WorkinG With rational nUMbers
Math Grade 7 Unit 1
Working with Rational Numbers
Exercises
Grade 7 Unit 1: Working with Rational Numbers
CONTENTS
Exercises Exercises
LESSON 1: UP, UP, AND AWAY! ������������������������������������������������������������������� 5 LESSON 2: MODELING INTEGER ADDITION ���������������������������������������� 6 LESSON 3: SUBTRACTION AS “TAKING AWAY” ���������������������������������� 9 LESSON 4: SUBTRACTION AS DISTANCE ��������������������������������������������� 13 LESSON 5: ADDING AND SUBTRACTING �������������������������������������������� 15 LESSON 6: PROPERTIES OF OPERATIONS ���������������������������������������������� 17 LESSON 7: PUTTING IT TOGETHER 1 ����������������������������������������������������� 19 LESSON 11: MULTIPLYING INTEGERS ������������������������������������������������������ 20 LESSON 12: PROVING RULES FOR MULTIPLYING ������������������������������� 22 LESSON 13: DIVIDING ������������������������������������������������������������������������������������ 24 LESSON 14: THE DISTRIBUTIVE PROPERTY ������������������������������������������ 26 LESSON 15: RATIONAL NUMBERS ����������������������������������������������������������� 28 LESSON 16: PUTTING IT TOGETHER 2 �������������������������������������������������� 30 answers
LESSON 2: MODELING INTEGER ADDITION �������������������������������������� 32 LESSON 3: SUBTRACTION AS “TAKING AWAY” �������������������������������� 34 LESSON 4: SUBTRACTION AS DISTANCE ��������������������������������������������� 36 LESSON 5: ADDING AND SUBTRACTING �������������������������������������������� 38
Grade 7 Unit 1: Working with Rational Numbers
CONTENTS
Exercises answers
LESSON 6: PROPERTIES OF OPERATIONS ���������������������������������������������� 39 LESSON 11: MULTIPLYING INTEGERS ������������������������������������������������������ 41 LESSON 12: PROVING RULES FOR MULTIPLYING ������������������������������� 43 LESSON 13: DIVIDING ������������������������������������������������������������������������������������44 LESSON 14: THE DISTRIBUTIVE PROPERTY ������������������������������������������ 45 LESSON 15: RATIONAL NUMBERS ����������������������������������������������������������� 47
Grade 7 Unit 1: Working with Rational Numbers
Lesson 1: UP, UP, AND AWAY!
Exercises
• Write your wonderings about working with rational numbers. • Write a goal stating what you plan to accomplish in this unit. • Based on your previous work, write three things you will do differently during this unit to increase your success.
Grade 7 Unit 1: Working with Rational Numbers
Lesson 2: MODELING INTEGER ADDITION
Exercises
Exercises 1. 7 + –7 = A –14 B –7 C 0 D 14 2. Use this number line to show 3 + (–7).
–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0
3. Use this number line to show –4 + (–2).
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9 10
Grade 7 Unit 1: Working with Rational Numbers
Lesson 2: MODELING INTEGER ADDITION
10 9 8 7 6 5 4 3 2 1 0 –1 –2 –3 –4 –5 –6 –7 –8 –9 –10
4. –4 + 11 = 5. –9 + (–16) = 6. –8 + 3 = 7. 15 + 4 = 8. –15 + 32 = 9. –38 + (–56) = 10. 13 + (–13) =
Exercises
Grade 7 Unit 1: Working with Rational Numbers
Lesson 2: MODELING INTEGER ADDITION
Exercises
11. The temperature at 6 a.m. was –7°F. By noon, the temperature had risen by 10°F. At noon, what was the temperature in degrees Fahrenheit?
12. A 40 ft utility pole was vertically placed in a hole 6 ft deep. What is the height of the portion of the pole that is above ground?
13. You can use a negative number to represent a situation about debt. For example, Marcus owes his father $7. This situation can be represented as –7. a. Marcus borrows $5 from his sister. Write an equation to represent his total debt (that is, the total amount he owes his father and his sister).
b. Marcus earns $10 shoveling his neighbor’s driveway. He uses all of the $10 to pay back some of his debt. Write an equation to represent what the amount of his new debt will be.
14. Describe a situation that can be represented by the equation –6 + 6 = 0.
15. Describe a situation that can be represented by the equation –10 + (–10) = –20.
16. Describe a situation that can be represented by the equation 25 + (–5) = 20.
Challenge Problem 17. For what integer values of a and b is |a + b| = |a| + |b|?
Grade 7 Unit 1: Working with Rational Numbers
Lesson 3: SUBTRACTION AS “TAKING AWAY”
Exercises
Exercises 1. 3 – 4 = A –7 B –1 C 1 D 7 2. Use this number line to show –4 – (–8).
