UNIT 1
The Number System
MODULE
1
Adding and Subtracting Integers 7.NS.1, 7.NS.1b, 7.NS.1c, 7.NS.1d, 7.NS.3, 7.EE.3
2 Multiplying and
MODULE MODULE
Dividing Integers 7.NS.2, 7.NS.2a, 7.NS.2b, 7.NS.2c, 7.NS.3, 7.EE.3
1 3 Rational Numbers
MODULE MODULE
7.NS.1, 7.NS.1a, 7.NS.1b, 7.NS.1c, 7.NS.1d, 7.NS.2, 7.NS.2a, 7.NS.2b, 7.NS.2c, 7.NS.2d, 7.NS.3, 7.EE.3
MATH IN CAREERS
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Urban Planner An urban planner creates plans for urban, suburban, and rural communities and makes recommendations about locations for infrastructure, such as buildings, roads, and sewer and water pipes. Urban planners perform cost-benefit analysis of projects, use measurement and geometry when they design the layout of infrastructure, and use statistics and mathematical models to predict the growth and future needs of a population. If you are interested in a career as an urban planner, you should study these mathematical subjects: • Algebra • Trigonometry • Geometry • Statistics Research other careers that require using measurement, geometry, and mathematical modeling. ACTIVITY At the end of the unit, check out how urban planners use math. Unit 1
1
Unit
Project
Preview
It’s Okay to Be Negative! In the Unit Project at the end of this unit you will do research to find real-world examples of negative rational numbers. Then you will write four problems involving the numbers you find. To successfully complete the Unit Project you’ll need to master these skills: • • • •
Add rational numbers. Subtract rational numbers. Multiply rational numbers. Divide rational numbers.
1. Name some real-world situations that could involve negative numbers.
Tracking Your Learning Progression This unit addresses important California Common Core Standards in the Critical Areas of applying understanding of operations to rational numbers and solving problems involving algebraic expressions. Domain 7.NS The Number System Cluster Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. The unit also supports additional standards. Domain 7.EE Expressions and Equations Cluster Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
2
Unit 1 Preview
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2. Write (but do not solve) a problem involving these facts: The low temperatures for four consecutive days were -15 °F, -12 °F, -17 °F, and -9 °C.
Adding and Subtracting Integers ?
MODULE
1
LESSON 1.1
ESSENTIAL QUESTION
Adding Integers with the Same Sign
How can you use addition and subtraction of integers to solve real-world problems?
7.NS.1, 7.NS.1b, 7.NS.1d
LESSON 1.2
Adding Integers with Different Signs 7.NS.1, 7.NS.1b
LESSON 1.3
Subtracting Integers 7.NS.1, 7.NS.1c
LESSON 1.4
Applying Addition and Subtraction of Integers © Houghton Mifflin Harcourt Publishing Company • Image Credits: © Peter Haigh/Digital Vioion/Getty Images
7.NS.1, 7.NS.1d, 7.NS.3, 7.EE.3
Real-World Video
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Death Valley contains the lowest point in North America, elevation –282 feet. The top of Mt. McKinley, elevation 20,320 feet, is the highest point in North America. To find the difference between these elevations, you can subtract integers.
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Math On the Spot
Animated Math
Personal Math Trainer
Go digital with your write-in 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.
3
Are YOU Ready? Personal Math Trainer
Complete these exercises to review skills you will need for this module.
Understand Integers EXAMPLE
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Online Practice and Help
Decide whether the integer is positive or negative: descended → negative Write the integer.
A diver descended 20 meters. -20
Write an integer to represent each situation.
1. an elevator ride down 27 stories
2. a $700 profit
3. 46 degrees below zero
4. a gain of 12 yards
Whole Number Operations EXAMPLE
3 15
245 - 28
24 5 28 __ 217
245 - 28 = 217
Think: 8>5 Regroup 1 ten as 10 ones. 1 ten + 5 ones = 15 ones Subtract: 15 - 8 = 7
Find the sum or difference.
6.
183 + 78 _
7.
677 -288 _
8.
1,188 + 902 __
2,647 -1,885 __
Locate Points on a Number Line EXAMPLE
-5
0
5
Graph +2 by starting at 0 and counting 2 units to the right. Graph -5 by starting at 0 and counting 5 units to the left.
Graph each number on the number line.
9.
10. -4
7 -10
4
Unit 1
-5
11. -9 0
12. 4 5
10
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5.
Reading Start-Up Visualize Vocabulary Use the ✔ words to fill in the ovals on the graphic. You may put more than one word in each oval.
Understanding Integers
Vocabulary Review Words difference (diferencia) integers (enteros) ✔ negative number (número negativo) ✔ opposites (opuestos) ✔ positive number (número positivo) sum (suma) ✔ whole number (número entero)
Preview Words
-50, 50 50
-50
absolute value (valor absoluto) additive inverse (inverso aditivo) expression (expresión) model (modelo)
Understand Vocabulary Complete the sentences using the preview words.
1. The
of a number gives its distance from zero.
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2. The sum of a number and its
is zero.
Active Reading 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 processes. Refer to your finished booklet as you work on assignments and study for tests.
Module 1
5
GETTING READY FOR
Adding and Subtracting Integers Understanding the standards and the vocabulary terms in the standards will help you know exactly what you are expected to learn in this module.
7.NS.1 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.
Key Vocabulary additive inverse (inverso aditivo) The opposite of a number.
What It Means to You You will learn how to use models to add and subtract integers with the same sign and with different signs. EXAMPLE 7.NS.1
You will learn how to use models to add and subtract integers with the same sign and with different signs. 4 + (-7)
+(-7) 4
-5 -4 -3 -2 -1
0 1 2 3 4 5
Start at 0. Move right 4 units. Then move left 7 units.
4 + (-7) = -3
Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.
Key Vocabulary integer (entero) A member of the set of whole numbers and their opposites.
What It Means to You You will learn that subtracting an integer is the same as adding its additive inverse. EXAMPLE 7.NS.1c
Find the difference between 3,000 °F and -250 °F, the temperatures the space shuttle must endure. 3,000 - (-250) 3,000 + 250 = 3,250 The difference in temperatures the shuttle must endure is 3,250 °F.
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6
Unit 1
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7.NS.1c
LESSON
1.1 ?
Adding Integers with the Same Sign
7.NS.1 Apply and extend previous understandings of addition and subtraction to add... rational numbers; represent addition... on a…vertical number line diagram. Also
7.NS.1b, 7.NS.1d
ESSENTIAL QUESTION How do you add integers with the same sign?
EXPLORE ACTIVITY 1
7.NS.1
Modeling Sums of Integers with the Same Sign You can use colored counters to add positive integers and to add negative integers.
=1 = -1
Model with two-color counters.
A 3+4 3 positive counters 4 positive counters
total number of counters
How many counters are there in total? What is the sum and how do you find it?
B - 5 + (- 3)
Math Talk
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Mathematical Practices
5 negative counters 3 negative counters
total number of counters
What does the color of each row of counters represent?
How many counters are there in total? Since the counters are negative integers, what is the sum?
Reflect 1.
Communicate Mathematical Ideas When adding two numbers with the same sign, what sign do you use for the sum?
Lesson 1.1
7
EXPLORE ACTIVITY 2
7.NS.1, 7.NS.1b
Adding on a Number Line Just as you can add positive integers on a number line, you can add negative integers. The temperature was 2 °F below zero. The temperature drops by 5 °F. What is the temperature now?
8
8
6
6
4
4
B Mark the initial temperature on the number line.
2
2
C A drop in temperature of 5° is like adding -5° to the temperature.
0
0
-2
-2
-4
-4
-6
-6
-8
-8
A What is the initial temperature written as an integer?
Count on the number line to find the final temperature. Mark the temperature now on the number line.
D What is the temperature written as an integer?
The temperature is above / below
zero. Temperature (˚F)
8
2.
What If? Suppose the temperature is -1 °F and drops by 3 °F? Explain how to use the number line to find the new temperature.
3.
Communicate Mathematical Ideas How would using a number line to find the sum 2 + 5 be different from using a number line to find to find the sum - 2 + (- 5)?
4.
Analyze Relationships What are two other negative integers that have the same sum as - 2 and - 5?
Unit 1
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Reflect
Adding Integers with a Common Sign To add integers with the same sign, add the absolute values of the integers and use the sign of the integers for the sum. Math On the Spot
EXAMPL 1 EXAMPLE Add -7 + (-6). STEP 1
7.NS.1, 7.NS.1d
The signs of both integers are the same.
Find the absolute values. The absolute value is always | -7 | = 7 | -6 | = 6 positive or zero.
STEP 2
Find the sum of the absolute values: 7 + 6 = 13
STEP 3
Use the sign of the integers to write the sum. -7 + (-6) = -13
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The sign of each integer is negative.
Math Talk
Mathematical Practices
Can you use the same procedure you use to find the sum of two negative integers to find the sum of two positive numbers? Explain.
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Reflect 5.
Communicate Mathematical Ideas Does the Commutative Property of Addition apply when you add two negative integers? Explain.
6.
Critical Thinking Choose any two negative integers. Is the sum of the integers less than or greater than the value of either of the integers? Will this be true no matter which integers you choose? Explain.
YOUR TURN Find each sum. 7.
-8 + (-1) =
9.
-48 + (-12) =
10.
-32 + (-38) =
11.
109 + 191 =
12.
-40 + (-105) =
13.
-150 + (-1500) =
14.
-200 + (-800) =
8.
-3 + (-7) =
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Lesson 1.1
9
Guided Practice Find each sum. (Explore Activity 1) 1. -5 + (-1)
2. -2 + (-7)
a. How many counters are there?
a. How many counters are there?
b. Do the counters represent positive or
b. Do the counters represent positive or
negative numbers?
negative numbers?
c. -5 + (-1) =
c. -2 + (-7) =
Model each addition problem on the number line to find each sum. (Explore Activity 2) 3. -5 + (-2) =
4. -1 + (-3) =
-8 -7 -6 -5 -4 -3 -2 -1
0
5. -3 + (-7) =
0 1 2 3
6. -4 + (-1) =
- 10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2
7. -2 + (-2) = -5 -4 -3 -2 -1
-5 -4 -3 -2 -1
-5 -4 -3 -2 -1
0 1 2 3
8. -6 + (-8) = 0 1 2 3
- 16
- 12
-8
-4
0
Find each sum. (Example 1) 10. -1 + (-10) =
11. -9 + (-1) =
12. -90 + (-20) =
13. -52 + (-48) =
14. 5 + 198 =
15. -4 + (-5) + (-6) =
16. -50 + (-175) + (-345) =
?
ESSENTIAL QUESTION CHECK-IN
17. How do you add integers with the same sign?
10
Unit 1
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9. -5 + (-4) =
Name
Class
Date
1.1 Independent Practice
Personal Math Trainer
7.NS.1, 7.NS.1b, 7.NS.1d
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18. Represent Real-World Problems Jane and Sarah both dive down from the surface of a pool. Jane first dives down 5 feet, and then dives down 3 more feet. Sarah first dives down 3 feet, and then dives down 5 more feet. a. Multiple Representations Use the number line to model the equation -5 + (-3) = -3 + (-5). 2 0
20. A football team loses 3 yards on one play and 6 yards on another play. Write a sum of negative integers to represent this situation. Find the sum and explain how it is related to the problem.
21. When the quarterback is sacked, the team loses yards. In one game, the quarterback was sacked four times. What was the total sack yardage? Game
-2
Sack yardage
-4
Online Practice and Help
1
2
3
4
-14
-5
-12
-23
-6 -8
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b. Does the order in which you add two integers with the same sign affect the sum? Explain.
a. Write an equation that models the change to the temperature in Jonestown since 1:00.
19. A golfer has the following scores for a 4-day tournament. Day Score
22. Multistep The temperature in Jonestown and Cooperville was the same at 1:00. By 2:00, the temperature in Jonestown dropped 10 degrees, and the temperature in Cooperville dropped 6 degrees. By 3:00, the temperature in Jonestown dropped 8 more degrees, and the temperature in Cooperville dropped 2 more degrees.
1
2
3
4
-3
-1
-5
-2
b. Write an equation that models the change to the temperature in Cooperville since 1:00.
What was the golfer’s total score for the tournament? c. Where was it colder at 3:00, in Jonestown or Cooperville?
Lesson 1.1
11
23. Represent Real-World Problems Julio is playing a trivia game. On his first turn, he lost 100 points. On his second turn, he lost 75 points. On his third turn, he lost 85 points. Write a sum of three negative integers that models the change to Julio’s score after his first three turns.
FOCUS ON HIGHER ORDER THINKING
Work Area
24. Multistep On Monday, Jan made withdrawals of $25, $45, and $75 from her savings account. On the same day, her twin sister Julie made withdrawals of $35, $55, and $65 from her savings account. a. Write a sum of negative integers to show Jan’s withdrawals on Monday. Find the total amount Jan withdrew.
b. Write a sum of negative integers to show Julie’s withdrawals on Monday. Find the total amount Julie withdrew.
c. Julie and Jan’s brother also withdrew money from his savings account on Monday. He made three withdrawals and withdrew $10 more than Julie did. What are three possible amounts he could have withdrawn?
25. Communicate Mathematical Ideas Why might you want to use the Commutative Property to change the order of the integers in the following sum before adding?
26. Critique Reasoning The absolute value of the sum of two different integers with the same sign is 8. Pat says there are three pairs of integers that match this description. Do you agree? Explain.
12
Unit 1
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-80 + (-173) + (-20)
LESSON
1.2 ?
Adding Integers with Different Signs
7.NS.1 Apply and extend previous understandings of addition... to add... rational numbers; represent addition... on a horizontal... number line diagram. Also 7.NS.1b
ESSENTIAL QUESTION How do you add integers with different signs?
EXPLORE ACTIVITY 1
7.NS.1, 7.NS.1b
Adding on a Number Line To find the sum of integers with the same sign, such as 3 + 2, you can start at 3 and move | 2 | = 2 units in the positive direction.
3+2=5
-3 -2 -1
0 1 2 3 4 5
-3 -2 -1
0 1 2 3 4 5
The sum of 3 + 2 is the number that is |2| units from 3 in the positive direction.
To find the sum of integers with different signs, such as 3 + (-2), you can start at 3 and move | -2 | = 2 units in the negative direction.
3 + (−2) = 1
The sum of 3 + (-2) is the number that is |-2| units from 3 in the negative direction.
Model each sum on a number line.
A Model 4 + (-3). Start at 4. Move 3 units to the left, or in the negative direction.
0 1 2 3 4 5 6 7 8
4 + (-3) =
B Model -7 + 5. © Houghton Mifflin Harcourt Publishing Company
Start at or in the
. Move 5 units to the
,
-8 -7 -6 -5 -4 -3 -2 -1
0
direction. -7 + 5 =
C Model 6 + (-6). Start at the
. Move , or in the
units to
0 1 2 3 4 5 6 7 8
direction.
6 + (-6) =
Reflect 1.
Make a Prediction Predict the sum of -2 + 2. Explain your prediction and check it using the number line.
-5 -4 -3 -2 -1
0 1 2 3
Lesson 1.2
13
EXPLORE ACTIVITY 2
7.NS.1, 7.NS.1b
Modeling Sums of Integers with Different Signs You can use colored counters to model adding integers with different signs. When you add a positive integer (yellow counter) and a negative integer (red counter), the result is 0. One red and one yellow counter form a zero pair. 1 + (-1) = 0
Model and find each sum using counters. Part A is modeled for you. For Part B, follow the steps to model and find the sum using counters.
A Model 3 + (-2). The value of a zero pair is 0. Adding or subtracting 0 to any number does not change its value.
Start with 3 positive counters to represent 3. Add 2 negative counters to represent adding -2. Form zero pairs. What is left when you remove the zero pairs? counter Find the sum: 3 + (-2) =
B Model -6 + 3. Start with
counters to represent
Add
counters to represent adding
. .
Form zero pairs. What is left when you remove the zero pairs?
Find the sum: -6 + 3 =
Reflect 2.
14
Make a Prediction Kyle models a sum of two integers. He uses more negative (red) counters than positive (yellow) counters. What do you predict about the sign of the sum? Explain.
Unit 1
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counters
EXPLORE ACTIVITY 2 (cont’d)
Model and find each sum using counters. 3. 5 + (-1)
4. 4 + (-6)
5. 1 + (-7)
6. 3 + (-4)
Adding Integers You have learned how to add integers with the same signs and how to add integers with different signs. The table below summarizes the rules for adding integers. Adding Integers
Math On the Spot
Examples
Same signs
Add the absolute values of the integers. Use the common sign for the sum.
Different signs
Subtract the lesser absolute value 3 + (-5) = -2 from the greater absolute value. Use the sign of the integer with the -10 + 1 = -9 greater absolute value for the sum.
A number and its opposite
The sum is 0. The opposite of any number is called its additive inverse.
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3+5=8 -2 + (-7) = -9
4 + (-4) = 0 -11 + 11 = 0
EXAMPL 1 EXAMPLE
7.NS.1, 7.NS.1b
Find each sum.
A -11 + 6
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| -11 |
- |6| = 5
-11 + 6 = -5
B
( -37 )
+ 37
( -37 )
+ 37 = 0
Subtract the lesser absolute value from the greater. Use the sign of the number with the greater absolute value. The sum of a number and its opposite is 0.
