A because they looked only at the endpoints under the arrow and didn’t pay attention to the direction in which the arrow was pointing. Remind them that they need to analyze art or diagrams carefully to fully understand them.
a  5 = 11
2. For which equation is y = 7 a solution?
A 6x = 42
6 C _ x = 42
B 42  x = 6
D 6 + x = 42
A t = 19
C
B t = 108
D t = 684
t = 120
8. The area of a rectangular deck is 680 square feet. The deck’s width is 17 feet. What is its length?
A 7y = 1 B y  26 = 19 C
6. Jeordie spreads out a rectangular picnic blanket with an area of 42 square feet. Its width is 6 feet. Which equation could you use to find its length?
7. What is a solution to the equation 6t = 114?
D a + 5 = 11
Avoid Common Errors
Online Assessment and Intervention
y+7=0
y D _ = 14 2
A 17 feet
C
B 20 feet
D 51 feet
40 feet
3. Which is an equation? A 17 + x
C
20x = 200
B 45 ÷ x
D 90  x
4. The number line below represents which equation?
9. Sylvia earns $7 per hour at her afterschool job. One week she worked several hours and received a paycheck for $91. Write and solve an equation to find the number of hours in which Sylvia would earn $91.
.
1
3
0
0
0
0
0
0
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
3 + 7 = 4
4
4
4
4
4
4
D 3  7 = 4
5
5
5
5
5
5
6
6
6
6
6
6
7
7
7
7
7
7
8
8
8
8
8
8
9
9
9
9
9
9
5  4  3  2  1
0 1 2 3 4 5
A 4 + 7 = 3 B 4  7 = 3 C
5. Becca hit 7 more home runs than Beverly. Becca hit 21 home runs. How many home runs did Beverly hit?
344
Gridded Response
A 3
C
B 14
D 28
© Houghton Mifflin Harcourt Publishing Company
Texas Testing Tip Instead of solving equations directly, students can work backward by substituting answers into given equations to see if they are correct. Item 7 Students can try substituting the answers in the equation in order. When they try choice A, they should find that the result is a true equation, indicating that this is the correct solution.
my.hrw.com
21
Unit 4
Texas Essential Knowledge and Skills Items
Grade 6 TEKS
Mathematical Process TEKS
1
6.9.A
6.1.A
2
6.10.B
6.1.C
3
6.7.B
6.1.F
4*
6.3.D, 6.9.A, 6.9.B
6.1.D
5
6.10.A
6.1.A, 6.1.F
6
6.10.A
6.1.A, 6.1.F
7
6.10.B
6.1.C
8
6.10.A
6.1.A, 6.1.F
9
6.10.A, 6.10.B
6.1.A, 6.1.F
* Item integrates mixed review concepts from previous modules or a previous course.
Equations and Relationships
344
Inequalities and Relationships ?
ESSENTIAL QUESTION How can you use inequalities and relationships to solve realworld problems?
MODULE
You can model realworld problems with inequalities, then use algebraic rules to solve the inequalities.
13
LESSON 13.1
Writing Inequalities 6.9.A, 6.9.B, 6.10.B
LESSON 13.2
Addition and Subtraction Inequalities 6.9.B, 6.9.C, 6.10
LESSON 13.3
Multiplication and Division Inequalities with Positive Numbers 6.9.B, 6.9.C, 6.10
LESSON 13.4
Multiplication and Division Inequalities with Rational Numbers
© Houghton Mifflin Harcourt Publishing Company
6.9.B, 6.10.A, 6.10.B
RealWorld Video
my.hrw.com
my.hrw.com
345
Module 13
Some rides at amusement parks indicate a minimum height required for riders. You can model all the heights that are allowed to get on the ride with an inequality.
my.hrw.com
Math On the Spot
Animated Math
Personal Math Trainer
Go digital with your writein student edition, accessible on any device.
Scan with your smart phone to jump directly to the online edition, video tutor, and more.
Interactively explore key concepts to see how math works.
Get immediate feedback and help as you work through practice sets.
345
Are YOU Ready?
Are You Ready? Use the assessment on this page to determine if students need intensive or strategic intervention for the module’s prerequisite skills. 2 1
Understand Integers EXAMPLE
Response to Intervention
735 Write an integer to represent each situation.
Intervention
–75
Enrichment
Access Are You Ready? assessment online, and receive instant scoring, feedback, and customized intervention or enrichment.
Online Assessment and Intervention
my.hrw.com
Skills Intervention worksheets
Differentiated Instruction
• Skill 33 Understand Integers
• Challenge worksheets
• Skill 59 Solve Multiplication Equations
2. a football player’s 3. spending $1,200 4. a climb of 2,400 gain of 9 yards on a flat screen feet 2,400 9 TV 1,200
Integer Operations EXAMPLE
Online and Print Resources
• Skill 47 Integer Operations
Online Assessment and Intervention
Decide whether the integer is positive or negative: into the ground → negative Write the integer.
A water well was drilled 735 feet into the ground.
1. a loss of $75
Personal Math Trainer
my.hrw.com
3 × 8 = 24 30 ÷ (5) = 6
The product or quotient of two integers is positive if the signs of the integers are the same.
7 × (4) = 28 72 ÷ 9 = 8
The product or quotient of two integers is negative if the signs of the integers are different.
Find the product or quotient.
PREAP PREAP
Extend the Math Lesson Activities in TE
5. 6 × 9
54
9. 3 × (7)
21 10.
6. 15 ÷ (5) 64 ÷ 8
3 8
7. 8 × 6
48
11. 8 × (2)
16
8. 100 ÷ (10) 10 12. 32 ÷ 2
16 © Houghton Mifflin Harcourt Publishing Company
3
Personal Math Trainer
Complete these exercises to review skills you will need for this chapter.
Assess Readiness
Solve Multiplication Equations 3 _ h = 15 4
EXAMPLE
4 _ _ · 3 h = 15 · _43 3 4 ·4 _____ h = 15 3
Write the equation. 3 . Multiply both sides by the h is multiplied by __ 4 4 , to isolate reciprocal, __ the variable. 3
h = 20
Simplify.
14. _35 n = 21
35
Solve. 13. 9p = 108
346
12
15. _47 k = 84
147
3 16. __ e = 24 20
160
Unit 4
PROFESSIONAL DEVELOPMENT VIDEO my.hrw.com
Author Juli Dixon models successful teaching practices as she explores inequality concepts in an actual sixthgrade classroom.
Online Teacher Edition Access a full suite of teaching resources online—plan, present, and manage classes and assignments.
Professional Development
ePlanner Easily plan your classes and access all your resources online.
my.hrw.com
Interactive Answers and Solutions Customize answer keys to print or display in the classroom. Choose to include answers only or full solutions to all lesson exercises.
Interactive Whiteboards Engage students with interactive whiteboardready lessons and activities. Personal Math Trainer: Online Assessment and Intervention Assign automatically graded homework, quizzes, tests, and intervention activities. Prepare your students with updated, TEKSaligned practice tests.
Inequalities and Relationships
346
DO NOT EDITChanges must be made through "File info" CorrectionKey=B
Reading StartUp
Reading StartUp Have students complete the activities on this page by working alone or with others.
Visualize Vocabulary Use the ✔ words to complete the graphic.
Visualize Vocabulary The graphic organizer helps students review vocabulary associated with evaluating expressions. If time allows, brainstorm other vocabulary that can be added to the chart.
>, <
4x + 4 = 12; x = 2
greater than, less than
solution
3x  5
Understand Vocabulary
algebraic expression
Use the following explanation to help students learn the preview words. Inequalities are similar to equations in that they represent a statement relating two expressions with a symbol. Equations use an equal sign, while inequalities use an inequality symbol.
Students can use these reading and notetaking strategies to help them organize and understand new concepts and vocabulary. c.4.D Use prereading supports such as graphic organizers, illustrations, and
pretaught topicrelated vocabulary to enhance comprehension of written text.
✔ algebraic expression (expresión algebraica) evaluating (evaluar) ✔ greater than (mayor que) ✔ less than (menor que) like terms (términos semejantes) ✔ numerical expression (expresión numérica) properties of operations (propiedades de las operaciones) ✔ solution (solución) term (término, en una expresión)
Preview Words Match the term on the left to the correct expression on the right.
© Houghton Mifflin Harcourt Publishing Company
Integrating the ELPS
6×4
numerical expression
Review Words
Understand Vocabulary
The solution of an inequality is a value or values that make the inequality true.
Active Reading
Evaluating Expressions
Vocabulary
1. solution of an inequality
A. A value or values that make the inequality true.
2. coefficient
B. A specific number whose value does not change.
3. constant
C. The number that is multiplied by the variable in an algebraic expression.
coefficient (coeficiente) constant (constante) solution of an inequality (solución de una desigualdad) variable (variable)
Active Reading
Additional Resources Differentiated Instruction • Reading Strategies ELL
TwoPanel Flip Chart Create a twopanel flip chart to help you understand the concepts in this module. Label one flap “Adding and Subtracting Inequalities.” Label the other flap “Multiplying and Dividing Inequalities.” As you study each lesson, write important ideas under the appropriate flap.
Module 13
6_MTXESE051676_U4MO13.indd 347
Grades 6–8 TEKS Before Students understand: • operations with rational numbers • properties of operations: inverse, identity, commutative, associative, and distributive properties
347
Module 13
28/01/14 10:02 PM
InCopy Notes
InDesign Notes
1. This is a list
1. This is a list
In this module Students will learn how to: • write onevariable, onestep inequalities to represent constraints or conditions within problems • model and solve onevariable, onestep inequalities that represent problems • write corresponding realworld problems given onevariable, onestep inequalities
347
After Students will learn how to: • write onevariable, twostep inequalities to represent realworld problems • write a realworld problem to represent a onevariable, twostep inequality • solve onevariable, twostep inequalities
MODULE 13
Unpacking the TEKS
Unpacking the TEKS Understanding the TEKS and the vocabulary terms in the TEKS will help you know exactly what you are expected to learn in this module.
Use the examples on this page to help students know exactly what they are expected to learn in this module.
6.9.B Represent solutions for onevariable, onestep equations and inequalities on number lines.
Texas Essential Knowledge and Skills Content Focal Areas Expressions, equations, and relationships—6.9 The student applies mathematical process standards to use equations and inequalities to represent situations. Expressions, equations, and relationships—6.10 The student applies mathematical process standards to use equations and inequalities to solve problems.
What It Means to You You will learn to graph the solution of an inequality on a number line.
Key Vocabulary
UNPACKING EXAMPLE 6.9.B
equation (ecuación) A mathematical sentence that shows that two expressions are equivalent.
The temperature in a walkin freezer must stay under 5 °C. Write and graph an inequality to represent this situation.
inequality (desigualdad) A mathematical sentence that shows the relationship between quantities that are not equal. solution of an inequality (solución de una desigualdad) A value or values that make the inequality true.
Write the inequality. Let t represent the temperature in the freezer. The temperature must be less than 5 °C. t<5 Graph the inequality. 0
5
10
6.10.A
c.4.F Use visual and contextual support … to read gradeappropriate content area text … and develop vocabulary … to comprehend increasingly challenging language.
Model and solve onevariable, onestep equations and inequalities that represent problems, including geometric concepts.
You can model and solve a onevariable, onestep inequality. UNPACKING EXAMPLE 6.10.A
Donny buys 3 binders and spends more than $9. How much did he spend on each binder? Let x represent the cost of one binder.
Go online to see a complete unpacking of the .
Number of binders · Cost of a binder > Total cost of binders
x
>
Use algebra tiles to model 3x > 9 and solve the inequality. x>3 Donny spent more than $3 on each binder.
+ + +
3 Visit my.hrw.com to see all the unpacked.
my.hrw.com my.hrw.com
348
Grade 6 TEKS
What It Means to You
Lesson 13.1
Lesson 13.2
·
9
>
© Houghton Mifflin Harcourt Publishing Company • Image Credits: Image Source/Corbis
Integrating the ELPS
+ + + + + + + + +
Unit 4
Lesson 13.3
Lesson 13.4
6.9.A Write onevariable, onestep equations and inequalities to represent constraints or conditions within problems. 6.9.B Represent solutions for onevariable, onestep equations and inequalities on number lines. 6.9.C Write corresponding realworld problems given onevariable, onestep equations or inequalities. 6.10.A Model and solve onevariable, onestep equations and inequalities that represent problems, including geometric concepts. 6.10.B Determine if the given value(s) make(s) onevariable, onestep equations or inequalities true.
Inequalities and Relationships
348
LESSON
13.1 Writing Inequalities Texas Essential Knowledge and Skills The student is expected to: Expressions, equations, and relationships—6.9.A Write onevariable, onestep equations and inequalities to represent constraints or conditions within problems. Expressions, equations, and relationships—6.9.B
Engage
ESSENTIAL QUESTION How can you use inequalities to represent realworld constraints or conditions? Sample answer: You can choose a letter to represent the variable value in the situation and then use one of the inequality symbols to describe its range of values.
Motivate the Lesson Ask: If you know the lowest and highest temperatures recorded for yesterday, how could you describe yesterday’s temperature at any given time of day? For example, what could you say about the temperature at noon? Begin the Explore Activity to find out.
Represent solutions for onevariable, onestep equations and inequalities on number lines. Expressions, equations, and relationships—6.10.B Determine if the given value(s) make(s) onevariable, onestep equations or inequalities true.
Mathematical Processes 6.1.E Create and use representations to organize, record, and communicate mathematical ideas.
Explore EXPLORE ACTIVITY Engage with the Whiteboard
Have students graph 2 °F on the number line on the whiteboard and then graph 1 °F, 0 °F, 3 °F, 5 °F, and 6 °F in a different color on the same number line. Then have them write inequalities comparing each of the temperatures from B to 2 °F on the whiteboard. Students will see that all five temperatures are greater than 2 °F. Ask students to compare some of the numbers to the left of 2 as well, so that they use both the > and < symbols.
Explain ADDITIONAL EXAMPLE 1 Graph the solutions of each inequality. Check the solutions. A b ≥ 4 5
0
5
5
Sample check: 3 > 5 Interactive Whiteboard Interactive example available online my.hrw.com
349
Lesson 13.1
Mathematical Processes Remind students that when a variable is less than a given number, all the values to the left of the given number on the number line make that inequality true. They are all solutions of the inequality. For example, if x < 4, then every number less than 4 is a solution. Mathematical Processes • What is the difference between the graph of y ≤ 4 and the graph of y < 4? In the first graph, 4 is included in the solution set. In the second graph, 4 is not included in the solution set.
B 3 > s 0
Focus on Communication
Questioning Strategies
Sample check: 1 ≥ 4
5
EXAMPLE 1
• What is the difference between using a solid circle and an open circle? A solid circle includes the number at that point; an open circle does not. • How do you know when to use a solid circle or an open circle? A solid circle is used to represent ≤ or ≥ on a graph; an open circle is used to represent < or > on a graph. • How can you check that the graph of an inequality is correct? Pick a point on the shaded portion of the graph. Any point selected should make the inequality a true inequality.
LESSON
Expressions, equations, and relationships— 6.9.A Write … inequalities to represent constraints or conditions within problems. Also 6.9.B, 6.10.B.
13.1 Writing Inequalities ESSENTIAL QUESTION
Math On the Spot my.hrw.com
How can you use inequalities to represent realworld constraints or conditions?
EXAMPLE 1
Yes; yes; both numbers make the inequality true.
6.9.A
EXPLORE ACTIVITY
Using Inequalities to Describe Quantities You can use inequality symbols with variables to describe quantities that can have many values. Symbol
Meaning
Word Phrases
<
Is less than
Fewer than, below
>
Is greater than
More than, above
≤
Is less than or equal to
At most, no more than
≥
Is greater than or equal to
At least, no less than
A y ≤ 3
© Houghton Mifflin Harcourt Publishing Company
STEP 1
Draw a solid circle at 3 to show that 3 is a solution.
STEP 2
Shade the number line to the left of 3 to show that numbers less than 3 are solutions.
Mathematical Processes
Is 4 _14 a solution of y ≤ 3? Is 5.6?
5 4 3 2 1
STEP 3
Draw an empty circle at 1 to show that 1 is not a solution.
STEP 2
Shade the number line to the right of 1 to show that numbers greater than 1 are solutions. 5 4 3 2 1
are located to the right of 2.
STEP 3
D How many other numbers have the same relationship to 2 as the temperatures in B ? Give some examples.
D
Use an open circle for an inequality that uses > or <.
0 1 2 3 4 5
Check your answer. Substitute 2 for m. 1<2 1 is less than 2, so 2 is a solution.
Reflect
infinitely many; any number greater than 2; sample answer: 1, 2, 8.5, 10
1.
on a number
How is x < 5 different from x ≤ 5?
For x < 5, 5 is not a solution and is not included in the graph. For x ≤ 5, 5 is a solution and is included
a ray extending to the right with its open endpoint at 2 >
4 is less than 3, so 4 is a solution.
STEP 1
They are all greater than 2. All of the temperatures
D
Check your solution.
4 ≤ 3
C How do the temperatures in B compare to 2? How can you see this relationship on the number line?
Complete this inequality: x
0 1 2 3 4 5
B 1
0 1 2 3 4 5 6 7 8
F Let x represent all the possible answers to
Use a solid circle for an inequality that uses ≥ or ≤.
Choose a number that is on the shaded section of the number line, such as 4. Substitute 4 for y.
B The temperatures 0 °F, 3 °F, 6 °F, 5 °F, and 1 °F have also been recorded in Florida. Graph these temperatures on the number line.
E Suppose you could graph all of the possible answers to line. What would the graph look like?
6.9.B
Graph the solutions of each inequality. Check the solutions.
Math Talk
A The lowest temperature ever recorded in Florida was 2 °F. Graph this temperature on the number line. 8 7 6 5 4 3 2 1
A solution of an inequality that contains a variable is any value of the variable that makes the inequality true. For example, 7 is a solution of x > 2, since 7 > 2 is a true statement.
© Houghton Mifflin Harcourt Publishing Company
?
Graphing the Solutions of an Inequality
in the graph.
.
2 Lesson 13.1
349
350
Unit 4
PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.E, which calls for students to “create and use representations to… communicate mathematical ideas.” In the Explore Activity and the Examples, students use number lines, word expressions, and mathematical symbols to express inequalities. They use graphs to represent inequalities and to determine if a given number is a solution.
Math Background Inequalities have a number of properties, including the Transitive Property of Inequality. This property states that for any real numbers a, b, c, if a > b and b > c, then a > c. if a < b and b < c, then a < c. if a > b and b = c, then a > c. if a < b and b = c, then a < c. This may seem like common sense, but the Transitive Property is not necessarily true in daily life. If Team A defeats Team B, and Team B defeats Team C, you can’t assume that Team A will defeat Team C. Writing Inequalities
350
ADDITIONAL EXAMPLE 2 A Write an inequality that represents the phrase “y minus 3 is less than or equal to 5.” Then graph the inequality. y  3 ≤ 5 5
0
5
B The temperature of the river is greater than 1 °C. Write and graph an inequality to represent this situation. t > 1 5
0
5
Interactive Whiteboard Interactive example available online my.hrw.com
YOUR TURN Avoid Common Errors If students have trouble determining which side of the number line to shade, remind them that the inequality sign always points to the lesser of two numbers. Since t ≤ 4, they should shade the number line to the left of 4, because the numbers decrease to the left on a number line.
EXAMPLE 2 Focus on Reasoning
Mathematical Processes Point out to students that sometimes a problem may provide clues and facts that you must use to find a solution. Encourage them to use logical reasoning to solve this kind of problem. For example, in A, tell students that the first step is to identify key words or phrases that indicate operations or relationships. Then they can proceed to write an equation.
Questioning Strategies
Mathematical Processes • In A, how do you know which operation to use to write the inequality? The word sum indicates addition. • In A, how do you know which inequality symbol to use? The phrase greater than indicates the symbol >. • In B, how do you know which inequality symbol to use? The phrase keeps the temperature below 5 °C tells me that the temperature is less than 5, which indicates that the symbol < should be used.
Animated Math Modeling Inequalities Students model inequalities using interactive algebra tiles.
YOUR TURN Avoid Common Errors
Exercise 3 Some students may read the problem quickly and use > instead of ≥. Encourage them to begin by underlining the key words or phrases before trying to graph the inequality.
my.hrw.com
Elaborate Talk About It Summarize the Lesson Ask: How can you make sure you have graphed an inequality correctly? Test one of the values on the number line. If it makes the inequality true, the number line is correct.
GUIDED PRACTICE Engage with Whiteboard For Exercise 3, have students begin by underlining the key words or phrases on the whiteboard and then write an equation, using the same method shown in Example 2. Finally, ask students to graph the inequality on the number line.
Avoid Common Errors Exercises 1 and 2 Some students may shade in the wrong direction when they attempt to graph the solution set of an inequality, such as 1 ≤ x. Reading 1 ≤ x as “x is greater than or equal to 1” serves as a reminder to shade to the right. Exercise 4 Some students may have difficulty determining which inequality symbol to use. Encourage them to begin by underlining key words or phrases before graphing the inequality.
351
Lesson 13.1
YOUR TURN
YOUR TURN
Graph the solution of the inequality t ≤ 4.  10  9  8  7  6  5  4  3  2  1
0 1 2 3 4 5 6 7 8 9 10
Personal Math Trainer
Personal Math Trainer
Online Assessment and Intervention
3.
1 + y ≥ 3; y = 1 is not a solution because 1 + 1 is not
Online Assessment and Intervention
my.hrw.com
greater than or equal to 3.
my.hrw.com
Writing Inequalities
Write an inequality that represents the phrase the sum of 1 and y is greater than or equal to 3 . Check to see if y = 1 is a solution.
Write and graph an inequality to represent each situation.
You can write an inequality to model the relationship between an algebraic expression and a number. You can also write inequalities to represent certain realworld situations.
The highest temperature in February was 6 °F.
5.
Each package must weigh more than 2 ounces.
Math On the Spot
6.9.A, 6.10.B
A Write an inequality that represents the phrase the sum of y and 2 is greater than 5. Draw a graph to represent the inequality. STEP 1
y+2
>
Animated Math
© Houghton Mifflin Harcourt Publishing Company
Guided Practice 1. Graph 1 ≤ x. Use the graph to determine which of these numbers are solutions of the inequality: 1, 3, 0, 1 (Explore Activity and Example 1)
Use an open circle at 3 and shade to the right of 3.
3, 1
0 1 2 3 4 5
Check your solution by substituting a number greater than 3, such as 4, into the original inequality. 4+2>5
0 1 2 3 4 5
5 4 3 2 1
0 1 2 3 4 5
3. Write an inequality that represents the phrase “the sum of 4 and x is less than 6.” Draw a graph that represents the inequality, and check your solution. (Example 2)
5 4 3 2 1
0 1 2 3 4 5
5 4 3 2 1
0 1 2 3 4 5
4+x<6 4. During hibernation, a garter snake’s body temperature never goes below 3 °C. Write and graph an inequality that represents this situation. (Example 2)
6 is greater than 5, so 4 is a solution.
B To test the temperature rating of a coat, a scientist keeps the temperature below 5 °C. Write and graph an inequality to represent this situation.
Let t be temperature in °C; t ≥ 3
Write the inequality. Let t represent the temperature in the lab. t<5
STEP 2
5 4 3 2 1
2. Graph 3 > z. Check the graph using substitution. (Example 1)
Substitute 4 for y.
6>5
STEP 1
0 1 2 3 4 5 6 7 8 9 10 11 12
my.hrw.com
5
Graph the solution. For y + 2 to have a value greater than 5, y must be a number greater than 3.
5 4 3 2 1
STEP 3
2 1
w>2
Write the inequality. The sum of y and 2 is greater than 5.
STEP 2
t≤6
0 1 2 3 4 5 6 7 8 9 10 11 12
my.hrw.com
EXAMPL 2 EXAMPLE
4.
? ?
The temperature must be less than 5 °C.
Graph the inequality.
ESSENTIAL QUESTION CHECKIN
© Houghton Mifflin Harcourt Publishing Company
2.
5. Write an inequality to represent this situation: Nina wants to take at least $15 to the movies. How did you decide which inequality symbol to use?
d ≥ 15, where d represents dollars. “At least” means
0 1 2 3 4 5 6 7 8 9 10
she wants to take $15 or more than $15. Lesson 13.1
351
352
Unit 4
DIFFERENTIATE INSTRUCTION Critical Thinking
Cooperative Learning
Additional Resources
Have students work together to consider absolutevalue inequalities, such as  x  < 2.
Have students work in groups of 4. Have each group make a set of inequality symbol cards, a variable card, and 6 number cards (3 negative numbers and 3 positive numbers). Have the students take turns using the cards to create an inequality, such as this one:
Differentiated Instruction includes: • Reading Strategies • Success for English Learners ELL • Reteach • Challenge PREAP
First have students find numbers that make the inequality true. Then have them use the numbers to sketch a graph of what they think the solution should be. 5
2
0
2
5
x
≥
4.5
Then have the groups record the inequalities and graph them on a number line.
Writing Inequalities
352
Personal Math Trainer Online Assessment and Intervention
Online homework assignment available
Evaluate GUIDED AND INDEPENDENT PRACTICE 6.9.A, 6.9.B, 6.10.B
my.hrw.com
13.1 LESSON QUIZ 6.9.A Graph each inequality. 1. a ≤ 2 2. n < 4
Concepts & Skills
Practice
Explore Activity Using Inequalities to Describe Quantities
Exercise 1
Example 1 Graphing the Solutions of an Inequality
Exercises 1–2, 6–11, 16–19
Example 2 Writing Inequalities
Exercises 3–4, 12–15, 16–19
3. h > 1.5 4. t ≤ 3 Write an inequality that matches the number line model. Use x for the variable. 5. 6. 7. 8.
10
5
0
5
0
5
5
0
5
5
0
5
Lesson Quiz available online my.hrw.com
Answers
2. 3. 4.
5
0
5
5
0
5
5
0
5
5
0
5
5. x ≤ 5 6. x > 1 7. x ≥ 0.5 8. x < 2 9. x ≥ 2.5
353
Depth of Knowledge (D.O.K.)
Lesson 13.1
Mathematical Processes
6
2 Skills/Concepts
1.F Analyze relationships
7–15
2 Skills/Concepts
1.D Multiple representations
16
2 Skills/Concepts
1.F Analyze relationships
17–19
2 Skills/Concepts
1.A Everyday life
20
3 Strategic Thinking
1.G Explain and justify arguments
21–22
3 Strategic Thinking
1.F Analyze relationships
Additional Resources
9. The weight of a package is at least 2.5 pounds. Write an inequality to represent this situation.
1.
Exercise
Differentiated Instruction includes: • Leveled Practice Worksheets
DO NOT EDITChanges must be made through “File info” CorrectionKey=A
DO NOT EDITChanges must be made through “File info” CorrectionKey=B
Class
Name
Date
Write and graph an inequality to represent each situation.