–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0
3. Use this number line to show –1 – 6.
1
2
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9 10
Grade 7 Unit 1: Working with Rational Numbers
Lesson 3: SUBTRACTION AS “TAKING AWAY”
10 9 8 7 6 5 4 3 2 1 0 –1 –2 –3 –4 –5 –6 –7 –8 –9 –10
4. 5 – 11 = 5. 9 – (–6) = 6. –7 – 6 = 7. –3 – (–3) = 8. 15 – 32 = 9. –42 – (–26) = 10. 85 – 93 =
Exercises
Grade 7 Unit 1: Working with Rational Numbers
Lesson 3: SUBTRACTION AS “TAKING AWAY”
Exercises
11. The temperature was 4°F at 11 p.m. Overnight the temperature fell by 9°F. What was the morning temperature in degrees Fahrenheit?
12. Sophie’s credit card statement said she owed $75. Then the credit card company said it make a mistake and took away $15 of this debt. a. Write a subtraction expression to represent this situation. b. How much does Sophie owe now?
13. Atoms contain particles called protons and electrons. A proton has a positive charge and an electron has a negative charge. A neutral atom has a total charge of 0. Sometimes an atom gains or loses 1 or more electrons and becomes a charged particle called an ion. a. An oxygen atom has 8 protons, with a combined charge of 8, and 8 electrons, with a combined charge of –8. Write an equation to show that the total charge for an oxygen atom is 0.
b. Suppose the oxygen atom from Exercise 13a gains 2 electrons. What would be the charge of the resulting oxygen ion? Write an equation to show your answer.
c. An aluminum atom has 13 protons and 13 electrons. Suppose an aluminum atom loses 3 of its electrons. What would be the charge of the resulting ion? Write an equation to show your answer.
14. A hiker is in a canyon, at an elevation of –14 ft (that is, 14 ft below sea level). She descends 37 more ft. What is her elevation now?
Grade 7 Unit 1: Working with Rational Numbers
Lesson 3: SUBTRACTION AS “TAKING AWAY”
Exercises
15. Describe a situation that can be represented by the equation –6 – 6 = –12.
16. Describe a situation that can be represented by the equation 10 – 15 = –5.
Challenge Problem 17. For what integer values of a and b is |a – b| = a – b?
Grade 7 Unit 1: Working with Rational Numbers
Lesson 4: SUBTRACTION AS DISTANCE
Exercises
Exercises For Exercises 1–5, use this number line. –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0
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9 10
1. What is the distance between –5 and 0? 2. What is the distance between 5 and –7? 3. What is the distance between –8 and –3? 4. Explain how to use subtraction to find the distance between 6 and –4. 5. Explain how to use subtraction to find the distance between –8 and –1. 6. What is the distance between –43 and 72 on a number line? 7. What is the distance between –561 and –305 on a number line? 8. If a – b = –12, what is the value of b – a? 9. When does |a – b| = –(a – b)? 10. In the state of California, the highest point is located on Mount Whitney and the lowest point is located in Death Valley. Mount Whitney has an elevation of 14,494 ft. Death Valley has an elevation of –282 ft. What is the difference in elevation between the two locations?
11. In Marcus’s town, the school, the library, and the park are all on the same street. The school is 13 blocks west of the park and the library is 5 blocks west of the park. What is the distance between the library and the school?
Grade 7 Unit 1: Working with Rational Numbers
Lesson 4: SUBTRACTION AS DISTANCE
Exercises
12. Karen’s house, the fire station, and the grocery store are all on the same street. The fire station is 8 blocks north of Karen’s house. The grocery store is 12 blocks south of Karen’s house. How far apart are the fire station and the grocery store?
13. The highest and lowest recorded temperatures in the state of Iowa are 118°F and –47°F. What is the difference between these two temperatures?
Challenge Problem 14. A rectangle has vertices (–2, –6), (4, –6), (–2, –9), and (4, –9). What is the area of the rectangle?
Grade 7 Unit 1: Working with Rational Numbers
Lesson 5: ADDING AND SUBTRACTING
Exercises
Exercises 1. 3
1 1 + –3 = 2 2
2. –0.64 + 1.7 = 3. 4
2 3 –7 = 3 5
4. –5.03 – (–2.92) = 5.
4 3 + – = 7 4
6. –0.5 – 0.8 = 7. –
5 5 + – = 8 6
8. Maya owes her mother $26.75. a. Maya borrows another $5.50 from her mother to buy a ticket to the school play. Write an equation to represent Maya’s total debt to her mother.
b. Maya earned $12 babysitting and used all of that money to repay some of her debt to her mother. Write an equation to represent Maya’s new debt.
9. The city of Chicago has a biking and walking path along its lakefront. a. Jack is on the path at the Oak Street Beach. The north end of the path is 7.55 mi from him and the south end of the path is 10.76 mi from him. What is the total length of the path?
b. Doggie Beach is 3.21 mi north of Jack. Foster Beach is 6.55 mi north of him. How far is it from Doggie Beach to Foster Beach?
c. The Shedd Aquarium is 3.29 mi south of Daniel. The Theater on the Lake is 1.81 mi north of him. What is the distance between the aquarium and the theater?