Math Talk
Mathematical Practices
Give an example of two integers with different signs whose sum is a positive number. How did you choose the integers?
YOUR TURN Find each sum. 7. -51 + 23 = 9. 13 + (-13) =
8. 10 +
( -18 )
=
10. 25 + (-26) =
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Lesson 1.2
15
Guided Practice Use a number line to find each sum. (Explore Activity 1) 2. -2 + 7 =
1. 9 + (-3) =
2 3 4 5 6 7 8 9 10
3. -15 + 4 =
- 18
- 16
-3 -2 -1
0 1 2 3 4 5
4. 1 + (-4) =
- 14
- 12
- 10
-5 -4 -3 -2 -1
0 1 2 3
Circle the zero pairs in each model. Find the sum. (Explore Activity 2) 5. -4 + 5 =
6. -6 + 6 =
7. 2 + (-5) =
8. -3 + 7 =
Find each sum. (Example 1) 10. 7 + (-5) =
11. 5 + (-21) =
12. 14 + (-14) =
13. 0 + (-5) =
14. 32 + (-8) =
?
ESSENTIAL QUESTION CHECK-IN
15. Describe how to find the sums -4 + 2 and -4 + ( -2 ) on a number line.
16
Unit 1
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9. -8 + 14 =
Name
Class
Date
1.2 Independent Practice
Personal Math Trainer
7.NS.1, 7.NS.1b
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Online Practice and Help
Find each sum. 16. -15 + 71 =
17. -53 + 45 =
18. -79 + 79 =
19. -25 + 50 =
20. 18 + (-32) =
21. 5 + (-100) =
22. -12 + 8 + 7 =
23. -8 + (-2) + 3 =
24. 15 + (-15) + 200 =
25. -500 + (-600) + 1200 =
26. A football team gained 9 yards on one play and then lost 22 yards on the next. Write a sum of integers to find the overall change in field position. Explain your answer.
27. A soccer team is having a car wash. The team spent $55 on supplies. They earned $275, including tips. The team’s profit is the amount the team made after paying for supplies. Write a sum of integers that represents the team’s profit.
28. As shown in the illustration, Alexa had a negative balance in her checking account before depositing a $47.00 check. What is the new balance of Alexa’s checking account?
Accounts
Regular Checking
Sign Out
$$ Search transactions Available Balance
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29. The sum of two integers with different signs is 8. Give two possible integers that fit this description.
-$47.00
30. Multistep Bart and Sam played a game in which each player earns or loses points in each turn. A player’s total score after two turns is the sum of his points earned or lost. The player with the greater score after two turns wins. Bart earned 123 points and lost 180 points. Sam earned 185 points and lost 255 points. Which person won the game? Explain.
Lesson 1.2
17
FOCUS ON HIGHER ORDER THINKING
Work Area
31. Critical Thinking Explain how you could use a number line to show that -4 + 3 and 3 + (-4) have the same value. Which property of addition states that these sums are equivalent?
32. Represent Real-World Problems Jim is standing beside a pool. He drops a weight from 4 feet above the surface of the water in the pool. The weight travels a total distance of 12 feet down before landing on the bottom of the pool. Explain how you can write a sum of integers to find the depth of the water.
34. Analyze Relationships You know that the sum of -5 and another integer is a positive integer. What can you conclude about the sign of the other integer? What can you conclude about the value of the other integer? Explain.
18
Unit 1
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33. Communicate Mathematical Ideas Use counters to model two integers with different signs whose sum is positive. Explain how you know the sum is positive.
LESSON
1.3 Subtracting Integers ?
7.NS.1c Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). … Also 7.NS.1
ESSENTIAL QUESTION How do you subtract integers?
EXPLORE ACTIVITY 1
7.NS.1
Modeling Integer Subtraction You can use counters to find the difference of two integers. In some cases, you may need to add zero pairs. Model and find each difference using counters.
1 + (-1) = 0
A Model -4 - (-3). Start with 4 negative counters to represent -4. Take away 3 negative counters to represent subtracting -3. What is left? Find the difference: -4 - (-3) =
B Model 6 - (-3). Start with 6 positive counters to represent 6. You need to take away 3 negative counters, so add 3 zero pairs.
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Take away 3 negative counters to represent subtracting -3. What is left? Find the difference: 6 - (-3) =
C Model -2 - (-5). Start with
counters.
You need to take away Take away
counters, so add
zero pairs.
counters.
What is left? Find the difference: -2 - (-5) =
Lesson 1.3
19
EXPLORE ACTIVITY 1 (cont’d)
Reflect 1.
Communicate Mathematical Ideas Suppose you want to model the difference -4 - 7. Do you need to add zero pairs? If so, why? How many should you add? What is the difference?
EXPLORE ACTIVITY 2
7.NS.1, 7.NS.1c
Subtracting on a Number Line To model the difference 5 - 3 on a number line, you start at 5 and move 3 units to the left. Notice that you model the sum 5 + (-3) in the same way. Subtracting 3 is the same as adding its opposite, -3.
5 - 3 = 5 + (-3)
-1
0 1 2 3 4 5
You can use the fact that subtracting a number is the same as adding its opposite to find a difference of two integers. Find each difference on a number line.
A Find -1 - 5 on a number line. Rewrite subtraction as addition of the opposite.
Start at
and move
units to the left.
The difference is
-8 -7 -6 -5 -4 -3 -2 -1
0
-8 -7 -6 -5 -4 -3 -2 -1
0
B Find -7 - (-3). Rewrite subtraction as addition of the opposite. -7 - (-3) = -7 + Start at
and move
The difference is
20
Unit 1
units to the
.
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-1 - 5 = - 1 +
EXPLORE ACTIVITY 2 (cont’d)
Reflect 2.
Communicate Mathematical Ideas Describe how to find 5 - (-8) on a number line. If you found the difference using counters, would you get the same result? Explain.
Subtracting Integers by Adding the Opposite You can use the fact that subtracting an integer is the same as adding its opposite to solve problems.
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EXAMPL 1 EXAMPLE
7.NS.1c, 7.NS.1
The temperature on Monday was -5 °C. By Tuesday the temperature rose to -2 °C. Find the change in temperature. Animated Math
STEP 1
Write a subtraction expression.
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final temperature - Monday’s temperature = change in temperature
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-2 °C - (-5 °C) STEP 2
Math Talk
Find the difference. -2 - (-5) = -2 + 5 -2 + 5 = 3
Mathematical Practices
To subtract -5, add its opposite, 5. Use the rule for adding integers.
Why does it make sense that the change in temperature is a positive number?
The temperature increased by 3 °C.
Reflect 3.
What If? In Example 1, the temperature rose by 3 °C. Suppose it fell from -2 °C to -10 °C. Predict whether the change in temperature would be positive or negative. Then subtract to find the change.
Lesson 1.3
21
YOUR TURN Personal Math Trainer Online Practice and Help
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Find each difference. 4. -7 - 2 =
5. -1 - (-3) =
6. 3 - 5 =
7. -8 - (-4) =
Guided Practice Explain how to find each difference using counters. (Explore Activity 1) 1. 5 - 8 =
2. -5 - (-3) =
Use a number line to find each difference. (Explore Activity 2) 3. -4 - 5 = -4 +
=
-9 -8 -7 -6 -5 -4 -3 -2 -1
4. 1 - 4 = 1 + 0
-4 -3 -2 -1
= 0 1 2 3 4
5. 8 - 11 =
6. -3 - (-5) =
7. 15 - 21 =
8. -17 - 1 =
9. 0 - (-5) =
10. 1 - (-18) =
11. 15 - 1 =
12. -3 - (-45) =
13. 19 - (-19) =
14. -87 - (-87) =
?
ESSENTIAL QUESTION CHECK-IN
15. How do you subtract an integer from another integer without using a number line or counters? Give an example.
22
Unit 1
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Solve. (Example 1)
Name
Class
Date
1.3 Independent Practice
Personal Math Trainer
7.NS.1, 7.NS.1c
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16. Theo had a balance of -$4 in his savings account. After making a deposit, he has $25 in his account. What is the overall change to his account?
17. As shown, Suzi starts her hike at an elevation below sea level. When she reaches the end of the hike, she is still below sea level at -127 feet. What was the change in elevation from the beginning of Suzi’s hike to the end of the hike? Current Elevation: –225 feet
Online Practice and Help
20. A scientist conducts three experiments in which she records the temperature of some gases that are being heated. The table shows the initial temperature and the final temperature for each gas. Gas
Initial Final Temperature Temperature
A
-21 °C
-8 °C
B
-12 °C
12 °C
C
-19 °C
-15 °C
a. Write a difference of integers to find the overall temperature change for each gas. Gas A:
Gas B:
Gas C:
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18. The record high January temperature in Austin, Texas, is 90 °F. The record low January temperature is -2 °F. Find the difference between the high and low temperatures.
b. What If? Suppose the scientist performs an experiment in which she cools the three gases. Will the changes in temperature be positive or negative for this experiment? Why?
19. Cheyenne is playing a board game. Her score was -275 at the start of her turn, and at the end of her turn her score was -425. What was the change in Cheyenne’s score from the start of her turn to the end of her turn?
Lesson 1.3
23
21. Analyze Relationships For two months, Nell feeds her cat Diet Chow brand cat food. Then for the next two months, she feeds her cat Kitty Diet brand cat food. The table shows the cat’s change in weight over 4 months. Cat’s Weight Change (oz) Diet Chow, Month 1
-8
Diet Chow, Month 2
-18
Kitty Diet, Month 3
3
Kitty Diet, Month 4
-19
Which brand of cat food resulted in the greatest weight loss for Nell’s cat? Explain.
FOCUS ON HIGHER ORDER THINKING
Work Area
22. Represent Real-World Problems Write and solve a word problem that can be modeled by the difference -4 - 10.
24. Draw Conclusions When you subtract one negative integer from another, will your answer be greater than or less than the integer you started with? Explain your reasoning and give an example.
25. Look for a Pattern Find the next three terms in the pattern 9, 4, −1, −6, −11, … . Then describe the pattern.
24
Unit 1
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23. Explain the Error When Tom found the difference -11 - (-4), he got -15. What might Tom have done wrong?
Applying Addition and Subtraction of Integers
LESSON
1.4 ?
ESSENTIAL QUESTION
7.NS.3 Solve real-world and mathematical problems involving the four operations with rational numbers. Also
7.NS.1, 7.NS.1d, 7.EE.3
How do you solve multistep problems involving addition and subtraction of integers?
Solving a Multistep Problem You can use what you know about adding and subtracting integers to solve a multistep problem. Math On the Spot
EXAMPL 1 EXAMPLE
7.NS.3, 7.NS.1
A seal is swimming in the ocean 5 feet below sea level. It dives down 12 feet to catch some fish. Then, the seal swims 8 feet up towards the surface with its catch. What is the seal’s final elevation relative to sea level? STEP 1
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5 Sea Level
0 –5
Write an expression. • The seal starts at 5 feet below the surface, so its initial position is -5 ft.
–10
– 12 +8
–15
Starts -5
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STEP 2
Swims Dives + up down -
12
+
–20
8
Add or subtract from left to right to find the value of the expression. -5 - 12 + 8 = -17 + 8 = -9
This is reasonable because the seal swam farther down than up.
The seal’s final elevation is 9 feet below sea level.
YOUR TURN 1.
Anna is in a cave 40 feet below the cave entrance. She descends 13 feet, then ascends 18 feet. Find her new position relative to the cave entrance.
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Lesson 1.4
25
Applying Properties to Solve Problems You can use properties of addition to solve problems involving integers. Math On the Spot
EXAMPLE 2
Problem Solving
7.NS.1d, 7.NS.3, 7.EE.3
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Irene has a checking account. On Monday she writes a $160 check for groceries. Then she deposits $125. Finally she writes another check for $40. What was the total change in the amount in Irene’s account? Analyze Information
My Notes
When Irene deposits money, she adds that amount to the account. When she writes a check, that money is deducted from the account. Formulate a Plan
Use a positive integer for the amount Irene added to the account. Use negative integers for the checks she wrote. Find the sum. -160 + 125 + (-40) Justify and Evaluate Solve
Add the amounts to find the total change in the account. Use properties of addition to simplify calculations. -160 + 125 + (-40) = -160 + (-40) + 125 = -200 + 125
Commutative Property Associative Property
= -75 The amount in the account decreased by $75. Justify and Evaluate
Reflect 2.
Communicative Mathematical Ideas Describe a different way to find the change in Irene’s account.
YOUR TURN Personal Math Trainer Online Practice and Help
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26
Unit 1
3.
Alex wrote checks on Tuesday for $35 and $45. He also made a deposit in his checking account of $180. Find the overall change in the amount in his checking account.
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Irene’s account has $75 less than it did before Monday. This is reasonable because she wrote checks for $200 but only deposited $125.
Comparing Values of Expressions Sometimes you may want to compare values obtained by adding and subtracting integers.
EXAMPL 3 EXAMPLE
Math On the Spot
Problem Solving
7.NS.3, 7.EE.3
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The Tigers, a football team, must gain 10 yards in the next four plays to keep possession of the ball. The Tigers lose 12 yards, gain 5 yards, lose 8 yards, and gain 14 yards. Do the Tigers maintain possession of the ball? Analyze Information
When the team gains yards, add that distance. When the team loses yards, subtract that distance. If the total change in yards is greater than or equal to 10, the team keeps possession of the ball. Formulate a Plan
- 12 + 5 - 8 + 14 Justify and Evaluate Solve
-12 + 5 - 8 + 14 -12 + 5 + (- 8) + 14
To subtract, add the opposite.
-12 + (- 8) + 5 + 14
Commutative Property
(-12 + (- 8)) + (5 + 14)
Associative Property
-20 + 19 = -1 -1 < 10
Compare to 10 yards.
Math Talk
Mathematical Practices
What does it mean that the football team had a total of -1 yard over four plays?
The Tigers gained less than 10 yards, so they do not maintain possession.
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Justify and Evaluate
The football team gained 19 yards and lost 20 yards for a total of -1 yard.
YOUR TURN 4.
Jim and Carla are scuba diving. Jim started out 10 feet below the surface. He descended 18 feet, rose 5 feet, and descended 12 more feet. Then he rested. Carla started out at the surface. She descended 20 feet, rose 5 feet, and descended another 18 feet. Then she rested. Which person rested at a greater depth? Explain.
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Lesson 1.4
27
Guided Practice Write an expression. Then find the value of the expression. (Examples 1, 2, 3) 1. Tomas works as an underwater photographer. He starts at a position that is 15 feet below sea level. He rises 9 feet, then descends 12 feet to take a photo of a coral reef. Write and evaluate an expression to find his position relative to sea level when he took the photo.
2. The temperature on a winter night was -23 °F. The temperature rose by 5 °F when the sun came up. When the sun set again, the temperature dropped by 7 °F. Write and evaluate an expression to find the temperature after the sun set.
3. Jose earned 50 points in a video game. He lost 40 points, earned 87 points, then lost 30 more points. Write and evaluate an expression to find his final score in the video game.
Find the value of each expression. (Example 2) 4. -6 + 15 + 15 =
5. 9 - 4 - 17 =
6. 50 - 42 + 10 =
7. 6 + 13 + 7 - 5 =
8. 65 + 43 - 11 =
9. -35 - 14 + 45 + 31 =
Determine which expression has a greater value. (Example 3)
11. 21 - 3 + 8 or -14 + 31 - 6
?
ESSENTIAL QUESTION CHECK-IN
12. Explain how you can find the value of the expression -5 + 12 + 10 - 7.
28
Unit 1
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10. -12 + 6 - 4 or -34 - 3 + 39
Name
Class
Date
1.4 Independent Practice
Personal Math Trainer
7.NS.1, 7.NS.1d, 7.NS.3, 7.EE.3
13. Sports Cameron is playing 9 holes of golf. He needs to score a total of at most 15 over par on the last four holes to beat his best golf score. On the last four holes, he scores 5 over par, 1 under par, 6 over par, and 1 under par. a. Write and find the value of an expression that gives Cameron’s score for 4 holes of golf.
b. Is Cameron’s score on the last four holes over or under par?
c. Did Cameron beat his best golf score?
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14. Herman is standing on a ladder that is partly in a hole. He starts out on a rung that is 6 feet under ground, climbs up 14 feet, then climbs down 11 feet. What is Herman’s final position, relative to ground level?
15. Explain the Error Jerome finds the value of the expression 3 - 6 + 5 by first rewriting the expression as 3 - 5 + 6. Explain what is wrong with Jerome’s approach.
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Online Practice and Help
16. Lee and Barry play a trivia game in which questions are worth different numbers of points. If a question is answered correctly, a player earns points. If a question is answered incorrectly, the player loses points. Lee currently has -350 points. a. Before the game ends, Lee answers a 275-point question correctly, a 70-point question correctly, and a 50-point question incorrectly. Write and find the value of an expression to find Lee’s final score.
b. Barry’s final score is 45. Which player had the greater final score?
17. Multistep Rob collects data about how many customers enter and leave a store every hour. He records a positive number for customers entering the store each hour and a negative number for customers leaving the store each hour. Entering
Leaving
1:00 to 2:00
30
-12
2:00 to 3:00
14
-8
3:00 to 4:00
18
-30
a. During which hour did more customers leave than arrive?
b. There were 75 customers in the store at 1:00. The store must be emptied of customers when it closes at 5:00. How many customers must leave the store between 4:00 and 5:00?