13.1 Independent Practice
Personal Math Trainer
6.9.A, 6.9.B, 6.10.B
10
Online Assessment and Intervention
my.hrw.com
0.03, 0, 1.5, _12
5
9. x ≥ 9 10. n > 2.5 11. 4 _12 >x
© Houghton Mifflin Harcourt Publishing Company
x>6
13.
x ≤ 3
14.
x < 1.5
15.
x ≥ 3.5
14
15
16
17
18
0
1
2
3
t < 3.5
3
2
1
19
20
4
5
g > 150
50 100 150 200 250 300
0 1 2 3 4 5 6 7 8 9 10 11 12
 10  9  8  7  6  5  4  3  2  1
0
1
2
3
4
 12  11  10  9  8  7  6  5  4  3  2  1
0
1
2
20. Communicate Mathematical Ideas Explain how to graph the inequality 8 ≥ y.
Sample answer: Make a solid circle at 8 because of the
5
4
3
2
1
0
1
2
2.5 3
4
5
8
7
6
5
4
3
2
1
0
1
2
4 12
 10  9  8  7  6  5  4  3  2  1
inequality symbol, greater than or equal to. Then shade in the numbers to the left of 8, which are the numbers that make y less than 8.
0 1 2 3 4 5 6 7 8 9 10
21. Represent RealWorld Problems The number line shows an inequality. Describe a realworld situation that the inequality could represent.
0 1 2 3 4 5 6 7 8 9 10
Sample answer: Steve has more than $2.75 in his wallet.
0  10  9  8  7  6  5  4  3  2  1
5
4
3
2
1
0
5
4
3
2
1
0
1
2
3
4
Lesson 13.1
6_MTXESE051676_U4M13L1.indd 353
353
354
10/29/12 12:14 PM
D x is less than or equal to 2 or x is greater than or equal to 3; 2 ≥ x or x ≥ 3
 10  9  8  7  6  5  4  3  2  1
0 1 2 3 4 5 6 7 8 9 10
 10  9  8  7  6  5  4  3  2  1
0 1 2 3 4 5 6 7 8 9 10
C
D
Unit 4
6_MTXESE051676_U4M13L1.indd 354
Activity available online
Activity The four graphs at right show constraints on A both ends of the graph. Challenge students to describe each graph in words and with an inequality statement. Tell them that graphs C and D describe the solutions for a single variable and that they should use the word “or” B to describe these situations.
C x is less than 2 or x is greater than 3; 2 > x or x>3
5
or fractional parts of students.
No; 46 is not greater than or equal to 48.
B x is greater than 2 and is less than or equal to 3; 2 < x ≤ 3
4
not make sense to have negative numbers of students
b. Can a child who is 46 inches tall ride the roller coaster? Explain.
A x is greater than 2 and less than 3: 2 < x < 3
3
3, and 4 as solutions is correct. In this example, it does
c ≥ 48
PREAP
2
Sample answer: The number line that shows only 0, 1, 2,
38 40 42 44 46 48 50 52 54 56 58
EXTEND THE MATH
1
22. Critique Reasoning Natasha is trying to represent the following situation with a number line model: There are fewer than 5 students in the cafeteria. She has come up with two possible representations, shown below. Which is the better representation, and why?
5
16. A child must be at least 48 inches tall to ride a roller coaster. a. Write and graph an inequality to represent this situation.
Work Area
FOCUS ON HIGHER ORDER THINKING
Write an inequality that matches the number line model. 12.
13
© Houghton Mifflin Harcourt Publishing Company
8. 7 < h
4
0 2 1
12
19. The goal of the fundraiser is to make more than $150.
Graph each inequality. 7. t ≤ 8
11
18. The temperature is less than 3.5 °F.
6. Which of the following numbers are solutions to x ≥ 0? 5, 0.03, 1, 0, 1.5, 6, _12
s ≥ 14.5
17. The stock is worth at least $14.50.
28/01/14 10:12 PM
my.hrw.com
5
0
5
5
0
5
5
0
5
5
0
5
Writing Inequalities
354
LESSON
13.2 Addition and Subtraction Inequalities Texas Essential Knowledge and Skills The student is expected to: Expressions, equations, and relationships—6.10.A Model and solve onevariable, onestep inequalities that represent problems, including geometric concepts. Expressions, equations, and relationships—6.9.B Represent solutions for onevariable, onestep equations and inequalities on number lines. Expressions, equations, and relationships—6.9.C Write corresponding realworld problems given onevariable, onestep equations or inequalities. Expressions, equations, and relationships—6.10.B Determine if the given value(s) make(s) onevariable, onestep equations or inequalities true.
Engage
ESSENTIAL QUESTION How can you solve an inequality involving addition or subtraction? Sample answer: You can use the Addition and Subtraction Properties of Inequality: add or subtract the same amount from both sides to isolate the variable.
Motivate the Lesson Ask: You’ve used algebra tiles to model an equation. Do you think you can use algebra tiles to model an inequality? Take a guess. Begin the Explore Activity to find out how.
Explore EXPLORE ACTIVITY Engage with the Whiteboard
Have a student use algebra tiles to make a model of the equation x + 5 = 8 on the whiteboard. Then have another student make a model of the inequality x + 5 ≥ 8 on the whiteboard. Then have all students graph the results on number lines. Discuss with the students how the two models are similar and how they are different. Emphasize the differences.
Mathematical Processes 6.1.D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.
Explain EXAMPLE 1 Focus on Math Connection
ADDITIONAL EXAMPLE 1 Solve each inequality. Graph and check the solution. A x + 3 ≥ 2 5
x ≥ 5 0
B 6 < y 5
5
Questioning Strategies
Mathematical Processes • In A, Step 1, why do you subtract 5 from both sides? The inequality has + 5 in it. To isolate the variable, you need to add the inverse, which is the same as subtracting 5.
y > 1
5
0
Mathematical Processes Solving inequalities is very similar to solving equations. Remind students that they used the properties of equality and inverse operations to solve equations in an earlier module. Similarly, you can use the Addition and Subtraction Properties of Inequality and inverse operations to solve inequalities. Show how solving x + 5 = 12 is similar to solving x + 5 < 12.
5
Interactive Whiteboard Interactive example available online my.hrw.com
• In B, Step 2, why do you shade to the right when the inequality is ≤? Because the variable is on the righthand side of the inequality, you need to read the inequality starting with the variable so that you will know how to shade the graph correctly. If you read the inequality 11 ≤ y as “y is greater than or equal to 11,” you will see that you need to shade to the right. • How can you check that the graph of an inequality is correct? Pick a point on the shaded portion of the graph. Any point selected should make the inequality a true inequality.
355
Lesson 13.2
DO NOT EDITChanges must be made through “File info” CorrectionKey=B
LESSON
13.2 ?
DO NOT EDITChanges must be made through “File info” CorrectionKey=B
Addition and Subtraction Inequalities
ESSENTIAL QUESTION
Expressions, equations, and relationships—6.10.A Model and solve onevariable, onestep… inequalities that represent problems. Also 6.9.B, 6.9.C, 6.10.B.
Addition and Subtraction Properties of Inequality Addition Property of Inequality
Math On the Spot my.hrw.com
How can you solve an inequality involving addition or subtraction?
EXPLORE ACTIVITY
Using Properties of Inequalities
6.10.A
EXAMPLE 1
Modeling OneStep Inequalities
A x + 5 < 12
On a day in January in Watertown, NY, the temperature was 5 °F at dawn. By noon it was at least 8 °F. By how many degrees did the temperature increase?
STEP 1
Increase in temperature
≥
Solve the inequality. x + 5 < 12
A Let x represent the increase in temperature. Write an inequality. +
6.9.B, 6.10.B
Solve each inequality. Graph and check the solution.
You can use algebra tiles to model an inequality involving addition.
Temperature at dawn
Subtraction Property of Inequality
You can add the same number to You can subtract the same number both sides of an inequality and the from both sides of an inequality inequality will remain true. and the inequality will remain true.
8
5 ____ 5 ____ x < 17
Use the Subtraction Property of Inequality. Subtract 5 from both sides.
STEP 2
Graph the solution.
STEP 3
Check the solution. Substitute a solution from the shaded part of your number line into the original inequality. ? 18 + 5 < 12 Substitute 18 for x into x + 5 < 12
5
x
+
B The model shows 5 + x ≥ 8. How many tiles must you remove from each side to isolate x on one side of the inequality? Circle these tiles.
5
≥
8
+ + + + + 5
+ +
x
≥ ≥
C What values of x make this inequality true? Graph the solution of the inequality on the number line. x≥
3
5 4 3 2 1
8
Mathematical Processes
What would it tell you if the inequality is false when you check the solution?
No; Yes; It can only increase by an amount greater than or equal to 3.
0 1 2 3 4 5
Reflect 1.
Math Talk
+ + + + + + + +
Analyze Relationships How is solving the inequality 5 + x ≥ 8 like solving the equation 5 + x = 8? How is it different?
To solve both, you subtract 5 from both sides. But there is only one solution for the equation and
Math Talk
The number line is shaded in the wrong direction, and the number you chose is not a solution.
13 < 12 STEP 1
8 ≤y  3 + 3 + 3 _ _ 11 ≤ y
Lesson 13.2
355
28/01/14 10:39 PM
356
Use the Addition Property of Inequality. Add 3 to both sides. You can rewrite 11 ≤ y as y ≥ 11.
STEP 2
Graph the solution.
STEP 3
Check the solution. Substitute a solution from the shaded part of your number line into the original inequality.
infinitely many solutions for the inequality.
6_MTXESE051676_U4M13L2.indd 355
Solve the inequality.
Mathematical Processes
Could the temperature have increased by 2 degrees by noon? Could it have increased by 5 degrees? Explain.
The inequality is true.
B 8≤ y3
5 6 7 8 9 10 11 12 13 14 15
? 8 ≤ 12  3
Substitute 12 for y in 8 ≤ y  3
8≤ 9
The inequality is true.
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Janusz Wrobel/ Alamy
20 19 18 17 16 15 14 13 12 11 10
Unit 4
6_MTXESE051676_U4M13L2.indd 356
1/29/14 11:06 PM
PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.D, which calls for students to “communicate mathematical ideas… using multiple representations…as appropriate.” In the Explore Activity, students use algebra tiles and a number line to model solving a onestep inequality. In the Examples, students represent inequalities symbolically, graph them on a number line, and write word problems involving realworld situations that correspond to specific inequalities.
Math Background The Commutative and Associative properties hold for addition but not for subtraction, as shown below. Caution students not to use either property when simplifying inequalities that involve subtraction. Commutative Property: Subtraction: 5  3 ≠ 3  5 Associative Property: Subtraction: 5  (3  1) ≟ (5  3)  1 52≟21 3≠1
Addition and Subtraction Inequalities
356
YOUR TURN Avoid Common Errors
Mathematical Processes Exercise 3 Some students may shade in the wrong direction when they attempt to graph the solution set of an inequality with the variable on the righthand side of the inequality symbol. Remind students that reading the inequality beginning with the variable will help them understand which side of the number line to shade.
ADDITIONAL EXAMPLE 2 Write a realworld problem that can be described by 14 < x + 7. Sample: A cook has 7 pounds of potatoes. He needs more than 14 pounds for dinner. What inequality describes the amount of potatoes the cook needs to buy? Interactive Whiteboard Interactive example available online my.hrw.com
EXAMPLE 2 Questioning Strategies
Mathematical Processes • What is the first step in writing a realworld problem to describe an inequality? First, analyze the numbers and relationships given in the inequality. Then think of realworld situations for which those numbers would make sense. • Are there any situations where the negative numbers on the number line would make sense? Yes. For example, it would make sense to use negative numbers for a situation involving changes in temperature.
Connect Vocabulary ELL
Mathematical Processes To help students understand the suggested comparison, relate the words “no more than 60 pounds” to the inequality symbol and the number 60 in the inequality. You may want to draw a simple sketch of a dog in a box on a weight scale to help students visualize the problem.
YOUR TURN Connect to Daily Life
Mathematical Processes Some students may need help conceiving inequality situations. You may want to make a list of various types of situations that students can use for inspiration. Invite the class to help you make the list.
Elaborate Talk About It Summarize the Lesson Ask: How does solving an addition inequality compare with solving an addition equation? The process is essentially the same: subtracting the same amount from each side to isolate the variable.
GUIDED PRACTICE Engage with the Whiteboard For Exercise 1, have a student circle the tiles that need to be removed from each side of the inequality on the whiteboard. Then ask the student to write the inequality and show how to “remove” the three tiles from the lefthand side of the inequality. Finally, have the student show how the solution matches the model. For Exercises 2–3, have students make models to represent the inequality on the whiteboard and explain how to solve it, using the same method that was used for Exercise 1.
Avoid Common Errors Exercises 2–5 Watch for students who incorrectly apply one of the Properties of Inequality. When this is the case, have the student circle the variable and the operation sign associated with it. Remind the student to apply the opposite operation to both sides of the inequality.
357
Lesson 13.2
DO NOT EDITChanges must be made through “File info” CorrectionKey=B
DO NOT EDITChanges must be made through “File info” CorrectionKey=B
YOUR TURN
YOUR TURN
Solve each inequality. Graph and check the solution. 2. y  5 ≥ 7
y≥  2
5 4 3 2 1
0 1 2
Personal Math Trainer
3. 21 > 12 + x
Online Assessment and Intervention
x< 9
3 4 5
0 1 2 3 4 5 6 7
Personal Math Trainer
5. Write a realworld problem that can be modeled by x  13 > 20. Solve your problem and tell what values make sense for the situation.
Check students’ problems.
Online Assessment and Intervention
my.hrw.com
my.hrw.com
8 9 10
Guided Practice
Interpreting Inequalities as Comparisons
1. Write the inequality shown on the model. Circle the tiles you would remove from each side and give the solution. (Explore Activity)
You can write a realworld problem for a given inequality. Examine each number and mathematical operation in the inequality.
EXAMPL 2 EXAMPLE
Inequality: Math On the Spot 6.9.C
my.hrw.com
Write a realworld problem for the inequality 60 ≥ w + 5. Then solve the inequality. STEP 1
Solution:
3+x≤5
+ + +
x≤2
+
≤
+ + + + +
Solve each inequality. Graph and check the solution. (Example 1)
Examine each part of the inequality.
2. x + 4 ≥ 9
x≥5
3. 5 > z  3
z<8
w is the unknown quantity.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Lew Robertson/Corbis
STEP 2
STEP 3
8 9 10
t>7
0 1 2 3 4 5 6 7
60 is greater than or equal to a number added to 5.
4. t + 5 > 12
Write a comparison that the inequality could describe. June’s dog will travel to a dog show in a pet carrier. The pet carrier weighs 5 pounds. The total weight of the pet carrier and the dog must be no more than 60 pounds. What inequality describes the weight of June’s dog?
6. Write a realworld problem that can be represented by the inequality y  4 < 2. Solve the inequality and tell whether all values in the solution make sense for the situation. (Example 2)
0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10
Sample answer: After Chad used a gift card to buy a $4 souvenir,
Solve the inequality.
he had less than $2 left on the card. How much money was on the
60 ≥ w + 5 5 ____ 55 ≥ w
5 ____
card before the purchase?; y < 6. Only values between 4 and 6 make sense, since these values result in positive dollar values.
June’s dog currently weighs ≤ 55 pounds.
? ?
Reflect
4. If you were to graph the solution, would all points on the graph make sense for the situation?
ESSENTIAL QUESTION CHECKIN
7. Explain how to solve 7 + x ≥ 12. Tell what property of inequality you would use.
No; the dog’s weight cannot be 0 or a negative value,
Use the Subtraction Property of Inequality to subtract 7 from both sides of
so only positive numbers make sense.
the inequality. The answer is any number greater than or equal to 5. Lesson 13.2
6_MTXESE051676_U4M13L2.indd 357
5. y  4 < 2
8 9 10
y<6
© Houghton Mifflin Harcourt Publishing Company
0 1 2 3 4 5 6 7
5 is added to w.
357
28/01/14 10:53 PM
358
Unit 4
6_MTXESE051676_U4M13L2.indd 358
28/01/14 10:57 PM
DIFFERENTIATE INSTRUCTION Cooperative Learning
Number Sense
Additional Resources
Let students work in small groups to describe a situation from science that suggests inequalities. Topics may include comparing speeds, temperatures, weights, and so on. Have each group write an inequality for their situation and share the result with the class. Sample answers: The weight of sample A < 2.5 grams; the weight of sample A > the weight of sample B.
Start with the solution to an inequality, x > 3 Discuss with students that many different addition and subtraction inequalities have this same solution. Demonstrate that by “working backward” they can create an inequality that has this same solution. x + 4 > 3 + 4 x+4>1 Ask students to write three additional inequalities that have the solution x > 3. Then record them in a list for the class.
Differentiated Instruction includes: • Reading Strategies • Success for English Learners ELL • Reteach • Challenge PREAP
Addition and Subtraction Inequalities
358
Personal Math Trainer Online Assessment and Intervention
Online homework assignment available
Evaluate GUIDED AND INDEPENDENT PRACTICE 6.10.A, 6.9.B, 6.9.C, 6.10.B
my.hrw.com
13.2 LESSON QUIZ 6.10.A Solve each inequality. Graph and check the solution. 1. x + 20 ≥ 7
Concepts & Skills
Practice
Explore Activity Modeling OneStep Inequalities
Exercise 1
Example 1 Using Properties of Inequalities
Exercises 2–5, 8–11, 17
Example 2 Interpreting Inequalities as Comparisons
Exercises 6, 12–17
2. y  30 ≤ 32 3. 16 < z 14 4. 2.5 > t + 3.5
Exercise
Write and solve an inequality for each situation. 5. Jeremy’s goal is to earn at least $80 toward a week at soccer camp. He has earned $32 so far. How much more does he need to earn? 6. The maximum weight an airline allows for a suitcase is 50 pounds. Ella’s suitcase weighs 8 pounds empty. a. If Ella meets the requirements, what could be the weight of the contents of her suitcase?
Depth of Knowledge (D.O.K.)
8–11
2 Skills/Concepts
1.D Multiple representations
12–16
2 Skills/Concepts
1.A Everyday life
17
3 Strategic Thinking
1.G Explain and justify arguments
18
3 Strategic Thinking
1.F Analyze relationships
19
3 Strategic Thinking
1.G Explain and justify arguments
20
3 Strategic Thinking
1.F Analyze relationships
Additional Resources Differentiated Instruction includes: • Leveled Practice Worksheets
b. Do all values in the solution make sense for the situation? Explain. Lesson Quiz available online my.hrw.com
Answers 1.
30
25
20
60
65
0
5
x ≥ 27 2.
55
y ≤ 62 3.
5
z > 2 (or 2 < z) 4.
10
5
t < 6 (or 6 > t) 5. m + 32 ≥ 80; m ≥ 48
359
Lesson 13.2
0
Mathematical Processes
6. a. w + 8 ≤ 50; w ≤ 42; b. No. A negative value would make no sense, and 0 would mean that the suitcase is empty.
DO NOT EDITChanges must be made through “File info” CorrectionKey=A
DO NOT EDITChanges must be made through “File info” CorrectionKey=B
Name
Class
Date
13.2 Independent Practice
17. Multistep The table shows Marco’s checking account activity for the first week of June.
Personal Math Trainer
6.9.B, 6.9.C, 6.10.A, 6.10.B
my.hrw.com
a. Marco wants his total deposits for the month of June to exceed $1,500. Write and solve an inequality to find how much more he needs to deposit to meet this goal.
Online Assessment and Intervention
x > 50
y ≥ 10
8
7
6
5
3
z ≤ 11
11. 15 ≥ z + 26  15  14  13  12  11  10  9
4
8
7
6
Yes; he has to spend less than $356.68. He spent $93.32 in the first week. $93.32(3) = $279.96, which is less than $356.68.
2
FOCUS ON HIGHER ORDER THINKING
18. Critique Reasoning Kim solved y  8 ≤ 10 and got y ≤ 2. What might Kim have done wrong?
5
Write an inequality to solve each problem.
Kim added 8 to the left side of the inequality and subtracted
12. The water level in the aquarium’s shark tank is always greater than 25 feet. If the water level decreased by 6 feet during cleaning, what was the water level before the cleaners took out any water?
8 from the right side of the inequality to isolate y. She should have added 8 to both sides of the inequality.
© Houghton Mifflin Harcourt Publishing Company
w  6 > 25; w > 31 feet
19. Critical Thinking José solved the inequality 3 > x + 4 and got x < 1. Then, to check his solution, he substituted 2 into the original inequality to check his solution. Since his check worked, he believes that his answer is correct. Describe another check José could perform that will show his solution is not correct. Then explain how to solve the inequality.
13. Danny has at least $15 more than his big brother. Danny’s big brother has $72. How much money does Danny have?
72 ≤ d  15; d ≥ 87 dollars
Substitute 0 for x in the original inequality. 0 is less than
14. The vet says that Ray’s puppy will grow to be at most 28 inches tall. Ray’s puppy is currently 1 foot tall. How much more will the puppy grow?
1 but is not a solution. To solve the inequality, subtract 4
12 + x ≤ 28; x ≤ 16 inches
from both sides to get x <1.
15. Pierre’s parents ordered some pizzas for a party. 4.5 pizzas were eaten at the party. There were at least 5_12 whole pizzas left over. How many pizzas did Pierre’s parents order?
20. Look for a Pattern Solve x + 1 > 10, x + 11 > 20, and x + 21 > 30. Describe a pattern. Then use the pattern to predict the solution of x + 9,991 > 10,000.
p  4.5 ≥ 5.5; p ≥ 10 pizzas
x > 9 for each inequality; in each case the number
16. To get a free meal at his favorite restaurant, Tom needs to spend $50 or more at the restaurant. He has already spent $30.25. How much more does Tom need to spent to get his free meal?
x + 30.25 ≥ 50
added to x is 9 less than the number on the right side of each inequality, so x > 9 is the solution.
x ≥ $19.75
Lesson 13.2
359
6_MTXESE051676_U4M13L2.indd 359
EXTEND THE MATH
360
29/10/12 3:41 PM
You may wish to present the model shown at the right for x  2 ≤ 4. Remind students that x  2 and x + (2) are equivalent. Also remind them of the concept of zero pairs. Have students write an explanation of the way to use algebra tiles to solve subtraction inequalities and be prepared to demonstrate the method to other students.
Unit 4
6_MTXESE051676_U4M13L2.indd 360
Activity available online
PREAP
Activity Challenge pairs of students to use what they know about adding and subtracting integers to model subtraction inequalities with algebra tiles.
Work Area
© Houghton Mifflin Harcourt Publishing Company
 12  11  10  9
$22.82
c. There are three weeks left in June. If Marco spends the same amount in each of these weeks that he spent during the first week, will he meet his goal of spending less than $450 for the entire month? Justify your answer.
8 9 10
10. y  5 ≥ 15
$24.00
Purchase – Water bill
x + 93.32 < 450; x < 356.68; less than $356.68
y≥8
0 1 2 3 4 5 6 7
$46.50
Purchase – Movie Theatre
b. Marco wants his total purchases for the month to be less than $450. Write and solve an inequality to find how much more he can spend and still meet this goal.
0 10 20 30 40 50 60 70 80 90 100
9. 193 + y ≥ 201
$520.45
Purchase – Grocery Store
x + 520.45 > 1500; x > 979.55; more than $979.55
Solve each inequality. Graph and check the solution. 8. x  35 > 15
Deposit – Paycheck
28/01/14 11:00 PM
my.hrw.com
x + (2) ≤ 4
+
− −
≤
+ +
+ +
x2+2≤4+2
+
− −
+ +
≤
+ +
≤
+ +
+ +
+ +
+ +
+ +
x≤6
+
Addition and Subtraction Inequalities
360
LESSON
13.3
Multiplication and Division Inequalities with Positive Numbers
Texas Essential Knowledge and Skills The student is expected to: Expressions, equations, and relationships—6.10.A Model and solve onevariable, onestep inequalities that represent problems, including geometric concepts. Expressions, equations, and relationships—6.9.B
Engage ESSENTIAL QUESTION How can you solve an inequality involving multiplication or division with positive numbers? Sample answer: You can use the Multiplication and Division Properties of Inequality: multiply or divide each side by the same positive amount to isolate the variable.
Motivate the Lesson
Ask: If you get a monthly allowance of $36, how could you use the inequality 4x ≤ 36 to plan a weekly budget? Begin the Explore Activity to find out how to solve multiplication inequalities.
Represent solutions for onevariable, onestep equations and inequalities on number lines. Expressions, equations, and relationships—6.9.C Write corresponding realworld problems given onevariable, onestep equations or inequalities. Expressions, equations, and relationships—6.10.B Determine if the given value(s) make(s) onevariable, onestep equations or inequalities true.
Explore EXPLORE ACTIVITY Connect Multiple Representations Mathematical Processes In the activity, the realworld problem is represented first by an inequality in words, then as an inequality in symbols. The inequality is modeled and solved with algebra tiles. Finally, the solution is shown on a number line. Discuss with students how each representation leads to the next, and discuss the advantages and disadvantages of each type of representation.
Mathematical Processes 6.1.B Use a problemsolving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problemsolving process and the reasonableness of the solution.
ADDITIONAL EXAMPLE 1 Solve each inequality. Graph and check the solution. A 7x > 28
x>4
0 y B __5 ≤ 30
5
10
150
155
Interactive Whiteboard Interactive example available online my.hrw.com
361
Lesson 13.3
EXAMPLE 1 Focus on Math Connections
Mathematical Processes Present the equation 12x = 24. Encourage students to solve for x, x = 2. The solution to the equation is one point on the number line. The solution to the inequality, x < 2, is a section of the number line, in this case, everything to the left of 2.
Questioning Strategies
Mathematical Processes • In A, Step 1, why do you divide by 12? To isolate x, it is necessary to “remove” the 12. Since x is multiplied by 12, using the inverse operation, division, will “undo” the multiplication. • In A, Step 2, 24 ÷ 12 is 2. Why isn’t 2 a solution? Since the inequality sign is the less than sign, the solution is less than 2. • When graphing inequalities, how do you know when to use a solid circle or an open circle? A solid circle is used to represent ≤ or ≥ on a graph. An open circle is used to represent < or > on a graph.
y ≤ 150
145
Explain
Engage with the Whiteboard For A, have a volunteer make a model with algebra tiles on the whiteboard. Have the student solve the inequality, using the same method that was used in B of the Explore Activity. Then have students compare and contrast the model with the number line provided in the example.