Grade 7 Unit 1: Working with Rational Numbers
Lesson 5: ADDING AND SUBTRACTING
Exercises
10. In December at the Amundsen-Scott South Pole Station, the mean daily low temperature is –29.3°C and the mean daily high temperature is –26.5°C. What is the difference between these two temperatures?
Challenge Problem 11. Write a word problem that requires adding a negative number and a positive number.
Grade 7 Unit 1: Working with Rational Numbers
Lesson 6: PROPERTIES OF OPERATIONS
Exercises
Exercises 1. What property tells us that 6
1 1 +0=6 ? 2 2
A Additive identity property B Inverse property of addition C Commutative property of addition D Associative property of addition
2. What is the additive inverse of
7 ? 9
3. What is the additive inverse of –0.053?
4. What is the additive inverse of –a?
In Exercises 5 and 6, choose the property that justifies the step. Next to each step, there should be a drop-down menu from which students choose from the following: Additive Identity Property, Additive Inverse Property, Commutative Property of Addition, Associative Property of Addition 2 1 5 2 1 5 5. 4 + 3 + 6 = 4 + 3 + 6 8 3 3 8 3 3 5 = 4 + 10 8 = 14
5 8
6. (3.8 – 4.5) – 3.8 = (3.8 + (–4.5)) + (–3.8) = (–4.5 + 3.8) + (–3.8) = –4.5 + (3.8 + (–3.8)) = –4.5 + 0 = –4.5
Grade 7 Unit 1: Working with Rational Numbers
Lesson 6: PROPERTIES OF OPERATIONS
Exercises
7. Find the error(s) in Marcus’s calculation. Calculate the correct value. 5 + 4 – (–5) = 5 + (–5) – 4 = 0 – 4 = 4 – 0 =4
8. Find the error(s) in Lucy’s calculation. Calculate the correct value. 12 – (12 + 17) = (12 – 12) + 17 = 0 + 17 = 17
9. Find the error(s) in Jack’s calculation. Calculate the correct value. 20 – 12.5 + 7.5 = 20 – (12.5 + 7.5) = 20 – 20 =0 In Exercises 10–12, simplify the expression. 4 3 2 10. 3 + 1 + 2 − 5 9 5
4 9
11. 16 – 9 – 7 + 20 – 9
12. (–0.09 + 0.73) + (–1 + 0.27)
13. Circle all equivalent expressions. 5 – 12 + 8 – 3
3 – 8 + 12 – 5
8 – 12 + 5 – 3
Challenge Problem 14. Use properties to prove that the expressions you selected in Exercise 12 are equivalent.
Grade 7 Unit 1: Working with Rational Numbers
Lesson 7: Putting It Together 1
Exercises
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Read through your work on the Self Check task and think about your other work in this lesson.
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Write what you have learned.
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What would you do differently if you were starting the Self Check task now?
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Record your ideas. Keep track of any strategies you have learned.
•
Complete any exercises that you have not finished from this unit.
Grade 7 Unit 1: Working with Rational Numbers
Lesson 11: MULTIPLYING INTEGERS
Exercises
Exercises 1. 2(– 7) = A –14 B –7 C 7 D 14 2. Use this number line to show 4(–2).
–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0
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3. Find each of these products. a. 4 • –9 = b. 3 • –9 = c. 2 • –9 = d. 1 • –9 = e. 0 • –9 = 4. Describe the pattern you see in the equations in Exercise 3.
5. Extend the pattern you described in Exercise 4 to find these products: a. –1 • –9= b. –2 • –9= c. –3 • –9= 6. Use a pattern like the one used in Exercise 5 to show that –4 • –4 = 16.
7. –4 • 11 = 8. –9(–8) = 9. 8(–3) =
Grade 7 Unit 1: Working with Rational Numbers
Lesson 11: MULTIPLYING INTEGERS
Exercises
10. –15(–20) = 11. 33(3) = 12. (–2)(–2)(–2) = 13. (–3)(3)(–3) = 14. The temperature at midnight is 0°F. The temperature drops 2°F every hour for the next 7 hours. Write an equation for the temperature at 7 a.m.
15. Sophie saves $5 of her allowance each week. a. What does the expression 20 • 5 represent in this situation? b. What does the expression -6 dot symbol 5 represent in this situation?
16. Karen spends $9 of her savings each week to go to the movies. a. What does the expression 6(–9) represent in this situation? b. What does the expression –2(–9) represent in this situation?
Challenge Problem 17. Consider the expression a(b + 5), where a and b are integers. a. When will the value of the expression be positive? b. When will the value of the expression be negative?