Lesson 1.4
29
The table shows the changes in the values of two friends’ savings accounts since the previous month. June
July
August
Carla
-18
22
-53
Leta
-17
-22
18
18. Carla had $100 in her account in May. How much money does she have in her account in August? 19. Leta had $45 in her account in May. How much money does she have in her account in August? 20. Analyze Relationships Whose account had the greatest decrease in value from May to August? FOCUS ON HIGHER ORDER THINKING
Work Area
21. Represent Real-World Problems Write and solve a word problem that matches the diagram shown. -9 -8 -7 -6 -5 -4 -3 -2 -1 0
23. Draw Conclusions An expression involves subtracting two numbers from a third number. Under what circumstances will the value of the expression be negative? Give an example.
30
Unit 1
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22. Critical Thinking Mary has $10 in savings. She owes her parents $50. She does some chores and her parents pay her $12. She also gets $25 for her birthday from her grandmother. Does Mary have enough money to pay her parents what she owes them? If not, how much more money does she need? Explain.
MODULE QUIZ
Ready
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1.1 Adding Integers with the Same Sign
Online Practice and Help
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Add. 1. -8 + (-6)
2. -4 + (-7)
3. -9 + (-12)
1.2 Adding Integers with Different Signs Add. 4. 5 + (-2)
5. -8 + 4
6. 15 + (-8)
8. -3 - (-4)
9. 11 - (-12)
1.3 Subtracting Integers Subtract. 7. 2 - 9
1.4 Applying Addition and Subtraction of Integers 10. A bus makes a stop at 2:30, letting off 15 people and letting on 9. The bus makes another stop ten minutes later to let off 4 more people. How many more or fewer people are on the bus after the second stop compared to the number of people on the bus before the 2:30 stop?
11. Cate and Elena were playing a card game. The stack of cards in the middle had 24 cards in it to begin with. Cate added 8 cards to the stack. Elena then took 12 cards from the stack. Finally, Cate took 9 cards from the stack. How many cards were left in the stack?
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ESSENTIAL QUESTION 12. Write and solve a word problem that can be modeled by addition of two negative integers.
Module 1
31
MODULE 1 MIXED REVIEW
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Assessment Readiness
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Online Practice and Help
1. Look at each expression. Does it have the same value as -6 - 4? Select Yes or No for expressions A–C. A. -6 + (-4) B. -4 + (-6) C. 6 + (-4)
Yes Yes Yes
No No No
2. Choose True or False for A–C. A. x = 4 is the solution for x + 4 = 0. B. x = 24 is the solution for _3x = 8. C. x = 6 is the solution for 6x = 1
True True True
False False False
3. At 3:00 a.m., the temperature is –5 °F. Between 3:00 a.m. and 6:00 a.m., the temperature drops by 12 °F. Between 6:00 a.m. and 9:00 a.m., the temperature rises by 4 °F. What is the temperature at 9:00 a.m.? Explain how you solved this problem.
32
Round
Sherri’s Points
1
35
-10
2
-20
15
3
-5
15
Unit 1
Darren’s Points
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4. Sherri and Darren are playing a board game. The table shows the number of points each player scores in 3 rounds. If the player with the greater total score wins, who is the winner? Explain how you know.
Multiplying and Dividing Integers ?
2
MODULE
LESSON 2.1
ESSENTIAL QUESTION
Multiplying Integers
How can you use multiplication and division of integers to solve real-world problems?
7.NS.2, 7.NS.2a
LESSON 2.2
Dividing Integers 7.NS.2, 7.NS.2b, 7.NS.3
LESSON 2.3
Applying Integer Operations 7.NS.2a, 7.NS.2c,
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7.NS.3, 7.EE.3
Real-World Video The giant panda is an endangered animal. For some endangered species, the population has made a steady decline. This can be represented by my.hrw.com multiplying integers with different signs.
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Math On the Spot
Animated Math
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Go digital with your write-in 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.
33
Are YOU Ready? Personal Math Trainer
Complete these exercises to review skills you will need for this module.
Multiplication Facts EXAMPLES
7×9= 7 × 9 = 63 12 × 10 = 12 × 10 = 120
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Online Practice and Help
Use patterns. When you multiply 9 by a number 1 through 9, the digits of the product add up to 9. 6+3=9 Products of 10 end in 0.
Multiply.
1. 9 × 3
2. 7 × 10
3. 9 × 8
4. 15 × 10
5. 6 × 9
6. 10 × 23
7. 9 × 9
8. 10 × 20
Division Facts EXAMPLE
48 ÷ 6 = 48 ÷ 6 = 8
Think: 6 times what number equals 48? 6 × 8 = 48 So, 48 ÷ 6 = 8
Divide.
9. 54 ÷ 9
10. 42 ÷ 6
11. 24 ÷ 3
12. 64 ÷ 8
13. 90 ÷ 10
14. 56 ÷ 7
15. 81 ÷ 9
16. 110 ÷ 11
EXAMPLE
32 - 2(10 - 7)2 32 - 2(3)2 32 - 2(9) 32 - 18 14
To evaluate, first operate within parentheses. Next, simplify exponents. Then multiply and divide from left to right. Finally add and subtract from left to right.
Evaluate each expression.
34
Unit 1
17. 12 + 8 ÷ 2
18. 15 -(4 + 3) × 2
19. 18 -(8 - 5)2
20. 6 + 7 × 3 - 5
21. 9 + (22 + 3)2 × 2
22. 6 + 5 - 4 × 3 ÷ 2
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Order of Operations
Reading Start-Up Visualize Vocabulary Use the ✔ words to complete the chart. You may put more than one word in each box. ÷, or put into equal groups
×, or repeated addition
Multiplying and Dividing Whole Numbers 4 × 1=4
Vocabulary Review Words ✔ divide (dividir) ✔ dividend (dividendo) ✔ divisor (divisor) integers (enteros) ✔ multiply (multiplicar) negative number (número negativo) operation (operación) opposites (opuestos) positive number (número positivo) ✔ product (producto) ✔ quotient (cociente)
32 ÷ 4 = 8
Understand Vocabulary Complete the sentences using the review words.
1. A is a number that is less than 0. A is a number that is greater than 0.
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2. Division problems have three parts. The part you want to divide into groups is called
3.
the
. The number that is divided into another number is called
the
. The answer to a division problem is called the
.
are all whole numbers and their opposites.
Active Reading Double-Door Fold Create a double-door fold to help you understand the concepts in this module. Label one flap “Multiplying Integers” and the other flap “Dividing Integers.” As you study each lesson, write important ideas under the appropriate flap. Include information that will help you remember the concepts later when you look back at your notes. Module 2
35
GETTING READY FOR
Multiplying and Dividing Integers Understanding the standards and the vocabulary terms in the standards will help you know exactly what you are expected to learn in this module.
7.NS.2a
Key Vocabulary integer (entero) A member of the set of whole numbers and their opposites.
What It Means to You You will use your knowledge of multiplication of whole numbers and addition of negative numbers to multiply integers. EXAMPLE 7.NS.2a
Show that (-1)(-1) = 1. 0 = -1(0)
Multiplication property of 0
0 = -1(-1 + 1)
Addition property of opposites
0 = (-1)(-1) + (-1)(1)
Distributive Property
0 = (-1)(-1) + (-1)
Multiplication property of 1
So, (-1)(-1) = 1.
Definition of opposites
In general, a negative number times a negative number is always a positive number.
7.NS.2b Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, p (-p) p ___ then -( _q ) = ___ q = (-q) . Interpret quotients of rational numbers by describing real-world contexts.
What It Means to You You will use your knowledge of division of whole numbers and multiplication of integers to divide integers. EXAMPLE 7.NS.2b
The temperature in Fairbanks, Alaska, dropped over four consecutive hours from 0° F to -44 °F. If the temperature dropped the same amount each hour, how much did the temperature change each hour? -44 ___ = -11 4
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36
Unit 1
The quotient of -44 and 4 is the same as the negative quotient of 44 and 4. A negative number divided by a positive number is negative.
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Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.
LESSON
2.1 Multiplying Integers ?
7.NS.2 Apply and extend previous understandings of multiplication and division... to multiply ... rational numbers. Also 7.NS.2a
ESSENTIAL QUESTION How do you multiply integers?
EXPLORE ACTIVITY 1
7.NS.2, 7.NS.2a
Multiplying Integers Using a Number Line You can use a number line to see what happens when you multiply a positive number by a negative number.
A Henry made three withdrawals of $2 each from his savings account. What was the change in his balance? Find 3(-2). To graph -2, you would start at 0 and move )+(
3(-2) means (
)+(
To graph 3(-2), start at 0 and move 2 units to the left The result is
units to the left. ). +(-2) +(-2) +(-2)
times. -8 -7 -6 -5 -4 -3 -2 -1
.
0
The change in Henry’s balance was
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B Lisa plays a video game in which she loses points. She loses 3 points 2 times. What is her score? Find 2(-3). 2(-3) means ( )+( Show this on the number line. Lisa has a score of
.
). -8 -7 -6 -5 -4 -3 -2 -1
0
Reflect 1.
What do you notice about the product of two integers with different signs?
Lesson 2.1
37
EXPLORE ACTIVITY 2
7.NS.2, 7.NS.2a
Modeling Integer Multiplication Counters representing positive and negative numbers can help you understand how to find the product of two negative integers.
= +1 = -1
Find the product of -3 and -4. Write (-3)(-4) as -3(-4), which means the opposite of 3(-4). STEP 1
Use negative counters to model 3(-4). 3 groups of -4
STEP 2
Make the same model using positive counters to find the opposite of 3(-4). The opposite of 3 groups of -4
STEP 3
Translate the model into a mathematical expression: (-3)(-4) = .
Reflect 2.
What do you notice about the sign of the product of two negative integers?
3.
Make a Conjecture What can you conclude about the sign of the product of two integers with the same sign?
38
Unit 1
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The product of -3 and -4 is
Multiplying Integers The product of two integers with opposite signs is negative. The product of two integers with the same sign is positive. The product of 0 and any other integer is 0. You can use the Multiplication property of 0 and the Distributive Property to show that a negative number times a negative number is always a positive number.
Math On the Spot my.hrw.com
Show that (-1)(-1) = 1. 0 = -1(0)
Multiplication property of 0
0 = -1(-1 + 1)
Addition property of opposites
0 = (-1)(-1) + (-1)(1)
Distributive Property
0 = (-1)(-1) + (-1)
Multiplication property of 1
So, (-1)(-1) = 1.
Definition of opposites
EXAMPL 1 EXAMPLE
7.NS.2
A Multiply: (13)(-3). STEP 1
Determine the sign of the product. 13 is positive and -3 is negative. Since the numbers have opposite signs, the product will be negative.
STEP 2
Animated Math my.hrw.com
Find the absolute values of the numbers and multiply them. | 13 |
= 13
| -3 |
=3
13 × 3 = 39 STEP 3
Assign the correct sign to the product. 13(-3) = -39
The product is -39.
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B Multiply: (-5)(-8). STEP 1
Determine the sign of the product. -5 is negative and -8 is negative. Since the numbers have the same sign, the product will be positive.
STEP 2
Find the absolute values of the numbers and multiply them. | -5 |
=5
| -8 |
=8
5 × 8 = 40 STEP 3
Mathematical Practices
Assign the correct sign to the product. (-5)(-8) = 40
The product is 40.
C Multiply: (-10)(0). (-10)(0) = 0
Math Talk
Compare the rules for finding the product of a number and zero and finding the sum of a number and 0.
One of the factors is 0, so the product is 0. Lesson 2.1
39
YOUR TURN Personal Math Trainer Online Practice and Help
my.hrw.com
Find each product. 4.
-3(5)
5.
(-10)(-2)
6.
0(-22)
7.
8(4)
Guided Practice Find each product. (Explore Activity 2 and Example 1) 1. -1(9)
2. 14(-2)
3. (-9)(-6)
4. (-2)(50)
5. (-4)(15)
6. -18(0)
7. (-7)(-7)
8. -15(9)
9. (8)(-12)
10. -3(-100)
11. 0(-153)
12. -6(32)
13. Flora made 7 withdrawals of $75 each from her bank account. What was the overall change in her account? (Example 1)
15. The temperature dropped 2 °F every hour for 6 hours. What was the total number of degrees the temperature changed in the 6 hours? (Explore Activity 1)
?
ESSENTIAL QUESTION CHECK-IN
16. Explain the process for finding the product of two integers.
40
Unit 1
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14. A football team lost 5 yards on each of 3 plays. Explain how you could use a number line to find the team’s change in field position after the 3 plays. (Explore Activity 1)
Name
Class
Date
2.1 Independent Practice 7.NS.2
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17. Critique Reasoning Lisa used a number line to model –2(3). Does her number line make sense? Explain why or why not. +(-3)
+(-3)
-8 -7 -6 -5 -4 -3 -2 -1
Online Practice and Help
20. Adam is scuba diving. He descends 5 feet below sea level. He descends the same distance 4 more times. What is Adam’s final elevation?
0
21. The price of jeans was reduced $6 per week for 7 weeks. By how much did the price of the jeans change over the 7 weeks? 18. Represent Real-World Problems Mike got on an elevator and went down 3 floors. He meant to go to a lower level, so he stayed on the elevator and went down 3 more floors. How many floors did Mike go down altogether?
Solve. Show your work.
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19. When Brooke buys lunch at the cafeteria, money is withdrawn from a lunch account. The table shows amounts withdrawn in one week. By how much did the amount in Brooke’s lunch account change by the end of that week?
Lunch Account Week 1
Lunch
Cost
Balance $28
Monday
Pizza
$4
Tuesday
Fish Tacos
$4
Wednesday
Spaghetti
$4
Thursday
Sandwich
$4
Chicken
$4
Friday
22. Casey uses some of his savings on batting practice. The cost of renting a batting cage for 1 hour is $6. He rents a cage for 9 hours in each of two months. What is the change in Casey’s savings after two months?
23. Volunteers at Sam’s school use some of the student council’s savings for a special project. They buy 7 backpacks for $8 each and fill each backpack with paper and pens that cost $5. By how much did the student council’s savings change because of this project?
Lesson 2.1
41
24. Communicate Mathematical Ideas Describe a real-world situation that can be represented by the product 8(–20). Then find the product and explain what the product means in terms of the real-world situation.
25. What If? The rules for multiplying two integers can be extended to a product of 3 or more integers. Find the following products by using the Associative Property to multiply 2 numbers at a time. a. 3(3)(–3)
b. 3(–3)(–3)
c. –3(–3)(–3)
d. 3(3)(3)(–3)
e. 3(3)(–3)(–3)
f.
3(–3)(–3)(–3)
g. Make a Conjecture Based on your results, complete the following statements: When a product of integers has an odd number of negative factors, then the sign of the product is
.
When a product of integers has an even number of negative factors, then the sign of the product is
.
FOCUS ON HIGHER ORDER THINKING
Work Area
26. Multiple Representations The product of three integers is –3. Determine all of the possible values for the three factors.
28. Justify Reasoning The sign of the product of two integers with the same sign is positive. What is the sign of the product of three integers with the same sign? Explain your thinking.
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Unit 1
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27. Analyze Relationships When is the product of two integers less than or equal to both of the two factors?
LESSON
2.2 Dividing Integers ?
7.NS.2 Apply and extend previous understandings of multiplication and division... to…divide rational numbers. Also 7.NS.2b, 7.NS.3
ESSENTIAL QUESTION How do you divide integers?
EXPLORE ACTIVITY
7.NS.2, 7.NS.3
A diver needs to descend to a depth of 100 feet. She wants to do it in 5 equal descents. How far should she travel in each descent?
A Use the number line at the right to find how far the diver should travel in each of the 5 descents. -100 = ? B To solve this problem, you can set up a division problem: _____
C Rewrite the division problem as a multiplication problem. Think: Some number multiplied by 5 equals -100.
0 -10
× ? = -100
-20
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D Remember the rules for integer multiplication. If the product is negative, one of the factors must be negative. Since positive, the unknown factor must be
E You know that 5 ×
positive / negative.
= 100. So, using the rules for integer
multiplication you can say that 5 × The diver should descend
is
-30 -40 -50 -60 -70 -80 -90
= -100.
-100
feet in each descent.
-110
F Use the process you just learned to find each of the quotients below. 14 ___ = -7
-36 ____ = -9
-55 ____ = 11
-45 ____ = -5
Reflect 1.
Make a Conjecture Make a conjecture about the quotient of two integers with different signs. Make a conjecture about the quotient of two integers with the same sign.
Lesson 2.2
43
Dividing Integers Math On the Spot my.hrw.com
You used the relationship between multiplication and division to make conjectures about the signs of quotients of integers. You can use multiplication to understand why division by zero is not possible. Think about the division problem below and its related multiplication problem. 5÷0=?
0×?=5
The multiplication sentence says that there is some number times 0 that equals 5. You already know that 0 times any number equals 0. This means division by 0 is not possible, so we say that division by 0 is undefined.