DO NOT EDITChanges must be made through “File info” CorrectionKey=B
LESSON
13.3
DO NOT EDITChanges must be made through “File info” CorrectionKey=A
Multiplication and Division Inequalities with Positive Numbers
Expressions, equations, and relationships—6.10.A Model and solve onevariable, onestep inequalities that represent problems. Also 6.9.B, 6.9.C, 6.10.B.
Solving Inequalities Involving Multiplication and Division Math On the Spot
You can use properties of inequality to solve inequalities involving multiplication and division with positive integers.
my.hrw.com
ESSENTIAL QUESTION
Multiplication and Division Properties of Inequality How can you solve an inequality involving multiplication or division with positive numbers?
• You can multiply both sides of an inequality by the same positive number and the inequality will remain true. • You can divide both sides of an inequality by the same positive number and the inequality will remain true.
6.10.A
EXPLORE ACTIVITY
Modeling OneStep Inequalities
EXAMPLE 1
You can use algebra tiles to solve inequalities that involve multiplying positive numbers.
Solve each inequality. Graph and check the solution.
A 12x < 24
Dominic is buying school supplies. He buys 3 binders and spends more than $9. How much did he spend on each binder?
STEP 1
A Let x represent the cost of one binder. Write an inequality. Number of binders
3
•
Cost of a binder
B The model shows the inequality from There are
3
A
3
Mathematical Processes
9
Are all negative numbers solutions to 12x < 24? Explain.
.
+ + +
>
+ + + + + + + + +
Yes; because 12 times a negative number is a negative number, and all negative numbers are less than 24.
3
C What values make the inequality you wrote in true? Graph the solution of the inequality.
A
x>3
5 4  3  2  1
0 1 2 3 4 5
Reflect 1. Analyze Relationships Is 3.25 a solution of the inequality you wrote in A ? If so, does that solution make sense for the situation?
Yes, 3.25 is a solution. Dominic may have paid $3.25 for each binder.
Divide both sides by 12.
x<2
Math Talk
equal groups.
How many units are in each group?
Solve the inequality. 12x __ ___ < 24 12 12
9
xtiles, so draw circles
to separate the tiles into © Houghton Mifflin Harcourt Publishing Company
>
x
•
>
6.9.B, 6.10.B
Use an open circle to show that 2 is not a solution.
STEP 2
Graph the solution.
STEP 3
Check the solution by substituting a solution from the shaded part of the graph into the original inequality. ? Substitute 0 for x in the original inequality. 12(0) < 24
5 4  3  2  1
0 < 24
y B _3 ≥ 5
STEP 1
0 1 2 3 4 5
The inequality is true.
Solve the inequality. 3 ( _3 ) ≥ 3(5) y
Multiply both sides by 3.
y ≥ 15
Use a closed circle to show that 15 is a solution.
STEP 2
Graph the solution.
STEP 3
Check the solution by substituting a solution from the shaded part of the graph into the original inequality. 18 ? __ ≥5 Substitute 18 for x in the original inequality. 3
© Houghton Mifflin Harcourt Publishing Company
?
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
2. Represent RealWorld Problems Rewrite the situation in represent the inequality 3x < 9.
A
to
Dominic is buying school supplies. He buys 3 binders and spends less than $9. How much did he spend on each binder?
6≥5 Lesson 13.3
6_MTXESE051676_U4M13L3.indd 361
InCopy Notes 1. This is a list
361
362
1/29/14 11:13 PM
InDesign Notes 1. This is a list
PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.B, which calls for students to “use a problemsolving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problemsolving process and the reasonableness of the solution.” Students directly apply each of these steps of the problemsolving model in Example 2, to determine the possible side lengths of a square flag.
The inequality is true.
Unit 4
6_MTXESE051676_U4M13L3.indd 362
InCopy Notes
10/29/12 7:05 PM
InDesign Notes
1. This is a list Bold, Italic, Strickthrough.
1. This is a list
Math Background As with addition and subtraction, not all properties that hold for multiplication hold for division. For example, the Commutative and Associative properties hold for multiplication but not for division. Caution students to avoid rearranging or regrouping numbers that are being divided. Commutative Property: Division: 15 ÷ 3 ≠ 3 ÷ 15 Associative Property: Division: 50 ÷ (10 ÷ 5) ≟ (50 ÷ 10) ÷ 5 50 ÷ 2 ≟ 5 ÷ 5 25 ≠ 1
Multiplication and Division Inequalities with Positive Numbers
362
YOUR TURN Avoid Common Errors Exercise 4 Some students may divide both sides by 4 instead of multiplying. Rewriting the equation as z ÷ 4 = 11 may help these students see which operation they should use.
ADDITIONAL EXAMPLE 2 Kayla earned more than $50 babysitting. Her mother paid her $4 an hour to babysit her little brother. Write and solve an inequality to find the possible number of hours Kayla babysat. 4h > 50; h > 12.5; Kayla babysat more than 12.5 hours. Interactive Whiteboard Interactive example available online my.hrw.com
EXAMPLE 2 Focus on Reasoning
Mathematical Processes Point out to students that sometimes a problem may provide clues and facts that you must use to find a solution. Encourage students to begin by identifying the important information. They can underline or circle the information in the problem statement.
Questioning Strategies
Mathematical Processes • What information is given in the problem? Because the flag being made is a square, all the sides have the same length, and the perimeter needs to be 22 inches or longer. • How do you find the perimeter of a figure? Perimeter = sum of the lengths of the sides
Focus on Modeling Mathematical Processes Have students draw a square to represent the square flag. Then have students label each of the sides of the square with an x. Guide them to see that the four sides (x + x + x + x) are the same as the 4x shown in the inequality in the Solve step. c.2.I.4 ELL Encourage class discussion to develop the scenario in Reflect. English learners will benefit from hearing and participating in classroom discussions.
Integrating the ELPS
YOUR TURN Focus on Critical Thinking Mathematical Processes Point out to students that the unknown is the total weight of sand needed for 6 paperweights. Help students to see that the total weight divided by 6 is equivalent to the weight of a single paperweight.
Elaborate Talk About It Summarize the Lesson Ask: How do you solve a division inequality? Multiply both sides of the inequality by the same number to isolate the variable.
GUIDED PRACTICE Engage with the Whiteboard For Exercise 1, have students circle the groups of tiles on the whiteboard and then write the inequality and the solution below the model. Have students make a number line to show the solution. Ask the class to compare and contrast the model with the number line.
Avoid Common Errors Exercises 2–3 Watch for students who incorrectly apply one of the Properties of Inequality. Remind them that they should apply the opposite operation to both sides of the inequality. Exercise 4 If students have difficulty writing the inequality, encourage them to make a model to help them understand the situation better.
363
Lesson 13.3
DO NOT EDITChanges must be made through “File info” CorrectionKey=B
DO NOT EDITChanges must be made through “File info” CorrectionKey=A
YOUR TURN Solve each inequality. Graph and check the solution. 3. 5x ≥ 100 4. _4z < 11
Reflect
Personal Math Trainer
x ≥ 20
15 16 17 18 19 20 21 22 23 24 25
z < 44
40 41 42 43 44 45 46 47 48 49 50
5. Represent RealWorld Problems Write and solve a realworld problem for the inequality 4x ≤ 60.
Online Assessment and Intervention
Sample problem: Cy is making a square flag. He wants
my.hrw.com
the perimeter to be no more than 60 inches. What are the possible side lengths? x ≤ 15 inches
Solving RealWorld Problems
YOUR TURN
You can use multiplication and division inequalities to model and solve realworld problems.
EXAMPL 2 EXAMPLE
Personal Math Trainer Math On the Spot
Problem Solving
6.10.A
Online Assessment and Intervention
my.hrw.com
6. A paperweight must weigh less than 4 ounces. Brittany wants to make 6 paperweights using sand. Write and solve an inequality to find the possible weight of the sand she needs.
w __ < 4; w < 24; Brittany must use less than 24 oz of sand. 6
my.hrw.com
Cy is making a square flag. He wants the perimeter to be at least 22 inches. Write and solve an inequality to find the possible side lengths.
Guided Practice 1. Write the inequality shown on the model. Circle groups of tiles to show the solution. Then write the solution. (Explore Activity)
Formulate a Plan
Inequality:
Write and solve a multiplication inequality. Use the fact that the perimeter of a square is 4 times its side length.
Solution:
© Houghton Mifflin Harcourt Publishing Company
Let x represent a side length.
4x __ __ ≥ 22 4 4
Divide both sides by 4.
x ≥ 5.5
2. 8y < 320
35 36 37 38 39 40 41 42 43 44 45
The side lengths must be greater than or equal to 5.5 in.
? ?
Check the solution by substituting a value in the solution set in the original inequality. Try x = 6.
30 31 32 33 34 35 36 37 38 39 40
ESSENTIAL QUESTION CHECKIN
Use the Division Property of Inequality to divide both sides by 5 to get x < 8.
The statement is true.
Cy’s flag could have a side length of 6 inches.
To check, substitute a number from the solution into the original inequality. Lesson 13.3
364
363
6_MTXESE051676_U4M13L3.indd 363
1. This is a list
r ≥ 33
5. Explain how to solve and check the solution to 5x < 40 using properties of inequalities.
Substitute 6 for x.
InCopy Notes
3. _3r ≥ 11
+ + + + + + + +
b __ ≥ 14; b ≥ 84; Karen had at least 84 books. 6
Justify and Evaluate
24 ≥ 22
y < 40
<
4. Karen divided her books and put them on 6 shelves. There were at least 14 books on each shelf. How many books did she have? Write and solve an inequality to represent this situation. (Example 2)
Cy’s flag should have a side length of 5.5 inches or more.
? 4(6) ≥ 22
+ +
Solve each inequality. Graph and check the solution. (Example 1)
Justify and Evaluate Solve
4x ≥ 22
2x < 8 x<4
© Houghton Mifflin Harcourt Publishing Company
Analyze Information
Find the possible lengths of 1 side of a square that has a perimeter of at least 22 inches.
10/29/12 7:06 PM
Unit 4
6_MTXESE051676_U4M13L3.indd 364
InCopy Notes
InDesign Notes 1. This is a list
1. This is a list Bold, Italic, Strickthrough.
DIFFERENTIATE INSTRUCTION
28/01/14 11:13 PM
InDesign Notes 1. This is a list
Cooperative Learning
Critical Thinking
Additional Resources
Have students work in pairs. Give each pair an index card. Have students write an inequality word problem on the card, then exchange problems with another pair and work together to solve the problems.
Let students work together to decide which of the following statements, a + c > b + c or ac > bc, are always true if a, b, and c are real numbers and a > b. If the statement is not always true, have students give an example to show that the statement is false.
Differentiated Instruction includes: • Reading Strategies • Success for English Learners ELL • Reteach • Challenge PREAP
If students have trouble writing word problems, suggest that they use the word problems in the Independent Practice as a template. They can change the names, numbers, and/or the direction of the inequality to create new problems.
a + c > b + c is always true; ac > bc is not always true. a = 2, b = 1, c = 3 2‧3<1‧3
6 < 3, false
Multiplication and Division Inequalities with Positive Numbers
364
Personal Math Trainer Online Assessment and Intervention
Online homework assignment available
Evaluate GUIDED AND INDEPENDENT PRACTICE 6.10.A, 6.9.B, 6.9.C, 6.10.B
my.hrw.com
13.3 LESSON QUIZ 6.10.A Solve each inequality. Graph and check the solution. 1. 4x ≥ 12 y 2. __3 ≤ 6
Concepts & Skills
Practice
Explore Activity Modeling OneStep Inequalities
Exercise 1
Example 1 Solving Inequalities Involving Multiplication and Division with Positive Integers
Exercises 2–3, 10–11, 15–18
Example 2 Solving RealWorld Problems
Exercises 4, 6–9, 12–14
3. 2p > 24 4. __8t < 20 Write and solve an inequality for each problem. 5. Kellin pays more than $40 a day to rent a canoe. What amount might he pay to rent the canoe for a 7day trip? 6. Amanda earns $15 an hour. She needs at least $90 to buy a computer desk. How many hours does she need to work to buy the desk?
Exercise
Depth of Knowledge (D.O.K.)
6–9
2 Skills/Concepts
1.B Problemsolving model
10–11
2 Skills/Concepts
1.D Multiple representations
12–14
2 Skills/Concepts
1.B Problemsolving model
15–18
2 Skills/Concepts
1.D Multiple representations
19–22
2 Skills/Concepts
1.A Everyday life
23–24
3 Strategic Thinking
1.F Analyze relationships
25
3 Strategic Thinking
1.B Problemsolving model
Lesson Quiz available online my.hrw.com
Answers 1. 0
5
10
15
20
15
20
160
165
x≥3 2. 10
y ≤ 18 3. 10
p > 12 4. 155
t < 160 w 5. __ > 40; w > 280; more than $280 7
6. 15h ≥ 90; h ≥ 6
365
Lesson 13.3
Mathematical Processes
Additional Resources Differentiated Instruction includes: • Leveled Practice Worksheets
DO NOT EDITChanges must be made through “File info” CorrectionKey=B
DO NOT EDITChanges must be made through “File info” CorrectionKey=B
Class
Date
The sign shows some prices at a produce stand.
13.3 Independent Practice 6.9.B, 6.9.C, 6.10.A, 6.10.B
my.hrw.com
Write and solve an inequality for each problem.
22. The produce buyer for a local restaurant wants to buy more than 30 lb of onions. The produce buyer at a local hotel buys exactly 12 pounds of spinach. Who spends more at the produce stand? Explain.
numbers don’t make sense.
The restaurant produce buyer; he will spend more than
14. Multistep Lina bought 4 smoothies at a health food store. The bill was less than $16.
$37.50; the hotel’s produce buyer will spend $36.
a. Write and solve an inequality to represent the cost of each smoothie.
8. In a litter of 7 kittens, each kitten weighs more than 3.5 ounces. Find the possible total weight of the litter. w __ 7 > 3.5; w > 24.5; more than
4s < 16; s < 4; the cost of each
_r 5
24. Represent RealWorld Problems Write and solve a word problem that can be represented with 240 ≤ 2x.
Sample answer: Jake and his brother want to earn at
less than $4.
least $240. If each brother earns the same amount, how
c. Graph the values that make sense for this situation on the number line.
Solve each inequality. Graph and check the solution.
1
x≤6
p 15. __ ≤ 30 13
0 1 2 3 4 5 6 7 8 9 10
t>0
16. 2t > 324 17. 12y ≥ 1
0 1 2 3 4 5
x 18. ___ < 11 9.5
much money does each brother have to earn?; x ≥ 120;
each brother must earn at least $120.
0 1 2 3 4 5
Solve each inequality.
25. Persevere in Problem Solving A rectangular prism has a length of 13 inches and a width of _12 inch. The volume of the prism is at most 65 cubic inches. Find all possible heights of the prism. Show your work.
p ≤ 390
()
13 ∙ _12 ∙ h ≤ 65; 6.5h ≤ 65; h < 10; all heights greater
t > 162 1 y ≥ __ 12
than 0 inches but no more than 10 inches
x < 104.5 Lesson 13.3
6_MTXESE051676_U4M13L3.indd 365
365
366
28/01/14 11:16 PM
InCopy Notes
InDesign Notes
EXTEND THE MATH
Unit 4
6_MTXESE051676_U4M13L3.indd 366
InCopy Notes
1. This is a list
Activity available online A Mystery Rectangle
There are only 3 possible rectangles.
Clue 2: Each dimension must be a whole number.
Perimeter
21
24
90
22
25
94
23
26
98
1. This is a list
my.hrw.com
Activity Challenge students to translate Clues 1, 3, and 4 into inequalities or equations. Direct them to use the clues together to find all the possible dimensions of the rectangle. List them in the table below.
Length
28/01/14 11:19 PM
InDesign Notes
1. This is a list Bold, Italic, Strickthrough.
PREAP
Width
is the
side and divided by 5 on the right side.
amount greater than $0 and
of Will’s backyard is at least 11 feet.
2 __ . What 25
r ≤ 2; the student may have multiplied by 5 on the left
Each smoothie can cost any
15.5w ≥ 170.5; w ≥ 11; the width
_2 and 5
23. Critique Reasoning A student solves ≤ gets r ≤ correct solution? What mistake might the student have made?
b. What values make sense for this situation? Explain.
9. To cover his rectangular backyard, Will needs at least 170.5 square feet of sod. The length of Will’s yard is 15.5 feet. What are the possible widths of Will’s yard?
Work Area
FOCUS ON HIGHER ORDER THINKING
smoothie was less than $4.
24.5 ounces.
© Houghton Mifflin Harcourt Publishing Company
No; 1.25x ≤ 3; x ≤ 2.4 so 2.4 pounds of onions are the most
Florence can buy. 2.4 < 2.5, so she cannot buy 2.5 pounds.
negative amount so negative
14x ≥ 84; x ≥ 6; at least 6 hours
1. This is a list
21. Florence wants to spend no more than $3 on onions. Will she be able to buy 2.5 pounds of onions? Explain.
No; Steve would not pay a
7. Tamar needs to make at least $84 at work on Tuesday to afford dinner and a movie on Wednesday night. She makes $14 an hour at her job. How many hours does she need to work on Tuesday?
5 4  3  2  1
at most $2.75
13. If you were to graph the solution for exercise 12, would all points on the graph make sense for the situation? Explain.
to 7 inches
11. _2t > 0
20. Gary has enough money to buy at most 5.5 pounds of potatoes. How much money does Gary have?
r __ < 32; r < 992; less than $992 31
6s ≤ 42; s ≤ 7; less than or equal
10. 10x ≤ 60
3_13 pounds
Online Assessment and Intervention
12. Steve pays less than $32 per day to rent his apartment. August has 31 days. What are the possible amounts Steve could pay for rent in August?
6. Geometry The perimeter of a regular hexagon is at most 42 inches. Find the possible side lengths of the hexagon.
Price per Pound Produce $1.25 Onions $0.99 Yellow Squash $3.00 Spinach $0.50 Potatoes
19. Tom has $10. What is the greatest amount of spinach he can buy?
Personal Math Trainer
© Houghton Mifflin Harcourt Publishing Company
Name
Clue 1: A rectangle has a perimeter that is less than 100 units. 2w + 2l < 100 Clue 3: The width must be less than 25, but greater than 20. w > 20; w < 25 Clue 4: The length must be 3 units longer than the width. l=w+3
Multiplication and Division Inequalities with Positive Numbers
366
LESSON
13.4
Multiplication and Division Inequalities with Rational Numbers
Texas Essential Knowledge and Skills The student is expected to: Expressions, equations, and relationships—6.9.B Represent solutions for onevariable, onestep equations and inequalities on number lines. Expressions, equations, and relationships—6.10.A Model and solve onevariable, onestep equations and inequalities that represent problems, including geometric concepts. Expressions, equations, and relationships—6.10.B Determine if the given value(s) make(s) onevariable, onestep equations or inequalities true.
Mathematical Processes 6.1.F Analyze mathematical relationships to connect and communicate mathematical ideas.
Engage ESSENTIAL QUESTION How do you solve inequalities that involve multiplication and division of integers? Sample answer: You multiply or divide each side of the inequality to isolate the variable. If the integer is negative, you must reverse the direction of the inequality symbol.
Motivate the Lesson Ask: Have you ever had a friend ask you a question using doublenegative language, such as “You don’t want to not go, do you?” Well, double negatives can change things around. Begin the Explore Activity and find out how multiplying or dividing both sides of an inequality by a negative number can change things.
Explore EXPLORE ACTIVITY Engage with the Whiteboard Have students draw number lines on the whiteboard to graph each example in A. First, have them graph the given inequality. Then have them graph the inequality after multiplying each side by the same number in a different color. For each example, ask if the original symbol still makes the inequality true after multiplying. If it doesn’t, ask what symbol would make the inequality true.
Explain ADDITIONAL EXAMPLE 1 Solve each inequality. Graph and check the solution. A 3x < 18 10
B __6 > 7 y
35
x > 6 5
0
45
my.hrw.com
Lesson 13.4
Questioning Strategies
Mathematical Processes
number you need to use to multiply both sides of the inequality
Interactive Whiteboard Interactive example available online
367
ELL c.1.F Remind students that the word reverse means to turn backward in position or direction. Connect this meaning to reversing the inequality, where the point of the symbol changes direction.
Connect Vocabulary
y • In B, why might you rewrite __3 as ( __13 )y before multiplying? to make it easier to see what
y < 42 40
EXAMPLE 1
• How can you check that the graph of an inequality is correct? Pick a point on the shaded portion of the graph. Any point selected should make the inequality a true inequality.
DO NOT EDITChanges must be made through "File info" CorrectionKey=A
LESSON
13.4
DO NOT EDITChanges must be made through "File info" CorrectionKey=B
Multiplication and Division Inequalities with Rational Numbers
Expressions, equations, and relationships—6.9.B Represent solutions for onestep inequalities on number lines. Also 6.10.A, 6.10.B
Multiplication and Division Properties of Inequality Math On the Spot
Recall that you can multiply or divide both sides of an inequality by the same positive number, and the statement will still be true.
my.hrw.com
ESSENTIAL QUESTION
Multiplication and Division Properties of Inequality How do you solve inequalities that involve multiplication and division of integers?
EXPLORE ACTIVITY
• If you multiply or divide both sides of an inequality by the same negative number, you must reverse the inequality symbol for the statement to still be true.
6.10.A
Investigating Inequality Symbols
EXAMPLE 1
You have seen that multiplying or dividing both sides of an inequality by the same positive number results in an equivalent inequality. How does multiplying or dividing both sides by the same negative number affect an inequality?
Solve each inequality. Graph and check the solution.
My Notes
A Complete the tables. New inequality
New inequality is true or false?
3<4
2
true
2 ≥ 3
6< 8
3
© Houghton Mifflin Harcourt Publishing Company
5>2
1
8 > 10
8
6 ≥ 9
5 > 2
false
x < 13
Divide each side by:
New inequality
New inequality is true or false?
4<8
4
1< 2
true
4 ≤ 3
false
4 ≥ 5
3
16 ≤ 12
4
15 > 5
5
3 > 1
Divide both sides by 4. Reverse the inequality symbol.
4x 52 ____ < ___ 4 4
false
64 > 80
Solve the inequality. 4x > 52
true
Inequality
12 ≥ 15
A 4x > 52 STEP 1
Multiply each side by:
Inequality
6.9.B, 6.10.B
STEP 2
Graph the solution.
STEP 3
Check your answer using substitution. ? Substitute 15 for x in 4x > 52. 4(15) > 52 60 > 52
y
15 14 13 12 11 10 9
8
The statement is true.
B  _3 < 5
true
STEP 1
Solve the inequality. y
 _3 < 5
false
( )
y 3  _3 > 3(5)
Multiply both sides by 3. Reverse the inequality symbol.
y > 15
B What do you notice when you multiply or divide both sides of an inequality by the same negative number?
The inequality is no longer true. C How could you make each of the multiplication and division inequalities that were not true into true statements?
STEP 2
Graph the solution.
STEP 3
Check your answer using substitution. ? 18 <  ___ 5 3 6 < 5
Reverse the inequality symbol.
Lesson 13.4
6_MTXESE051676_U4M13L4.indd 367
367
368
29/10/12 6:35 PM
10 11 12 13 14 15 16 17 18 19 20
© Houghton Mifflin Harcourt Publishing Company
?
y Substitute 18 for y in  __ < 5. 3 The inequality is true.
Unit 4
6_MTXESE051676_U4M13L4.indd 368
1/29/14 11:16 PM
PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.F, which calls for students to “analyze mathematical relationships to connect and communicate mathematical ideas.” In the Explore Activity, students use tables and numerical inequalities to explore the effects of multiplying or dividing an inequality by a negative number. Students examine examples and counterexamples, leading to the conclusion that the inequality symbol must be reversed when multiplying or dividing both sides of the inequality by a negative number.
Math Background The solution set also can be described by using interval notation. In this notation, parentheses indicate that the endpoints are not included in the solution set. Brackets indicate that the endpoints are included in the solution set. Suppose a and b are any real numbers: Inequality
Interval notation
a
(a, b)
a
(a, b] [a, b) [a, b]
Multiplication and Division Inequalities with Rational Numbers
368
YOUR TURN Avoid Common Errors Exercise 2 Some students may divide both sides by 6 instead of multiplying. Rewriting the equation as 7 ≥ t ÷ (6) may help these students see which operation they should use.
ADDITIONAL EXAMPLE 2 During the month, Jenna uses the dining commons at the college for dinner. Each dinner costs $12. Each time she uses her dining commons card, her balance changes by 12. Last month the balance change was an amount greater than or equal to 324. How many times did she use her dining commons card? 12n ≥ 324; n ≤ 27; Jenna used the dining commons card 27 or fewer times. Interactive Whiteboard Interactive example available online my.hrw.com
EXAMPLE 2 Focus on Reasoning
Mathematical Processes Point out to students that sometimes a problem may provide clues and facts that must be used to find a solution. Encourage students to begin by identifying the important information. They can underline or circle the information in the problem statement.
Questioning Strategies
Mathematical Processes • Why is “40 feet below sea level” written as a negative number? Because the words “below sea level” indicate a negative direction, down. • What is the unknown in this situation? The unknown is the time it took the submersible to reach its final elevation of more than 40 feet below sea level.
Focus on Modeling Mathematical Processes It may help students to understand the problem better if they draw a vertical number line to represent this situation. They can label the descent in intervals of 5 feet for each second.
YOUR TURN Focus on Modeling Mathematical Processes To help students visualize the $35 being deducted each month, draw a vertical number line from 0 to 315. Show the decreases in multiples of 35 for each month. Help students understand that what they need to find is the number of months it takes to go no farther than 315.
Elaborate Talk About It Summarize the Lesson Ask: How do you know when to reverse the inequality symbol when solving an inequality and when not to reverse it? If you multiply or divide both sides by a negative number, you must reverse the inequality symbol. If you multiply or divide by a positive number, the inequality symbol stays the same.
GUIDED PRACTICE Engage with the Whiteboard For Exercises 1–2, have students solve and graph each inequality on the whiteboard, showing all their work. Then have students explain how they knew which operation to use to solve each inequality.
Avoid Common Errors Exercises 3–5 Watch for students who may forget to reverse the inequality symbol when multiplying or dividing by a negative number. Remind them that they should always check their solution. Exercise 5 Remind students that “not colder than 80 °C” means that the temperature is at least 80 °C or warmer. Warmer temperatures are above 80° on a thermometer or to the right of 80 on a number line.
369
Lesson 13.4
DO NOT EDITChanges must be made through "File info" CorrectionKey=B
DO NOT EDITChanges must be made through "File info" CorrectionKey=B
YOUR TURN
YOUR TURN
Solve each inequality. Graph and check the solution. 1. 10y < 60 2. 7 ≥  __t 6
y > 6
t ≥ 42
 10  9  8  7  6  5  4  3  2  1
0 1
47 46 45 44 43 42 41 40
Personal Math Trainer Online Assessment and Intervention
my.hrw.com
3. Every month, $35 is withdrawn from Tom’s savings account to pay for his gym membership. He has enough savings to withdraw no more than $315. For how many months can Tony pay for his gym membership?