Grade 7 Unit 1: Working with Rational Numbers
Lesson 12: PROVING RULES FOR MULTIPLYING
Exercises
Exercises 1 1. –1 • = 2 2. –1 • (–4.09) = In Problem 3, a drop-down menu should be included next to each step. The menu should contain these choices: Multiplicative Identity Property, Multiplication Property of –1, Associative Property of Multiplication, and Commutative Property of Multiplication. 1 1 1 • 3, which is . Select a property These steps show that – • –3 equals 3 9 that justifies each step. 9 1 1 − • −3 = −1• − 11• (•−−1• 3) −1• 11 • ( −1• 3) 911 • −33 = −1• 911 • ( −1• 3) = 9 9− − − 99 •• − −33 = = − −1• −1• 1• 9 •• (( − 1• 33)) 9 1 99 1 = −1• • −1 • 3= −1• 1 • −1 • 3 9 = −1• 911 • −1 • 3 = =− −1• −11 •• 33 1• 99 •• − 9 1 1 = −1• −1• • 3= −1• −1• 1 • 3 = −1• 1• −1• 1• 911 • 3 9 = =− −1• − −1• 99 •• 33 9 1 = ( −1• −1) • • 3=( −1• −1) • 11 • 3 9 =( −1• −1) • 911 • 3 = (( − −1• 1• − −11)) •• 9 •• 33 = 9 1 1 1 9 = 1• • 3 = 1• 9 = 1• 911 ••• 333 = 1• = 1• 99 • 3 1 11 9 = •3 = 1 •3 = 9 91 • 3 = = 99 1•• 33 =9 • 3 9 3. Which property tells you that –1(–3 + 7) = (–1)(–3) + (–1)(7)? A Associative property of multiplication B Additive identity property C Multiplication property of –1 D Distributive property 4. 18 • –0.1 = 5. –
3 2 •– = 4 3
Grade 7 Unit 1: Working with Rational Numbers
Lesson 12: PROVING RULES FOR MULTIPLYING
Exercises
6. –1.2 • 0.09 = 7. 7 • –45.98 = 8.
1 1 1 1 •– • •– = 2 3 4 5
9. Maya begins a hike at an elevation of 0 ft. Each half hour, her elevation decreases by 13 ft. What is her elevation after 2 hr?
Challenge Problem 10. Explain why the product of a positive number and a negative number is negative.
Grade 7 Unit 1: Working with Rational Numbers
Lesson 13: DIVIDING
Exercises
Exercises 1. –1 ÷ 1 = 2. Write a division expression to represent this situation: Lucy’s book is due to the library today, but she has not finished reading it. She will owe the library 12¢ for every day her book is late. If she does not return the book, after how many days will she owe the library 72¢? 3. Explain how you can use the rules for finding the sign of a product to find the sign of a quotient.
4. 34 ÷ –17 = 5. –42 ÷ –6 = 6. –9 ÷ 18 = 7. –
5 5 ÷ = 12 6
8. –0.4 ÷ –0.02 = 9. 55 ÷ –0.5 = 10. Is it possible for two numbers to have a product that is positive and a quotient that is negative? Explain.
11. Sophie hiked from an elevation of –8 ft to an elevation of –72 ft in 16 min. On average, how much did her elevation change each minute?
Grade 7 Unit 1: Working with Rational Numbers
Lesson 13: DIVIDING
Exercises
Challenge Problem 12. Find numbers a and b that satisfy the given criteria. If it is impossible to find numbers a and b for the given criteria, explain why. a. b. c. d. e. f.
a • b > a ÷ b and a and b are both positive. a • b < a ÷ b and a and b are both positive. a • b > a ÷ b and a and b are both negative. a • b < a ÷ b and a and b are both negative. a • b > a ÷ b, a is negative, and b is positive. a • b < a ÷ b, a is negative, and b is positive.
Grade 7 Unit 1: Working with Rational Numbers
Lesson 14: THE DISTRIBUTIVE PROPERTY
Exercises
Exercises 1. Which expression is equivalent to 3 • 5.7 + 3 • 4.3? A 3 • 5.7 • 4.3 B 3 • 5.7 + 4.3 C 5.7(3 + 4.3) D 3(5.7 + 4.3) 2. A popsicle costs $0.88. Jack wants to buy a popsicle for himself and his 4 cousins. Explain how he can use the distributive property to calculate the total cost.
For Exercises 3–5, find the product. Use the distributive property to simplify the calculation. 3. 11($7.95) 2 4. 4 5 9 5. 3.09 • 12 3 1 1 4 • •4 + 5•− • 4 592 12 5 For Exercises 9 6–9, simplify each expression. Use the properties of addition and multiplication to simplify the calculations. 6.
3 1 1 4 • •4 + 5•− • 4 9 12 5
7. 24 • 137 + 63 • 24 8. (5.4 – 6.3) ÷ 9 9. –0.007 • 0.4 • –0.02 • 10 • 1,000 • 100
Grade 7 Unit 1: Working with Rational Numbers
Lesson 14: THE DISTRIBUTIVE PROPERTY 10. Which expression has the same value as 12 – (6 – 9)? A 12 – 6 – 9 B 12 – 6 + 9 C 12 + 6 – 9 D 12 – (9 – 6)
Challenge Problem 11. Solve each equation. Use the properties of operations to help you. 2 • x • 12 = –32 3 b. –5x + 11x = 54
a. –
c. –2(x + 3) = 10 d. –8x + 64 = 0
Exercises
Grade 7 Unit 1: Working with Rational Numbers
Lesson 15: RATIONAL NUMBERS
Exercises
Exercises 1. Show that 2 is a rational number by writing it as a ratio of integers.