EXAMPLE 1 My Notes
7.NS.2
A Divide: 24 ÷ (-3) STEP 1
Determine the sign of the quotient. 24 is positive and -3 is negative. Since the numbers have opposite signs, the quotient will be negative.
STEP 2
Divide. 24 ÷ (-3) = -8
B Divide: -6 ÷ (-2) STEP 1
Determine the sign of the quotient. -6 is negative and -2 is negative. Since the numbers have the same sign, the quotient will be positive.
STEP 2
Divide: -6 ÷ (-2) = 3
STEP 1
Determine the sign of the quotient. The dividend is 0 and the divisor is not 0. So, the quotient is 0.
STEP 2
Divide: 0 ÷ (-9) = 0
YOUR TURN Personal Math Trainer Online Practice and Help
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44
Unit 1
Find each quotient. 2.
0 ÷ (-6)
3.
38 ÷ (-19)
4.
-13 ÷ (-1)
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C Divide: 0 ÷ (-9)
Using Integer Division to Solve Problems You can use integer division to solve real-world problems. For some problems, you may need to perform more than one step. Be sure to check that the sign of the quotient makes sense for the situation.
EXAMPL 2 EXAMPLE
Math On the Spot my.hrw.com
7.NS.3, 7.NS.2
Jake answers questions in two different online Olympic trivia quizzes. In each quiz, he loses points when he gives an incorrect answer. The table shows the points lost for each wrong answer in each quiz and Jake’s total points lost in each quiz. In which quiz did he have more wrong answers? Olympic Trivia Quiz
STEP 1
Points lost for each wrong answer
Total points lost
Winter Quiz
-3 points
-33 points
Summer Quiz
-7 points
-56 points
Find the number of incorrect answers in the winter quiz. -33 ÷ (-3) = 11
STEP 2
Find the number of incorrect answers Jake gave in the summer quiz. -56 ÷ (-7) = 8
STEP 3
Divide the total points lost by the number of points lost per wrong answer.
Divide the total points lost by the number of points lost per wrong answer.
Math Talk
Mathematical Practices
What is the sign of each quotient in Steps 1 and 2? Why does this make sense for the situation?
Compare the numbers of wrong answers.
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11 > 8, so Jake had more wrong answers in the winter quiz.
YOUR TURN 5.
A penalty in Meteor-Mania is -5 seconds. A penalty in Cosmic Calamity is -7 seconds. Yolanda had penalties totaling -25 seconds in a game of Meteor-Mania and -35 seconds in a game of Cosmic Calamity. In which game did Yolanda receive more penalties? Justify your answer.
Personal Math Trainer Online Practice and Help
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Lesson 2.2
45
Guided Practice Find each quotient. (Example 1) -14 1. ____ 2
2. 21 ÷ (-3)
26 3. ____ -13
4. 0 ÷ (-4)
-45 5. ____ -5
6. -30 ÷ (10)
-11 7. ____ -1
8. -31 ÷ (-31)
0 9. ___ -7
-121 10. _____ -11
11. 84 ÷ (-7)
500 12. ____ -25
13. -6 ÷ (0)
-63 14. ____ -21
Write a division expression for each problem. Then find the value of the expression. (Example 2) 15. Clark made four of his truck payments late and was fined four late fees. The total change to his savings from late fees was -$40. How much was one late fee?
16. Jan received -22 points on her exam. She got 11 questions wrong out of 50 questions. How much was Jan penalized for each wrong answer?
18. Louisa’s savings change by -$9 each time she goes bowling. In all, it changed by -$99 during the summer. How many times did she go bowling in the summer?
?
ESSENTIAL QUESTION CHECK-IN
19. How is the process of dividing integers similar to the process of multiplying integers?
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Unit 1
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17. Allen’s score in a video game was changed by -75 points because he missed some targets. He got -15 points for each missed target. How many targets did he miss?
Name
Class
Date
2.2 Independent Practice 7.NS.2, 7.NS.2b, 7.NS.3
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Online Practice and Help
20. Walter buys a bus pass for $30. Every time he rides the bus, money is deducted from the value of the pass. He rode 12 times and $24 was deducted from the value of the pass. How much does each bus ride cost?
21. Analyze Relationships Elisa withdrew $20 at a time from her bank account and withdrew a total of $140. Francis withdrew $45 at a time from his bank account and withdrew a total of $270. Who made the greater number of withdrawals? Justify your answer.
22. Multistep At 7 p.m. last night, the temperature was 10 °F. At 7 a.m. the next morning, the temperature was -2 °F. a. By how much did the temperature change from 7 p.m. to 7 a.m.?
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b. The temperature changed by a steady amount overnight. By how much did it change each hour?
23. Analyze Relationships Nola hiked down a trail at a steady rate for 10 minutes. Her change in elevation was -200 feet. Then she continued to hike down for another 20 minutes at a different rate. Her change in elevation for this part of the hike was -300 feet. During which portion of the hike did she walk down at a faster rate? Explain your reasoning.
24. Write a real world description to fit the expression -50 ÷ 5.
Lesson 2.2
47
25. Communicate Mathematical Ideas Two integers, a and b, have different signs. The absolute value of integer a is divisible by the absolute value of integer b. Find two integers that fit this description. Then decide if the product of the integers is greater than or less than the quotient of the integers. Show your work.
Determine if each statement is true or false. Justify your answer. 26. For any two nonzero integers, the product and quotient have the same sign.
27. Any nonzero integer divided by 0 equals 0.
FOCUS ON HIGHER ORDER THINKING
Work Area
28. Multi-step A perfect score on a test with 25 questions is 100. Each question is worth the same number of points. a. How many points is each question on the test worth?
number of questions Fred answered incorrectly. 29. Persevere in Problem Solving Colleen divided integer a by -3 and got 8. Then she divided 8 by integer b and got -4. Find the quotient of integer a and integer b. 30. Justify Reasoning The quotient of two negative integers results in an integer. How does the value of the quotient compare to the value of the original two integers? Explain.
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Unit 1
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b. Fred got a score of 84 on the test. Write a division sentence using negative numbers where the quotient represents the
LESSON
2.3 ?
Applying Integer Operations
7.NS.3 Solve real-world and mathematical problems involving the four operations with rational numbers. Also
7.NS.2a, 7.NS.2c, 7.EE.3
ESSENTIAL QUESTION How can you use integer operations to solve real-world problems?
Using the Order of Operations with Integers The order of operations applies to integer operations as well as positive number operations. Perform multiplication and division first, and then addition and subtraction. Work from left to right in the expression.
EXAMPL 1 EXAMPLE
Problem Solving
Math On the Spot my.hrw.com
7.NS.2c, 7.NS.2a
Hannah made four withdrawals of $20 from her checking account. She also wrote a check for $215. By how much did the amount in her checking account change? Analyze Information
You need to find the total change in Hannah’s account. Since withdrawals and writing a check represent a decrease in her account, use negative numbers to represent these amounts. Formulate a Plan
Write a product to represent the four withdrawals. -20 + (-20) + (-20) + (-20) = 4(-20)
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Add -215 to represent the check that Hannah wrote. 4(-20) + (-215) Justify and Evaluate Solve
Evaluate the expression to find by how much the amount in the account changed. 4(-20) - 215 = -80 - 215 = -295
Multiply first. Then subtract.
The amount in the account decreased by $295. Justify and Evaluate
The value -295 represents a decrease of 295 dollars. This makes sense, since withdrawals and writing checks remove money from the checking account. Lesson 2.3
49
YOUR TURN Personal Math Trainer Online Practice and Help
1. Reggie lost 3 spaceships in level 3 of a video game. He lost 30 points for each spaceship. When he completed level 3, he earned a bonus of 200 points. By how much did his score change?
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2. Simplify: -6(13) - 21
Using Negative Integers to Represent Quantities Math On the Spot my.hrw.com
You can use positive and negative integers to solve problems involving amounts that increase or decrease. Sometimes you may need to use more than one operation.
EXAMPLE 2
7.NS.3, 7.EE.3
Three brothers each have their own savings. They borrow $72 from their parents for concert tickets. Each brother must pay back an equal share of this amount. Also, the youngest brother owes his parents $15. By how much will the youngest brother’s savings change after he pays his parents? STEP 1
Determine the signs of the values and the operations you will use. Write an expression.
Math Talk
Mathematical Practices
Suppose the youngest brother has $60 in savings. How much will he have left after he pays his parents what he owes? STEP 2
Since an equal share of the $72 will be paid back, use division to determine 3 equal parts of -72. Then add -15 to one of these equal parts. Change to youngest brother’s savings = (-72) ÷ 3 + (-15) Evaluate the expression. (-72) ÷ 3 + (-15) = -24 + (-15) = -39
Divide. Add.
The youngest brother’s savings will decrease by $39.
Reflect 3.
50
Unit 1
What If? Suppose there were four brothers in Example 2. How much would the youngest brother need to pay?
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Since the money is being paid back, it will decrease the amount in each brother’s savings. Use -72 and -15.
YOUR TURN Simplify each expression. 4. (-12) ÷ 6 + 2 6. 40 ÷ (-5) + 30
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5. -87 ÷ (-3) -9
Online Practice and Help
7. -39 ÷ 3 -15
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Comparing Values of Expressions Often, problem situations require making comparisons between two values. Use integer operations to calculate values. Then compare the values. Math On the Spot
EXAMPL 3 EXAMPLE
7.NS.3, 7.EE.3
Jill and Tony play a board game in which they move counters along a board. Jill moves her counter back 3 spaces four times, and then moves her counter forward 6 spaces. Tony moves his counter back 2 spaces three times, and then moves his player forward 3 spaces one time. Find each player’s overall change in position. Who moved farther? STEP 1
STEP 2
Find each player’s overall change in position. Jill: 4(-3) + 6 = -12 + 6 = - 6
Jill moves back 6 spaces.
Tony: 3(-2) + 3 = - 6 + 3 = -3
Tony moves back 3 spaces.
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Math Talk
Mathematical Practices
Why do you compare absolute values in Step 2?
Compare the numbers of spaces moved by the players. | -6 |
> | -3 |
Compare absolute values.
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Jill moves farther back than Tony.
YOUR TURN 8. Amber and Will are in line together to buy tickets. Amber moves back by 3 places three times to talk to friends. She then is invited to move 5 places up in line. Will moved back by 4 places twice, and then moved up in line by 3 places. Overall, who moved farther back in line?
Evaluate each expression. Circle the expression with the greater value. 9. (-10) ÷ 2 - 2 =
10. 42 ÷ (-3) + 9 =
(-28) ÷ 4 + 1 =
(-36) ÷ 9 - 2 =
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Lesson 2.3
51
Guided Practice Evaluate each expression. (Example 1) 1. -6(-5) + 12
2. 3(-6) - 3
3. -2(8) + 7
4. 4(-13) + 20
5. (-4)(0) - 4
6. -3(-5) - 16
Write an expression to represent the situation. Evaluate the expression and answer the question. (Example 2) 7. Bella pays 7 payments of $5 each to a game store. She returns one game and receives $20 back. What is the change to the amount of money she has?
8. Ron lost 10 points seven times playing a video game. He then lost an additional 100 points for going over the time limit. What was the total change in his score?
9. Ned took a test with 25 questions. He lost 4 points for each of the 6 questions he got wrong and earned an additional 10 points for answering a bonus question correctly. How many points did Ned receive or lose overall?
10. Mr. Harris has some money in his wallet. He pays the babysitter $12 an hour for 4 hours of babysitting. His wife gives him $10, and he puts the money in his wallet. By how much does the amount in his wallet change?
11. -3(-2) + 3 13. -7(5) - 9
?
3(- 4) + 9 -3(20) + 10
12. -8(-2) - 20 14. -16(0) - 3
ESSENTIAL QUESTION CHECK-IN
15. When you solve a problem involving money, what can a negative answer represent?
52
Unit 1
3(-2) + 2 - 8(-2) - 3
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Compare the values of the two expressions using <, =, or >. (Example 3)
Name
Class
Date
2.3 Independent Practice
Personal Math Trainer
7.NS.2a, 7.NS.2c, 7.NS.3, 7.EE.3
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Online Practice and Help
Evaluate each expression. 16. -12(-3) + 7
17. -42 ÷ (-6) + 5 - 8
18. 10(- 60) - 18
19. (-11)(-7) + 5 - 82
20. 35 ÷ (-7) + 6
21. -13(-2) - 16 - 8
22. Multistep Lily and Rose are playing a game. In the game, each player starts with 0 points and the player with the most points at the end wins. Lily gains 5 points two times, loses 12 points, and then gains 3 points. Rose loses 3 points two times, loses 1 point, gains 6 points, and then gains 7 points. a. Write and evaluate an expression to find Lily’s score.
b. Write and evaluate an expression to find Rose’s score.
c. Who won the game?
Write an expression from the description. Then evaluate the expression. 23. 8 less than the product of 5 and -4
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24. 9 more than the quotient of -36 and -4.
25. Multistep Arleen has a gift card for a local lawn and garden store. She uses the gift card to rent a tiller for 4 days. It costs $35 per day to rent the tiller. She also buys a rake for $9. a. Find the change to the value on her gift card.
b. The original amount on the gift card was $200. Does Arleen have enough left on the card to buy a wheelbarrow for $50? Explain.
Lesson 2.3
53
26. Carlos made up a game where, in a deck of cards, the red cards (hearts and diamonds) are negative and the black cards (spades and clubs) are positive. All face cards are worth 10 points, and number cards are worth their value. a. Samantha has a king of hearts, a jack of diamonds, and a 3 of spades. Write an expression to find the value of her cards.
b. Warren has a 7 of clubs, a 2 of spades, and a 7 of hearts. Write an expression to find the value of his cards.
c. If the greater score wins, who won?
d. If a player always gets three cards, describe two different ways to receive a score of 7.
FOCUS ON HIGHER ORDER THINKING
Work Area
28. Critique Reasoning Jim found the quotient of two integers and got a positive integer. He added another integer to the quotient and got a positive integer. His sister Kim says that all the integers Jim used to get this result must be positive. Do you agree? Explain.
29. Persevere in Problem Solving Lisa is standing on a dock beside a lake. She drops a rock from her hand into the lake. After the rock hits the surface of the lake, the rock’s distance from the lake’s surface changes at a rate of -5 inches per second. If Lisa holds her hand 5 feet above the lake’s surface, how far from Lisa’s hand is the rock 4 seconds after it hits the surface?
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Unit 1
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27. Represent Real-World Problems Write a problem that the expression 3(-7) - 10 + 25 = -6 could represent.
MODULE QUIZ
Ready
Personal Math Trainer
2.1 Multiplying Integers
Online Practice and Help
Find each product.
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1. (-2)(3)
2. (-5)(-7)
3. (8)(-11)
4. (-3)(2)(-2)
5. The temperature dropped 3 °C every hour for 5 hours. Write an integer that represents the change in temperature.
2.2 Dividing Integers Find each quotient. -63 6. ____ 7
-15 7. ____ -3
8. 0 ÷ (-15)
9. 96 ÷ (-12)
10. An elephant at the zoo lost 24 pounds over 6 months. The elephant lost the same amount of weight each month. Write an integer that represents the change in the elephant’s weight each month.
2.3 Applying Integer Operations Evaluate each expression. 11. (-4)(5) + 8
12. (-3)(-6) - 7
13. -27 ÷ 9 - 11
-24 - (-2) 14. ____ -3
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ESSENTIAL QUESTION 15. Write and solve a real-world problem that can be represented by the expression ( –3 )( 5 ) + 10.
Module 2
55
MODULE 2 MIXED REVIEW
Personal Math Trainer
Assessment Readiness
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Online Practice and Help
1. Look at each expression. Does it have a value of 0? Select Yes or No for expressions A–C. A. B. C.
28 ___ +7 -4
(-4)(-7) + 28 -28 -4 - ( ____ 7 )
Yes Yes Yes
No No No
2. An addition problem is shown below. 2 + (-3) Choose True or False for the accurate representations of the solution. A.
True
False
B.
True
False
True
False
-4 - 3 - 2 - 1
C. 2 + (–2) + (–1)
0 1 2 3 4
4. An elevator starts at the 6th floor. It goes up 7 floors twice, and then it goes down 10 floors. A second elevator starts on the 20th floor. It goes down 5 floors 3 times, and then it goes up 12 floors. Which elevator ends on a higher floor? Explain how you know.
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Unit 1
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3. A diver descends from an elevation of 6 feet below sea level to an elevation of 98 feet below sea level in 4 equal intervals. What was the diver’s change in elevation in each interval? Explain how you solved this problem.
Rational Numbers ?
MODULE
ESSENTIAL QUESTION How can you use rational numbers to solve real-world problems?
3
LESSON 3.1
Rational Numbers and Decimals 7.NS.2b, 7.NS.2d
LESSON 3.2
Adding Rational Numbers 7.NS.1a, 7.NS.1b, 7.NS.1d, 7.NS.3
LESSON 3.3
Subtracting Rational Numbers 7.NS.1, 7.NS.1c
LESSON 3.4
Multiplying Rational Numbers 7.NS.2, 7.NS.2a, © Houghton Mifflin Harcourt Publishing Company • Image Credits: Diego Barbieri/Shutterstock.com
7.NS.2c
LESSON 3.5
Dividing Rational Numbers 7.NS.2, 7.NS.2b, 7.NS.2c
LESSON 3.6
Real-World Video In many competitive sports, scores are given as decimals. For some events, the judges’ scores are averaged to give the athlete’s final score.