Personal Math Trainer Online Assessment and Intervention
m ≤ 9; Tom can pay for no more than 9 months of his
my.hrw.com
gym membership using this account.
Solving a RealWorld Problem
EXAMPL 2 EXAMPLE
Problem Solving
Guided Practice Math On the Spot my.hrw.com
6.10.A
z ≤ 3
1. 7z ≥ 21
A marine submersible descends more than 40 feet below sea level. As it descends from sea level, the change in elevation is 5 feet per second. For how many seconds does it descend?
2. __t > 5 4
t < 20
t >5 4. ___
Rewrite the question as a statement. • Find the number of seconds that the submersible decends below sea level.
10
10 9  8  7  6  5  4  3  2  1 50 40 30 20 10
x < 6
3. 11x < 66
Analyze Information
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Jeffrey L. Rotman/Peter Arnold Inc/Getty Images
Solve each inequality. Graph and check the solution. (Explore Activity and Example 1)
t < 50
0
10
20
30
40
8 7  6  5  4  3  2  1
0
50
0 1 2
100 90 80 70 60 50 40 30 20 10
0
5. For a scientific experiment, a physicist must make sure that the temperature of a metal does not get colder than 80 °C. The metal begins the experiment at 0 °C and is cooled at a steady rate of 4 °C per hour. How long can the experiment run? (Example 2)
List the important information: • The final elevation is greater than 40 feet below sea level or < 40 feet. • The rate of descent is 5 feet per second.
a. Let t represent time in hours. Write an inequality. Use the fact that the rate of change in temperature times the number of seconds equals the final temperature.
Formulate a Plan
Write and solve an inequality. Use this fact:
4 · t ≥ 80
Rate of change in elevation × Time in seconds = Total change in elevation
b. Solve the inequality in part a. How long will it take the physicist to change the temperature of the metal?
20 or fewer hours
Justify and Evaluate Solve
c. The physicist has to repeat the experiment if the metal gets cooler than 80 °C. How many hours would the physicist have to cool the metal for this to happen?
more than 20 hours
5t < 40 5t > ____ 40 ____ 5 5 t> 8
Rate of change × Time < Maximum elevation Divide both sides by 5. Reverse the inequality symbol.
? ?
The submersible descends for more than 8 seconds.
6. Suppose you are solving an inequality. Under what circumstances do you reverse the inequality symbol?
Justify and Evaluate
when you divide or multiply both sides by a negative number
Check your answer by substituting a value greater than 8 seconds in the original inequality. ? Substitute 9 for t in the inequality 5t < 40. 5(9) < 40 45 < 40
ESSENTIAL QUESTION CHECKIN
© Houghton Mifflin Harcourt Publishing Company
Although elevations below sea level are represented by negative numbers, we often use absolute value to describe these elevations. For example, 50 feet relative to sea level might be described as 50 feet below sea level.
The statement is true. Lesson 13.4
6_MTXESE051676_U4M13L4.indd 369
369
28/01/14 11:28 PM
370
Unit 4
6_MTXESE051676_U4M13L4.indd 370
28/01/14 11:33 PM
DIFFERENTIATE INSTRUCTION Cooperative Learning
Modeling
Additional Resources
Have students work in pairs to explain to each other how to solve multiplication and division inequalities. Have one student explain how to solve an inequality that involves multiplying or dividing by a positive number, and have the other student explain how to solve an inequality that involves multiplying or dividing by a negative number. Then have each pair work with specific inequalities. Invite several pairs of students to share their explanations with the class.
Draw a number line with 1 and 3 graphed.
Differentiated Instruction includes: • Reading Strategies • Success for English Learners ELL • Reteach • Challenge PREAP
Sample inequalities for a pair of students:
Use similar steps to show that dividing both sides of a numerical inequality by 1 will change the sign as well.
3x ≥ 60 and __6x < 4
Ask if it is correct to write 1 < 3. Yes
Multiply both numbers by 1 and graph the products. Ask if it is correct to write 1 < 3. No
Why? 1 is farther to the right than 3. Ask if it is correct to write 1 > 3. Yes
What changed? the direction of the inequality sign
Multiplication and Division Inequalities with Rational Numbers
370
Personal Math Trainer Online Assessment and Intervention
Online homework assignment available
Evaluate GUIDED AND INDEPENDENT PRACTICE 6.9.B, 6.10.A, 6.10.B
my.hrw.com
13.4 LESSON QUIZ 6.9.B Solve each inequality. Graph and check the solution. 1. __3x > 7 2. 10z ≤ 20 3. 6t ≥ 54
Concepts & Skills
Practice
Explore Activity Investigating Inequalities Involving Multiplication and Division of Integers
Exercises 1–4, 7–12, 17
Example 1 Multiplication and Division Properties of Inequality
Exercises 1–4, 7–12, 18–24
Example 2 Solving a RealWorld Problem
Exercises 5, 13–16
4. __2r < 7 5. Melissa’s rent check is lost in the mail. There is a $9 late fee charged to Melissa’s account for every day the rent is late. Her account started out at $0, and after the late fees showed a balance that was less than $36. How many days late was the check? Lesson Quiz available online my.hrw.com
Answers 1. x < 21 15
20
25
0
5
10
5
15
10
3. t ≤ 9 15
4. r > 14 20
5. 9d < 36; d > 4; The check was more than 4 days late.
371
Depth of Knowledge (D.O.K.)
Lesson 13.4
Mathematical Processes
2 Skills/Concepts
1.D Multiple representations
3 Strategic Thinking
1.G Explain and justify arguments
14–16
2 Skills/Concepts
1.A Everyday life
17
2 Skills/Concepts
1.F Analyze relationships
18–23
2 Skills/Concepts
1.C Select tools
24
2 Skills/Concepts
1.C Select tools
25
3 Strategic Thinking
1.F Analyze relationships
26
3 Strategic Thinking
1.E Create and use representations
7–12 13
Additional Resources Differentiated Instruction includes: • Leveled Practice Worksheets
2. z ≤ 2 5
Exercise
Name
Class
Date
Solve each inequality.
13.4 Independent Practice 6.9.B, 6.10.A, 6.10.B
my.hrw.com
Solve each inequality. Graph and check your solution. q q≤7 7.  __ ≥ 1 7 0 1 2 3 4 5 6 7 8 9 10
x > 5
10 9  8  7  6  5  4  3  2  1
y 9. 0.5 ≤ __ 8
11. 12 > 2x
© Houghton Mifflin Harcourt Publishing Company
0
2
x>4 1 x < __ 20
21. 0.4 < x
x ≥ 161 x < 0.4
x ≤ 30 23.  ___ 0.8
x ≥ 24
24. Use the order of operations to simplify the left side of the inequality below. What values of x make the inequality a true statement?  _12 (32 + 7)x > 32
 _12 (32 + 7)x > 32;  _12 (9 + 7)x > 32;  _12(16)x > 32; 8x > 32; x < 4. The solution would be all values less than 4.
FOCUS ON HIGHER ORDER THINKING
Work Area
25. Counterexamples John says that if one side of an inequality is 0, you don’t have to reverse the inequality symbol when you multiply or divide both sides by a negative number. Find an inequality that you can use to disprove John’s statement. Explain your thinking.
10 seconds or more
x < 6
If you divide both sides of 7z ≥ 0 by 7, you get z ≥ 0.
0 1 2
x≥3
x ≤ 0.5 12.  __ 6 5 4  3  2  1
14. A veterinarian tells Max that his cat should lose no more than 30 ounces. The veterinarian suggests that the cat should lose 7 ounces or less per week. What is the shortest time in weeks and days it would take Max’s cat to lose the 30 ounces?
15. The elevation of an underwater cave is 120 feet relative to sea level. A submarine descends to the cave. The submarine’s rate of change in elevation is no greater than 12 feet per second. How long will it take to reach the cave?
r < 6
8 7  6  5  4  3  2  1
8
1 22. 4x < __ 5
0
y≥4
10 9  8  7  6  5  4  3  2  1
Online Assessment and Intervention
4 weeks and 2 days
0 1 2 3 4 5 6 7 8 9 10
10. 36 < 6r
x <  __ 1 20.  __
x ≤ 23 19.  __ 7
0 1 2 3 4 5
13. Multistep Parav is playing a game in which he flips a counter that can land on either a 6 or a 6. He adds the point values of all the flips to find his total score. To win, he needs to get a score less than 48. a. Assuming Parav only gets 6s when he flips the counter, how many times does he have to flip the counter?
6x < 48; x > 8
16. The temperature of a freezer is never greater than 2 °C. Yesterday the temperature was 10 °C, but it increased at a steady rate of 1.5 °C per hour. How long in hours and minutes did the temperature increase inside the freezer?
This is incorrect because if you choose a value from the possible solutions, such as z = 1, and substitute it into the original equation, you get 7 ≥ 0, which is not true.
less than 5 hours and 20 minutes 26. Communicate Mathematical Thinking Van thinks that the answer to 3x < 12 is x < 4. How would you convince him that his answer is incorrect?
17. Explain the Error A student's solution to the inequality 6x > 42 was x > 7. What error did the student make in the solution? What is the correct answer?
b. Suppose Parav flips the counter and gets five 6s and twelve 6s when he plays the game. Does he win? Explain.
No; his score is 5(6) +
I would remind him of the properties of inequality that we learned. I would show him that if you substitute 5 into the original inequality, the statement is not true.
The student did not reverse the inequality sign. The answer
Therefore his answer is not correct. Then I would show
should be x < 7.
him that the several solutions from x > 4 do hold true.
© Houghton Mifflin Harcourt Publishing Company
8. 12x < 60
x ≤ 9
18. 18 ≤ 2x
Personal Math Trainer
12(6) = 42, which is not less than 48. Lesson 13.4
EXTEND THE MATH
PREAP
Activity Present the statements at the right. Challenge students to make the statements true by filling in the blanks with <, >, or =. For each choice, students should give an example to demonstrate that the statement is true. 1. > example: a = 3, b = 4, c = 2 (3)(2) > (4)(2); 6 > 8 2. < example: x = 5, y = 3, z = 2 (5)(2) < (3)(2); 10 < 6 3. = example: n = 3, m = 2, p = 0 (3)(0) = (2)(0); 0 = 0
371
372
Activity available online
Unit 4
my.hrw.com
Algebraically Speaking 1. If a < b and c < 0, then ac
bc.
2. If x < y and z > 0, then xz
yz.
3. If n < m and p = 0, then np
mp.
4. If f < 0 and g < 0, then fg
0.
4. > example: f = 6, g = 2 (6)(2) > 0; 12 > 0
Multiplication and Division Inequalities with Rational Numbers
372
DO NOT EDITChanges must be made through “File info” CorrectionKey=B
MODULE QUIZ
Ready to Go On?
Ready
Assess Mastery
13.1 Writing Inequalities
Personal Math Trainer Online Assessment and Intervention
Write an inequality to represent each situation, then graph the solutions.
Use the assessment on this page to determine if students have mastered the concepts and standards covered in this module.
1. There are fewer than 8 gallons of gas in the tank. 0 1 2 3 4 5 6 7 8 9 10
3
Response to Intervention
2 1
0 1 2 3 4 5 6 7 8 9 10
Enrichment
 10  9  8  7  6  5  4  3  2  1
Access Ready to Go On? assessment online, and receive instant scoring, feedback, and customized intervention or enrichment.
Personal Math Trainer
v ≤ 4
my.hrw.com
Solve each inequality. Graph the solution. 4. c  28 > 32
Online and Print Resources Differentiated Instruction
Differentiated Instruction
• Reteach worksheets
• Challenge worksheets
• Reading Strategies • Success for English Learners ELL
ELL
0 1 2 3 4 5 6 7 8 9 10
13.2 Addition and Subtraction Inequalities
10 9  8  7  6  5  4  3  2  1
5. 0
v + 17 ≤ 20
0 1 2 3 4 5 6 7 8 9 10
6. Today’s high temperature of 80 °F is at least 16 ° warmer than yesterday’s high temperature. What was yesterday’s high temperature? 80
PREAP
Extend the Math PREAP Lesson activities in TE
Additional Resources Assessment Resources includes: • Leveled Module Quizzes
≥ 16 + T; T ≤ 64 °F
13.3, 13.4 Multiplication and Division Inequalities © Houghton Mifflin Harcourt Publishing Company
Online Assessment and Intervention
p≥3
2. There are at least 3 pieces of gum left in the pack.
3. The valley was at least 4 feet below sea level.
Intervention
my.hrw.com
f<8
Solve each inequality. Graph the solution. 8. __a2 < 4
7. 7f ≤ 35 0 1 2 3 4 5 6 7 8 9 10
10 9  8  7  6  5  4  3  2  1
0 1 2 3 4 5 6 7 8 9 10 k <3 10. ___ 3
9. 25g ≥ 150 0
10 9  8  7  6  5  4  3  2  1
0
Module 13
6_MTXESE051676_U4M13RT.indd 373
Texas Essential Knowledge and Skills Lesson
Exercises
13.1
1–3
6.9.A, 6.9.B, 6.10.B
13.2
4–6
6.9.B, 6.9.C, 6.10.A, 6.10.B
13.3
7–10
6.9.B, 6.9.C, 6.10.A, 6.10.B
13.4
7–10
6.9.B, 6.10.A, 6.10.B
373
Module 13
TEKS
373
29/01/14 12:17 AM
Personal Math Trainer
MODULE 13 MIXED REVIEW
Texas Test Prep
Texas Test Prep
Item 6 Students can look at each graph and test the ending point to see if it makes the inequality true. Then, using the ≤ symbol, students can identify that choice B is the correct answer.
Selected Response 1. Em saves at least 20% of what she earns each week. If she earns $140 each week for 4 weeks, which inequality describes the total amount she saves? A t > 112 B t ≥ 112 C
t < 28
D t ≤ 28
2. Which number line represents the inequality r > 6?
Avoid Common Errors
A
Item 1 If students do not read the item carefully, they may find the amount saved per week rather than the total amount saved. Remind them to read all items fully and carefully. Item 7 When students divide $150 by $12, they will get 12.5. Many students might insert this answer, not realizing that the question asks for full weeks, which requires the answer to be rounded up to 13. Remind students to read the questions carefully and be sure to understand what is being asked.
0 1 2 3 4 5 6 7 8 9 10
5. The number line below represents the solution to which inequality? 0 1 2 3 4 5 6 7 8 9 10 m A __ > 2.2 4
C
m __ > 2.5 3
B 2m < 17.6
D 5m > 40
6. Which number line shows the solution to w  2 ≤ 8? A
0 1 2 3 4 5 6 7 8 9 10
B
0 1 2 3 4 5 6 7 8 9 10
C
0 1 2 3 4 5 6 7 8 9 10
B
0 1 2 3 4 5 6 7 8 9 10
Online Assessment and Intervention
D
0 1 2 3 4 5 6 7 8 9 10
C
0 1 2 3 4 5 6 7 8 9 10
Gridded Response
D
0 1 2 3 4 5 6 7 8 9 10
3. For which inequality below is z = 3 a solution? A z+5≥9
7. Hank needs to save at least $150 to ride the bus to his grandparent’s home. If he saves $12 a week, what is the least number of weeks he needs to save?
B z+5>9
3
0
0
0
0
0
0
D z+5<8
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
4
4
4
4
4
4
5
5
5
5
5
5
6
6
6
6
6
6
7
7
7
7
7
7
8
8
8
8
8
8
9
9
9
9
9
9
4. What is the solution to the inequality −6x < −18? A x>3 B x<3 C
x≥3
D x≤3
374
.
1
z+5≤8
C
© Houghton Mifflin Harcourt Publishing Company
Texas Testing Tip Students can use the given solutions to work backwards and find a solution. Item 3 Students are given the solution of 3 in the main problem. Instead of solving every problem to see which would give 3 as a true solution, the students can substitute 3 into each inequality. The choice with the true statement is the correct answer.
my.hrw.com
Unit 4
Texas Essential Knowledge and Skills Items
Grade 6 TEKS
Mathematical Process TEKS
1
6.10.A
6.1.A, 6.1.F
2
6.9.B
6.1.D, 6.1.E
3
6.10.B
6.1.D
4
6.10.A
6.1.D
5
6.9.B, 6.10.A
6.1.D
6
6.9.B, 6.10.A
6.1.D, 6.1.E
7*
6.3.D, 6.9.A, 6.10.A
6.1.A
* Item integrates mixed review concepts from previous modules or a previous course.
Inequalities and Relationships
374
Relationships in Two Variables ?
ESSENTIAL QUESTION How can you use relationships in two variables to solve realworld problems?
MODULE
You can use tables, graphs, and equations in two variables to model realworld problems, then use algebraic methods to solve the problems.
14
LESSON 14.1
Graphing on the Coordinate Plane 6.11
LESSON 14.2
Independent and Dependent Variables in Tables and Graphs 6.6.A, 6.6.C
LESSON 14.3
Writing Equations from Tables 6.6.B, 6.6.C
LESSON 14.4
© Houghton Mifflin Harcourt Publishing Company • Image Credits: Blickwinkel/Alamy
Representing Algebraic Relationships in Tables and Graphs 6.6.A, 6.6.B, 6.6.C
RealWorld Video
my.hrw.com
my.hrw.com
375
Module 14
A twovariable equation can represent an animal’s distance over time. A graph can display the relationship between the variables. You can graph two or more animals’ data to visually compare them.
my.hrw.com
Math On the Spot
Animated Math
Personal Math Trainer
Go digital with your writein student edition, accessible on any device.
Scan with your smart phone to jump directly to the online edition, video tutor, and more.
Interactively explore key concepts to see how math works.
Get immediate feedback and help as you work through practice sets.
375
Are YOU Ready?
Are You Ready? Use the assessment on this page to determine if students need intensive or strategic intervention for the module’s prerequisite skills. 2 1
Multiplication Facts EXAMPLE
Response to Intervention
1. 7 × 6
Enrichment
my.hrw.com
5.
Skills Intervention worksheets
Differentiated Instruction
• Skill 36 Multiplication Facts
• Challenge worksheets
42
2. 10 × 9
90
3. 13 × 12
6.
x
1
2
3
4
y
7
14
21
28
156
4. 8 × 9
72
x
1
2
3
4
y
7
8
9
10
y is 7 times x.
Online and Print Resources
• Skill 69 Graph Ordered Pairs (First Quadrant)
Use a related fact you know. 7 × 7 = 49 Think: 8 × 7 = (7 × 7) + 7 = 49 + 7 = 56
8×7=
Write the rule for each table.
Access Are You Ready? assessment online, and receive instant scoring, feedback, and customized intervention or enrichment.
Online Assessment and Intervention
Online Assessment and Intervention
Multiply.
Intervention
Personal Math Trainer
my.hrw.com
7.
y is 6 more than x.
x
1
2
3
4
y
5
10
15
20
8.
x
0
4
8
12
y
0
2
4
6
y is 5 times x.
PREAP
y is onehalf x.
Graph Ordered Pairs (First Quadrant)
Extend the Math PREAP Lesson Activities in TE
EXAMPLE
10 8
B
D
6 4
A
Start at the origin. Move 9 units right. Then move 5 units up. Graph point A(9, 5).
C
2 O
© Houghton Mifflin Harcourt Publishing Company
3
Personal Math Trainer
Complete these exercises to review skills you will need for this chapter.
Assess Readiness
E 2
4
6
8 10
Graph each point on the coordinate grid above.
9. B (0, 8)
376
10. C (2, 3)
11. D (6, 7)
12. E (5, 0)
Unit 4
PROFESSIONAL DEVELOPMENT VIDEO my.hrw.com
Author Juli Dixon models successful teaching practices as she explores graphing in the coordinate plan in an actual sixthgrade classroom.
Online Teacher Edition Access a full suite of teaching resources online—plan, present, and manage classes and assignments.
Professional Development
ePlanner Easily plan your classes and access all your resources online.
my.hrw.com
Interactive Answers and Solutions Customize answer keys to print or display in the classroom. Choose to include answers only or full solutions to all lesson exercises.
Interactive Whiteboards Engage students with interactive whiteboardready lessons and activities. Personal Math Trainer: Online Assessment and Intervention Assign automatically graded homework, quizzes, tests, and intervention activities. Prepare your students with updated, TEKSaligned practice tests.
Relationships in Two Variables
376
DO NOT EDITChanges must be made through "File info" CorrectionKey=B
Reading StartUp
Reading StartUp Have students complete the activities on this page by working alone or with others.
Parts of the Algebraic Expression 14 + 3x Definition
Understand Vocabulary Use the following explanations to help students learn the preview words. A coordinate plane is formed by two number lines that intersect at right angles. Coordinate planes are used in geographic maps and for locating images on computer screens.
Students can use these reading and notetaking strategies to help them organize and understand new concepts and vocabulary.
Mathematical Representation
Review Word
A specific number whose value does not change
14
constant
A number that is multiplied by a variable in an algebraic expression
3
coefficient
A letter or symbol used to represent an unknown
x
variable
Preview Words
Understand Vocabulary Complete the sentences using the preview words.
1. The numbers in an ordered pair are © Houghton Mifflin Harcourt Publishing Company
Integrating the ELPS
✔ coefficient (coeficiente) ✔ constant (constante) equation (ecuación) negative number (número negativo) positive number (número positivo) scale (escala) ✔ variable (variable)
Use the ✔ words to complete the chart.
The chart helps students review vocabulary associated with algebraic expressions. Write additional expressions on the board and have students identify the parts of each expression.
Active Reading
Review Words
Visualize Vocabulary
Visualize Vocabulary
The two lines that make a coordinate plane are called the axes. The xaxis is the horizontal number line that runs left to right on the coordinate plane. The yaxis is the vertical line that runs up and down on the coordinate plane.
Vocabulary
coordinate plane 2. A lines that intersect at right angles.
coordinates
.
is formed by two number
axes (ejes) coordinate plane (plano cartesiano) coordinates (coordenadas) dependent variable (variable dependiente) independent variable (variable independiente) ordered pair (par ordenado) origin (origen) quadrants (cuadrantes) xaxis (eje x) xcoordinate (coordenada x) yaxis (eje y) ycoordinate (coordenada y)
Active Reading
c.4.D Use prereading supports such as graphic organizers, illustrations, and pretaught topicrelated vocabulary to enhance comprehension of written text.
Additional Resources
Layered Book Before beginning the module, create a layered book to help you learn the concepts in this module. Label each flap with lesson titles from this module. As you study each lesson, write important ideas such as vocabulary and formulas under the appropriate flap. Refer to your finished layered book as you work on exercises from this module.
Differentiated Instruction • Reading Strategies ELL
Module 14
6_MTXESE051676_U4MO14.indd 377
29/01/14 12:23 AM
Grades 6–8 TEKS Before Students understand: • how to recognize the difference between additive and multiplicative numerical patterns given in a table or graph • how to graph a relationship on a number line • how to identify and locate ordered pairs of whole numbers in the first quadrant
377
Module 14
In this module Students will learn to: • identify independent and dependent quantities from tables and graphs • write an equation that represents the relationship between independent and dependent quantities from a table • represent a given situation using verbal descriptions, tables, graphs, and equations in the form y = kx or y=x+b • graph points in all four quadrants using ordered pairs of rational numbers
377
After Students will connect: • tables and verbal descriptions with a linear relationship • graphs and equations with a linear relationship • ordered pairs with an equation
MODULE 14
Unpacking the TEKS
Unpacking the TEKS Understanding the TEKS and the vocabulary terms in the TEKS will help you know exactly what you are expected to learn in this module.
Use the examples on this page to help students know exactly what they are expected to learn in this module.
6.6.B Write an equation that represents the relationship between independent and dependent quantities from a table.
Texas Essential Knowledge and Skills Content Focal Areas
Key Vocabulary equation (ecuación) A mathematical sentence that shows that two expressions are equivalent.
Expressions, equations, and relationships—6.6 The student applies mathematical process standards to use multiple representations to describe algebraic relationships.
What It Means to You You will learn to write an equation that represents the relationship in a table. UNPACKING EXAMPLE 6.6.B
Emily has a dogwalking service. She charges a daily fee of $7 to walk a dog twice a day. Create a table that shows how much Emily earns for walking 1, 6, 10, and 15 dogs. Write an equation that represents the situation. Dogs walked
1
6
10
15
Earnings ($)
7
42
70
105
Earnings is 7 times the number of dogs walked. Let the variable e represent earnings and the variable d represent the number of dogs walked.
Measurement and data—6.11 The student applies mathematical process standards to use coordinate geometry to identify locations on a plane.
e=7×d
6.6.C Represent a given situation using verbal descriptions, tables, graphs, and equations in the form y = kx or y = x + b.
Go online to see a complete unpacking of the .
What It Means to You You can use words, a table, a graph, or an equation to model the same mathematical relationship.
Key Vocabulary
UNPACKING EXAMPLE 6.6.C
coordinate plane (plano cartesiano) A plane formed by the intersection of a horizontal number line called the xaxis and a vertical number line called the yaxis.
The equation y = 4x represents the total cost y for x games of miniature golf. Make a table of values and a graph for this situation.
Visit my.hrw.com to see all the unpacked.
my.hrw.com
Number of games, x Total cost ($), y
1
2
3
4
4
8
12
16
y
Total cost ($)
c.4.F Use visual and contextual support … to read gradeappropriate content area text … and develop vocabulary … to comprehend increasingly challenging language.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: PhotoDisc/Getty Images
Integrating the ELPS
20 16 12 8 4 x O
2
4
6
8
Number of games my.hrw.com
378
Grade 6 TEKS
Lesson 14.1
Lesson 14.2
Unit 4
Lesson 14.3
Lesson 14.4
6.6.A Identify independent and dependent quantities from tables and graphs. 6.6.B Write an equation that represents the relationship between independent and dependent quantities from a table. 6.6.C Represent a given situation using verbal descriptions, tables, graphs, and equations in the form y = kx or y = x + b. 6.11 Graph points in all four quadrants using ordered pairs of rational numbers.
Relationships in Two Variables
378
LESSON
14.1 Graphing on the Coordinate Plane Engage
Texas Essential Knowledge and Skills
ESSENTIAL QUESTION
The student is expected to:
How do you locate and name points in the coordinate plane? Sample answer: Points in the coordinate plane are located and named by their locations from the origin along the xaxis first, followed by the yaxis. The order of the coordinates is important.
Measurement and data—6.11 Graph points in all four quadrants using ordered pairs of rational numbers.
Motivate the Lesson Ask: Have you ever tried to find a city or town by its location on a map grid? Maps are like a coordinate plane. Begin Example 1 to find out how to locate a point on a coordinate plane.