For Exercises 2–7, write each number as a ratio of integers. 2. 0.03 3. –9 4. 28% 5. 4.7 6. 0.4444 7. 0.3
For Exercises 8–11, write each number as a terminating or repeating decimal. 8. – 9.
2 9
10. 1 11.
7 8
3 20
5 12
Grade 7 Unit 1: Working with Rational Numbers
Lesson 15: RATIONAL NUMBERS
Exercises
12. a. Write each of these fractions as a decimal. 1 = 9 1 = 99 1 = 999 1 9, 999 = b. Describe the pattern you see in your results for Exercise 12a.
c. Predict the decimal form of
1 . Check your prediction by dividing. 99, 999
Challenge Problem 13. a. Use a calculator to find the decimal equivalents of all the unit fractions from 1 1 to . 2 12
b. What do you think determines whether the decimal form of a unit fraction will terminate or repeat? Test your idea on other unit fractions. (Hint: consider the factors of the denominators)
Grade 7 Unit 1: Working with Rational Numbers
Lesson 16: PUTTING IT TOGETHER 2
Exercises
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Read through your work on the Self Check task and think about your other work in this lesson.
•
Write what you have learned.
•
What would you do differently if you were starting the Self Check task now?
•
Record your ideas. Keep track of any strategies you have learned.
•
Complete any exercises that you have not finished from this unit.
Math Grade 7 Unit 1
Working with Rational Numbers
ANSWERS Exercises FOR EXERCISES
Grade 7 Unit 1: Working with Rational Numbers
Lesson 2: MODELING INTEGER ADDITION
answers
ANSWERS 1. C 0 2. –4
–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0
3. –6 10 9 8 7 6 5 4 3 2 1 0 –1 –2 –3 –4 –5 –6 –7 –8 –9 –10
4. 7 5. –25
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9 10
Grade 7 Unit 1: Working with Rational Numbers
Lesson 2: MODELING INTEGER ADDITION
answers
6. –5 7. 19 8. 17 9. –94 10. 0 11. At noon, the temperature was 3°F. –7 + 10 = 3 12. The distance from the ground to the top of the pole is 34 ft. 40 + –6 = 34 13. a. An equation that represents his total debt is –7 + (–5) = –12. b. An equation that represents his new debt is –12 + 10 = –2. 14. Answers will vary. Ask a classmate to read your situation. Possible answer: The temperature was –6°F. Then it rose by 6°F. Now the temperature is 0°F. 15. Answers will vary. Ask a classmate to read your situation. Possible answer: Lucy has a balance of –$10 on her credit card (that is, her debt is $10). Then she charges another item that costs $10. Lucy now has a balance of –$20 on her credit card. That is, her debt is $20. 16. Answers will vary. Ask a classmate to read your situation. Possible answer: Jack’s checking account balance was $25. Then the bank charged him a $5 fee. Now his balance is $20.
Challenge Problem 17. For the equation |a + b| = |a| + |b|, the integer values for a and b are both positive or both negative, or at least one of the integers is 0.
Grade 7 Unit 1: Working with Rational Numbers
Lesson 3: SUBTRACTION AS “TAKING AWAY”
answers
ANSWERS 1. B –1 2. 4
–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0
3. –7 10 9 8 7 6 5 4 3 2 1 0 –1 –2 –3 –4 –5 –6 –7 –8 –9 –10
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9 10
Grade 7 Unit 1: Working with Rational Numbers
Lesson 3: SUBTRACTION AS “TAKING AWAY”
answers
4. –6 5. 15 6. –13 7. 0 8. –17 9. –16 10. –8 11. The morning temperature is –5°F. 4 – 9 = 4 + (–9) = –5 12. a. A subtraction expression that represents this situation is –75 – (–15). b. Sophie now owes $60. –75 – (–15) = –75 + 15 = –60 13. a. An equation that shows the total charge is 8 + (–8) = 0.° b. The charge of the resulting oxygen ion would be –2. 0 + (–2) = –2. c. The charge of the resulting aluminum ion would be 3. 13 + (–13) – (–3) = 0 + 3 = 3 14. Her elevation is now –51 ft. –14 – 37 = –14 + (–37) = –51 15. Answers will vary. Ask a classmate to read your situation. Possible answer: The temperature at midnight was –6°F. By 2:00 a.m., the temperature had dropped another 6°F to –12°F. 16. Answers will vary. Ask a classmate to read your situation. Possible answer: Jack had $10 in his checking account. Then he wrote a check for $15. Now his balance is –$5.
Challenge Problem 17. |a – b| = a – b when a ≥ b.