Applying Rational Number Operations 7.NS.3, 7.EE.3
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Math On the Spot
Animated Math
Personal Math Trainer
Go digital with your write-in 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.
57
Are YOU Ready? Personal Math Trainer
Complete these exercises to review skills you will need for this module.
Multiply Fractions EXAMPLE
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1
3 _ _ × 49 8
1
3 _ 3 __ _ × 49 = ___ × 94 8 8 2
= _16
Online Practice and Help
Divide by the common factors.
3
Simplify.
Multiply. Write the product in simplest form. 9 1. __ × _76 14
2.
3 _ _ × 47 5
3.
11 __ __ × 10 8 33
4.
4 _ ×3 9
Operations with Fractions EXAMPLE
10 2 __ 7 _ ÷ 10 = _25 × __ 5 7
Multiply by the reciprocal of the divisor.
2 10 = _25 × __ 7
Divide by the common factors.
= _47
Simplify.
1
Divide.
5. _12 ÷ _14
13 6. _38 ÷ __ 16
9. _35 ÷ _56
23 10. _14 ÷ __ 24
14 7. _25 ÷ __ 15
16 8. _49 ÷ __ 27
11. 6 ÷ _35
12. _45 ÷ 10
EXAMPLE
50 - 3(3 + 1)2 50 - 3(4)2 50 - 3(16) 50 - 48 2
To evaluate, first operate within parentheses. Next simplify exponents. Then multiply and divide from left to right. Finally add and subtract from left to right.
Evaluate each expression.
58
Unit 1
13. 21 - 6 ÷ 3
14. 18 + (7 - 4) × 3
15. 5 + (8 - 3)2
16. 9 + 18 ÷ 3 + 10
17. 60 - (3 - 1)4 × 3
18. 10 - 16 ÷ 4 × 2 + 6
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Order of Operations
Reading Start-Up Visualize Vocabulary Use the ✔ words to complete the graphic. You can put more than one word in each section of the triangle.
Integers
Vocabulary Review Words integers (enteros) ✔ negative numbers (números negativos) pattern (patrón) ✔ positive numbers (números positivos) ✔ whole numbers (números enteros)
45
Preview Words
2, 24, 108
-2, -24, -108
additive inverse (inverso aditivo) opposite (opuesto) rational number (número racional) repeating decimal (decimal periódico) terminating decimal (decimal finito)
Understand Vocabulary Complete the sentences using the preview words.
1. A decimal number for which the decimals come to an end is a
© Houghton Mifflin Harcourt Publishing Company
decimal.
2. The , or , of a number is the same distance from 0 on a number line as the original number, but on the other side of 0.
Active Reading Layered Book Before beginning the module, create a layered book to help you learn the concepts in this module. At the top of the first flap, write the title of the module, “Rational Numbers.” Label the other flaps “Adding,” “Subtracting,” “Multiplying,” and “Dividing.” As you study each lesson, write important ideas, such as vocabulary and processes, on the appropriate flap.
Module 3
59
GETTING READY FOR
Rational Numbers Understanding the Standards and the vocabulary terms in the Standards will help you know exactly what you are expected to learn in this module.
7.NS.3 Solve real-world and mathematical problems involving the four operations with rational numbers.
Key Vocabulary rational number (número racional) Any number that can be expressed as a ratio of two integers.
What It Means to You You will add, subtract, multiply, and divide rational numbers. EXAMPLE 7.NS.3
-15 · _23 - 12 ÷ 1 _13 4 15 _ 12 ÷ _ - __ · 2 - __ 3 1 3 1
Write as fractions.
3 15 _ 12 · _ - __ · 2 - __ 4 1 3 1
To divide, multiply by the reciprocal.
5
3
·2 ·3 _____ _____ - 15 - 12 1· 4 1·3
Simplify.
10 - __ - _91 = -10 - 9 = -19 1
Multiply.
1
1
7.NS.3
What It Means to You You will solve real-world and mathematical problems involving the four operations with rational numbers. EXAMPLE 7.NS.3
In 1954, the Sunshine Skyway Bridge toll for a car was $1.75. In 2012, the toll was _57 of the toll in 1954. What was the toll in 2012? 1.75 · _57 = 1_34 · _57 = _74 · _57
Write the decimal as a fraction. Write the mixed number as an improper fraction.
1
7· 5 = _____ 4·7
Simplify.
= _54 = 1.25
Multiply, then write as a decimal.
1
The Sunshine Skyway Bridge toll for a car was $1.25 in 2012. Visit my.hrw.com to see all CA Common Core Standards explained. my.hrw.com
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Unit 1
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Solve real-world and mathematical problems involving the four operations with rational numbers.
LESSON
3.1 ?
Rational Numbers and Decimals
7.NS.2d Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats. Also 7.NS.2b
ESSENTIAL QUESTION How can you convert a rational number to a decimal?
EXPLORE ACTIVITY
7.NS.2b, 7.NS.2d
Describing Decimal Forms of Rational Numbers A rational number is a number that can be written as a ratio of two integers a and b, where b is not zero. For example, _47 is a rational 37 number, as is 0.37 because it can be written as the fraction ___ 100 .
A Use a calculator to find the equivalent decimal form of each fraction.__ Remember that numbers that repeat can be written as 0.333… or 0.3. Fraction
1 _ 4
5 _ 8
2 _ 3
2 _ 9
12 __ 5
Decimal Equivalent
0.2
0.875
B Now find the corresponding fraction of the decimal equivalents given in the last two columns in the table. Write the fractions in simplest form.
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C Conjecture What do you notice about the digits after the decimal point in the decimal forms of the fractions? Compare notes with your neighbor and refine your conjecture if necessary.
Reflect 1.
Consider the decimal 0.101001000100001000001…. Do you think this decimal represents a rational number? Why or why not?
2.
Do you think a negative sign affects whether or not a number is a rational number? Use -_85 as an example.
Lesson 3.1
61
EXPLORE ACTIVITY (cont’d)
3.
Do you think a mixed number is a rational number? Explain.
Writing Rational Numbers as Decimals Math On the Spot my.hrw.com
You can convert a rational number to a decimal using long division. Some decimals are terminating decimals because the decimals come to an end. Other decimals are repeating decimals because one or more digits repeat infinitely.
EXAMPLE 1
7.NS.2d
Write each rational number as a decimal. 5 A - __ 16 Divide 5 by 16. Add a zero after the decimal point. Subtract 48 from 50. Use the grid to help you complete the long division.
Add zeros in the dividend and continue dividing until the remainder is 0. 5 The decimal equivalent of - __ 16 is - 0.3125.
0. 3 1 ⎯ 1 6 ⟌ 5. 0 0 −4 8 2 0 - 1 6 4 - 3
2 5 0 0
0. 3 9 ⎯ 3 3 ⟌ 1 3. 0 0 −9 9 3 1 0 -2 9 7 1 3 -9 3 -2
3 9 0 0
0 2 8 0 - 8 0 0
Divide 13 by 33. Add a zero after the decimal point. Subtract 99 from 130. Use the grid to help you complete the long division.
Math Talk
Mathematical Practices
Do you think that decimals that have repeating patterns always have the same number of digits in their pattern? Explain.
You can stop dividing once you discover a repeating pattern in the quotient. Write the quotient with its repeating pattern and indicate that the repeating numbers continue. 13 The decimal equivalent of __ ___ 33 is 0.3939…, or 0.39.
62
Unit 1
0 9 1 0 9 7 1 3
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13 B __ 33
YOUR TURN Write each rational number as a decimal. 4.
- _47
5.
1 _ 3
6.
Personal Math Trainer
9 - __ 20
Online Practice and Help
my.hrw.com
Writing Mixed Numbers as Decimals You can convert a mixed number to a decimal by rewriting the fractional part of the number as a decimal. Math On the Spot
EXAMPL 2 EXAMPLE
7.NS.2d
Shawn rode his bike 6 _34 miles to the science museum. Write 6 _34 as a decimal. STEP 1
Rewrite the fractional part of the number as a decimal. 0.75 ⎯ 4⟌ 3.00 -28 20 -20 0
STEP 2
Science
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My Notes
Museum
Divide the numerator by the denominator. 6_34 mi
Rewrite the mixed number as the sum of the whole part and the decimal part. 6 _34 = 6 + _34
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= 6 + 0.75
= 6.75
YOUR TURN 7.
The change ($) in a stock value was -2 _34 per share. Write -2 _34 as a decimal. -2 _34 = Is the decimal equivalent a terminating or repeating decimal?
8.
Yvonne bought a watermelon that weighed 7 _13 pounds. Write 7 _13 as a decimal. 7 _13 = Is the decimal equivalent a terminating or repeating decimal?
Personal Math Trainer Online Practice and Help
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Lesson 3.1
63
Guided Practice Write each rational number as a decimal. Then tell whether each decimal is a terminating or a repeating decimal. (Explore Activity and Example 1) 1. _35 =
89 2. - ___ = 100
4 3. __ = 12
25 4. __ = 99
5. - _79 =
9 6. - __ = 25
1 7. __ = 25
25 8. - ___ = 176
12 9. ____ = 1,000
Write each mixed number as a decimal. (Example 2) 1= 10. - 11 __ 6
9 = 11. 2 ___ 10
23 = 12. - 8 ____ 100
3 = 13. 7 ___ 15
3 = 14. 54 ___ 11
1 = 15. - 3 ___ 18
16. Maggie bought 3 _23 lb of apples to make some apple pies. What is the weight of the apples written as a decimal? (Example 2) 3 _23 =
?
17. Harry’s dog lost 2 _78 pounds. What is the change in the dog’s weight written as a decimal? (Example 2) -2 _78 =
ESSENTIAL QUESTION CHECK-IN
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3 18. Tom is trying to write __ 47 as a decimal. He used long division and divided until he got the quotient 0.0638297872, at which point he stopped. Since 3 the decimal doesn’t seem to terminate or repeat, he concluded that __ 47 is not rational. Do you agree or disagree? Why?
64
Unit 1
Name
Class
Date
3.1 Independent Practice
Personal Math Trainer
7.NS.2b, 7.NS.2d
Use the table for 19–23. Write each ratio in the form __ba and then as a decimal. Tell whether each decimal is a terminating or a repeating decimal. 19. basketball players to football players
20. hockey players to lacrosse players
21. polo players to football players
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Online Practice and Help
Team Sports Sport Baseball Basketball Football Hockey Lacrosse Polo Rugby Soccer
Number of Players 9 5 11 6 10 4 15 11
22. lacrosse players to rugby players
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23. football players to soccer players
24. Look for a Pattern Beth said that the ratio of the number of players in any sport to the number of players on a lacrosse team must always be a terminating decimal. Do you agree or disagree? Why?
25. The change in the water level at the lake was - 4 _78 inches for the month. a. What is - 4 _78 written as an improper fraction? b. What is - 4 _78 written as a decimal? c. Communicate Mathematical Ideas If the water level at the lake continued to change at the same rate for 3 months in a row, explain how you could estimate the total change in the water level at the end of the 3 month period.
Lesson 3.1
65
26. Vocabulary A rational number can be written as the ratio of one to another and can be represented by a repeating or
decimal.
5 7 feet tall. Ben is 5 __ feet tall. Which of the 27. Problem Solving Marcus is 5 __ 16 24 two boys is taller? Justify your answer.
28. Represent Real-World Problems If one store is selling _34 of a bushel of apples for $9, and another store is selling _23 of a bushel of apples for $9, which store has the better deal? Explain your answer.
FOCUS ON HIGHER ORDER THINKING
Work Area
29. Analyze Relationships You are given a fraction in simplest form. The numerator is not zero. When you write the fraction as a decimal, it is a repeating decimal. Which numbers from 1 to 10 could be the denominator?
31. Look for a Pattern Look at the decimal 0.121122111222.… If the pattern continues, is this a repeating decimal? Explain.
66
Unit 1
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30. Communicate Mathematical Ideas Julie got 21 of the 23 questions on her math test correct. She got 29 of the 32 questions on her science test correct. On which test did she get a higher score? Can you compare 29 21 __ the fractions __ 23 and 32 by comparing 29 and 21? Explain. How can Julie compare her scores?
LESSON
3.2 ?
Adding Rational Numbers
7.NS.1d Apply properties of operations as strategies to add and subtract rational numbers. Also 7.NS.1a,
7.NS.1b, 7.NS.3
ESSENTIAL QUESTION How can you add rational numbers?
Adding Rational Numbers with the Same Sign To add rational numbers with the same sign, apply the rules for adding integers. The sum has the same sign as the sign of the rational numbers.
Math On the Spot my.hrw.com
EXAMPL 1 EXAMPLE
7.NS.1b
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A Malachi hikes for 2.5 miles and stops for lunch. Then he hikes for 1.5 more miles. How many miles did he hike altogether? STEP 1
Use positive numbers to represent the distance Malachi hiked.
STEP 2
Find 2.5 + 1.5.
STEP 3
Start at 2.5.
STEP 4
Move 1.5 units to the right because the second addend is positive.
-5 -4 -3 -2 -1
0 1 2 3 4 5
The result is 4. Malachi hiked 4 miles.
B Kyle pours out _34 liter of liquid from a beaker. Then he pours out another _12 liter of liquid. What is the overall change in the amount of liquid in the beaker? STEP 1
Use negative numbers to represent amounts the change each time Kyle pours liquid from the beaker.
STEP 2
Find - _3 + (- _1 ).
STEP 3
Start at -_34.
STEP 4
4
2
-2
-1
0
Move | -_12 | = _12 unit to the left because the second addend is negative. The result is -1_14. The amount of liquid in the beaker has decreased by 1_14 liters. Lesson 3.2
67
Reflect 1.
Explain how to determine whether to move right or left on the number line when adding rational numbers.
YOUR TURN Use a number line to find each sum. Personal Math Trainer Online Practice and Help
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2. 3 + 1_12 = 0
1
2
3
4
5
-4
-3
-2
3. -2.5 + (-4.5) = -7
-6
-5
-1
0
Adding Rational Numbers with Different Signs my.hrw.com
EXAMPLE 2
7.NS.1b
A During the day, the temperature increases by 4.5 degrees. At night, the temperature decreases by 7.5 degrees. What is the overall change in temperature? STEP 1
Use a positive number to represent the increase in temperature and a negative number to represent a decrease in temperature.
STEP 2
Find 4.5 + (-7.5).
STEP 3
Start at 4.5.
STEP 4
Move | -7.5 | = 7.5 units to the left because the second addend is negative.
-5 -4 -3 -2 -1
0 1 2 3 4 5
The result is -3. The temperature decreased by 3 degrees overall.
68
Unit 1
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Math On the Spot
To add rational numbers with different signs, find the difference of their absolute values. Then use the sign of the rational number with the greater absolute value.
B Ernesto writes a check for $2.50. Then he deposits $6 in his checking account. What is the overall increase or decrease in the account balance? STEP 1
Use a positive number to represent a deposit and a negative number to represent a withdrawal or a check.
STEP 2
Find -2.5 + 6.
STEP 3
Start at -2.5.
STEP 4
Animated Math my.hrw.com
-5 -4 -3 -2 -1
0 1 2 3 4 5
My Notes
Move | 6 | = 6 units to the right because the second addend is positive . The result is 3.5. The account balance will increase by $3.50.
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Reflect 4.
Do -3 + 2 and 2 + (-3) have the same sum? Does it matter if the negative number is the first addend or the second addend?
5.
Make a Conjecture Do you think the sum of a negative number and a positive number will always be negative? Explain your reasoning.
YOUR TURN Use a number line to find each sum. 6. 7.
8.
-8 + 5 =
( )=
1 _ + -_34 2
-8 -7 -6 -5 -4 -3 -2 -1
-1
0
0 1 2
1
-1 + 7 = -3 -2 -1
0 1 2 3 4 5 6 7 8
Personal Math Trainer Online Practice and Help
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Lesson 3.2
69
Finding the Additive Inverse The opposite, or additive inverse, of a number is the same distance from 0 on a number line as the original number, but on the other side of 0. Zero is its own additive inverse.
Math On the Spot my.hrw.com
Math Talk
EXAMPLE 3
7.NS.1a, 7.NS.1b, 7.NS.1d
A A football team loses 3.5 yards on their first play. On the next play, they gain 3.5 yards. What is the overall increase or decrease in yards?
Mathematical Practices
Explain how to use a number line to find the additive inverse, or opposite, of -3.5.
STEP 1
Use a positive number to represent the gain in yards and a negative number to represent the loss in yards.
STEP 2
Find -3.5 + 3.5.
STEP 3
Start at -3.5.
My Notes STEP 4
-5 -4 -3 -2 -1
0 1 2 3 4 5
Move | 3.5 | = 3.5 units to the right, because the second addend is positive. The result is 0. This means the overall change is 0 yards.
Addition Property of Opposites
YOUR TURN Use a number line to find each sum. 9.
( )
2 _12 + -2 _12 =
-5 -4 -3 -2 -1
11. Personal Math Trainer Online Practice and Help
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70
Unit 1
0 1 2 3 4 5
10.