Mathematical Processes 6.1.E Create and use representations to organize, record, and communicate mathematical ideas.
Explore
ADDITIONAL EXAMPLE 1
Engage with the Whiteboard
Identify the coordinates of each point. Name the quadrant where each point is located.
To introduce students to the fourquadrant coordinate plane, sketch a simple “treasure map” with a coordinate plane on the whiteboard. Mark a point for “Start” at the origin and a point for “Treasure” in Quadrant I. Ask students to draw a path to the treasure using the grid lines and then to describe the path in words, such as, “Walk east 3 steps. Then walk north 5 steps.” Repeat several times with new coordinates for the “Treasure.”
y 2
B
x 4
O
2
2
4
A
2
EXAMPLE 1 Focus on Communication
Mathematical Processes Point out to students that coordinates describe a location in relation to the origin, so it is important to always start at the origin when identifying the coordinates of a point.
Point A: (3, 2), Quadrant IV; Point B: (4, 1), Quadrant II Interactive Whiteboard Interactive example available online my.hrw.com
Students may give incorrect coordinates for a point because they transposed the x and ycoordinates. Remind students that the xcoordinate is the first number in an ordered pair.
y
EXAMPLE 2
2
2
R 2
x
4
P
Interactive Whiteboard Interactive example available online my.hrw.com
379
Lesson 14.1
ELL c.2.C Some students may have difficulty remembering what the x and ycoordinates mean in an ordered pair. Encourage students to think of plotting points as physical movements, run and jump. The xcoordinate tells how far to run to the right or left, and the ycoordinate tells how far to jump up or down. So, when plotting points, students should always “run before they jump.”
Connect Vocabulary
O
2
Mathematical Processes • Is point (2, 3) the same as point (3, 2)? Explain. No, they are not the same point. Point (2, 3) lies 2 units to the right of the origin and 3 units up, while point (3, 2) lies 3 units to the right and 2 units up.
Avoid Common Errors
P(0, 2), Q(4, 1.5), R(3.5, 0)
4
Questioning Strategies
YOUR TURN
ADDITIONAL EXAMPLE 2 Graph and label each point on the coordinate plane.
Q
Explain
Questioning Strategies
Mathematical Processes • Describe how graphing the point (0, 3) is similar to graphing the point (3, 0). How is it different? Sample answer: They are similar because you start at the origin and move three units to graph each point. They are different because in (0, 3), you move down from the origin. In (3, 0), you move left from the origin.
14.1 ?
Graphing on the Coordinate Plane
Measurement and data—6.11 Graph points in all four quadrants using ordered pairs of rational numbers.
Reflect 1.
If both coordinates of a point are negative, in which quadrant is the
2.
Describe the coordinates of all points in Quadrant I.
point located?
ESSENTIAL QUESTION
Both coordinates are positive.
How do you locate and name points in the coordinate plane? 3.
Naming Points in the Coordinate Plane
•
The horizontal axis is called the xaxis.
•
The vertical axis is called the yaxis.
•
The point where the axes intersect is called the origin.
•
The two axes divide the coordinate plane into four quadrants.
left of the origin. The xcoordinate in (3, 5) is 3 which
y
The two number lines are called the axes.
Math On the Spot
6 Quadrant II
Quadrant I
YOUR TURN
2
xaxis x
6
4
2
O
2 4 Origin
2 Quadrant III
will be to the right of the origin.
my.hrw.com
4
yaxis
Personal Math Trainer
6
Online Assessment and Intervention
Quadrant IV
4
my.hrw.com
4.
© Houghton Mifflin Harcourt Publishing Company
EXAMPL 1 EXAMPLE
Point A is 1 unit left of the origin, and 5 units down. It has xcoordinate 1 and ycoordinate 5, written (1, 5). It is located in Quadrant III. Point B is 2 units right of the origin, and 3 units up. It has xcoordinate 2 and ycoordinate 3, written (2, 3). It is located in Quadrant I.
Math On the Spot my.hrw.com
2
(1, 3); III
x 4
2
O
2
4
2 H 4
G
EXAMPLE 2
6.11 y 4
Point A is 5 units left and 2 units up from the origin.
x O
H
F
2
Graph and label each point on the coordinate plane. A(5, 2), B(3, 1.5), C(0, 3)
B
2 2
F
(2, 4); II (3, 2); I
4
Points that are located on the axes are not located in any quadrant. Points on the xaxis have a ycoordinate of 0, and points on the yaxis have an xcoordinate of 0.
y
4
(4, 4); IV
E
Graphing Points in the Coordinate Plane
6.11
4
G E
5.
The numbers in an ordered pair are called coordinates. The first number is the xcoordinate and the second number is the ycoordinate.
y
Identify the coordinates of each point. Name the quadrant where each point is located.
6
An ordered pair is a pair of numbers that gives the location of a point on a coordinate plane. The first number tells how far to the right (positive) or left (negative) the point is located from the origin. The second number tells how far up (positive) or down (negative) the point is located from the origin.
Identify the coordinates of each point. Name the quadrant where each point is located.
Communicate Mathematical Ideas Explain why (3, 5) represents a different location than (3, 5).
The xcoordinate in (3, 5) is 3 which will be to the
A coordinate plane is formed by two number lines that intersect at right angles. The point of intersection is 0 on each number line. •
III
© Houghton Mifflin Harcourt Publishing Company
LESSON
4
Point B is 3 units right and 1.5 units up from the origin. Graph the point halfway between (3, 1) and (3, 2).
2 4 A
Point C is 3 units down from the origin. Graph the point on the yaxis. Lesson 14.1
379
380
A
2
B x
4
2
O 2
2
4
C
4
Unit 4
PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.E, which calls for students to “create and use representations to organize, record, and communicate mathematical ideas.” Students use coordinate planes to locate points. Then students solve a realworld problem on a coordinate plane in which the scale on each axis represents a realworld situation. In this way, students are able to connect a coordinate plane to the real world.
Math Background The concept of the rectangular coordinate system is generally credited to French mathematician and philosopher René Descartes and, therefore, is sometimes referred to as the Cartesian plane. Every point on the plane can be located because all real numbers, not just integers, are used. The points represented by integer coordinates are sometimes called lattice points.
Graphing on the Coordinate Plane
380
YOUR TURN Avoid Common Errors Students may graph the points incorrectly by using the x and ycoordinates in the wrong order. Remind students to run before they jump.
ADDITIONAL EXAMPLE 3
EXAMPLE 3
The graph shows the location of a fountain, a slide, and a sandbox in a park. The scale on each axis represents yards. Give the coordinates for the sandbox and use them to describe the location of the sandbox relative to the fountain.
Focus on Math Connections
Mathematical Processes Point out to students that the coordinate plane also indicates directions. The xaxis points east (to the right) and west (to the left) and the yaxis points north (up) and south (down). For Example 3, have students add the correct north, south, east, and west labels to the axes of the coordinate plane.
Questioning Strategies
Mathematical Processes • Describe the direction you would go from Gary’s house to Jen’s house. I would travel east to go from Gary’s house to Jen’s house.
y
• What would the coordinates of Gary’s house be if he lived 30 miles directly east of Jen? Explain. (35, 15); Jen lives at (5, 15), so 30 miles directly east would be (35, 15).
20
10
x
Fountain 20
Slide
O
10
10
10
20
Sandbox
20
The sandbox is at (10, 15). This is 10 yards east and 15 yards south of the fountain. Interactive Whiteboard Interactive example available online my.hrw.com
YOUR TURN Focus on Math Connections
Mathematical Processes Show students how to translate directions to movements by using the axes. Remind students that “20 miles south” is on the yaxis below the origin and that “20 miles west” is on the xaxis to the left of the origin. Both movements are in a negative direction, so the coordinates of Ted’s home are (20, 20). Since Ned lives “50 miles directly north of Ted’s house,” only the ycoordinate changes. Because north is a positive direction, 20 + 50 = 30. So, Ned’s home is located at (20, 30).
Elaborate Talk About It Summarize the Lesson Have students complete a graphic organizer that shows the number of each quadrant and the signs of the coordinates in each quadrant. y
Q __ II Q __ I  __) + (__, + __) + (__,
x
O
III IV Q __ Q __  __)  (__, + __) (__,
GUIDED PRACTICE Engage with the Whiteboard For Exercises 1–2, have students fill in the blanks by identifying the coordinates of each point on the whiteboard. Then have them name the quadrant where each point is located. Also ask students to describe how they would graph each point.
Avoid Common Errors Exercises 3–4, 6–7 Students may graph the points incorrectly by using the x and ycoordinates in the wrong order. Remind students to run before they jump.
381
Lesson 14.1
DO NOT EDITChanges must be made through “File info” CorrectionKey=B
DO NOT EDITChanges must be made through “File info” CorrectionKey=A
YOUR TURN
6.
P(4, 2)
7.
Q(3, 2.5)
8.
R(4.5, 5)
9. 10.
S(4, 5)
4
P 4
2
T
Online Assessment and Intervention
x
2
O
2
left
1. Point A is 5 units
my.hrw.com
4
1 unit
2
up
S
2
2. Point B is
T(2.5, 0)
and
3
.
2
A 4
(2, 3)
. It is in quadrant
IV
2
D
O
2
x
4
2 B
4
.
Each grid square is _12 unit on a side.
y
Gary
Jen
20 10
x
City O 10
Each grid square is 5 miles on a side.
10
Gary’s house is at (25, 15), which is 25 miles west and 15 miles north of the city.
20
20
N E
W
( (
S
)
1
O 1 2
1
2
B
The first number, the xcoordinate, tells how many units to the right or left the point is located from the origin. The second number, the ycoordinate, tells how many units up or down the point is located from the origin.
? ?
How are north, south, east, and west represented on the graph in Example 3?
ESSENTIAL QUESTION CHECKIN
9. Give the coordinates of a point that could be in each of the four quadrants, a point on the xaxis, and an point on the yaxis.
YOUR TURN
Sample answers: Quadrant I: (2, 5); Quadrant II
Use the graph in the Example. Ted lives 20 miles south and 20 miles west of the city represented on the graph in Example 3. His brother Ned lives 50 miles north of Ted’s house. Give the coordinates of each brother’s house.
(3, 5); Quadrant III (3, 3); Quadrant IV
Personal Math Trainer
(5, 2). xaxis: (3, 0); yaxis: (0, 5).
Online Assessment and Intervention
my.hrw.com
Lesson 14.1
6_MTXESE051676_U4M14L1.indd 381
x 2
Mathematical Processes
Jen’s house is located 6 grid squares to the right of Gary’s house. Since each grid square is 5 miles on a side, her house is 6 · 5 = 30 miles east of Gary’s.
Ted (20, 20), Ned (20, 30)
2 1
8. Vocabulary Describe how an ordered pair represents a point on a coordinate plane. Include the terms xcoordinate, ycoordinate, and origin in your answer.
Math Talk
B Describe the location of Jen’s house relative to Gary’s house.
)
6. Plot point A at _12 , 2 . 7. Plot point B at 2 _12 , 2 .
North and south are the positive and negative directions along the yaxis; east and west are the positive and negative directions on the xaxis.
10
y
A
5. Describe the scale of the graph.
6.11
20
4. Point D at (5, 0)
For 5–7, use the coordinate plane shown. (Example 3)
Math On the Spot my.hrw.com
EXAMPL 3 EXAMPLE
© Houghton Mifflin Harcourt Publishing Company
II
units right of the origin
3. Point C at (3.5, 3)
The scale of an axis is the number of units that each grid line represents. So far, the graphs in this lesson have a scale of 1 unit, but graphs frequently use other units.
11.
. It is in quadrant
Graph and label each point on the coordinate plane above. (Example 2)
Reading Scales on Axes
A Use the scale to describe Gary’s location relative to the city.
(5, 1)
4
C
units down from the origin.
Its coordinates are
The graph shows the location of a city. It also shows the location of Gary’s and Jen’s houses. The scale on each axis represents miles.
y
of the origin and
from the origin.
Its coordinates are
4
R
Identify the coordinates of each point in the coordinate plane. Name the quadrant where each point is located. (Example 1)
Personal Math Trainer
Q
© Houghton Mifflin Harcourt Publishing Company
Graph and label each point on the coordinate plane.
Guided Practice
y
381
29/01/14 12:30 AM
382
Unit 4
6_MTXESE051676_U4M14L1.indd 382
10/30/12 9:55 AM
DIFFERENTIATE INSTRUCTION Curriculum Integration
Cooperative Learning
Additional Resources
Have students draw coordinate grid lines on maps of Texas. Instruct students to draw the x and yaxes through the state capital and the other lines at __12 inch increments above, below, to the left, and to the right of the axes. Have the students label the grid lines, beginning with the axes, with the appropriate numbers and give coordinates for various cities and towns on the map.
Have students work in three teams to play coordinate tictactoe. Use a coordinate plane that is 5 units from the origin in all directions. One player on each team alternates calling out the coordinates of a point. Another player on each team locates the point and marks it on the coordinate plane. The first team to place three marks in an uninterrupted row horizontally, vertically, or diagonally wins the round.
Differentiated Instruction includes: • Reading Strategies • Success for English Learners ELL • Reteach • Challenge PREAP
Graphing on the Coordinate Plane
382
Personal Math Trainer Online Assessment and Intervention
Evaluate GUIDED AND INDEPENDENT PRACTICE
Online homework assignment available
6.11
my.hrw.com
14.1 LESSON QUIZ 6.11 Use the coordinate plane shown. Each unit represents 1 city block. y Henry
Concepts & Skills
Practice
Example 1 Naming Points in the Coordinate Plane
Exercises 1–2, 10–13
Example 2 Graphing Points in the Coordinate Plane
Exercises 3–4, 10–13
Example 3 Reading Scales on Axes
Exercises 5–7, 14–15
4 2
x 4
2
O
Emma 2 Ice cream 4 shop
2
4
Library
1. Write the ordered pairs that represent Henry and the library. 2. Describe Henry’s location relative to the library. 3. Henry wants to meet his friend Emma at an ice cream shop before they go to the library. The ice cream shop is 7 blocks west of the library. Plot and label a point representing the ice cream shop. What are the coordinates of the point? 4. Emma describes her current location: “I’m directly west of the library, halfway to the ice cream shop.” Plot and label a point representing Emma’s location. What are the coordinates of the point? Lesson Quiz available online my.hrw.com
Answers
1. Henry (3, 4), Library (2, 3) 2. Henry is 5 blocks west and 7 blocks north of the library. 3. (5, 3) 4. (1.5, 3)
383
Lesson 14.1
Exercise
Depth of Knowledge (D.O.K.)
Mathematical Processes
10
2 Skills/Concepts
1.A Everyday life
11
3 Strategic Thinking
1.F Analyze relationships
12–13
2 Skills/Concepts
1.A Everyday life
14–15
2 Skills/Concepts
1.D Multiple representations
16
3 Strategic Thinking
1.F Analyze relationships
17
3 Strategic Thinking
1.G Explain and justify arguments
18–19
3 Strategic Thinking
1.F Analyze relationships
Additional Resources Differentiated Instruction includes: • Leveled Practice Worksheets
Class
Date
14.1 Independent Practice 6.11
my.hrw.com
17. Critical Thinking Choose scales for the coordinate plane shown so that you can graph the points J(2, 40), K(3, 10), L(3, 40), M(4, 50), and N(5, 50). Explain why you chose the scale for each axis.
Online Assessment and Intervention
For 10–13, use the coordinate plane shown. Each unit represents 1 kilometer. 10. Write the ordered pairs that represent the location of Sam and the theater.
FOCUS ON HIGHER ORDER THINKING
Personal Math Trainer
Theater
Sam: (4, 2); Theater: (3, 5)
4
11. Describe Sam’s location relative to the theater.
Sam is 3 km south and 7 km east of the
Beth 4
2
theater.
Sam
2 2
S
(3, 4)
40
L
down (in a negative direction). 19. Represent RealWorld Problems Zach graphs some ordered pairs in the coordinate plane. The xvalues of the ordered pairs represent the number of hours since noon, and the yvalues represent the temperature at that time.
y V
© Houghton Mifflin Harcourt Publishing Company
N
4
direction) along the xaxis. Then count 12 grid squares
For 14–15, use the coordinate plane shown. 14. Find the coordinates of points T, U, and V.
U 1.0
T (0.75, 1.0); U (0.75, 1.25); V (0.75, 1.25)
a. In which quadrants could Zach graph points? Explain your thinking.
0.5
Quadrants I and IV; time is always positive, but
x 1.0 0.5 O
0.5
temperatures can be positive or negative. In
1.0
0.5
W
16. Explain the Error Janine tells her friend that ordered pairs that have an xcoordinate of 0 lie on the xaxis. She uses the origin as an example. Describe Janine’s error. Use a counterexample to explain why Janine’s statement is false.
each grid square. On the
2
Count 18 grid squares to the right (in a positive
(3, 0.5)
W(0.75, 1.0)
K
18. Communicate Mathematical Ideas Edgar wants to plot the ordered pair (1.8, 1.2) on a coordinate plane. On each axis, one grid square equals 0.1. Starting at the origin, how can Edgar find (1.8, 1.2)?
13. Beth describes her current location: “I’m directly south of the theater, halfway to the restaurant.” Plot and label a point representing Beth’s location. What are the coordinates of the point?
15. Points T, U, and V are the vertices of a rectangle. Point W is the fourth vertex. Plot point W and give its coordinates.
I used a scale of 1 unit for
4 2 20
O
ycoordinates ranged from 50 to 50.
E
W
Restaurant
20
The xcoordinates ranged from 5 to 3, and the
N
4
J
units for each grid square.
4
2
12. Sam wants to meet his friend Beth at a restaurant before they go to the theater. The restaurant is 9 km south of the theater. Plot and label a point representing the restaurant. What are the coordinates of the point?
40
yaxis I used a scale of 10
x O
M
x
Sample answer: On the xaxis
y
Work Area y
quadrants I and IV, the xcoordinate is always positive,
T
1.0
but the ycoordinate can be positive or negative.
b. In what part of the world and at what time of year might Zach collect data so that the points he plots are in Quadrant IV?
Janine is describing points that lie on the yaxis. Ordered pairs that lie on the xaxis have a ycoordinate of 0. The origin lies on the x and yaxis. Any
Sample answer: in a region with a cold climate during
other point with an xcoordinate of 0 lies on the yaxis such as (0, 3).
the winter
Lesson 14.1
EXTEND THE MATH
PREAP
383
Activity available online
© Houghton Mifflin Harcourt Publishing Company
Name
384
Unit 4
my.hrw.com
Activity Plot the points for each set of ordered pairs below. Then connect the points in the order shown to reveal a figure. Name the figure and find its area. Set 1: (2, 5), (2, 1), (3, 1), (3, 5) Set 2: (4, 3), (6, 3), (6, 4) Set 3: (1, 3), (4, 3), (4, 2), (1, 2) Write the coordinates for another set of points that form a figure. Find its area. Then challenge a classmate to draw the figure and find its area. Set 1: rectangle, A = 30 square units Set 2: triangle, A = 35 square units Set 3: square, A = 25 square units
Graphing on the Coordinate Plane
384
LESSON
14.2
Independent and Dependent Variables in Tables and Graphs
Texas Essential Knowledge and Skills The student is expected to: Expressions, equations, and relationships—6.6.A Identify independent and dependent quantities from tables and graphs. Expressions, equations, and relationships—6.6.C Represent a given situation using verbal descriptions, tables, graphs, and equations in the form y = kx or y = x + b.
Mathematical Processes 6.1.D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.
Engage ESSENTIAL QUESTION How can you identify independent and dependent quantities from tables and graphs? Sample answer: The dependent variable is the quantity that depends on the other variable. On a graph, the independent variable is shown on the horizontal axis and the dependent variable is shown on the vertical axis.
Motivate the Lesson Ask: What is the relationship between the amount of time a person works and the amount of money that person earns? Begin the Explore Activity to find out what independent and dependent quantities are and how to recognize them.
Explore EXPLORE ACTIVITY 1 Connect to Vocabulary
ELL
Have students describe the meaning of the following phrases: Sample answers are given. • independently wealthy doesn’t need to work for money • Independence Day day of freedom • working independently doesn’t need help to do a job • dependent child needs a parent • insulindependent needs insulin daily • dependent clause cannot stand alone in a sentence Then ask students to define independent and dependent variables. An independent variable stands alone and isn’t changed by the other variables. A dependent variable depends on or is changed by another variable.
Explain EXPLORE ACTIVITY 2 Connect Multiple Representations Mathematical Processes Have students complete this table, which represents the situation about the art teacher and clay, to reinforce that both a table and a graph can represent this relationship. Clay bought by teacher (lb)
0
10
20
30
Clay available for classes (lb)
20
30
40
50
Engage with the Whiteboard Ask a student volunteer to locate the point on the graph that shows the 50 pounds of clay that is available for the art class. Have the volunteer draw a line from the yaxis to the point and a line from the point to the xaxis. Have students repeat this process for several more values.
385
Lesson 14.2
DO NOT EDITChanges must be made through “File info” CorrectionKey=B
DO NOT EDITChanges must be made through “File info” CorrectionKey=A
LESSON
14.2 ?
Independent and Dependent Variables in Tables and Graphs
ESSENTIAL QUESTION
EXPLORE ACTIVITY (cont’d)
Expressions, equations, and relationships—6.6.A Identify independent and dependent quantities from tables and graphs. Also 6.6.C.
Reflect
1. Analyze Relationships Describe how the value of the independent variable is related to the value of the dependent variable. Is the relationship additive or multiplicative?
The value of y is always 50 times the value of x; multiplicative. 2. What are the units of the independent variable and of the dependent variable?
How can you identify independent and dependent quantities from tables and graphs?
independent variable: hours; dependent variable: miles.
6.6.A
3. A rate is used in the equation. What is the rate?
Identifying Independent and Dependent Quantities from a Table
50 miles per hour
Many realworld situations involve two variable quantities in which one quantity depends on the other. The quantity that depends on the other quantity is called the dependent variable, and the quantity it depends on is called the independent variable.
EXPLORE ACTIVITY 2
A freight train moves at a constant speed. The distance y in miles that the train has traveled after x hours is shown in the table. 0
1
2
3
Distance y (mi)
0
50
100
150
Identifying Independent and Dependent Variables from a Graph In Explore Activity 1, you used a table to represent a relationship between an independent variable (time) and a dependent variable (distance). You can also use a graph to show a relationship of this sort.
A What are the two quantities in this situation?
time and distance
An art teacher has 20 pounds of clay but wants to buy more clay for her class. The amount of clay x purchased by the teacher and the amount of clay y available for the class are shown on the graph.
© Houghton Mifflin Harcourt Publishing Company
Which of these quantities depends on the other?
Distance depends on time. time, x
What is the independent variable?
A If the teacher buys 10 more pounds of clay, how many
distance, y
What is the dependent variable?
B How far does the train travel each hour?
pounds will be available for the art class?
50 miles
=
Distance traveled per hour
=
50
↓ y
·
↓
30 lb; find the point on the graph with a
Time (hours)
ycoordinate of 50. Then find the xcoordinate of this point, which is 30.
x
Lesson 14.2
6_MTXESE051676_U4M14L2.indd 385
385
10/30/12 11:31 AM
386
Clay Used in Art Class
lb
How can you use the graph to find this information?
↓ ·
30
If the art class has a total of 50 pounds of clay available, how many pounds of clay did the teacher buy?
The relationship between the distance traveled by the train and the time in hours can be represented by an equation in two variables. Distance traveled (miles)
© Houghton Mifflin Harcourt Publishing Company • Image Credits: © Hill Street Studios/Corbis
Time x (h)
6.6.A
y
Clay available for classes (Ib)
EXPLORE ACTIVITY 1
80 60 40 20 O
x 20 40 60 80
Clay bought by teacher (Ib)
Unit 4
6_MTXESE051676_U4M14L2.indd 386
29/01/14 12:37 AM
PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.D, which calls for students to “communicate mathematical ideas… using multiple representations…as appropriate.” Students use tables, graphs, equations, and language to describe and model relationships between independent and dependent variables. In this way, students use multiple representations to model realworld situations involving independent and dependent variables.
Math Background Although the term function is not mentioned in this lesson, the tables in the lesson represent functions. A function is a rule that relates two quantities so that each input value corresponds to one output value exactly. When y is a function of x, x is called the independent variable and y is called the dependent variable. Whenever a value is assigned to x, a value is automatically assigned to y by an applicable rule or correspondence.
Independent and Dependent Variables in Tables and Graphs
386
EXPLORE ACTIVITY 2 CONTINUED Questioning Strategies
Mathematical Processes • Why does the graph show only Quadrant I? Negative amounts do not make sense in this situation, so the values and the graph are limited to positive x and yvalues. • Why does the graph start at (0, 20)? The art teacher had 20 pounds of clay to start with.
• As the xvalue is increasing, what is happening to the yvalue? The yvalue is also increasing.
ADDITIONAL EXAMPLE 1 A The table below shows a relationship between two variables, x and y. Describe a possible situation the table could represent. Describe the independent and dependent variables in this situation. Independent variable, x
1
2
Dependent variable, y
8
16 24 32
3
4
Sample answer: The table could represent the amount a person earns at a rate of $8 per hour. The independent variable, x, is the number of hours the person works. The dependent variable, y, is the total earnings. B The graph below shows a relationship between two variables, x and y. Describe a possible situation the graph could represent. Describe the independent and dependent variables. y 6
2
x 2
4
6
Sample answer: The graph could represent the progress of a rock climber, starting at a 2foot height and continuing at a pace of 1 foot every second. The independent variable is the number of seconds, and the dependent variable is the total number of feet climbed after x seconds. Interactive Whiteboard Interactive example available online my.hrw.com
387
Engage with the Whiteboard Have a volunteer sketch a graph of the relationship shown in the table in A. Have a second volunteer make a table of the relationship shown in the graph in B. This will help students to see that both a table and a graph can represent the same relationship.
Connect Multiple Representations Mathematical Processes Point out to students that each of these situations can be represented by a verbal description, a table, a graph, or an equation. Questioning Strategies • How can the relationship in A be represented by an equation? The table begins with a yvalue of 10, so the yvalue always will be 10 units greater than the xvalue. Then, as x increases by 1, y also increases by 1, resulting in the equation y = x + 10. • How would you describe the relationship in A? Explain. The relationship is an additive relationship because the value of y is always 10 units greater than the value of x. • How can the relationship in B be represented by an equation? The graph begins at the origin, so both variables begin at 0. Then, as x increases by 1, y increases by 12, resulting in the equation y = 12x. • How could you check that the equation is correct for either A or B? Pick a point from either the table or the graph and substitute it into the equation. The result should be a true equation.