Grade 7 Unit 1: Working with Rational Numbers
Lesson 4: SUBTRACTION AS DISTANCE
answers
ANSWERS 1. 5 units 2. 12 units 3. 5 units 4. To find the distance between 6 units and –4 units, find the difference. 6 – (–4) = 6 + 4 = 10 5. To find the distance between –8 units and –1 unit, find the absolute value of the difference. |–8 – (–1)| = |–8 + 1| = |–7| = 7 6. The distance on a number line is 115 units. 7. The distance on a number line is 256 units. 8. The value of b – a is 12. 9. The equation |a – b| = –(a – b) is true when a ≤ b. 10. The difference is 14,776 ft.
Grade 7 Unit 1: Working with Rational Numbers
Lesson 4: SUBTRACTION AS DISTANCE
answers
11. The distance is 8 blocks. The school, library, and park have the coordinates of –13, –5, and 0, respectively. The distance from the school to the library in blocks is shown with this equation: |–13 – (–5)| = |–13 + 5| = |–8| = 8 12. The distance is 20 blocks. Lana’s house, the fire station, and the grocery store have the coordinates of 0, 8, and –12, respectively. The distance from the fire station to the grocery store is shown with this equation: 8 – (–12) = 8 + 12 = 20 13. The difference is 165°F.
Challenge Problem 14. The area is 18 square units. The vertices (–2, –6) and (4, –6) are on the same horizontal line. To find the length of the side connecting them, find the difference of their x-coordinates: 4 – (–2) = 4 + 2 = 6 The vertices (–2, –6) and (–2, –9) are on the same vertical line. To find the length of the side between them, find the difference of their y-coordinates: |–9 – (–6)| = |–9 + 6| = |–3| = 3 The area is then 3 • 6 = 18 square units.
Grade 7 Unit 1: Working with Rational Numbers
Lesson 5: ADDING AND SUBTRACTING
answers
ANSWERS 1. 0
2. 1.06
3. –2
14 15
4. –2.11
5. –
5 28
6. –1.3
35 24 35 11 or −1 (as mixed number) 7. − 24 24 11 −1 24 8. a. Maya’s total debt is represented by the equation –$26.75 + (–$5.50) = –$32.25. −
b. Maya’s new debt is represented by the equation –$32.25 + $12 = –$20.25.
9. a. The total length of the path is 18.31 mi. b. It is 3.34 mi from Doggie Beach to Foster Beach. c. The distance between the aquarium and the theater is 5.1 mi.
10. The difference is 2.8°C.
Challenge Problem 11. Answers will vary. Ask a classmate to check your work.
Grade 7 Unit 1: Working with Rational Numbers
Lesson 6: PROPERTIES OF OPERATIONS
answers
ANSWERS 1. A Additive identity property 2. –
7 9
3. 0.053 4. a 5. Associative Property of Addition 6. Commutative property of addition Associative Property of Addition Inverse property of addition Additive identity property 7. Marcus switched the order of 4 and –5 in 4 – (–5), and he switched the order of 4 and 0 in 0 – 4. Switching the order is incorrect because subtraction is not commutative. Here is a correct calculation: 5 + 4 – (–5)
= 5 + 4 + 5 = 14
8. Lucy tried to apply the associative property to group 12 – 12 together. However, the associative property works only with addition. Here is a correct calculation: 12 – (12 + 17) = 12 – 29 = 12 + (–29) = –17 9. Jack tried to apply the associative property to group 12.5 + 7.5 together. However, the associative property only works with addition. Here is a correct calculation: 20 – 12.5 + 7.5 = 20 + (–12.5) + 7.5 = 7.5 + 7.5 = 15 10. 7 11. 11
Grade 7 Unit 1: Working with Rational Numbers
Lesson 6: PROPERTIES OF OPERATIONS
answers
12. –0.09 13. Equivalent expressions are 5 – 12 + 8 – 3 and 8 – 12 + 5 – 3.
Challenge Problem 14. Possible answer: 5 – 12 + 8 – 3 = 5 + (–12) + 8 – 3 = 5 + 8 + (–12) – 3
Commutative property of addition
= 5 + (8 + (–12)) – 3
Associative property of addition
= (8 + (–12)) + 5 – 3
Commutative property of addition
= 8 – 12 + 5 – 3
Grade 7 Unit 1: Working with Rational Numbers
Lesson 11: MULTIPLYING INTEGERS
answers
ANSWERS 1. C –14 2. 4(–2) = 8 –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0
1
2
3
4
5
6
7
8
9 10
3. a. –36 b. –27 c. –18 d. –9 e. 0 4. Each time the first factor (the number –9 is multiplied by) decreases by 1, the product increases by 9. 5. a. 9 b. 18 c. 27 6. Here is a possible way to use the pattern to find the answer: 3 • –4 = –12 2 • –4 = –8 1 • –4 = –4 0 • –4 = 0 –1 • –4 = 4 –2 • –4 = 8 –3 • –4 = 12 –4 • –4 = 16 7. –44 8. 72 9. –24 10. 300 11. 99
Grade 7 Unit 1: Working with Rational Numbers
Lesson 11: MULTIPLYING INTEGERS
answers
12. –8 13. 27 14. The temperature will be –14°F. 15. a. The expression represents the amount Sophie saves in 20 weeks. b. The expression represents the total amount of changes that Sophie makes to her savings by spending 6 weeks worth of allowance. 16. a. The expression represents the total amount her savings changes in 6 weeks. b. The expression represents the total amount she saves by not going to the movies for 2 weeks.