-4.5 + 4.5 =
-5 -4 -3 -2 -1
0 1 2 3 4 5
Kendrick adds _34 cup of chicken stock to a pot. Then he takes _34 cup of stock out of the pot. What is the overall increase or decrease in the amount of chicken stock in the pot?
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The sum of a number and its opposite, or additive inverse, is 0. This can be written as p + (-p) = 0.
Adding Three or More Rational Numbers Recall that the Associative Property of Addition states that if you are adding more than two numbers, you can group any of the numbers together. This property can help you add numbers with different signs.
EXAMPL 4 EXAMPLE
Math On the Spot my.hrw.com
7.NS.1d, 7.NS.3
Tina spent $5.25 on craft supplies to make friendship bracelets. She made $6.75 and spent an additional $3.25 for supplies on Monday. On Tuesday, she sold an additional $4.50 worth of bracelets. What was Tina’s overall profit or loss? STEP 1
Use negative numbers to represent the amount Tina spent and positive numbers to represent the money Tina earned.
STEP 2
Find -5.25 + 6.75 + (-3.25) + 4.50.
STEP 3
Group numbers with the same sign. -5.25 + (-3.25) + 6.75 + 4.50
Profit means the difference between income and costs is positive.
Commutative Property
(-5.25 + (-3.25)) + (6.75 + 4.50) Associative Property STEP 4
-8.50 + 11.25
Add the numbers inside the parentheses. Find the difference of the absolute values: 11.25 - 8.50
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2.75
Use the sign of the number with the greater absolute value. The sum is positive.
Tina earned a profit of $2.75.
YOUR TURN Find each sum. 12.
-1.5 + 3.5 + 2 =
13.
3_14 + (-2) + -2 _14 =
14.
-2.75 + (-3.25) + 5 =
15.
15 + 8 + (-3) =
( )
Personal Math Trainer Online Practice and Help
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Lesson 3.2
71
Guided Practice Use a number line to find each sum. (Example 1 and Example 2) 1. -3 + (-1.5) = -5 -4 -3 -2 -1
2. 1.5 + 3.5 =
(
3. _14 + _12 = -1
-5 -4 -3 -2 -1
0 1 2 3 4 5
0 1 2 3 4 5
)
4. -1_12 + -1 _12 = - 0.5
0
0.5
5. 3 + (-5) =
-5 -4 -3 -2 -1
-5 -4 -3 -2 -1
1
0 1 2 3 4 5
6. -1.5 + 4 =
-5 -4 -3 -2 -1
0 1 2 3 4 5
0 1 2 3 4 5
7. Victor borrowed $21.50 from his mother to go to the theater. A week later, he paid her $21.50 back. How much does he still owe her? (Example 3)
8. Sandra used her debit card to buy lunch for $8.74 on Monday. On Tuesday, she deposited $8.74 back into her account. What is the overall increase or decrease in her bank account? (Example 3)
Find each sum without using a number line. (Example 4)
( ) ( )
10. -3 + 1 _12 + 2 _12 =
11. -12.4 + 9.2 + 1 =
12. -12 + 8 + 13 =
13. 4.5 + (-12) + (-4.5) =
14. _14 + - _34 =
15. -4 _12 + 2 =
16.
?
( ) -8 + (-1 _18 ) =
ESSENTIAL QUESTION CHECK-IN
17. How can you use a number line to find the sum of -4 and 6?
72
Unit 1
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9. 2.75 + (-2) + (-5.25) =
Name
Class
Date
3.2 Independent Practice
Personal Math Trainer
7.NS.1a, 7.NS.1b, 7.NS.1d, 7.NS.3
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Online Practice and Help
18. Samuel walks forward 19 steps. He represents this movement with a positive 19. How would he represent the opposite of this number? 19. Julia spends $2.25 on gas for her lawn mower. She earns $15.00 mowing her neighbor’s yard. What is Julia’s profit? 20. A submarine submerged at a depth of -35.25 meters dives an additional 8.5 meters. What is the new depth of the submarine? 21. Renee hiked for 4 _34 miles. After resting, Renee hiked back along the same route for 3_14 miles. How many more miles does Renee need to hike to return to the place where she started? 22. Geography The average elevation of the city of New Orleans, Louisiana, is 0.5 m below sea level. The highest point in Louisiana is Driskill Mountain at about 163.5 m higher than New Orleans. How high is Driskill Mountain? 23. Problem Solving A contestant on a game show has 30 points. She answers a question correctly to win 15 points. Then she answers a question incorrectly and loses 25 points. What is the contestant’s final score?
Financial Literacy Use the table for 24–26. Kameh owns a bakery. He recorded the bakery income and expenses in a table. 24. In which months were the expenses greater than the income? Name the month and find how much money
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was lost. 25. In which months was the income greater than the expenses? Name the months and find how much money was gained.
Month
Income ($)
Expenses ($)
January
1,205
1,290.60
February
1,183
1,345.44
March
1,664
1,664.00
June
2,413
2,106.23
July
2,260
1,958.50
August
2,183
1,845.12
26. Communicate Mathematical Ideas If the bakery started with an extra $250 from the profits in December, describe how to use the information in the table to figure out the profit or loss of money at the bakery by the end of August. Then calculate the profit or loss.
Lesson 3.2
73
27. Vocabulary -2 is the
of 2.
28. The basketball coach made up a game to play where each player takes 10 shots at the basket. For every basket made, the player gains 10 points. For every basket missed, the player loses 15 points. a. The player with the highest score sank 7 baskets and missed 3. What was the highest score?
b. The player with the lowest score sank 2 baskets and missed 8. What was the lowest score?
c. Write an expression using addition to find out what the score would be if a player sank 5 baskets and missed 5 baskets.
FOCUS ON HIGHER ORDER THINKING
Work Area
29. Communicate Mathematical Ideas Explain the different ways it is possible to add two rational numbers and get a negative number.
31. Draw Conclusions Can you find the sum [5.5 + (-2.3)] + (-5.5 + 2.3) without performing any additions? Explain.
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Unit 1
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30. Explain the Error A student evaluated -4 + x for x = -9 _12 and got an answer of 5 _12. What might the student have done wrong?
LESSON
3.3 ?
Subtracting Rational Numbers
7.NS.1c Understand subtraction… as adding the additive inverse…. Show that the distance between two rational numbers…is the absolute value of their difference…. Also 7.NS.1
ESSENTIAL QUESTION How do you subtract rational numbers?
Subtracting Positive Rational Numbers To subtract rational numbers, you can apply the same rules you use to subtract integers. Math On the Spot
EXAMPL 1 EXAMPLE
7.NS.1
my.hrw.com
The temperature on an outdoor thermometer on Monday was 5.5 °C. The temperature on Thursday was 7.25 degrees less than the temperature on Monday. What was the temperature on Thursday? Subtract to find the temperature on Thursday. STEP 1
Find 5.5 - 7.25.
STEP 2
Start at 5.5.
STEP 3
Move | 7.25 | = 7.25 units to the left because you are subtracting a positive number.
- 6 -5 -4 - 3 - 2 - 1
0 1 2 3 4 5 6
The result is -1.75.
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The temperature on Thursday was -1.75 °C.
YOUR TURN Use a number line to find each difference. 1. -6.5 - 2 = - 9 - 8.5 - 8 - 7.5 - 7 - 6.5 - 6 - 5.5 - 5 - 4.5 - 4
2. 1 _12 - 2 =
3. -2.25 - 5.5 =
-1
0
1
2
3
4 Personal Math Trainer
- 10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1
0
Online Practice and Help
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Lesson 3.3
75
Subtracting Negative Rational Numbers To subtract negative rational numbers, move in the opposite direction on the number line. Math On the Spot my.hrw.com
EXAMPLE 2
7.NS.1
During the hottest week of the summer, the water level of the Muskrat River was _56 foot below normal. The following week, the level was _13 foot below normal. What is the overall change in the water level? Subtract to find the difference in water levels. STEP 1
Find - _13 - (- _56 ).
STEP 2
Start at - _13.
STEP 3
Move | -_56 | = _56 to the right because you are subtracting a negative number.
-1
0
1
The result is _12 . So, the water level changed 2_1 foot.
Reflect 4. Work with other students to compare addition of negative numbers on a number line to subtraction of negative numbers on a number line.
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5. Compare the methods used to solve Example 1 and Example 2.
YOUR TURN Use a number line to find each difference. 6. 0.25 - ( -1.50 ) = Personal Math Trainer Online Practice and Help
my.hrw.com
76
Unit 1
-1
7. -_12 - (-_34 ) =
0
1
-1
2
0
1
EXPLORE ACTIVITY 1
7.NS.1c
Adding the Opposite Joe is diving 2 _12 feet below sea level. He decides to descend 7 _12 more feet. How many feet below sea level is he? STEP 1
Use negative numbers to represent the number of feet below sea level.
STEP 2
Find -2 _12 - 7 _12.
STEP 3
Start at -2 _12.
STEP 4
Move | 7 _12 | = 7 _12 units to the
- 10 - 9
-8
-7 -6
-5
-4
because you are subtracting a
Reflect
-2
-1
0
number.
The result is -10 . Joe is
-3
sea level.
You move left on a number line to add a negative number. You move the same direction to subtract a positive number.
8. Compare the difference -3.5 - 5.8 to the sum -3.5 + (-5.8).
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9. Analyze Relationships Work with other students to explain how to change a subtraction problem into an addition problem.
Adding the Opposite To subtract a number, add its opposite. This can also be written as p - q = p + (-q).
Lesson 3.3
77
EXPLORE ACTIVITY 2
7.NS.1c
Finding the Distance between Two Numbers A cave explorer climbed from an elevation of -11 meters to an elevation of -5 meters. What vertical distance did the explorer climb? There are two ways to find the vertical distance.
A Start at
0
.
Count the number of units on the vertical number line up to -5. The explorer climbed
meters.
-2 -3 -4
This means that the vertical distance between -11 meters and -5 meters is
-1
meters.
B Find the difference between the two elevations and use absolute value to find the distance.
-5 -6 -7 -8 -9 - 10
-11 - (-5) =
- 11
Take the absolute value of the difference because distance traveled is always a nonnegative number.
The vertical distance is
meters.
Reflect 10. Does it matter which way you subtract the values when finding distance? Explain.
11. Would the same methods work if both the numbers were positive? What if one of the numbers were positive and the other negative?
Distance Between Two Numbers The distance between two values a and b on a number line is represented by the absolute value of the difference of a and b. Distance between a and b = | a - b | or | b - a |.
78
Unit 1
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| -11 - (-5) | =
Guided Practice Use a number line to find each difference. (Example 1, Example 2 and Explore Activity 1) 1. 5 - (-8) = 5 6 7 8 9 10 11 12 13 14 15
2. -3 _12 - 4 _12 = -9
-8
-7
-6
-5
-4
-3
3. -7 - 4 = -15 -14 -13 -12 -11 -10 -9
-8
-7
-6
-5
4. -0.5 - 3.5 = -6
-5
-4
-3
-2
-1
0
1
Find each difference. (Explore Activity 1) 5. -14 - 22 =
6. -12.5 - (-4.8) =
8. 65 - (-14) =
9. -_29 - (-3) =
7. _13 - (-_23 ) = 10. 24 _38 - (-54 _18 ) =
11. A girl is snorkeling 1 meter below sea level and then dives down another 0.5 meter. How far below sea level is the girl? (Explore Activity 1 12. The first play of a football game resulted in a loss of 12_12 yards. Then a penalty resulted in another loss of 5 yards. What is the total loss or gain? (Explore Activity 1)
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13. A climber starts descending from 533 feet above sea level and keeps going until she reaches 10 feet below sea level. How many feet did she descend? (Explore Activity 2) 14. Eleni withdrew $45.00 from her savings account. She then used her debit card to buy groceries for $30.15. What was the total amount Eleni took out of her account? (Explore Activity 1)
?
ESSENTIAL QUESTION CHECK-IN
15. Mandy is trying to subtract 4 - 12, and she has asked you for help. How would you explain the process of solving the problem to Mandy, using a number line?
Lesson 3.3
79
Name
Class
Date
3.3 Independent Practice
Personal Math Trainer
7.NS.1, 7.NS.1c
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Online Practice and Help
16. Science At the beginning of a laboratory experiment, the temperature of a substance is -12.6 °C. During the experiment, the temperature of the substance decreases 7.5 °C. What is the final temperature of the substance?
17. A diver went 25.65 feet below the surface of the ocean, and then 16.5 feet further down, he then rose 12.45 feet. Write and solve an expression to find the diver’s new depth.
Astronomy Use the table for problems 18–19. 18. How much deeper is the deepest canyon on Mars than the deepest canyon on Venus?
Elevations on Planets Lowest (ft) Highest (ft) Earth -36,198 29,035 Mars -26,000 70,000 Venus -9,500 35,000
20. A city known for its temperature extremes started the day at -5 degrees Fahrenheit. The temperature increased by 78 degrees Fahrenheit by midday, and then dropped 32 degrees by nightfall. a. What expression can you write to find the temperature at nightfall? b. What expression can you write to describe the overall change in temperature? Hint: Do not include the temperature at the beginning of the day since you only want to know about how much the temperature changed. c. What is the final temperature at nightfall? What is the overall change in temperature?
80
Unit 1
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19. Persevere in Problem Solving What is the difference between Earth’s highest mountain and its deepest ocean canyon? What is the difference between Mars’ highest mountain and its deepest canyon? Which difference is greater? How much greater is it?
21. Financial Literacy On Monday, your bank account balance was -$12.58. Because you didn’t realize this, you wrote a check for $30.72 for groceries. a. What is the new balance in your checking account? b. The bank charges a $25 fee for paying a check on a negative balance. What is the balance in your checking account after this fee? c. How much money do you need to deposit to bring your account balance back up to $0 after the fee? 22. Pamela wants to make some friendship bracelets for her friends. Each friendship bracelet needs 5.2 inches of string. a. If Pamela has 20 inches of string, does she have enough to make bracelets for 4 of her friends?
b. If so, how much string would she had left over? If not, how much more string would she need?
23. Jeremy is practicing some tricks on his skateboard. One trick takes him forward 5 feet, then he flips around and moves backwards 7.2 feet, then he moves forward again for 2.2 feet. a. What expression could be used to find how far Jeremy is from his starting position when he finishes the trick?
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b. How far from his starting point is he when he finishes the trick? Explain.
24. Esteban has $20 from his allowance. There is a comic book he wishes to buy that costs $4.25, a cereal bar that costs $0.89, and a small remote control car that costs $10.99. a. Does Esteban have enough to buy everything?
b. If so, how much will he have left over? If not, how much does he still need?
Lesson 3.3
81
FOCUS ON HIGHER ORDER THINKING
Work Area
25. Look for a Pattern Show how you could use the Commutative Property 5 7 __ __ to simplify the evaluation of the expression -16 - 4_1 - 16 .
26. Problem Solving The temperatures for five days in Kaktovik, Alaska, are given below. -19.6 °F, -22.5 °F, -20.9 °F, -19.5 °F, -22.4 °F Temperatures for the following week are expected to be approximately twelve degrees lower each day than the given temperatures. What are the highest and lowest temperatures expected for the corresponding 5 days next week?
28. Look for a Pattern Evan said that the difference between two negative numbers must be negative. Was he right? Use examples to illustrate your answer.
82
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27. Make a Conjecture Must the difference between two rational numbers be a rational number? Explain.
LESSON
3.4 ?
Multiplying Rational Numbers
7.NS.2 Apply and extend previous understandings of multiplication...and of fractions to multiply ...rational numbers. Also
7.NS.2a, 7.NS.2c
ESSENTIAL QUESTION How do you multiply rational numbers?
Multiplying Rational Numbers with Different Signs The rules for the signs of products of rational numbers with different signs are summarized below. Let p and q be rational numbers.
Math On the Spot my.hrw.com
Products of Rational Numbers Sign of Factor p
Sign of Factor q
Sign of Product pq
+
-
-
-
+
-
You can also use the fact that multiplication is repeated addition.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Sebastien Fremont/Fotolia
EXAMPL 1 EXAMPLE
7.NS.2, 7.NS.2a
Gina hiked down a canyon and stopped each time she descended 1 _ mile to rest. She hiked a total of 4 sections. What is her overall 2 change in elevation? STEP 1
Use a negative number to represent the change in elevation.
STEP 2
Find 4 -_21 .
STEP 3
Start at 0. Move _12 unit to the left 4 times.
( )
The result is -2. The overall change is -2 miles. - 3 Check:
-2
-1
0
Use the rules for multiplying rational numbers.
( ) ( )
4 -_12 = -_42
A negative times a positive equals a negative.
= -2 ✓
Simplify.
YOUR TURN 1.
Personal Math Trainer
Use a number line to find 2(-3.5).
Online Practice and Help
-8
-7
-6
-5
-4
-3
-2
-1
0
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Lesson 3.4
83
Multiplying Rational Numbers with the Same Sign The rules for the signs of products with the same signs are summarized below. Math On the Spot
Products of Rational Numbers
my.hrw.com
Sign of Factor p
Sign of Factor q
Sign of Product pq
+
+
+
-
-
+
You can also use a number line to find the product of rational numbers with the same signs.