Focus on Reasoning
Mathematical Processes Ask students to identify the independent and dependent quantities in the following situations. • A veterinarian must weigh an animal before determining the amount of medication it needs. independent quantity, weight of animal, dependent quantity: amount of medication
4
O
EXAMPLE 1
Lesson 14.2
• A company charges $10 per hour to rent a jackhammer. independent quantity: time, dependent quantity: cost c.4.D ELL Encourage English learners to use the active reading strategies presented at the beginning of the module.
Integrating the ELPS
DO NOT EDITChanges must be made through “File info” CorrectionKey=B
DO NOT EDITChanges must be made through “File info” CorrectionKey=A
Describing Relationships Between Independent and Dependent Variables
B What are the two quantities in this situation?
the clay bought by the teacher and the amount of clay available to the class Math On the Spot
Which of these quantities depends on the other?
my.hrw.com
The amount of clay available to the class depends on the amount of clay bought by the teacher.
Thinking about how one quantity depends on another helps you identify which quantity is the independent variable and which quantity is the dependent variable. In a graph, the independent variable is usually shown on the horizontal axis and the dependent variable on the vertical axis.
EXAMPLE 1
clay bought by teacher
A The table shows a relationship between two variables, x and y. Describe a possible situation the table could represent. Describe the independent and dependent variables in the situation.
clay available for the class
What is the dependent variable?
C The relationship between the amount of clay purchased by the teacher and the amount of clay available to the class can be represented by an equation in two variables. Amount of clay available (pounds) ↓ y
=
Amount of clay Current amount + purchased (pounds) of clay (pounds) ↓
=
+
20
x
2
3
11
12
13
B The graph shows a relationship between two variables. Describe a possible situation that the graph could represent. Describe the independent and dependent variables.
Reflect
© Houghton Mifflin Harcourt Publishing Company
1
10
The independent variable, x, is the number of days she has been adding money to her savings. The dependent variable, y, is her savings after x days.
value of x.
y 36 24 12
As x increases by 1, y increases by 12. The relationship O is multiplicative. The value of y is always 12 times the value of x.
4. In this situation, the same units are used for the independent and dependent variables. How is this different from the situation involving the train in the first Explore?
x 2
4
6
The graph could represent the number of eggs in cartons that each hold 12 eggs.
The other situation involves two different units
The independent variable, x, is the number of cartons. The dependent variable, y, is the total number of eggs.
(miles and hours). 5. Analyze Relationships Tell whether the relationship between the independent variable and the dependent variable is a multiplicative or an additive relationship.
Reflect 7. What are other possible situations that the table and graph in Example 1 could represent?
additive
Sample answer: Table: Paul has 10 DVDs and buys more. Independent: number of DVDs he buys; dependent:
6. What are the units of the independent variable, and what are the units of the dependent variable? ; dependent variable:
0
Dependent variable, y
The table could represent Jina’s savings if she starts with $10 and adds $1 to her savings every day.
↓
The value of y is always 20 units greater than the
pounds
Independent variable, x
As x increases by 1, y increases by 1. The relationship is additive. The value of y is always 10 units greater than the value of x.
D Describe in words how the value of the independent variable is related to the value of the dependent variable.
independent variable:
6.6.A
© Houghton Mifflin Harcourt Publishing Company
What is the independent variable?
number he has after he buys x DVDs. Graph: 12 photos
pounds
fit on each page of a yearbook. Independent: number of pages; dependent: total number of photos on x pages. Lesson 14.2
6_MTXESE051676_U4M14L2.indd 387
388
387
10/30/12 11:31 AM
Unit 4
6_MTXESE051676_U4M14L2.indd 388
29/01/14 12:41 AM
DIFFERENTIATE INSTRUCTION Curriculum Integration
Cooperative Learning
Additional Resources
Music: The notes you hear played by a musical instrument are an example of a dependent relationship. For example, a clarinet’s pitch at a particular moment depends on the number of holes covered by the musician. A harp’s pitch depends on the length of the string being plucked.
One way to remember which is the independent variable and which is the dependent variable is to use the names of the two variables in a sentence that makes sense. For example:
Differentiated Instruction includes: • Reading Strategies • Success for English Learners ELL • Reteach • Challenge PREAP
Dollars Earned depends on Hours Worked, but Hours Worked does not depend on Dollars Earned. So, Dollars Earned must be the dependent variable and Hours Worked must be the independent variable.
Independent and Dependent Variables in Tables and Graphs
388
YOUR TURN Avoid Common Errors If students have difficulty distinguishing between independent and dependent variables, remind them that the independent variable causes a change in the dependent variable, while the dependent variable could not cause a change in the independent variable.
Elaborate Talk About It Summarize the Lesson Ask: How do you know which is the dependent variable and which is the independent variable in a table or graph? In a table, the independent variable usually is represented by the variable x. The dependent variable usually is represented by the variable y. On a graph, the independent variable usually is shown on the horizontal axis and the dependent variable on the vertical axis.
GUIDED PRACTICE Engage with the Whiteboard For Exercises 1–2, have a student sketch the graph to represent the table on the whiteboard. Have the student explain how to know which quantity should be represented by the xaxis and which by the yaxis. For Exercise 3, have a student volunteer label the axes in the graph to represent the realworld situation suggested by the student.
Avoid Common Errors Exercise 3 If students have difficulty determining whether a relationship is additive or multiplicative, remind them that in a multiplicative relationship the graph will pass through the origin, but in an additive relationship the graph will not pass through the origin. Exercise 4 Remind students that if the independent variable is on the horizontal axis of a graph, the dependent variable is on the vertical axis of the graph.
389
Lesson 14.2
DO NOT EDITChanges must be made through “File info” CorrectionKey=B
DO NOT EDITChanges must be made through “File info” CorrectionKey=B
Guided Practice
YOUR TURN
8.
x
0
1
2
3
y
15
16
17
18
Personal Math Trainer Online Assessment and Intervention
1. A boat rental shop rents paddleboats for a fee plus an additional cost per hour. The cost of renting for different numbers of hours is shown in the table.
0
1
2
3
Cost ($)
10
11
12
13
What is the independent variable, and what is the dependent variable? How do you know? (Explore Activity 1)
my.hrw.com
Sample answer: Bridget’s grandmother gave her a
Time is the independent variable and cost is the
collection of 15 perfume bottles. Bridget adds one
dependent variable, because cost depends on the
bottle per week to the collection. The independent
number of hours rented.
variable is the number of weeks. The dependent variable
2. A car travels at a constant rate of 60 miles per hour. (Explore Activity 1)
is the number of perfume bottles in her collection. The
a. Complete the table.
value of y is always 15 units greater than the value of x. 9.
x
0
1
2
3
4
y
0
16
32
48
64
1
2
3
0
60
120
180
dependent variable. c. Describe how the value of the dependent variable is related to the value of the independent variable.
The value of y is always 60 times the value of x.
profit per Tshirt. The independent variable is the
Use the graph to answer the questions.
number of Tshirts he sells, and the dependent variable
3. Describe in words how the value of the dependent variable is related to the value of the independent variable. (Explore Activity 2)
is his profit in dollars. The value of y is always 16 times
The dependent variable is 5 times the value of the
the value of x.
independent variable.
y 18
4. Describe a realworld situation that the graph could represent. (Example 1)
12 6
O
0
Distance y (mi)
Time is the independent variable and distance is the
that are printed with funny slogans. He makes a $16
10.
Time x (h)
b. What is the independent variable, and what is the dependent variable?
Sample answer: Colin created a website to sell Tshirts
© Houghton Mifflin Harcourt Publishing Company
Time (hours)
4
20 10
O
x 2
4
6
Sample answer: The graph could represent the total
x 2
y 30
cost y of buying x carnival tickets for $5 each.
6
Sample answer: Tickets to the school musical cost $3
? ?
each. The independent variable is the number of tickets
ESSENTIAL QUESTION CHECKIN
5. How can you identify the dependent and independent variables in a realworld situation modeled by a graph?
purchased, and the dependent variable is the total cost. The value of y is always 3 times the value of x.
© Houghton Mifflin Harcourt Publishing Company
Describe a realworld situation that the variables could represent. Describe the relationship between the independent and dependent variables.
Sample answer: The dependent variable is the quantity that depends on the other variable. On a graph, the independent variable is usually shown on the horizontal axis and the dependent variable on the vertical axis. Lesson 14.2
6_MTXESE051676_U4M14L2.indd 389
389
29/01/14 12:46 AM
390
Unit 4
6_MTXESE051676_U4M14L2.indd 390
Independent and Dependent Variables in Tables and Graphs
29/01/14 12:49 AM
390
Personal Math Trainer Online Assessment and Intervention
Online homework assignment available
Evaluate GUIDED AND INDEPENDENT PRACTICE 6.6.A, 6.6.C
my.hrw.com
14.2 LESSON QUIZ 6.6.A The graph below shows the relationship between the number of tickets Lisa is ordering for a raffle and the cost. Use the graph to answer questions 1–3.
Total Cost (including postage)
Lisa’s Ticket Order
Concepts & Skills
Practice
Explore Activity 1 Identifying Independent and Dependent Quantities from a Table
Exercises 1–2, 7
Explore Activity 2 Identifying Independent and Dependent Variables from a Graph
Exercises 3, 6, 8
Example 1 Describing Relationships Between Independent and Dependent Variables
Exercises 4, 6
y 12
Exercise
8
Depth of Knowledge (D.O.K.)
Mathematical Processes
6
2 Skills/Concepts
1.A Everyday life
7
3 Strategic Thinking
1.F Analyze relationships
Number of Tickets
8–9
3 Strategic Thinking
1.G Explain and justify arguments
1. What are the dependent and independent variables?
10
3 Strategic Thinking
1.F Analyze relationships
4
x O
2
4
6
2. Is the relationship between the two variables additive or multiplicative? 3. Describe the relationship between the two quantities in words.
Additional Resources Differentiated Instruction includes: • Leveled Practice Worksheets
Use the table for question 4. x
0
1
2
3
4
y
0
7
14
21
28
4. Describe a possible situation that can be represented by the table. Identify the dependent and independent variables in this situation. Lesson Quiz available online my.hrw.com
Answers 1. Independent variable, x, is the number of tickets bought; dependent variable, y, is the total cost. 2. It is an additive relationship. 3. The total cost is the number of tickets bought plus $5 for postage for the tickets.
391
Lesson 14.2
4. Sample answer: Parking at the airport costs $7 per day. Independent variable, x, is the number of days a vehicle is parked; dependent variable, y, is the total cost for parking.
DO NOT EDITChanges must be made through “File info” CorrectionKey=B
Class
Date
14.2 Independent Practice
8. Ty borrowed $500 from his parents. The graph shows how much he owes them each month if he pays back a certain amount each month.
Personal Math Trainer
6.6.A, 6.6.C
a. How many hours did the soccer team practice before the season began?
6 hours b. What are the two quantities in this situation?
hours practiced during the season and total
Total practice time for year (hours)
my.hrw.com
6. The graph shows the relationship between the hours a soccer team practiced after the season started and their total practice time for the year.
a. Describe the relationship between the number of months and the amount Ty owes. Identify an independent and dependent variable and explain your thinking.
Online Assessment and Intervention
y 10
Ty starts out owing $500 and every
8 6 4 2
400 300 200 100
O
2
8 10
b. How long will it take Ty to pay back his parents?
10 months FOCUS ON HIGHER ORDER THINKING
d. Analyze Relationships Describe the relationship between the quantities in words.
Work Area
9. Error Analysis A discount store has a special: 8 cans of juice for a dollar. A shopper decides that since the number of cans purchased is 8 times the number of dollars spent, the cost is the independent variable and the number of cans is the dependent variable. Do you agree? Explain.
Total practice time for the year is 6 hours more than practice time during the season.
Sample answer: I disagree because the amount
e. Is the relationship between the variables additive or multiplicative? Explain.
a shopper pays depends on the number of cans
Additive; the total practice increases by 1 hour as the
purchased. So, the number of cans is the independent
practice time during the season increases by 1 hour.
variable, and cost is the dependent variable.
b. What is the independent variable?
6
owes.
dependent: total practice time for year
a. What is the dependent variable?
4
Months
months; dependent variable: amount he
4
independent: hours practiced during season;
7. Multistep Teresa is buying glitter markers to put in gift bags. The table shows the relationship between the number of gift bags and the number of glitter markers she needs to buy.
2
by $50; independent variable: number
x
O
c. What are the dependent and independent variables?
© Houghton Mifflin Harcourt Publishing Company
500
month the amount he owes decreases
Practice time during the season (hours)
practice time for year
Ty’s Loan Payments
Number of gift bags, x
0
1
2
3
Number of markers, y
0
5
10
15
10. Analyze Relationships Provide an example of a realworld relationship where there is no clear independent or dependent variable. Explain.
Sample answer: Andrea is 4 years older than Lisa. You
number of markers number of gift bags
could say that Andrea’s age depends on Lisa’s because you can add 4 to Lisa’s age. You can also say that Lisa’s
c. Describe the relationship between the quantities in words.
© Houghton Mifflin Harcourt Publishing Company
Name
Amount Ty owes (dollars)
DO NOT EDITChanges must be made through “File info” CorrectionKey=A
age depends on Andrea’s age because you can subtract
The number of glitter markers is 5 times the number of gift bags.
4 from Andrea’s age.
d. Is the relationship additive or multiplicative? Explain.
The relationship is multiplicative because y increases by a factor of 5 as x increases by 1. Lesson 14.2
6_MTXESE051676_U4M14L2.indd 391
EXTEND THE MATH
3/11/13 9:40 AM
PREAP
Introduce students to independent and dependent variables in situations that involve decimals or fractions. For example: Gina is charged $0.15 for each text message that she sends. 1. What is the independent variable? a a. number of texts sent b. charge per text c. total amount charged for texting
392
391
Unit 4
6_MTXESE051676_U4M14L2.indd 392
Activity available online 2. What is the dependent variable?
29/01/14 12:52 AM
my.hrw.com
c
a. number of texts sent b. charge per text c. total amount charged for texting 3. Write an equation that expresses the situation. Let b be the amount of Gina’s bill. Let s be the number of texts sent. b = 0.15s
Independent and Dependent Variables in Tables and Graphs
392
LESSON
14.3 Writing Equations from Tables Texas Essential Knowledge and Skills The student is expected to: Expressions, equations, and relationships—6.6.B Write an equation that represents the relationship between independent and dependent quantities from a table. Also 6.6.C
Engage
ESSENTIAL QUESTION How can you use an equation to show a relationship between two variables? Use a table to find the relationship between the two variables. Use that relationship to write an equation.
Motivate the Lesson Ask students to imagine a hot dog stand that charges $3 per hot dog. How much would 4 hot dogs cost? 30 hot dogs? Begin the Explore Activity to find out how to write an equation that will help you predict the total cost for any number of hot dogs.
Mathematical Processes 6.1.B Use a problemsolving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problemsolving process and the reasonableness of the solution.
Explore EXPLORE ACTIVITY Focus on Patterns
Mathematical Processes Point out to students that to write an equation from the data in the table, they need to look for a pattern in the data. First, they should look for changes in both the input values and the output values. Then they need to see how the changes are related. For example:
ADDITIONAL EXAMPLE 1
8=8·1 16 = 8 · 2 24 = 8 · 3
Write an equation that expresses y in terms of x. A x
1
2
3
4
5
y
3
6
9
12
15
x
2
4
6
8
10
y
7
9
11
13
15
So, the pattern is y = 8 · x, where x is the number of dogs walked and y is the amount of money earned.
y = 3x B
y=x+5
my.hrw.com
Animated Math Writing Equations from Tables Students generate patterns with an interactive model, record the data in a table, and write equations to represent the pattern. my.hrw.com
Lesson 14.3
EXAMPLE 1 Focus on Reasoning
Interactive Whiteboard Interactive example available online
393
Explain Mathematical Processes In A, point out to students that the yvalue is always less than the xvalue. Therefore, the operation in the equation must be subtraction, division, or multiplication by a factor that is less than 1. In B, point out to students that the yvalue is always more than the xvalue. Therefore, the operation in the equation must be addition or multiplication by a factor that is greater than 1.
Questioning Strategies
Mathematical Processes • For A, how can you write an equation that expresses x in terms of y? I compared the x and yvalues and found that each xvalue is twice the corresponding yvalue, which gives me the equation x = 2y.
YOUR TURN Engage with the Whiteboard For Exercises 2–5, have students write a pattern on the whiteboard for each table. Then have them use the pattern to write an equation to represent each table. Ask students to explain their reasoning.
Writing Equations from Tables
14.3 ?
ESSENTIAL QUESTION
Expressions, equations, and relationships—6.6.B Write an equation that represents the relationship between independent and dependent quantities from a table. Also 6.6.C.
Writing an Equation Based on a Table The relationship between two variables where one variable depends on the other can be represented in a table or by an equation. An equation expresses the dependent variable in terms of the independent variable.
Math On the Spot
When there is no realworld situation to consider, we usually say x is the independent variable and y is the dependent variable. The value of y depends on the value of x.
my.hrw.com
How can you use an equation to show a relationship between two variables?
EXAMPLE 1
6.6.B, 6.6.C
EXPLORE ACTIVITY
Write an equation that expresses y in terms of x.
Writing an Equation to Represent a RealWorld Relationship
Animated Math
A
my.hrw.com
Many realworld situations involve two variable quantities in which one quantity depends on the other. This type of relationship can be represented by a table. You can also use an equation to model the relationship.
1
2
3
5
10
20
Earnings
$8
$16
$24
$40
$80
$160
For 1 dog, Amanda earns 1 · 8 = $8. For 2 dogs, she earns 2 · 8 = $16.
© Houghton Mifflin Harcourt Publishing Company
3
4
5
1
1.5
2
2.5
Compare the x and yvalues to find a pattern.
Use the pattern to write an equation expressing y in terms of x. y = 0.5x
B
x
2
4
6
8
10
y
5
7
9
11
13
Compare the x and yvalues to find a pattern.
STEP 1
Each yvalue is 3 more than the corresponding xvalue.
Use the pattern to write an equation expressing y in terms of x.
STEP 2
Math Talk
Mathematical Processes
number of dogs walked.
y=x+3
How can you check that your equations are correct?
for each dog she walks.
YOUR TURN
C Write an equation that relates the number of dogs Amanda walks to the amount she earns. Let e represent earnings and d represent dogs.
For each table, write an equation that expresses y in terms of x. 2.
e=8·d D Use your equation to complete the table for 5, 10, and 20 walked dogs. E Amanda’s earnings depend on the number of dogs walked
x
12
11
10
y
10
9
8
3.
x
10
12
14
y
25
30
35
y=x2
y = 2.5x
. 4.
Reflect
Personal Math Trainer
1. What If? If Amanda changed the amount earned per dog to $11, what equation could you write to model the relationship between number of dogs walked and earnings?
2
STEP 2
Substitute each value of x in the equation. If the equation is correct, the result is the corresponding yvalue.
Each earnings amount is 8 times the corresponding 8
1 0.5
1 Each yvalue is __ , or 0.5 times, the corresponding xvalue. 2
A For each column, compare the number of dogs walked and earnings. What is the pattern?
B Based on the pattern, Amanda earns $
x y STEP 1
The table shows how much Amanda earns for walking 1, 2, or 3 dogs. Use the table to determine how much Amanda earns per dog. Then write an equation that models the relationship between number of dogs walked and earnings. Use your equation to complete the table. Dogs walked
6.6.B, 6.6.C
Online Assessment and Intervention
e = 11 · d
x
5
4
3
y
10
9
8
y=x+5
5.
x
0
1
2
y
0
2
4
© Houghton Mifflin Harcourt Publishing Company
LESSON
y = 2x
my.hrw.com
Lesson 14.3
393
394
Unit 4
PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.B, which calls for students to “use a problemsolving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problemsolving process and the reasonableness of the solution.” Each of these steps is explicitly used in solving a realworld problem in this lesson.
Math Background A rule that relates the x and yvalues in a table also can be called a relation. A relation can describe a function if, for each xvalue (input), there is only one yvalue (output). There are several different ways to describe the variables of a function: Independent Variable
Dependent Variable
xvalue
yvalue
Domain
Range
Input
Output
x
f(x)
Writing Equations from Tables
394
ADDITIONAL EXAMPLE 2 Meredith is playing a video game. She earns the same number of points for each alien she captures. She earned 750 points for capturing 5 aliens and 1,350 points for capturing 9 aliens. Write an equation to represent the relationship. Then solve the equation to find how many points Meredith will earn if she captures 27 aliens. p = 150a, where p represents the number of points and a represents the number of aliens captured; 4,050 points Interactive Whiteboard Interactive example available online my.hrw.com
EXAMPLE 2 Focus on Reasoning
Mathematical Processes Point out to students that sometimes a problem may provide clues and facts that you must use to find a solution. Encourage students to begin by identifying the important information. They can underline or circle the information in the problem statement.
Engage with the Whiteboard Have students extend the table on the whiteboard, continuing with sale prices of $400 through $1,200 in increments of $100. Then have students find the donation amount for these sale prices. After they have completed the table, ask students to identify any patterns in the table.
Questioning Strategies • What is the independent variable in this situation? the dependent variable? The independent variable is the sale price of a painting. The dependent variable is the amount donated to charity. • How else could you solve this problem? Write and solve a proportion to find the amount 50 75 x x of the donation if a painting sells for $1,200: ___ = ____ or ___ = ____ . 200 1,200 300 1,200
YOUR TURN Avoid Common Errors If students have difficulty identifying the independent and dependent variables, remind them to begin by using the given information to make a table and then look for a pattern.
Elaborate Talk About It Summarize the Lesson Ask: How can you use a table to write an equation that represents the relationship in the table? In the table, find the relationship between the independent and dependent variables. Then write the equation that represents the relationship.
GUIDED PRACTICE Engage with the Whiteboard For Exercises 1–2, have students write a pattern on the whiteboard for each table. Then have students use the pattern to write an equation to represent each table. Ask students to explain their reasoning. For Exercise 5, have students fill in the missing information in the table on the whiteboard. Then have them identify the pattern and write an equation.
Avoid Common Errors Exercises 1–4 Some students may write an equation that expresses x in terms of y instead of y in terms of x. Remind them that the form of the equation should be y = kx or y = x + b. Exercise 5 If students have difficulty identifying the independent and dependent variables, remind them to begin by using the given information to make a table and then look for a pattern.
395
Lesson 14.3
DO NOT EDITChanges must be made through “File info” CorrectionKey=B
DO NOT EDITChanges must be made through “File info” CorrectionKey=B
YOUR TURN
Using Tables and Equations to Solve Problems Problem Solving
EXAMPL 2 EXAMPLE
6. When Ryan is 10, his brother Kyle is 15. When Ryan is 16, Kyle will be 21. When Ryan is 21, Kyle will be 26. Complete the table for Ryan and Kyle. Write and solve an equation to find Kyle’s age when Ryan is 52.
Personal Math Trainer
You can use tables and equations to solve realworld problems. 6.6.B, 6.6.C
Math On the Spot
Online Assessment and Intervention
my.hrw.com
my.hrw.com
A certain percent of the sale price of paintings at a gallery will be donated to charity. The donation will be $50 if a painting sells for $200. The donation will be $75 if a painting sells for $300. Find the amount of the donation if a painting sells for $1,200.
Ryan
10
16
21
Kyle
15
21
26
k = r + 5; 57 years old
Analyze Information
Guided Practice
You know the donation amount when the sale price of a painting is $200 and $300. You need to find the donation amount if a painting sells for $1,200.
Write an equation to express y in terms of x. (Explore Activity, Example 1) 1.
Formulate a Plan
You can make a table to help you determine the relationship between sale price and donation amount. Then you can write an equation that models the relationship. Use the equation to find the unknown donation amount. 3. 200
300
Donation amount ($)
50
75
50 ÷ 2 50 25 ___ = ______ = ___ = 25% 200 200 ÷ 2 100
75 ÷ 3 75 25 ___ = _______ = ___ = 25% 300 300 ÷ 3 100
Write an equation. Let p represent the sale price of the painting. Let d represent the donation amount to charity. © Houghton Mifflin Harcourt Publishing Company
The donation amount is equal to 25% of the sale price. d = 0.25 · p Find the donation amount when the sale price is $1,200.
One way to determine the relationship between sale price and donation amount is to find the percent.
d = 0.25 · 1,200
p is the independent variable; its value does not depend on any other value. d is the dependent variable; its value depends on the price of the painting.
26
36
2.
x
0
1
2
3
y
0
4
8
12
y = 4x
x
4
6
8
10
y
7
9
11
13
4.
1
2
5
1.35
2.70
6.75
Number of songs = n; Cost =
? ?
x
12
24
36
48
y
2
4
6
8
y = _6x
1.35n $33.75
.
ESSENTIAL QUESTION CHECKIN
6. Explain how to use a table to write an equation that represents the relationship in the table.
Justify and Evaluate
✓
16
The total cost of 25 songs is
Substitute values from the table for p and d to check that they are solutions of the equation d = 0.25 · p. Then check your answer of $300 by substituting for d and solving for p.
✓
6
Total cost ($)
Simplify to find the donation amount.
d = 0.25 · p 300 = 0.25 · p p = 1,200
y
Songs downloaded
Substitute $1,200 for the sale price of the painting.
d = 0.25 · p d = 0.25 · 300 d = 75
40
y=x+3
When the sale price is $1,200, the donation to charity is $300.
d = 0.25 · p d = 0.25 · 200 d = 50
30
5. Jameson downloaded one digital song for $1.35, two digital songs for $2.70, and 5 digital songs for $6.75. Complete the table. Write and solve an equation to find the cost to download 25 digital songs. (Example 2)
d = 0.25 · p
d = 300
20
© Houghton Mifflin Harcourt Publishing Company
Sale price ($)
10
y=x4
Justify and Evaluate Solve
Make a table.
x
Compare the x and yvalues to find a pattern. Use the pattern to write an equation expressing y in terms of x.
✓ Lesson 14.3
6_MTXESE051676_U4M14L3.indd 395
395
396
29/01/14 1:11 AM
Unit 4
6_MTXESE051676_U4M14L3.indd 396
29/01/14 1:11 AM
DIFFERENTIATE INSTRUCTION Curriculum Integration
Cognitive Strategies
Additional Resources
Discuss the relationship between Celsius temperature and Kelvin temperature. Show students the following table and ask them to write an equation to convert from degrees Celsius to degrees Kelvin.
Some students may find it helpful to include a “Process” column in a table to help them identify patterns. Have students complete the table below.