Challenge Problem 17. a. The expression a(b + 5) will be positive when a is negative and b is less than –5 or when a is positive and b is greater than –5. b. The expression a(b + 5) will be negative when a is negative and b is greater than –5, or when a is positive and b is less than –5.
Grade 7 Unit 1: Working with Rational Numbers
Lesson 12: PROVING RULES FOR MULTIPLYING
answers
ANSWERS 1 1 1. –1 • = – 2 2 2. –1 • (–4.09) = 4.09
11 1 1 3. − −33 = = − −1• −1• 1• 9 •• (( − 1• 33)) − 9 •• − 9 9 11 =− −1• −11 •• 33 = 1• 9 •• − 9 11 =− −1• −1• = 1• − 1• 9 •• 33 9 11 = = (( − −1• −11)) •• •• 33 1• − 99 1 = 1• 1 •• 33 = 1• 99 1 == 11 ••33 = 99 • 3 9
[Multiplication property of –1] [Associative property of multiplication] [Commutative property of multiplication] [Associative property of multiplication] [Multiplication property of –1] [Multiplicative identity property]
4. D Distributive property 5. –1.8 6.
1 2
7. –0.108 8. –321.86 9.
1 120
10. Her elevation will be –52 ft (or 52 ft below sea level).
Challenge Problem 11. Possible answer: The negative number can be written as –1 times a positive number, so the product is the same as the product of –1 and two positive numbers, which is negative.
Grade 7 Unit 1: Working with Rational Numbers
Lesson 13: DIVIDING
answers
ANSWERS 1. –1 2. The division expression 72 ÷ 12 represents 72¢ ÷ 12¢ per day. 3. Possible answer:You can write the quotient as a product of the dividend and the 1 reciprocal of the divisor (that is, you can write a ÷ b as a • ). A reciprocal has the b same sign as the original number. So, the sign of the quotient will be the same as the sign of the product. 4. –2 5. 7 6.
−1 2
7. –
1 2
8. 20 9. –110 10. No, if two numbers have the same sign, their product and their quotient will both be positive. If they have different signs, their product and their quotient will both be negative. 11. Her elevation changed at a rate of –4 ft per minute: elevation change = –72 ft – (–8 ft) = –64 ft change per minute = –64 ft ÷ 16 min = –4 ft per minute
Challenge Problem 12. a. b. c. d. e. f.
Possible answer: a = 1 and b = 2 Possible answer: a = 1 and b = 0.5 Possible answer: a = –1 and b = –2 Possible answer: a = –1 and b = –0.5 Possible answer: a = –1 and b = 0.5 Possible answer: a = –1 and b = 2
Grade 7 Unit 1: Working with Rational Numbers
Lesson 14: THE DISTRIBUTIVE PROPERTY
answers
ANSWERS 1. D 3(5.7 + 4.3) 2. Possible answer: He can think of $0.88 as $1 minus $0.12. So, using the distributive property the total cost is 5 • $1 – 5 • $0.12, which is $5 – $0.60, or $4.40. 3. = 11($8 – $0.05) = $88 – $0.55 = $87.45
2 4. = 4 5 + 9 8 = 20 + 91 4 3 1 • 48 • + 5 • • − 4 9 5 12 = 20 9 5. = 12(3 + 0.09) = 36 + 1.08 2 = 37.08 45 + 9 3 1 4 1 •4 • + 5• •− 4 9 5 12 1 1 = + (– ) 3 3 =0
6. =
7. = 24(137 + 63) = 24 • 200 = 4,800
8. = (5.4 – 6.3) • =
5.4 6.3 – 9 9
= 0.6 – 0.7 = –0.1
1 9
Grade 7 Unit 1: Working with Rational Numbers
Lesson 14: THE DISTRIBUTIVE PROPERTY 9. = (–0.007 • 1,000) • (0.4 • 10) • (–0.02 • 100) = –7 • 4 • –2 = 56 10. B 12 – 6 + 9
Challenge Problem 11. Answers will vary. Ask a classmate to check your work. a. b. c. d.
x=4 x=9 x = –8 x=8
answers
Grade 7 Unit 1: Working with Rational Numbers
Lesson 15: RATIONAL NUMBERS
answers
ANSWERS 1.
2 1
2.
3 100
3.
–9 1
4.
28 7 , or 100 25
47 10 4, 444 1,111 , or 6. 10, 000 2, 500 5.