My Notes
EXAMPLE 2
7.NS.2, 7.NS.2a
Multiply -2(-3.5). STEP 1
First, find the product 2(-3.5). + ( - 3.5) -8
-7
-6
-5
+ ( - 3.5) -4
-3
-2
STEP 2
Start at 0. Move 3.5 units to the left two times.
STEP 3
The result is -7.
STEP 4
This shows that 2 groups of -3.5 equals -7.
-1
0
So, -2 groups of -3.5 must equal the opposite of -7.
0
1
2
3
4
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84
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7
8
-2(-3.5) = 7
Check:
Use the rules for multiplying rational numbers. -2(-3.5) = 7 A negative times a negative equals a positive.
Find -3(-1.25).
Personal Math Trainer Online Practice and Help
6
STEP 5
YOUR TURN 2.
5
-4
-3
-2
-1
0
1
2
3
4
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-8 -7 -6 -5 -4 -3 -2 -1
Multiplying More Than Two Rational Numbers If you multiply three or more rational numbers, you can use a pattern to find the sign of the product.
Math On the Spot my.hrw.com
EXAMPL 3 EXAMPLE
7.NS.2, 7.NS.2c
( )(-__12 )(-__35 ).
2 Multiply -__ 3
STEP 1
STEP 2
First, find the product of the first two factors. Both factors are negative, so their product will be positive.
(-_23 ) (-_12 ) = + (_23 · _12) = _13
STEP 3
Now, multiply the result, which is positive, by the third factor, which is negative. The product will be negative.
STEP 4
1 _ _ -35 3
STEP 5
( ) = _13 (-_35)
(-_23 )(-_12)(-_35) = -_15
Reflect
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3.
Look for a Pattern You know that the product of two negative numbers is positive, and the product of three negative numbers is negative. Write a rule for finding the sign of the product of n negative numbers.
Math Talk
Mathematical Practices
Suppose you find the product of several rational numbers, one of which is zero. What can you say about the product?
YOUR TURN Find each product. 4. 5. 6.
(-_34)(-_47)(-_23) (-_23)(-_34)(_45) (_23)(-__109 )(_56)
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Lesson 3.4
85
Guided Practice Use a number line to find each product. (Example 1 and Example 2)
( )
( )
1. 5 - _2 = 3
-5
-4
2. 3 - _1 = 4
-3
-2
-1
( )
-1
- 1.5
0
1
2
5. 4(-3) =
6. -1.8(5) =
8. 0.54(8) =
9. -5(-1.2) =
-4
-3
-2 -1
- 0.5
0
× _34 =
13. - _1 × 5 × _2 = 8 3
0
1
2
3
4
7. -2 (-3.4) = 10. -2.4(3) =
Multiply. (Example 3) 11. _1 × _2 × _3 = 2 3 4
-1
4. -_34 (-4) =
3. -3 - _47 =
-2
-2
0
( )( )
12. -_4 -_3 -_7 = 7 5 3
( )
( )
× -_73 =
( )( )
1 14. -_23 _2 -_67 =
16. In one day, 18 people each withdrew $100 from an ATM machine. What is the overall change in the amount of money in the ATM machine? (Example 1)
?
ESSENTIAL QUESTION CHECK-IN
17. Explain how you can find the sign of the product of two or more rational numbers.
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15. The price of one share of Acme Company declined $3.50 per day for 4 days in a row. What is the overall change in price of one share? (Example 1)
Name
Class
Date
3.4 Independent Practice 7.NS.2, 7.NS.2a, 7.NS.2c
18. Financial Literacy Sandy has $200 in her bank account. a. If she writes 6 checks for exactly $19.98, what expression describes the change in her bank account?
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Online Practice and Help
22. Multistep For Home Economics class, Sandra has 5 cups of flour. She made 3 batches of cookies that each used 1.5 cups of flour. Write and solve an expression to find the amount of flour Sandra has left after making the 3 batches of cookies.
b. What is her account balance after the checks are cashed?
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19. Communicating Mathematical Ideas Explain, in words, how to find the product of -4(-1.5) using a number line. Where do you end up?
23. Critique Reasoning In class, Matthew stated, “I think that a negative is like an opposite. That is why multiplying a negative times a negative equals a positive. The opposite of negative is positive, so it is just like multiplying the opposite of a negative twice, which is two positives.” Do you agree or disagree with this statement? What would you say in response to him?
20. Greg sets his watch for the correct time on Wednesday. Exactly one week later, he finds that his watch has lost 3 _14 minutes. If his watch continues to lose time at the same rate, what will be the overall change in time after 8 weeks?
21. A submarine dives below the surface, heading downward in three moves. If each move downward was 325 feet, where is the submarine after it is finished diving?
24. Kaitlin is on a long car trip. Every time she stops to buy gas, she loses 15 minutes of travel time. If she has to stop 5 times, how late will she be getting to her destination?
Lesson 3.4
87
25. The table shows the scoring system for quarterbacks in Jeremy’s fantasy football league. In one game, Jeremy’s quarterback had 2 touchdown passes, 16 complete passes, 7 incomplete passes, and 2 interceptions. How many total points did Jeremy’s quarterback score?
Quarterback Scoring Action Touchdown pass Complete pass
Points 6 0.5
Incomplete pass
−0.5
Interception
−1.5
FOCUS ON HIGHER ORDER THINKING
Work Area
26. Represent Real-World Problems The ground temperature at Brigham Airport is 12 °C. The temperature decreases by 6.8 °C for every increase of 1 kilometer above the ground. What is the overall change in temperature outside a plane flying at an altitude of 5 kilometers above Brigham Airport?
a
b
c
d
+
+
+
-
28. Reason Abstractly Find two integers whose sum is -7 and whose product is 12. Explain how you found the numbers.
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27. Identify Patterns The product of four numbers, a, b, c, and d, is a negative number. The table shows one combination of positive and negative signs of the four numbers that could produce a negative product. Complete the table to show the seven other possible combinations.
Dividing Rational Numbers
LESSON
3.5 ?
7.NS.2 Apply and extend previous understandings of multiplication and division and of fractions to…divide rational numbers. Also
7.NS.2b, 7.NS.2c
ESSENTIAL QUESTION How do you divide rational numbers?
EXPLORE ACTIVITY
7.NS.2b
Placement of Negative Signs in Quotients Quotients can have negative signs in different places. Let p and q be rational numbers.
Quotients of Rational Numbers Sign of Dividend p
Sign of Divisor q
p Sign of Quotient __ q
+
-
-
-
+
-
+
+
+
-
-
+
( )
12 ____ 12 , -12 , and - ___ equivalent? Are the rational numbers ___ -4 4 4
A Find each quotient. Then use the rules in the table to make sure the sign of the quotient is correct. -12 ____ = 4
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12 ___ = -4
( )
12 - __ = 4
B What do you notice about each quotient?
C The rational numbers are / are not equivalent. D Conjecture Explain how the placement of the negative sign in the rational number affects the sign of the quotients.
E If p and q are rational numbers and q is not zero, what do you know p p -p ___ about - __q , ___ q , and -q?
()
Lesson 3.5
89
EXPLORE ACTIVITY (cont’d)
Reflect Write two equivalent quotients for each expression. 14 1. ___ -7 -32 2. ____ -8
, ,
Quotients of Rational Numbers The rules for dividing rational numbers are the same as dividing integers. Math On the Spot
EXAMPLE 1
7.NS.2c
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Over 5 months, Carlos wrote 5 checks for a total of $323.75 to pay for his cable TV service. His cable bill is the same amount each month. What was the change in Carlos’s bank account each month to pay for cable? -323.75 Find the quotient: _______ 5
STEP 1
Use a negative number to represent the withdrawal from his account each month.
STEP 2
-323.75 Find _______ . 5
STEP 3
Determine the sign of the quotient. The quotient will be negative because the signs are different.
STEP 4
Divide. -323.75 _______ = -64.75 5
YOUR TURN Find each quotient. 2.8 3. ___ = -4
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90
Unit 1
4.
-6.64 _____ = -0.4
5.
5.5 -___ = 0.5
6. A diver descended 42.56 feet in 11.2 minutes. What was the diver’s average change in elevation per minute?
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Carlos withdrew $64.75 each month to pay for cable TV.
Complex Fractions
a __
A complex fraction is a fraction that has a fraction in its numerator, denominator, or both.
a __ c b ___ __ c =b÷d __ d
EXAMPL 2 EXAMPLE
7.NS.2c, 7.NS.3
7 __ 10 ____ . - _15
A Find
Math On the Spot my.hrw.com
My Notes
STEP 1
Determine the sign of the quotient. The quotient will be negative because the signs are different.
STEP 2
10 7 Write the complex fraction as division: ____ = __ ÷ -_15 1 10 _ -5
7 __
STEP 3
7 Rewrite using multiplication: __ × (-_51 ) 10
STEP 4
35 7 __ × -_51 = - __ 10 10
( )
= - _72
Multiply by the reciprocal.
Multiply. Simplify.
7 ___ 10 ____ = - _72 1 __ 5
B Maya wants to divide a _34 -pound box of trail mix into small bags. Each bag 1 will hold __ pound of trail mix. How many bags of trail mix can Maya fill? 12 3 _
STEP 1
4 Find ___ .
STEP 2
Determine the sign of the quotient. The quotient will be positive because the signs are the same.
1 __ 12
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3 _
STEP 3
4 1 Write the complex fraction as division: ___ = _3 ÷ __ .
STEP 4
12 Rewrite using multiplication: _43 × __ 1 .
STEP 5
3 __ 36 _ × 12 = __ 4 1 4 =9
1 __ 12
4
12
Multiply by the reciprocal.
Multiply. Simplify.
3 _
4 ___ =9 1 __ 12
Maya can fill 9 bags of trail mix. Personal Math Trainer
YOUR TURN 5 -__
7. ___68 = -__ 7
8.
5 -___ 12 ____ 2 = __ 3
9.
4 -__ 5 ___ 1 = __ 2
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Lesson 3.5
91
Guided Practice Find each quotient. (Explore Activity 1 and 2, Example 1)
( ) 1 _
0.72 1. ____ = -0.9
2.
56 3. ___ = -7
251 4. ___ ÷ -_38 = 4
75 5. ___ 1= _
-91 6. ____ = -13
-_73 ___ 7. 9 =
12 8. -____ = 0.03
-5
_ 4
5 -___ = 7 _ 5
( )
9. A water pail in your backyard has a small hole in it. You notice that it has drained a total of 3.5 liters in 4 days. What is the average change in water volume each day? (Example 1)
10. The price of one share of ABC Company decreased a total of $45.75 in 5 days. What was the average change of the price of one share per day? (Example 1)
11. To avoid a storm, a passenger-jet pilot descended 0.44 mile in 0.8 minute. What was the plane’s average change of altitude per minute? (Example 1)
ESSENTIAL QUESTION CHECK-IN
32 ÷ (-2) 12. Explain how you would find the sign of the quotient _________ . -16 ÷ 4
92
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?
Name
Class
Date
3.5 Independent Practice 7.NS.2, 7.NS.2b, 7.NS.2c 5 13. ___ = 2 -__ 8
( )
14. 5_13 ÷ -1_12 = -120 15. _____ = -6 4 -__ 5 ___ 16. 2 = -__ 3
17. 1.03 ÷ (-10.3) = -0.4 18. ____ = 80
19. 1 ÷ _95 = -1 ___ 4 ___ 20. 23 = ___ 24
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-10.35 21. ______ = -2.3
22. Alex usually runs for 21 hours a week, training for a marathon. If he is unable to run for 3 days, describe how to find out how many hours of training time he loses, and write the appropriate integer to describe how it affects his time.
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Online Practice and Help
23. The running back for the Bulldogs football team carried the ball 9 times for a total loss of 15_34 yards. Find the average change in field position on each run.
24. The 6:00 a.m. temperatures for four consecutive days in the town of Lincoln were -12.1 °C, -7.8 °C, -14.3 °C, and -7.2 °C. What was the average 6:00 a.m. temperature for the four days?
25. Multistep A seafood restaurant claims an increase of $1,750.00 over its average profit during a week where it introduced a special of baked clams. a. If this is true, how much extra profit did it receive per day?
b. If it had, instead, lost $150 per day, how much money would it have lost for the week?
c. If its total loss was $490 for the week, what was its average daily change?
26. A hot air balloon descended 99.6 meters in 12 seconds. What was the balloon’s average rate of descent in meters per second?
Lesson 3.5
93
27. Sanderson is having trouble with his assignment. His shown work is as follows: 3 -__ 4 12 3 _ ___ = -__ × 43 = - __ = -1 4 12 4 __ 3
However, his answer does not match the answer that his teacher gives him. What is Sanderson’s mistake? Find the correct answer.
28. Science Beginning in 1996, a glacier lost an average of 3.7 meters of thickness each year. Find the total change in its thickness by the end of 2012.
FOCUS ON HIGHER ORDER THINKING
Work Area
30. Construct an Argument Divide 5 by 4. Is your answer a rational number? Explain.
31. Critical Thinking Should the quotient of an integer divided by a nonzero integer always be a rational number? Why or why not?
94
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29. Represent Real-World Problems Describe a real-world situation that can be represented by the quotient -85 ÷ 15. Then find the quotient and explain what the quotient means in terms of the real-world situation.
LESSON
3.6 ?
Applying Rational Number Operations
ESSENTIAL QUESTION
7.EE.3 Solve … problems … with positive and negative rational numbers in any form … using tools strategically. Also
7.NS.3
How do you use different forms of rational numbers and strategically choose tools to solve problems?
Assessing Reasonableness of Answers Even when you understand how to solve a problem, you might make a careless solving error. You should always check your answer to make sure that it is reasonable.
EXAMPL 1 EXAMPLE
Math On the Spot my.hrw.com
7.EE.3, 7.NS.3
Jon is hanging a picture. He wants to center it horizontally on the wall. The picture is 32 _12 inches long, and the wall is 120 _34 inches long. How far from each edge of the wall should he place the picture? STEP 1
STEP 2
Find the total length of the wall not covered by the picture. Subtract the whole number parts and then the fractional 120 _34 - 32 _12 = 88 _14 in. parts.
120 34 in. 32 12 in.
Find the length of the wall on each side of the picture. 1 _ 2
( 88 _14 ) = 44 _18 in.
Jon should place the picture 44 _18 inches from each edge of the wall.
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STEP 3
Check the answer for reasonableness. The wall is about 120 inches long. The picture is about 30 inches long. The length of wall space left for both sides of the picture is about 120 - 30 = 90 inches. The length left for each side is about _1 (90) = 45 inches. 2 The answer is reasonable because it is close to the estimate.
YOUR TURN 1. A 30-minute TV program consists of three commercials, each 2_12 minutes long, and four equal-length entertainment segments. How long is each entertainment segment?
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Lesson 3.6
95
Using Rational Numbers in Any Form Math On the Spot my.hrw.com
My Notes
You have solved problems using integers, positive and negative fractions, and positive and negative decimals. A single problem may involve rational numbers in two or more of those forms.
EXAMPLE 2
Problem Solving
7.EE.3, 7.NS.3
Alana uses 1_14 cups of flour for each batch of blueberry muffins she makes. She has a 5-pound bag of flour that cost $4.49 and contains seventy-six 1 _ 4 -cup servings. How many batches can Alana make if she uses all the flour? How much does the flour for one batch cost? Analyze Information
Identify the important information. • Each batch uses 1_14 cups of flour. • Seventy-six _14 -cup servings of flour cost $4.49. Formulate a Plan
Use logical reasoning to solve the problem. Find the number of cups of flour that Alana has. Use that information to find the number of batches she can make. Use that information to find the cost of flour for each batch. Justify and Evaluate Solve
Number of cups of flour in bag: 76 × _14 cup per serving = 19 cups 1 as a Write 1__ 4 decimal.
cups of flour 1.25 cups total cups of flour ÷ _________ = 19 cups ÷ _______ 1 batch batch = 19 ÷ 1.25 = 15.2
Alana cannot make 0.2 batch. The recipe calls for one egg, and she cannot divide one egg into tenths. So, she can make 15 batches. Cost of flour for each batch: $4.49 ÷ 15 = $0.299, or about $0.30. Justify and Evaluate
A bag contains about 80 quarter cups, or about 20 cups. Each batch uses about 1 cup of flour, so there is enough flour for about 20 batches. A bag costs about $5.00, so the flour for each batch costs about $5.00 ÷ 20 = $0.25. The answers are close to the estimates, so the answers are reasonable.
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Number of batches Alana can make:
YOUR TURN 2. A 4-pound bag of sugar contains 454 one-teaspoon servings and costs $3.49. A batch of muffins uses _34 cup of sugar. How many batches can you make if you use all the sugar? What is the cost of sugar for each
Personal Math Trainer Online Practice and Help
batch? (1 cup = 48 teaspoons)
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Using Tools Strategically A wide variety of tools are available to help you solve problems. Rulers, models, calculators, protractors, and software are some of the tools you can use in addition to paper and pencil. Choosing tools wisely can help you solve problems and increase your understanding of mathematical concepts.
Math On the Spot my.hrw.com
EXAMPL 3 EXAMPLE
7.EE.3, 7.NS.3
The depth of Golden Trout Lake has been decreasing in recent years. Two years ago, the depth of the lake was 186.73 meters. Since then the depth has been changing at an average rate of -1_34 % per year. What is the depth of the lake today? STEP 1
Convert the percent to a decimal. −1_34 % = −1.75% = −0.0175
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STEP 2
STEP 3
STEP 4
Write the fraction as a decimal. Move the decimal point two places left.