Differentiated Instruction includes: • Reading Strategies • Success for English Learners ELL • Reteach • Challenge PREAP
Celsius (°C)
Kelvin (°K)
100
173
K = C + 273
x
Process
y
3
2 = 3 + 1
2
0 = 1 + 1
2
50
223
1
0
273
0
50
323
1
100
373
2
1 = 2 + 1 1=0+1 2=1+1 3=2+1
1 0 1 2 3
Each value of y is one more than the value of x. Writing Equations from Tables
396
Personal Math Trainer Online Assessment and Intervention
Online homework assignment available my.hrw.com
Evaluate GUIDED AND INDEPENDENT PRACTICE 6.6.B, 6.6.C
Concepts & Skills
Practice
Explore Activity Writing an Equation to Represent a RealWorld Relationship
Exercises 1–4, 8, 11
Write an equation that expresses y in terms of x.
Example 1 Writing an Equation Based on a Table
Exercises 1–4, 9–10
1.
Example 2 Using Tables and Equations to Solve Problems
Exercises 5, 11
14.3 LESSON QUIZ 6.6.B
x
1
2
3
4
5
y
5
10 15 20 25
x
10 20 30 40 50
y
7
2. 17 27 37 47
3. Jaime bought 2 puzzles for $5.00 and 3 puzzles for $7.50. Write and solve an equation to find the cost of 15 puzzles. 4. A submarine descends to 100 feet in 2 minutes and 250 feet in 5 minutes. Write and solve an equation to find the depth of the submarine in 8 minutes. Lesson Quiz available online my.hrw.com
Answers 1. y = 5x
2. y = x  3
3. c = 2.50p; $37.50
4. d = 50m; 400 feet
397
Lesson 14.3
Exercise
Depth of Knowledge (D.O.K.)
Mathematical Processes
7
3 Strategic Thinking
1.F Analyze relationships
8
2 Skills/Concepts
1.A Everyday life
9–10
3 Strategic Thinking
1.F Analyze relationships
11
3 Strategic Thinking
1.A Everyday life
12
3 Strategic Thinking
1.G Explain and justify arguments
13–14
3 Strategic Thinking
1.F Analyze relationships
15
3 Strategic Thinking
1.G Explain and justify arguments
Additional Resources Differentiated Instruction includes: • Leveled Practice Worksheets
DO NOT EDITChanges must be made through “File info” CorrectionKey=B
DO NOT EDITChanges must be made through “File info” CorrectionKey=A
Name
Class
Date
14.3 Independent Practice
12. Communicate Mathematical Ideas For every hour that Noah studies, his test score goes up 3 points. Explain which is the independent variable and which is the dependent variable. Write an equation modeling the relationship between hours studied h and the increase in Noah’s test score s.
Personal Math Trainer
6.6.B, 6.6.C
my.hrw.com
Online Assessment and Intervention
Independent: hours studied; its value does not depend on
7. Vocabulary What does it mean for an equation to express y in terms of x?
another variable; dependent: test score; its value depends on the number of hours that Noah studies; s = 3h
The variable y is on one side of the equation. The expression on the other side of the equation shows the relationship
FOCUS ON HIGHER ORDER THINKING
between x and y.
13. Make a Conjecture Compare the yvalues in the table to the corresponding xvalues. Determine whether there is an additive relationship or a multiplicative relationship between x and y. If possible, write an equation modeling the relationship. If not, explain why.
8. The length of a rectangle is 2 inches more than twice its width. Write an equation relating the length l of the rectangle to its width w.
l = 2w + 2
9. Look for a Pattern Compare the yvalues in the table to the corresponding xvalues. What pattern do you see? How is this pattern used to write an equation that represents the relationship between the x and yvalues? 20
24
28
32
y
5
6
7
8
© Houghton Mifflin Harcourt Publishing Company
2
4
6
8
8
16
24
32
3
5
7
3
6
8
21
yvalues and corresponding xvalues. 14. Represent RealWorld Problems Describe a realworld situation in which there is an additive or multiplicative relationship between two quantities. Make a table that includes at least three pairs of values. Then write an equation that models the relationship between the quantities.
10. Explain the Error A student modeled the relationship in the table with the equation x = 4y. Explain the student’s error. Write an equation that correctly models the relationship. x
1
Not possible; there is no consistent pattern between the
The yvalue is _14 of the xvalue. Write an equation that relates y to _14 of x.
y
x y
Sample answer: The distance Yasmine traveled in miles is equal to 50 times the number of hours she drove; d = 50 × t (multiplicative relationship). Time (h)
The student switched the variables; y = 4x
2
Distance (mi) 100
3
4
5
150
200
250
15. Critical Thinking Georgia knows that there is either an additive or multiplicative relationship between x and y. She only knows a single pair of data values. Explain whether Georgia has enough information to write an equation that models the relationship between x and y.
11. Multistep Marvin earns $8.25 per hour at his summer job. He wants to buy a video game system that costs $206.25. a. Write an equation to model the relationship between number of hours worked h and amount earned e.
No; with only one pair of values, Georgia cannot tell
e = 8.25h
© Houghton Mifflin Harcourt Publishing Company
x
Work Area
whether the relationship is additive or multiplicative, so
b. Solve your equation to find the number of hours Marvin needs to work in order to afford the video game system.
she cannot write an equation for the relationship.
206.25 = 8.25h; 25 = h; 25 hours
Lesson 14.3
6_MTXESE051676_U4M14L3.indd 397
10/30/12 12:23 PM
InCopy Notes 1. This is a list
398
397
Unit 4
6_MTXESE051676_U4M14L3.indd 398
29/01/14 1:12 AM
InDesign Notes
EXTEND THE MATH
1. This is a list
PREAP
Activity available online
my.hrw.com
Activity To introduce the idea of relationships that are not purely additive or multiplicative, have students find the yvalues in the following tables. Remind them to use the order of operations. Have them compare and contrast these relationships with additive and multiplicative relationships. 1. y = 2x + 1 3. y = __12 x + 3 x
0
1
2
3
x
0
2
4
6
y
1
3
5
7
y
3
4
5
6
4. y = __13 x 1
2. y = 3x  2 x
1
2
3
4
x
3
6
9
12
y
1
4
7
10
y
0
1
2
3
Writing Equations from Tables
398
LESSON
14.4
Representing Algebraic Relationships in Tables and Graphs
Texas Essential Knowledge and Skills The student is expected to: Expressions, equations, and relationships—6.6.C Represent a given situation using verbal descriptions, tables, graphs, and equations in the form y = kx or y = x + b. Expressions, equations, and relationships—6.6.A
Engage ESSENTIAL QUESTION How can you use verbal descriptions, tables, and graphs to represent algebraic relationships? Sample answer: You can make a table from the verbal description and then make a graph from the ordered pairs in the table. From a graph, you can make a table and write an equation.
Motivate the Lesson Ask: Have you ever thought about running in a marathon? Do you know how many kilometers you could run in an hour? in two hours? Begin Explore Activity 1 to find out how to make a table and a graph to estimate how far you could run in a given period of time.
Identify independent and dependent quantities from tables and graphs. Expressions, equations, and relationships—6.6.B Write an equation that represents the relationship between independent and dependent quantities from a table.
Mathematical Processes 6.1.D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.
Explore EXPLORE ACTIVITY 1 Connect Multiple Representations
Mathematical Processes Point out to students that the ordered pairs from each table are used to make the graphs. However, after the lines are drawn on the graphs, they represent a more complete picture of the relationship than the tables do because all positive real numbers, not just integers, are included in the graph.
Explain EXPLORE ACTIVITY 2 Focus on Math Connections
Mathematical Processes Point out to students that when finding an equation from a graph, it is easier to first make a table of values from the graph. Then they can look for a pattern for the equation.
Questioning Strategies
Mathematical Processes • Is the relationship additive or multiplicative? Explain how you know. The relationship is additive, because the line drawn through the points does not go through the origin. • Explain how you can find the entrance fee for the museum from the graph. The starting point of the graph is (0, 5). This ordered pair represents the cost of Cherise’s expenses at the museum without any purchases at the gift shop, so it represents the entrance fee, $5.
c.4.C ELL Be sure English learners understand the context in Explore Activity 2. You may want to discuss the terms “museum”, “souvenir”, and “entrance fee” before starting the activity.
Integrating the ELPS
Engage with the Whiteboard Ask a student volunteer to complete the table and identify the pattern. Then have the student write the equation to represent the total amount spent at the museum gift shop. Finally, discuss with the class what the independent and dependent variables are.
399
Lesson 14.4
Representing Algebraic Relationships in Tables and Graphs
?
ESSENTIAL QUESTION
How can you use verbal descriptions, tables, and graphs to represent algebraic relationships?
Writing an Equation from a Graph Cherise pays the entrance fee to visit a museum, then buys souvenirs at the gift shop. The graph shows the relationship between the total amount she spends at the museum and the amount she spends at the gift shop. Write an equation to represent the relationship.
A Read the ordered pairs from the graph. Use them to complete a table comparing total spent y to amount spent at the gift shop x.
6.6.C
EXPLORE ACTIVITY 1
Representing Algebraic Relationships Angie’s walking speed is 5 kilometers per hour, and May’s is 4 kilometers per hour. Use tables and graphs to show how the distance each girl walks is related to time.
A For each girl, make a table comparing time and distance. 0
1
2
3
4
Angie’s distance (km)
0
5
10
15
20
Time (h)
0
1
2
3
4
May’s distance (km)
0
4
8
12
16
For every hour May walks, she travels 4 km.
Distance (km)
Distance (km)
© Houghton Mifflin Harcourt Publishing Company
20 16 12 8 4 1
2
3
4
Time (h)
5
10
15
20
Total amount ($)
5
10
15
20
25
y 32
shop amount.
C Write an equation that expresses the total amount y in terms of the gift shop amount x.
28 24 20 16 12 8 4 x
O
4
8 12 16 20 24
Gift shop amount ($)
Reflect Math Talk
20
Mathematical Processes
16
2. Identify the dependent and independent quantities in this situation.
Dependent: total amount spent; independent: amount
Why does it make sense to connect the points in each graph?
12 8
spent at the gift shop
4 x
x O
5
y= x+5
May
y
0
The total amount is 5 more than the gift
B For each girl, make a graph showing her distance y as it depends on time x. Plot points from the table and connect them with a line. Angie
Gift shop amount ($)
B What is the pattern in the table? For every hour Angie walks, she travels 5 km.
Time (h)
y
6.6.C, 6.6.B
EXPLORE ACTIVITY 2
O
1
2
3
4
5
Time (h)
Reflect 1. Analyze Relationships How can you use the tables to determine which girl is walking faster? How can you use the graphs?
The girls can walk for fractional parts of an hour and can travel fractional parts of a kilometer.
3. Draw a line through the points in the graph. Find the point that represents Cherise spending $25 at the gift shop. Use this point to find the total she would spend if she spent $25 at the gift shop. Then use your equation from C to verify your answer.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: © Thinkstock/ Corbis
14.4
Expressions, equations, and relationships—6.6.C Represent a given situation using tables, graphs, and equations…. Also 6.6.A, 6.6.B
Total amount ($)
LESSON
(25, 30); $30; 30 = 25 + 5; 30 = 30
Compare the distances walked after the same amount of time; compare the steepness of the lines. Lesson 14.4
399
400
Unit 4
PROFESSIONAL DEVELOPMENT Integrate Mathematical Processes This lesson provides an opportunity to address Mathematical Process TEKS 6.1.D, which calls for students to “communicate mathematical ideas...using multiple representations…as appropriate.” Students use verbal descriptions to make tables and draw graphs that represent reallife situations. They also represent information from graphs by using tables and equations; and represent equations by using tables and graphs.
Math Background The equations in this lesson are linear equations. A linear equation is an equation whose solutions fall on a line on a coordinate plane. All solutions of a particular linear equation fall on the line, and all the points on the line are solutions of the equation. Linear equations have constant slope. In the slopeintercept form, y = mx + b, m is the slope and b is the yintercept. Linear equations fit into the general category of polynomial equations. Linear equations are called firstdegree equations, because the greatest power of x is 1.
Representing Algebraic Relationships in Tables and Graphs
400
ADDITIONAL EXAMPLE 1 Graph each equation.
Talk About It
A y=x+2
Check for Understanding Ask: Can the points on the graphs only have whole number coordinates? Explain your answer. No, the coordinates can be any pair of rational numbers that satisfies the equation. For example, (0.5, 1.5) is on the graph of y = x + 1.
y 6 4 2
x
O
EXAMPLE 1
2
4
6
B y = 3x
Questioning Strategies
Mathematical Processes • By looking at the graph, how can you tell if the relationship in A is additive or multiplicative? The relationship is additive because the line drawn through the points does not go through the origin. • By looking at the graph, how can you tell if the relationship in B is additive or multiplicative? The relationship is multiplicative because the line drawn through the points goes through the origin.
y 6 4 2 O
x 2
4
YOUR TURN Avoid Common Errors
6
Interactive Whiteboard Interactive example available online
Students may plot the points in the table but forget to draw a line connecting the points. Remind them that they must connect the points with a line for the graph to be correctly drawn.
my.hrw.com
Elaborate Talk About It Summarize the Lesson Ask: How can you use tables and graphs to represent algebraic relationships? You can make a table from the verbal description and then make a graph from the ordered pairs in the table. From a graph, you can make a table and write an equation.
GUIDED PRACTICE Engage with the Whiteboard For Exercises 1–2, have students complete the table and graph the points on the coordinate grid on the whiteboard. Then have another student identify the pattern and write the equation.
Avoid Common Errors Exercises 1–2 Students may plot the points in the table but forget to draw a line connecting the points. Remind them that they must connect the points with a line for the graph to be drawn correctly.
401
Lesson 14.4
Graphing an Equation
YOUR TURN
An ordered pair (x, y) that makes an equation like y = x + 1 true is called a solution of the equation. The graph of an equation represents all the ordered pairs that are solutions.
EXAMPL 1 EXAMPLE
4. Graph y = x + 2.5.
Personal Math Trainer Math On the Spot
Online Assessment and Intervention
my.hrw.com
my.hrw.com
6.6.C
Graph each equation.
y
x
x + 2.5 = y
(x, y)
0 1 2 3
0 + 2.5 = 2.5 1 + 2.5 = 3.5 2 + 2.5 = 4.5 3 + 2.5 = 5.5
(0, 2.5) (1, 3.5) (2, 4.5) (3, 5.5)
10 8 6 4 2 x
A y=x+1
O
STEP 1
Make a table of values. Choose some values for x and use the equation to find the corresponding values for y.
STEP 2
Plot the ordered pairs from the table. Mathematical Processes
Is the ordered pair (3.5, 4.5) a solution of the equation y = x + 1? Explain.
y
x
x+1=y
(x, y)
1+1=2
(1, 2)
8
10
2
2+1=3
(2, 3)
6
3
3+1=4
(3, 4)
4
4
4+1=5
(4, 5)
2
5
5+1=6
(5, 6)
O
x 1
2
3
4
5
B y = 2x
© Houghton Mifflin Harcourt Publishing Company
3
4
5
STEP 1
Make a table of values. Choose some values for x and use the equation to find the corresponding values for y.
STEP 2
Plot the ordered pairs from the table.
STEP 3
Draw a line through the plotted points to represent all of the ordered pair solutions of the equation.
Frank mows lawns in the summer to earn extra money. He can mow 3 lawns every hour he works. (Explore Activity 1 and Explore Activity 2) 1. Make a table to show the relationship between the number of hours Frank works, x, and the number of lawns he mows, y. Graph the relationship and write an equation.
Yes; (3.5, 4.5) is on the graph. You can also substitute for the variables in the equation to check.
Hours worked
Lawns mowed
0
0 3 6 9
1
2 3
y
10 8 6 4 2
y = 3x
x O
1
2
3
4
5
Hours worked
Graph y = 1.5x. (Example 1)
y 5
2. Make a table to show the relationship. x y
y
x
2x = y
(x, y)
1
2×1=2
(1, 2)
2
2×2=4
(2, 4)
6
3
2×3=6
(3, 6)
4
4
2×4=8
(4, 8)
2
5
2 × 5 = 10
(5, 10)
O
2 1
10
3. Plot the points and draw a line through them.
8
? ?
x 1
2
3
4
4 3
0 1 2 3 0 1.5 3 4.5
x O
1
2
3
4
5
ESSENTIAL QUESTION CHECKIN
© Houghton Mifflin Harcourt Publishing Company
Draw a line through the plotted points to represent all of the ordered pair solutions of the equation.
1
2
Guided Practice Math Talk
Lawns mowed
STEP 3
1
4. How can a table represent an algebraic relationship between two variables?
5
It shows pairs of values that satisfy the relationship.
Lesson 14.4
401
402
Unit 4
DIFFERENTIATE INSTRUCTION Cooperative Learning
Modeling
Additional Resources
Have students work in pairs to write an equation with two variables. Each equation should be an additive equation. Collect students’ equations and randomly redistribute them. Have the students make tables for the equations and write solutions of the equations as ordered pairs. Then have the students graph the equations.
Draw an equilateral triangle and a square, each with a side length of 6 inches, on the chalkboard. Have students find the perimeter of each. Ask students to come up with a formula for the perimeter of an equilateral triangle and a square. Then have students make a table and a graph for each formula.
Differentiated Instruction includes: • Reading Strategies • Success for English Learners ELL • Reteach • Challenge PREAP
Representing Algebraic Relationships in Tables and Graphs
402
Personal Math Trainer Online Assessment and Intervention
Evaluate GUIDED AND INDEPENDENT PRACTICE
Online homework assignment available
6.6.A, 6.6.C
my.hrw.com
14.4 LESSON QUIZ 6.6.C The graph shows the number of bracelets Olivia can make in an hour. Number of Bracelets
y
Concepts & Skills
Practice
Explore Activity 1 Representing Algebraic Relationships
Exercise 1
Explore Activity 2 Writing an Equation from a Graph
Exercises 1, 5–7
Example 1 Graphing an Equation
Exercises 2–3, 8–9, 11
8 6
Exercise
4
5–6
2
7 2
4
1.D Multiple representations
3 Strategic Thinking
1.F Analyze relationships
8–9
2 Skills/Concepts
1.D Multiple representations
10
3 Strategic Thinking
1.G Explain and justify arguments
11
3 Strategic Thinking
1.F Analyze relationships
12
3 Strategic Thinking
1.G Explain and justify arguments
13–14
3 Strategic Thinking
1.F Analyze relationships
6
Hours
1. Read the ordered pairs from the graph to make a table. 2. Write an equation to model the relationship. The equation y = x + 2 represents the total cost of doing x loads of laundry at a laundromat, including buying a box of detergent.
Additional Resources Differentiated Instruction includes: • Leveled Practice Worksheets
3. Make a table that represents the relationship between number of loads and total cost. 4. Make a graph showing the relationship. Lesson Quiz available online my.hrw.com
Answers Number of hours
0
1
2
Number of bracelets
0
4
8
2. y = 4x 3.
Number of loads
0
1
2
3
Total cost ($)
2
3
4
5
403
Lesson 14.4
y
4. Total Cost
1.
Mathematical Processes
2 Skills/Concepts
x O
Depth of Knowledge (D.O.K.)
6 4 2
x O
2
4
6
Number of Loads
DO NOT EDITChanges must be made through “File info” CorrectionKey=B
Class
Date
14.4 Independent Practice
Personal Math Trainer
6.6.A, 6.6.B, 6.6.C
my.hrw.com
11. Multistep The equation y = 9x represents the total cost y for x movie tickets. a. Make a table and a graph to represent the relationship between x and y.
Online Assessment and Intervention
Students at Mills Middle School are required to work a certain number of community service hours. Students may work additional hours beyond the requirement.
5
10
15
20
Total hours
20
25
30
35
40
50
Total (h)
0
Additional hours
30
3
4
5
Total cost ($), y
9
18
27
36
45
y 50 40 30 20 10
of tickets; the total cost depends on how many
20
O
2
Dependent: total cost; independent: number
40
10
6. Write an equation that expresses the total hours in terms of the additional hours.
1
b. Critical Thinking In this situation, which quantity is dependent and which is independent? Justify your answer.
y
5. Read the ordered pairs from the graph to make a table.
Number of tickets, x
2
3
4
5
Number of tickets
c. Multiple Representations Eight friends want to go see a movie. Would you prefer to use an equation, a table, or a graph to find the cost of 8 movie tickets? Explain how you would use your chosen method to find the cost.
Additional (h)
7. Analyze Relationships How many community service hours are students required to work? Explain.
Sample answer: an equation; substitute 8 for x in
20 hours; when 0 additional hours are worked,
y = 9x to get y = 9(8) = $72.
the total is 20 hours.
FOCUS ON HIGHER ORDER THINKING
Work Area
12. Critical Thinking Think about graphing the equations y = 5x and y = x + 500. Which line would be steeper? Why?
Beth is using a map. Let x represent a distance in centimeters on the map. To find an actual distance y in kilometers, Beth uses the equation y = 8x.
The graph of y = 5x would be steeper because y
8. Make a table comparing a distance on the map to the actual distance.
increases more rapidly for each value of x.
Map distance (cm)
1
2
3
4
5
Actual distance (km)
8
16
24
32
40
9. Make a graph that compares the map distance to the actual distance. 10. Critical Thinking The actual distance between Town A and Town B is 64 kilometers. What is the distance on Beth’s map? Did you use the graph or the equation to find the answer? Why?
8 cm; sample answer: I used the equation because the scales on the graph don’t extend far enough.
Actual distance (km)
© Houghton Mifflin Harcourt Publishing Company
x 1
tickets were purchased.
x 10 20 30 40 50
y = x + 20
O
13. Persevere in Problem Solving Marcus plotted the points (0, 0), (6, 2), (18, 6), and (21, 7) on a graph. He wrote an equation for the relationship. Find another ordered pair that could be a solution of Marcus’s equation. Justify your answer.
Sample answer: (30, 10); every yvalue is _13 of the xvalue.
y 50
So, 10 = _13 (30).
40 30 20 10 O
x 1
2
3
4
5
14. Error Analysis The cost of a personal pizza is $4. A drink costs $1. Anna wrote the equation y = 4x + 1 to represent the relationship between total cost y of buying x meals that include one personal pizza and one drink. Describe Anna’s error and write the correct equation.
Map distance (cm)
© Houghton Mifflin Harcourt Publishing Company
Name
Total cost ($)
DO NOT EDITChanges must be made through “File info” CorrectionKey=B
Anna’s equation does not show that every meal includes both a pizza and a drink; the correct equation is y = 5x.
Lesson 14.4
6_MTXESE051676_U4M14L4.indd 403
403
29/01/14 2:12 AM
EXTEND THE MATH
PREAP
404
Unit 4
6_MTXESE051676_U4M14L4.indd 404
Activity available online
29/01/14 2:18 AM
my.hrw.com
Activity Ask students if they know that you can find the approximate temperature by listening to crickets chirp? The chirp rate of a cricket varies by temperature. The hotter it is, the more chirps per minute. The temperature in °F can be found by multiplying the number of chirps in one minute by __14 and adding 40. Write the equation that represents this situation. Then make a table of values to find the temperature for 20, 40, 60, 80, and 100 chirps per minute. t = __14 c + 40, where t is the Fahrenheit temperature and c is the number of chirps in a minute c
20
40
60
80
100
t
45
50
55
60
65
Representing Algebraic Relationships in Tables and Graphs
404
MODULE QUIZ
Ready to Go On?
Ready
Assess Mastery
14.1 Graphing on the Coordinate Plane
Use the assessment on this page to determine if students have mastered the concepts and standards covered in this module.
Graph each point on the coordinate plane.
3
Response to Intervention
2 1
Differentiated Instruction
Differentiated Instruction
• Reteach worksheets
• Challenge worksheets
• Success for English Learners ELL
ELL
Additional Resources Assessment Resources includes: • Leveled Module Quizzes
E
2 6
6. F(4, 6)
2O
D
x 2
6
C
6
14.3 Writing Equations from Tables Write an equation that represents the data in the table. 8.
x
3
5
8
10
y
21
35
56
70
9.
x
5
10
15
20
y
17
22
27
32
y = x + 12
y = 7x
14.4 Representing Algebraic Relationships in Tables and Graphs
PREAP
Extend the Math PREAP Lesson activities in TE
A
independent: number of packages; dependent: total cost
Online and Print Resources
• Reading Strategies
2. B(3, 5) 4. D(3, 5)
my.hrw.com
B
6
7. Jon buys packages of pens for $5 each. Identify the independent and dependent variables in the situation.
Enrichment
Graph each equation. © Houghton Mifflin Harcourt Publishing Company
my.hrw.com
1. A(2, 4)
14.2 Independent and Dependent Variables in Tables and Graphs
Access Ready to Go On? assessment online, and receive instant scoring, feedback, and customized intervention or enrichment.
Online Assessment and Intervention
y
F
3. C(6, 4) 5. E(7, 2)
Intervention
Personal Math Trainer
Personal Math Trainer Online Assessment and Intervention
10. y = x + 3
11. y = 5x
8
40
6
30
4
20
2
O
10 2
4
6
8
O
2
4
6
8
ESSENTIAL QUESTION 12. How can you write an equation in two variables to solve a problem?
Decide which variable depends on the other. Use a table to find the relationship between the variables and write an equation.
Module 14
Texas Essential Knowledge and Skills Lesson
Exercises
14.1
1–6
6.11
14.2
7
6.6.A, 6.6.C
14.3
8–9
6.6.B, 6.6.C
14.4
10–11
6.6.A, 6.6.B, 6.6.C
405
Module 14
TEKS
405
Personal Math Trainer
MODULE 14 MIXED REVIEW
Texas Test Prep
Texas Testing Tip Some items are called contextbased items, which means the student has to examine each answer choice in order to determine the correct answer. Item 2 If students don’t remember that for every point in quadrant II the xcoordinate is a negative number and the ycoordinate is a positive number, they may need to plot each point to see that choice C is the correct answer.
Selected Response 1. What are the coordinates of point G on the coordinate grid below? y 4 2 x 4
Item 5 To find the point that the graph of y = 10 + x does not pass through, students may need to graph each point on a coordinate grid to see that choice C is the correct answer.
Item 4 Students often will get the independent and dependent quantities backward in problems, thereby choosing A for the answer. Remind students that the dependent quantity depends on the independent quantity. Therefore, the number of points earned depends on the number of prizes captured.
O
2
4
2
G
4
Avoid Common Errors Item 1 Students may forget what the first and second numbers in an ordered pair mean. Remind students that the first number is the xcoordinate and the second is the ycoordinate.