Find the depth of the lake after one year. Use a calculator to simplify the computations. 186.73 × (−0.0175) ≈ −3.27 meters
Find the change in depth.
186.73 − 3.27 = 183.46 meters
Find the new depth.
Find the depth of the lake after two years. 183.46 × (−0.0175) ≈ −3.21 meters
Find the change in depth.
183.46 − 3.21 = 180.25 meters
Find the new depth.
Math Talk
Mathematical Practices
How could you write a single expression for calculating the depth after 1 year? after 2 years?
Check the answer for reasonableness. The original depth was about 190 meters. The depth changed by about −2% per year. Because (−0.02)(190) = −3.8, the depth changed by about −4 meters per year or about −8 meters over two years. So, the new depth was about 182 meters. The answer is close to the estimate, so it is reasonable.
Lesson 3.6
97
YOUR TURN Personal Math Trainer
3. Three years ago, Jolene bought $750 worth of stock in a software company. Since then the value of her purchase has been increasing at an average rate of 12_35% per year. How much is the stock worth now?
Online Practice and Help
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Guided Practice 1. Mike hiked to Big Bear Lake in 4.5 hours at an average rate of 3_15 miles per hour. Pedro hiked the same distance at a rate of 3_35 miles per hour. How long did it take Pedro to reach the lake? (Example 1 and Example 2) STEP 1
Find the distance Mike hiked. 4.5 h ×
STEP 2
miles per hour =
miles
Find Pedro’s time to hike the same distance. miles ÷
miles per hour =
hours
2. Until this year, Greenville had averaged 25.68 inches of rainfall per year for more than a century. This year’s total rainfall showed a change of −2_38 % with respect to the previous average. How much rain fell this year? (Example 3) Use a calculator to find this year’s decrease to the nearest hundredth. inches × STEP 2
inches
Find this year’s total rainfall. inches −
?
≈
inches ≈
ESSENTIAL QUESTION CHECK-IN
3. Why is it important to consider using tools when you are solving a problem?
98
Unit 1
inches
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STEP 1
Name
Class
Date
3.6 Independent Practice
Personal Math Trainer
7.NS.3, 7.EE.3
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Online Practice and Help
Solve, using appropriate tools. 4. Three rock climbers started a climb with each person carrying 7.8 kilograms of climbing equipment. A fourth climber with no equipment joined the group. The group divided the total weight of climbing equipment equally among the four climbers. How much did each climber carry? 5. Foster is centering a photo that is 3_12 inches wide on a scrapbook page that is 12 inches wide. How far from each side of the page should he put the picture? 6. Diane serves breakfast to two groups of children at a daycare center. One box of Oaties contains 12 cups of cereal. She needs _13 cup for each younger child and _34 cup for each older child. Today’s group includes 11 younger children and 10 older children. Is one box of Oaties enough for everyone?
7. The figure shows how the yard lines on a football field are numbered. The goal lines are labeled G. A referee was standing on a certain yard line as the first quarter ended. He walked 41_34 yards to a yard line with the same number as the one he had just left. How far was the referee from the nearest goal
G 10 20 30 40 50 40 30 20 10 G
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Explain.
G 10 20 30 40 50 40 30 20 10 G
line? In 8–10, a teacher gave a test with 50 questions, each worth the same number of points. Donovan got 39 out of 50 questions right. Marci’s score was 10 percentage points higher than Donovan’s. 8. What was Marci’s score? Explain.
9. How many more questions did Marci answer correctly? Explain.
10. Explain how you can check your answers for reasonableness.
Lesson 3.6
99
(
)
19 . For 11–13, use the expression 1.43 × − ___ 37
11. Critique Reasoning Jamie says the value of the expression is close to −0.75. Does Jamie’s estimate seem reasonable? Explain.
12. Find the product. Explain your method.
13. Does your answer to Exercise 12 justify your answer to Exercise 11?
FOCUS ON HIGHER ORDER THINKING
Work Area
14. Persevere in Problem Solving A scuba diver dove from the surface of the 9 ocean to an elevation of -79__ feet at a rate of -18.8 feet per minute. After 10 spending 12.75 minutes at that elevation, the diver ascended to an elevation 9 of -28__ feet. The total time for the dive so far was 19_18 minutes. What was 10 the rate of change in the diver’s elevation during the ascent?
16. Represent Real-World Problems Describe a real-world problem you could solve with the help of a yardstick and a calculator.
100
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15. Analyze Relationships Describe two ways you could evaluate 37% of the sum of 27_35 and 15.9. Tell which method you would use and why.
MODULE QUIZ
Ready
Personal Math Trainer
3.1 Rational Numbers and Decimals
Online Practice and Help
Write each mixed number as a decimal. 1. 4 _15
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5 3. 5 __ 32
14 2. 12 __ 15
3.2 Adding Rational Numbers Find each sum. 4. 4.5 + 7.1 =
5. 5 _16 + (-3 _56 ) =
3.3 Subtracting Rational Numbers Find each difference. 6. - _18 - ( 6 _78 ) =
7. 14.2 - (-4.9 ) =
3.4 Multiplying Rational Numbers Multiply. 7 = 8. -4 ( __ 10 )
9. -3.2(-5.6 )( 4 ) =
3.5 Dividing Rational Numbers Find each quotient. 38 19 ÷ __ = 10. - __ 7 2
- 32.01 = 11. ______ -3.3
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3.6 Applying Rational Number Operations 12. Luis bought stock at $83.60. The next day, the price increased $15.35. This new price changed by -4 _34% the following day. What was the final stock price? Is your answer reasonable? Explain.
ESSENTIAL QUESTION 13. How can you use negative numbers to represent real-world problems?
Module 3
101
MODULE 3 MIXED REVIEW
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Assessment Readiness
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Online Practice and Help
1. Consider each expression. Is the value of the expression negative? Select Yes or No for expressions A–C. A. -_12 ÷ (-8) B. -_34 × _58 C. -0.7 - (-0.62)
Yes Yes Yes
No No No
2. Randall had $75 in his bank account. He made 3 withdrawals of $18 each. Choose True or False for each statement. A. The change in Randall’s balance is -$54. B. The account balance is equal to $75 - 3(-$18). C. Randall now has a negative balance.
True True True
False False False
4. A butcher has 10_34 pounds of ground beef that will be priced at $3.40 per pound. He divides the meat into 8 equal packages. To the nearest cent, what will be the price of each package? Explain how you know that your answer is reasonable.
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3. The water level in a lake was 12 inches below normal at the beginning of March. The water level decreased by 2_14 inches in March and increased by 1_58 inches in April. What was the water level compared to normal at the end of April? Explain how you solved this problem.
UNIT 1
Study Guide MODULE
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1
Review
Adding and Subtracting Integers
Key Vocabulary additive inverse (inverso aditivo)
ESSENTIAL QUESTION
How can you use addition and subtraction of integers to solve real-world problems? EXAMPLE 1 Add. A.
B.
−8 + (−7)
The signs of both integers are the same.
8 + 7 = 15
Find the sum of the absolute values.
−8 + (−7) = −15
Use the sign of integers to write the sum.
− 5 + 11
The signs of the integers are different.
| 11 |
− | −5 | = 6
−5 + 11 = 6
Greater absolute value − lesser absolute value. 11 has the greater absolute value, so the sum is positive.
EXAMPLE 2 The temperature Tuesday afternoon was 3 °C. Tuesday night, the temperature was −6 °C. Find the change in temperature. Find the difference −6 − 3. Rewrite as −6 + (−3).
−3 is the opposite of 3.
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−6 + (−3) = −9 The temperature decreased 9 °C.
EXERCISES Add. (Lessons 1.1, 1.2) 1. −10 + (−5)
2. 9 + (−20)
3. −13 + 32
5. 25 − (−4)
6. −3 − (−40)
Subtract. (Lesson 1.3) 4. −12 − 5
7. Antoine has $13 in his checking account. He buys some school supplies and ends up with $5 in his account. What was the overall change in Antoine’s account? (Lesson 1.4) Unit 1
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MODULE
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2
Multiplying and Dividing Integers
ESSENTIAL QUESTION
How can you use multiplication and division of integers to solve real-world problems? EXAMPLE 1 Multiply. A. (13)(−3)
B. (−5)(−8)
Find the sign of the product. The numbers have different signs, so the product will be negative. Multiply the absolute values. Assign the correct sign to the product. 13(−3) = −39
Find the sign of the product. The numbers have the same sign, so the product will be positive. Multiply the absolute values. Assign the correct sign to the product. (−5)(−8) = 40
EXAMPLE 2
EXAMPLE 3
Christine received −25 points on her exam for 5 wrong answers. How many points did Christine receive for each wrong answer?
Simplify: 15 + (−3) × 8 15 + (−24) −9
Divide −25 by 5.
Multiply first. Add.
The signs are different. The quotient is negative. Christine received −5 points for each wrong answer.
EXERCISES Multiply or divide. (Lessons 2.1, 2.2) 1. -9 × (-5)
2. 0 × (-10)
3. 12 × (-4)
4. -32 ÷ 8
5. -9 ÷ (-1)
6. -56 ÷ 8
8. 8 + (-20) × 3
9. 36 ÷ (-6) × -15
Simplify. (Lesson 2.3) 7. -14 ÷ 2 - 3
10. Tony bought 3 packs of pencils for $4 each and a pencil box for $7. Mario bought 4 binders for $6 each and used a coupon for $6 off. Write and evaluate expressions to find who spent more money. (Lesson 2.3)
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−25 ÷ 5 = −5
11. Sumaya is reading a book with 288 pages. She has already read 90 pages. She plans to read 20 more pages each day until she finishes the book. Determine how many days Sumaya will need to finish the book. In your answer, count part of a day as a full day. Show that your answer is reasonable.
MODULE
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3 1
Rational Numbers
Key Vocabulary
ESSENTIAL QUESTION
How can you use rational numbers to solve real-world problems? EXAMPLE 1
rational number (número racional) repeating decimal (decimal periódico) terminating decimal (decimal finito)
Eddie walked 1_23 miles on a hiking trail. Write 1_23 as a decimal. Use the decimal to classify 1_23 according to the number group(s) to which it belongs. 1_23 = _53
1.66 ⎯ 3⟌ 5.00 -3 20 -1 8 20 -18 2
2 Write 1__ as an improper 3 fraction.
Divide the numerator by the denominator.
_
The decimal equivalent of 1_23 is 1.66…, or 1.6. It is a repeating decimal, and therefore can be classified as a rational number.
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EXAMPLE 2 Find each sum or difference. A. -2 + 4.5 -5 -4 -3 -2 -1
0 1 2 3 4 5
Start at -2 and move 4.5 units to the right: -2 + 4.5 = 2.5. B. - _25 - (- _45 ) -1
0
1
Start at - _25 . Move | - _54 | = _45 unit to the right because you are subtracting a negative number: - _25 - (- _45 ) = _25 .
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EXAMPLE 3 1 - __ 2 . Find the product: 3(- __ 6 )( 5 )
3(- _16 ) = - _12
Find the product of the first two factors. One is positive and one is negative, so the product is negative.
- _12(- _25 ) = _15
Multiply the result by the third factor. Both are negative, so the product is positive.
3(- _16 )(- _25 ) = _15 EXAMPLE 4 15.2 Find the quotient: ____ . −2 15.2 ____ = -7.6 -2
The quotient is negative because the signs are different.
EXAMPLE 5 A lake’s level dropped an average of 3_45 inches per day for 21 days. A heavy rain then raised the level 8.25 feet, after which it dropped 9_12 inches per day for 4 days. Jayden says that overall, the lake level changed about -1_12 feet. Is this answer reasonable? Yes; the lake drops about 4 inches, or _13 foot, per day for 21 days, rises about 8 feet, then falls about _34 foot for 4 days: -_13(21) + 8 - _34 (4) = -7 + 8 - 3 = -2 feet.
1. _34
2. _82
11 3. __
4. _52
3
Find each sum or difference. (Lessons 3.2, 3.3)
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5. -5 + 9.5
6. _16 + (- _56 )
8. -3 - (-8)
9. 5.6 - (-3.1)
Unit 1
7. -0.5 + (-8.5) 10. 3_12 - 2_14
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EXERCISES Write each mixed number as a whole number or decimal. Classify each number according to the group(s) to which it belongs: rational numbers, integers, or whole numbers. (Lesson 3.1)
11. Jorge records his hours each day on a time sheet. Last week when he was ill, his time sheet was incomplete. If Jorge worked a total of 30.5 hours last week, how many hours are missing? Show your work. Then show that your answer is reasonable.
Mon
Tues
Wed
8
7_14
8_12
Thurs
Fri
Find each product or quotient. (Lessons 3.4, 3.5) 12. -9 × (-5)
13. 0 × (-7)
14. -8 × 8
-56 15. ____ 8
-130 16. _____ -5
34.5 17. ___ 1.5
18. - _25 (- _12 )(- _56 )
19. (_15 )(- _57 )(_34 )
20. Lei withdrew $50 from her bank account every day for a week. What was the change in her account in that week?
21. Dan is cutting 4.75-foot lengths of twine from a 240-foot spool of twine. He needs to cut 42 lengths and says that 40.5 feet of twine will remain. Show that this is reasonable.
22. Jackson works as a veterinary technician and earns $12.20 per hour.
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a. Jackson normally works 40 hours a week. In a normal week, what is his total pay before taxes and other deductions?
b. Last week, Jackson was ill and missed some work. His total pay before deductions was $372.10. Write and solve an equation to find the number of hours Jackson worked.
23. When Jackson works more than 40 hours in a week, he earns 1.5 times his normal hourly rate for each of the extra hours. Jackson worked 43 hours one week. What was his total pay before deductions? Justify your answer.
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Unit
Project
7.NS.1, 7.NS.2
It’s Okay to Be Negative! In 2012, film director James Cameron piloted a craft called the Deepsea Challenger to the deepest point in the Pacific Ocean, 6.8 miles below sea level. The descent took 2.6 hours. You can use this information and operations with negative rational numbers to find the average rate of descent. distance -6.8 mi rate = _______ = ______ ≈ -2.62 mi/h time 2.6 h
For this project, you can use any source in which you can find real-world data involving negative rational numbers. Create a presentation of four original real-world math problems involving negative rational numbers, including their solutions.
MATH IN CAREERS
ACTIVITY
Urban Planner Armand is an urban planner, and he has proposed a site for a new town library. The site is between city hall and the post office on Main Street. City hall
Library site
Post office
The distance between city hall and the post office is 6.5 miles. City hall is 1.25 miles closer to the library site than it is to the post office. Determine the distance from city hall to the library site and the distance from the post office to the library site.
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© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Alexis Rosenfeld / Science Source
You should have one problem for each operation: addition, subtraction, multiplication, and division. Your presentation should include the sources for the information that you used. Use the space below to write down any questions you have or important information from your teacher.
UNIT 1 MIXED REVIEW
Personal Math Trainer
Assessment Readiness
my.hrw.com
Online Practice and Help
1. Look at each number or expression. Is its value equal to the rational -7 number ___ ? 12 Select Yes or No. A. B. C.
7 ____ -12
-7 ÷ 12 0.583
Yes Yes Yes
No No No
2. Choose True or False for A–D to indicate whether the model can represent 4 - (-3). A.
True
False
B.
True
False
True True
False False
0 1 2 3 4 5 6 7 8
C. | 4 | - | -3 | D. 4 + 3
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3. A submarine starts at an elevation of –35 meters compared to sea level. It then makes 3 equal descents of 145 meters each. Is the final depth of the submarine greater than 500 meters? Explain how you solved this problem.
4. A website sells used comic books. The table shows how the price of Painted Tiger, Volume 1, has changed over the last 4 months. If the price started at $7.37, what is the current price? Explain how you know that your answer is reasonable.
Month
Change in Price ($)
1
+0.50
2
-0.38
3
+0.32
4
-0.12
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Performance Tasks 5. A trail has markers every _18 mile. Jody starts at the 2_14 -mile marker, hikes to the 4_38 -mile marker, and then hikes back to the 1_12-mile marker. Did Jody hike more than 4 miles? Explain.
6. The table shows the bills that Rosemary has this month. She begins the month with $54.30 in her bank account. a. Without any deposits, what would be her bank account balance after paying her bills? What does the sign of your answer indicate?
Bill
Amount ($)
Phone
27.56
Dance uniform
65.95
Dance shoes
55.00
b. Rosemary will babysit for a total of 14.5 hours this month. She earns $8 per hour. What would be the account balance after depositing all that she earns and paying all her bills?
7. The table shows the weekly change in the water level in a swimming pool. After 4 weeks, the pool owner adds water to return the water level back to the original level. Raising the water level by 1 inch requires 320 gallons. The water hose has a flow rate of 5.75 gallons of water per minute. Will it take less than an hour to fill the pool back to return the water level to the original level? Explain your reasoning. Week Change (in.)
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1
2
-_12
+_34
3
4
-1_18 -1_38
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c. Rosemary wants to buy a gift for her little brother this month. Should she buy a baseball glove for $22.95 or a T-shirt for $13.95? Use mathematics to justify your choice.