2
A (4, 3)
C
B (4, 3)
D (4, 3)
(4, 3)
2. A point is located in quadrant II of a coordinate plane. Which of the following could be the coordinates of that point? A (5, 7)
C
B (5, 7)
D (5, 7)
(5, 7)
3. Matt had 5 library books. He checked 1 additional book out every week without returning any books. Which equation describes the number of books he has, y, after x weeks? A y = 5x
C
y = 1 + 5x
B y= 5x
D y= 5+x
4. Stewart is playing a video game. He earns the same number of points for each prize he captures. He earned 1,200 points for 6 prizes, 2,000 points for 10 prizes, and 2,600 points for 13 prizes. Which is the dependent variable in the situation? A the number of prizes captured B the number of points earned C
my.hrw.com
Online Assessment and Intervention
5. Dwayne graphed the equation y = 10 + x. Which point does the graph not pass through? A (0, 10)
C
B (3, 13)
D (5, 15)
(8, 2)
6. Amy gets paid by the hour. Her little sister helps. As shown below, Amy gives her sister part of her earnings. Which equation represents Amy’s pay when her sister’s pay is $13? Amy’s pay in dollars
10
20
30
40
Sister’s pay in dollars
2
4
6
8
13 A y = __ 5 x B 13 = __ 5
C
5y = 13
D 13 = 5x
Gridded Response 7. Betty earns $7.50 per hour at a parttime job. Let x be the number of hours and y be the amount she earns. Betty makes a graph to show how x and y are related. If she earns $60, how many hours did she work?
8
.
0
0
0
0
0
0
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
4
4
4
4
4
4
5
5
5
5
5
5
6
6
6
6
6
6
7
7
7
7
7
7
8
8
8
8
8
8
9
9
9
9
9
9
© Houghton Mifflin Harcourt Publishing Company
Texas Test Prep
the number of hours
D the number of prizes available
406
Unit 4
Texas Essential Knowledge and Skills Items
Grade 6 TEKS
Mathematical Process TEKS
1
6.11
6.1.E
2
6.11
6.1.F
3
6.6.C
6.1.A
4
6.6.A
6.1.A
5
6.11
6.1.F
6
6.6.B
6.1.A, 6.1.E
7*
6.2.E, 6.6.C
6.1.A, 6.1.F
* Item integrates mixed review concepts from previous modules or a previous course.
Relationships in Two Variables
406
UNIT 4
Expressions, Equations, and Relationships
Additional Resources Personal Math Trainer my.hrw.com
Online Assessment and Intervention
Assessment Resources • Leveled Unit Tests: A, B, C, D • Performance Assessment
Study Guide Review Vocabulary Development Integrating the ELPS Encourage English learners to refer to their notes and the illustrated, bilingual glossary as they review the unit content. c.4.E Read linguistically accommodated content area material with a decreasing need for linguistic accommodations as more English is learned.
MODULE 10 Generating Equivalent Numerical Expressions 6.7.A
Key Concepts • A power is a number that is formed by repeated multiplication by the same factor. An exponent and a base can be used to write a power. (Lesson 10.1) • Factors are whole numbers that are multiplied to find a product. (Lesson 10.2) • To simplify an expression with more than one operation, there is a specific order in which to apply the operations. (Lesson 10.3)
MODULE 11 Generating Equivalent Algebraic Expressions 6.7.C, 6.7.D
Key Concepts • An algebraic expression is an expression that contains one or more variables and may also contain operation symbols, such as + or . A variable is a letter or symbol used to represent an unknown number. (Lesson 11.1) • To evaluate an expression, substitute a number for the variables and find the value of the expression. (Lesson 11.2) • To generate equivalent expressions, use the properties of operations to combine like terms. (Lesson 11.3)
407
Unit 4
DO NOT EDITChanges must be made through "File info" CorrectionKey=A
Study Guide MODULE MODULE
?
10
Review
Find the value of each power. (Lesson 10.1)
Generating Equivalent Numerical Expressions
base (base (en numeración)) exponent (exponente)
7. 75
power (potencia)
EXAMPLE 1 Find the value of each power.
0.9 = 0.9 × 0.9 = 0.81
MODULE MODULE
4 C. (_41 )
( ) ( )( )( )( )
Any number raised to the power of 0 is 1.
2
1 4 1 _ 1 _ 1 _ 1 1 _ _ ___ 4 = 4 4 4 4 = 256
?
180 = 1
EXAMPLE 2 Find the prime factorization of 60.
© Houghton Mifflin Harcourt Publishing Company
B. 27 ÷ 32 × 6
= 4 × 13 = 52
= 27 ÷ 9 × 6
32 = 9
Add.
=3×6
Divide.
Multiply.
= 18
Multiply.
3.62
2. 9 × 9 × 9 × 9
8 ___ 343
(_27 )
3
11
23 × 3 × 7
9. 168
22  (3 + 4) 12. __________ 12 ÷ 4 2
Generating Equivalent Algebraic Expressions
How can you generate equivalent algebraic expressions and use them to solve realworld problems?
21; 6, 14; 7, 12 3 Key Vocabulary algebraic expression (expresión algebraica) coefficients (coeficiente) constant (constante) equivalent expressions (expresiónes equivalente) evaluating (evaluar) term (término (en una expresión))
B. w  y 2 + 3w; w = 2, y = 6
2(52  9)
52 = 25
2  62 + 3(2)
62 = 36
= 2(16)
Subtract.
= 2  36 + 6
Multiply.
= 32
Multiply.
= 28
Add and subtract from left to right.
When w = 2 and y = 6, w  y 2 + 3w = 28.
EXAMPLE 2 Determine whether the algebraic expressions are equivalent: 5(x + 2) and 10 + 5x. 5(x + 2) = 5x + 10
= 10 + 5x
Distributive Property Commutative Property
5(x + 2) is equal to 10 + 5x. They are equivalent expressions.
EXERCISES Use exponents to write each expression. (Lesson 10.1) 1. 3.6 × 3.6
45
When x = 5, 2(x2 – 9) = 32.
EXAMPLE 3 Simplify each expression.
23 = 8
29
ESSENTIAL QUESTION
The prime factorization of 60 is 22 × 3 × 5.
= 4 × (8 + 5)
8. 29
A. 2(x2  9); x = 5
60 = 22 × 3 × 5
A. 4 × (23 + 5)
6.
EXAMPLE 1 Evaluate each expression for the given value of the variable.
60 = 2 × 2 × 3 × 5
2 60 2 30 3 15 5 5 1
52 × 3
11. 2 × 52 – (4 + 1)
B. 180
169
10. Eduardo is building a sandbox that has an area of 84 square feet. What are the possible whole number measurements for the length and width of the sandbox? (Lesson 10.2) 1, 84; 2, 42; 3, 28; 4,
order of operations (orden de las operaciones)
How can you generate equivalent numerical expressions and use them to solve realworld problems?
5. 132
Write the prime factorization of each number. (Lesson 10.2)
Key Vocabulary
ESSENTIAL QUESTION
A. 0.92
1
4. 120
© Houghton Mifflin Harcourt Publishing Company
UNIT 4
DO NOT EDITChanges must be made through “File info” CorrectionKey=B
(_45 )
3
94
3. _45 × _45 × _54
EXERCISES Write each phrase as an algebraic expression. (Lesson 11.1) 1. x subtracted from 15
Unit 4
6_MTXESE051676_U4EM.indd 407
407
29/11/12 12:11 PM
InCopy Notes
InDesign Notes
1. This is a list
1. This is a list
408
15  x
2. 12 divided by t
12 __ t
Unit 4
6_MTXESE051676_U4EM.indd 408
InCopy Notes 1. This is a list Bold, Italic, Strickthrough.
29/01/14 2:29 AM
InDesign Notes 1. This is a list
Expressions, Equations, and Relationships
408
MODULE 12 Equations and Relationships 6.7.B, 6.9.A, 6.9.B, 6.9.C, 6.10.A, 6.10.B
Key Concepts • An equation is a mathematical statement that two expressions are equal, which, if it includes a variable, has a solution. (Lesson 12.1) • Both sides of an equation remain equal after adding, or subtracting, the same number from both sides. (Lesson 12.2) • Both sides of an equation remain equal after multiplying, or dividing, both sides by the same number. (Lesson 12.3)
409
Unit 4
DO NOT EDITChanges must be made through “File info” CorrectionKey=B
DO NOT EDITChanges must be made through "File info" CorrectionKey=A
EXAMPLE 2
Write a phrase for each algebraic expression. (Lesson 11.1)
Solve each equation. Check your answer.
4. s + 7
67
7. s  5t + s2; s = 4, t = 1
Add 12 to both sides. ? Check: 22  12 = 10 Substitute.
33
(Lesson 12.1)
40 in2
1. 7x = 14; x = 3
13. 7x + 4(2x  6)
© Houghton Mifflin Harcourt Publishing Company
MODULE MODULE
?
12
yes
2. y + 13 = 4; y = 17
3. Don has three times as much money as his brother,
equivalent
d __ = 25 3
who has $25.
4. There are s students enrolled in Mr. Rodriguez’s class. There are 6 students absent and 18 students present
m2  2m  5
today.
15x  24
s  6 = 18
Solve each equation. Check your answer. (Lessons 12.2, 12.3)
Equations and Relationships
Key Vocabulary equation (ecuación) solution (solución)
ESSENTIAL QUESTION
12 is not a solution of r  5 = 17.
42 is a solution of _6x = 7.
InDesign Notes
1. This is a list
1. This is a list
q = 2.1
8. 3.5 + x = 7
x = 49
10. _27 = 2x
x = 3.5
x = _17
x – 12.50 = 34.25; $46.75
equation to solve the problem. (Lesson 12.3)
409
29/01/14 2:31 AM
InCopy Notes
7. 9q = 18.9
t = 48
12. Tom read 132 pages in 4 days. He read the same number of pages each day. How many pages did he read each day? Write and solve an
Unit 4
6_MTXESE051676_U4EM.indd 409
6. _4t = 12
the problem. (Lesson 12.2)
B. _6x = 7; x = 42 42 ? ____ = 7 Substitute. 6 7 = 7
p = 23
11. Sonia used $12.50 to buy a new journal. She has $34.25 left in her savings account. How much money did Sonia have before she bought the journal? Write and solve an equation to solve
EXAMPLE 1 Determine if the given value is a solution of the equation.
7 ≠ 17
5. p  5 = 18
9. 18 = x  31
How can you use equations and relationships to solve realworld problems?
A. r  5 = 17; r = 12 ? 12  5 = 17 Substitute.
no
Write an equation to represent the situation. (Lesson 12.1)
not equivalent
Combine like terms. (Lesson 11.3) 12. 3m  6 + m2  5m + 1
30 = 30
EXERCISES Determine whether the given value is a solution of the equation.
Determine if the expressions are equivalent. (Lesson 11.3)
11. 2.5(3 + x); 2.5x + 7.5
Divide both sides by 5. ? Check: 5(6) = 30 Substitute.
10 = 10
34
8. x  y3; x = 7, y = 3
9. The expression _12 (h)(b1 + b2) gives the area of a trapezoid, with b1 and b2 representing the two base lengths of a trapezoid and h representing the height. Find the area of a trapezoid with base lengths 4 in. and 6 in. and a height of 8 in. (Lesson 11.2)
10. 7 + 7x; 7(x + _17 )
p = 6
y = 22
6. 3(7 + x2); x = 2
25
5p 30 __ = ____ 5 5
+12 = +12
Evaluate each expression for the given value of the variable. (Lesson 11.2) 5. 8z + 3; z = 8
B. 5p = 30
A. y  12 = 10
the sum of s and 7
410
© Houghton Mifflin Harcourt Publishing Company
3. 8p
the product of 8 and p
4p = 132; 33 pages
Unit 4
6_MTXESE051676_U4EM.indd 410
InCopy Notes 1. This is a list Bold, Italic, Strickthrough.
29/11/12 12:11 PM
InDesign Notes 1. This is a list
Expressions, Equations, and Relationships
410
MODULE 13 Inequalities and Relationships 6.9.A, 6.9.B, 6.9.C, 6.10.A, 6.10.B
Key Concepts • An inequality is a mathematical statement that uses one of the following inequality symbols: greater than, >, less than, <, greater than or equal to, ≥, or less than or equal to, ≤. (Lesson 13.1) • All inequalities have many solutions. (Lesson 13.2) • Reverse the inequality symbol when multiplying or dividing both sides of an inequality by a negative number. (Lesson 13.4)
MODULE 14 Relationships in Two Variables 6.6.A, 6.6.B, 6.6.C, 6.11
Key Concepts • An ordered pair is a pair of numbers in the form (x, y) that gives the location of a point on a coordinate plane. (Lesson 14.1) • The quantity that depends on the other quantity is called the dependent variable, and the quantity it depends on is called the independent variable. (Lesson 14.2) • Tables and graphs can be used to represent the relationship between an independent and dependent variable. (Lesson 14.4)
411
Unit 4
?
13
Inequalities and Relationships
9.
Key Vocabulary solution of an inequality (solución de una desigualdad)
ESSENTIAL QUESTION
Sample answer: Juan’s dog lost 3 pounds and still weighs at least 11 pounds.
How can you use inequalities and relationships to solve realworld problems?
10. Omar wants a rectangular vegetable garden. He only has enough space to make the garden 5 feet wide, and he wants the area of the garden to be more than 80 square feet. Write and solve an inequality to find the possible lengths of the garden. (Lesson 13.3)
EXAMPLE 1 Write and graph an inequality to represent each situation. A. There are at least 5 gallons of water in an aquarium.
5ℓ > 80, ℓ > 16
B. The temperature today will be less than 35 °F.
g≥5
MODULE
t < 35
?
30 31 32 33 34 35 36 37 38 39 40
0 1 2 3 4 5 6 7 8 9 10
Solve each inequality. Graph and check your solutions. A. x  7 ≤ 2
B. 5y < 15 y>3
Add 7 to both sides.
Divide by 5. Reverse the symbol.
© Houghton Mifflin Harcourt Publishing Company
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
s < 2.5
2. Tina got a haircut, and her hair is still at least 15 inches long.
h ≥ 15
5. 9q > 10.8 7. _45 x < 8
q > 15
4. _4t ≤ 1
t ≤ 4
q > 1.2
6. 87 ≤ 25 + x
x ≥ 62
x > 10
8. 4 ≥ 0.5x
x≥8
origin (origen) quadrants (cuadrante) xaxis (eje x)
(4, 2) is in quadrant IV.
Quadrant I x
EXAMPLE 2 Tim is paid $8 more than the number of bags of peanuts he sells at the baseball stadium. The table shows the relationship between the money Tim earns and the number of bags of peanuts Tim sells. Identify the independent and dependent variables, and write an equation that represents the relationship.
10 11 12 13 14 15 16 17 18 19 20
Solve each inequality. Graph and check your solutions. (Lessons 13.2, 13.3, 13.4) 3. q  12 > 3
coordinates (coordenada) ordered pair (par ordenado)
̵5 ̵ 4 ̵3 ̵2 ̵1 O 1 2 3 4 5 ̵1 ̵2 (4, ̵2) ̵3 Quadrant III Quadrant IV ̵4 ̵5
(Lesson 13.1)
$2.50 per share.
axes (ejes)
(4, 2) is located 4 units to the right of the origin and 2 units down from the origin.
y 5 4 Quadrant II 3 2 1
Key Vocabulary coordinate plane (plano cartesiano)
EXAMPLE 1 Graph the point (4, 2) and identify the quadrant where it is located.
EXERCISES Write and graph an inequality to represent each situation.
1. Orange Tech’s stock is worth less than
Relationships in Two Variables
How can you use relationships in two variables to solve realworld problems?
0 1 2 3 4 5 6 7 8 9 10
5 6 7 8 9 10 11 12 13 14 15
14
ESSENTIAL QUESTION
EXAMPLE 2
x≤9
Write a realworld comparison that can be described by x  3 ≥ 11. (Lesson 13.2)
# of bags of peanuts, x
0
1
2
Money earned, y
8
9
10 11
3
© Houghton Mifflin Harcourt Publishing Company
MODULE
The number of bags is the independent variable, and the money Tim earns is the dependent variable. The equation y = x + 8 expresses the relationship between the number of bags Tim sells and the amount he earns.
Unit 4
411
412
Unit 4
Expressions, Equations, and Relationships
412
Unit 4 Performance Tasks The Performance Tasks provide students with the opportunity to apply concepts from this unit in realworld problem situations.
CAREERS IN MATH For more information about careers in mathematics as well as various mathematics appreciation topics, visit the American Mathematical Society at www.ams.org
CAREERS IN MATH Botanist In Performance Task Item 1, students can see how a botanist uses mathematics on the job.
SCORING GUIDES FOR PERFORMANCE TASKS 1. MATHEMATICAL PROCESSES Task
Possible Points (Total: 6)
a
1 point for correctly writing the expression 205 + 2d.
b
1 point for correctly setting the expression 205 + 2d = 235. 1 point for correct answer: 15 days
c
1 point for correct expression for Suntracker: 195 +2.5d. 1 point for determining the heights of each sunflower variety after 22 days: Suntracker: h = 195 + 2.5(22) = 250 Sunny Yellow: h = 205 + 2(22) = 249 1 point for stating that Suntracker is taller.
2. MATHEMATICAL PROCESSES Task
413
Unit 4
6.1.A, 6.1.F
6.1.A
Possible Points (Total: 6)
a
1 point for correctly defining a variable: Let w = the number of hours Vernon practiced soccer over the weekend. 1 point for the correct equation: 4__13 + w = 5__34 .
b
1 point for correctly finding the LCM, 12, and showing how to find it. Students can use a number line, a list of multiples, or prime factorization to find the LCM. 4 = 2 × 2; 3 = 3; LCM = 2 × 2 × 3 = 12
c
5 1 point for correctly solving the equation: w = 1__ . 12 1 point for showing steppedout solution to equation. 1 point for correctly interpreting the equation in terms of the problem: Vernon 5 practiced 1__ hours over the weekend. 12
DO NOT EDITChanges must be made through "File info" CorrectionKey=A
DO NOT EDITChanges must be made through “File info” CorrectionKey=B
Unit 4 Performance Tasks
EXERCISES Graph and label each point on the coordinate plane. (Lesson 14.1)
1.
y
1. (4, 4) 2. (3, 1)
(1, 4)
4
(4, 4)
2
3. (1, 4)
x ̵4
̵2
(3, 1)
O
2
CAREERS IN MATH Botanist Dr. Adama is a botanist. She measures the daily height of a particular variety of sunflower, Sunny Yellow, beginning when the sunflower is 60 days old. At 60 days, the height of the sunflower is 205 centimeters. Dr. Adama finds that the growth rate of this sunflower is 2 centimeters per day after the first 60 days.
a. Write an expression to represent the sunflower’s height d days after the 60th day.
4
̵2
205 + 2d
̵4
b. How many days after the 60th day does it take for the sunflower to reach 235 centimeters? Show your work.
235 = 205 + 2d; 30 = 2d; 15 = d
Use the graph to answer the questions. (Lesson 14.2)
It takes 15 days for the sunflower to reach 235 centimeters. c. Dr. Adama is studying a different variety of sunflower, Suntracker, which grows at a rate of 2.5 centimeters per day after the first 60 days. If this sunflower is 195 centimeters tall when it is 60 days old, write an expression to represent Suntracker’s height d days after the 60th day. Which sunflower will be taller 22 days after the 60th day? Explain how you found your answer.
8 6 4
Suntracker: 195 + 2.5d; after 22 days: Suntracker: h = 195 + 2.5(22)
2 O
2
4
6
8
Time (h)
= 250; Sunny Yellow: h = 205 + 2(22) = 249. Suntracker is taller.
10
2. Vernon practiced soccer 5_43 hours this week. He practiced 4_13 hours on weekdays and the rest over the weekend.
time
4. What is the independent variable?
a. Write an equation that represents the situation. Define your variable.
distance
5. What is the dependent variable?
Let w = the number of hours Vernon practiced soccer over the weekend; 4_13 + w = 5_34
© Houghton Mifflin Harcourt Publishing Company
6. Describe the relationship between the independent variable and the dependent variable.
b. What is the least common multiple of the denominators of 5_34 and 4_13 ? Show your work.
The dependent variable is 3 times the independent variable.
Using prime factorization: 4 = 2 × 2; 3 = 3; LCM = (2)(2)(3) = 12
7. Use the data on the table to write an equation to express y in terms of x. Then graph the equation. (Lessons 14.3, 14.4) x y
0
1
2 1
2
3
0
1
y= x2
c. Solve the equation and interpret the solution. Show your work.
5 9 9 4 4 4_13 + w = 5_34; 4__ + w = 5__ ; w = 5__  4__ = 1__ 12 12 12 12 12
y 4 2
(2, 0)
̵4
̵2
O ̵2 ̵4
2
5 hours over the weekend. Vernon practiced 1__ 12
(3, 1)
© Houghton Mifflin Harcourt Publishing Company
Distance (km)
10
x
4
(1, 1) (0, 2) Unit 4
6_MTXESE051676_U4EM.indd 413
413
29/11/12 12:12 PM
InCopy Notes
InDesign Notes
1. This is a list
1. This is a list
414
Unit 4
6_MTXESE051676_U4EM.indd 414
InCopy Notes 1. This is a list Bold, Italic, Strickthrough.
1/29/14 11:19 PM
InDesign Notes 1. This is a list
Expressions, Equations, and Relationships
414
UNIT 4
Expressions, Equations, and Relationships
Additional Resources Personal Math Trainer my.hrw.com
Online Assessment and Intervention
Assessment Resources • Leveled Unit Tests: A, B, C, D • Performance Assessment
MIXED REVIEW
Texas Test Prep Texas Testing Tip Students should always read each question carefully to identify key words or phrases, such as no more than, to help identify what the question is really asking. Item 5 Students should underline the phrase no more than; it will help them to realize that n cannot be larger than 7, but it can be equal to 7. This will help them choose the correct inequality. Item 7 Students who do not read the problem carefully may choose B because they see the word times. But reading carefully they will see the phrase “3 more times than,” which will reveal choice D as the correct answer.
Avoid Common Errors Item 8 Some students may fail to pay attention to the direction the graph is pointing. As a result, they may choose answer choice A or D, because they both include the number 4. 4 in the solution. Remind the students that the direction of the inequality is as important as its endpoint. Item 13 Some students may give 2 as their answer because they transposed the x and ycoordinates. Remind students that the ycoordinate is the second number in an ordered pair.
Texas Essential Knowledge and Skills Items
Grade 6 TEKS
Mathematical Process TEKS
1
6.7.A
6.1.D
2
6.7.A
6.1.F
3
6.7.C
6.1.A
4
6.9.B, 6.10.A
6.1.D
5
6.9.A
6.1.A, 6.1.F
6
6.10.B
6.1.F
7
6.9.A
6.1.A, 6.1.F
8
6.9.B, 6.10.A
6.1.D
9
6.6.A
6.1.A
10*
6.4.C, 6.4.E
6.1.F
11*
6.3.E
6.1.F
12
6.10.A
6.1.A
13
6.11
6.1.E
14
6.7.A, 6.7.C
6.1.A
* Item integrates mixed review concepts from previous modules or a previous course.
415
Unit 4
DO NOT EDITChanges must be made through “File info” CorrectionKey=B
DO NOT EDITChanges must be made through “File info” CorrectionKey=B
Personal Math Trainer
Texas Test Prep Selected Response 1. Which expression is equivalent to 2.3 × 2.3 × 2.3 × 2.3 × 2.3? A 2.3 × 5
2. Which operation should you perform first when you simplify 63 – (2 + 54 × 6) ÷ 5? A addition
3. Sheena was organizing items in a scrapbook. She took 25 photos and divided them evenly among p pages. Which algebraic expression represents the number of photos on each page? A p – 25
© Houghton Mifflin Harcourt Publishing Company
B 25 – p p __ C 25 25 D __ p
D 4 – 6 = –2
6. For which of the inequalities below is v = 4 a solution? A v+5≥9
̵4
5
6 C __ 15 18 __ D 45
A 1.8 centimeters B 11.4 centimeters C 13.7 centimeters D 114 centimeters
A j–9=3
Gridded Response
C j–3=9
12. The area of a rectangular mural is 84 square feet. The mural’s width is 7 feet. What is its length in feet?
B 3j = 9
5
10
m A __ > 1.1 4 m B __ < 1.2 3 C 2m < 8.8 D 5m > 22
Hot ! Tip
When possible, use logic to eliminate at least two answer choices.
Unit 4
6_MTXESE051676_U4EM.indd 415
O ̵2
11. One inch is 2.54 centimeters. About how many centimeters is 4.5 inches?
7. Sarah has read aloud in class 3 more times than Joel. Sarah has read 9 times. Which equation represents this situation?
0
̵2
D the number of stars available
2 _ A 5 __ B 12 25
D j+3=9
0
x ̵4
C the number of hours played
8. The number line below represents the solution to which inequality?
4. The number line below represents which equation?
G 2
10. Which ratio is not equivalent to the other three?
D v+5<8
D subtraction
C 4 + 6 = –2
A n<7
C v+5≤8
C multiplication
y 4
B the number of points earned
B v+5>9
B division
13. What is the ycoordinate of point G on the coordinate grid below?
A the number of stars picked up
D n≥7
D 2.35
B –2 – 6 = 4
5. No more than 7 copies of a newspaper are left in the newspaper rack. Which inequality represents this situation?
C n>7
C 25 × 35
A –2 + 6 = 4
9. Brian is playing a video game. He earns the same number of points for each star he picks up. He earned 2,400 points for 6 stars, 4,000 points for 10 stars, and 5,200 points for 13 stars. Which is the independent variable in the situation?
Online Assessment and Intervention
B n≤7
B 235
5
my.hrw.com
415
416
1
2
.
2
4
3
.
0
0
0
0
0
0
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
4
4
4
4
4
4
5
5
5
5
5
5
6
6
6
6
6
6
7
7
7
7
7
7
8
8
8
8
8
8
9
9
9
9
9
9
Gridded responses cannot be negative numbers. If you get a negative value, you likely made an error. Check your work!
Hot ! Tip
14. When traveling in Canada, Patricia converts the temperature given in degrees Celsius to a Fahrenheit temperature by using the expression 9x ÷ 5 + 32, where x is the Celsius temperature. Find the temperature in degrees Fahrenheit when it is 25 °C.
0
0
0
0
0
0
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
4
4
4
4
4
4
5
5
5
5
5
5
7
7
6
6
6
6
6
6
0
0
0
0
0
0
7
7
7
7
7
7
1
1
1
1
1
1
8
8
8
8
8
8
2
2
2
2
2
2
9
9
9
9
9
9
3
3
3
3
3
3
4
4
4
4
4
4
5
5
5
5
5
5
6
6
6
6
6
6
7
7
7
7
7
7
8
8
8
8
8
8
9
9
9
9
9
9
Unit 4
.
29/01/14 2:44 AM 6_MTXESE051676_U4EM.indd 416
InCopy Notes
InDesign Notes
1. This is a list
1. This is a list
InCopy Notes 1. This is a list Bold, Italic, Strickthrough.
© Houghton Mifflin Harcourt Publishing Company
Unit 4 MiXed ReVieW
29/01/14 2:46 AM
InDesign Notes 1. This is a list
Expressions, Equations, and Relationships
416