7.
1 3
8. –0.875 9. 0.2 10. 1.15 11. 0.416 12. a. 0.1 0.01 0.001 0.0001 b. Possible answer: Each decimal form is a repeating decimal. The repeating part is a number of 0s followed by 1. The number of 0s is 1 less than the number of 9s in the denominator.
c.
0.00001
Grade 7 Unit 1: Working with Rational Numbers
Lesson 15: RATIONAL NUMBERS
answers
Challenge Problem
13. a.
1 = 0.5 2 1 = 0.3 3 1 = 0.25 4 1 = 0.2 5 1 = 0.16 6 1 = 0.142857 7 1 = 0.125 8 1 = 0.1 9 1 = 0.1 10 1 = 0.09 11 1 = 0.083 12
b. Answers will vary. Unit fractions that have denominators of 2 and/or 5 as the only prime factors have decimal forms that terminate. All other unit fractions have decimal forms that repeat.
Working with Rational Numbers
Exercises
Grade 7 Unit 1: Working with Rational Numbers
CONTENTS
Exercises Exercises
LESSON 1: UP, UP, AND AWAY! ������������������������������������������������������������������� 5 LESSON 2: MODELING INTEGER ADDITION ���������������������������������������� 6 LESSON 3: SUBTRACTION AS “TAKING AWAY” ���������������������������������� 9 LESSON 4: SUBTRACTION AS DISTANCE ��������������������������������������������� 13 LESSON 5: ADDING AND SUBTRACTING �������������������������������������������� 15 LESSON 6: PROPERTIES OF OPERATIONS ���������������������������������������������� 17 LESSON 7: PUTTING IT TOGETHER 1 ����������������������������������������������������� 19 LESSON 11: MULTIPLYING INTEGERS ������������������������������������������������������ 20 LESSON 12: PROVING RULES FOR MULTIPLYING ������������������������������� 22 LESSON 13: DIVIDING ������������������������������������������������������������������������������������ 24 LESSON 14: THE DISTRIBUTIVE PROPERTY ������������������������������������������ 26 LESSON 15: RATIONAL NUMBERS ����������������������������������������������������������� 28 LESSON 16: PUTTING IT TOGETHER 2 �������������������������������������������������� 30 answers
LESSON 2: MODELING INTEGER ADDITION �������������������������������������� 32 LESSON 3: SUBTRACTION AS “TAKING AWAY” �������������������������������� 34 LESSON 4: SUBTRACTION AS DISTANCE ��������������������������������������������� 36 LESSON 5: ADDING AND SUBTRACTING �������������������������������������������� 38
Grade 7 Unit 1: Working with Rational Numbers
CONTENTS
Exercises answers
LESSON 6: PROPERTIES OF OPERATIONS ���������������������������������������������� 39 LESSON 11: MULTIPLYING INTEGERS ������������������������������������������������������ 41 LESSON 12: PROVING RULES FOR MULTIPLYING ������������������������������� 43 LESSON 13: DIVIDING ������������������������������������������������������������������������������������44 LESSON 14: THE DISTRIBUTIVE PROPERTY ������������������������������������������ 45 LESSON 15: RATIONAL NUMBERS ����������������������������������������������������������� 47
Grade 7 Unit 1: Working with Rational Numbers
Lesson 1: UP, UP, AND AWAY!
Exercises
• Write your wonderings about working with rational numbers. • Write a goal stating what you plan to accomplish in this unit. • Based on your previous work, write three things you will do differently during this unit to increase your success.
Grade 7 Unit 1: Working with Rational Numbers
Lesson 2: MODELING INTEGER ADDITION
Exercises
Exercises 1. 7 + –7 = A –14 B –7 C 0 D 14 2. Use this number line to show 3 + (–7).
–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0
3. Use this number line to show –4 + (–2).
1
2
3
4
5
6
7
8
9 10
Grade 7 Unit 1: Working with Rational Numbers
Lesson 2: MODELING INTEGER ADDITION
10 9 8 7 6 5 4 3 2 1 0 –1 –2 –3 –4 –5 –6 –7 –8 –9 –10
4. –4 + 11 = 5. –9 + (–16) = 6. –8 + 3 = 7. 15 + 4 = 8. –15 + 32 = 9. –38 + (–56) = 10. 13 + (–13) =
Exercises
Grade 7 Unit 1: Working with Rational Numbers
Lesson 2: MODELING INTEGER ADDITION
Exercises
11. The temperature at 6 a.m. was –7°F. By noon, the temperature had risen by 10°F. At noon, what was the temperature in degrees Fahrenheit?
12. A 40 ft utility pole was vertically placed in a hole 6 ft deep. What is the height of the portion of the pole that is above ground?
13. You can use a negative number to represent a situation about debt. For example, Marcus owes his father $7. This situation can be represented as –7. a. Marcus borrows $5 from his sister. Write an equation to represent his total debt (that is, the total amount he owes his father and his sister).
b. Marcus earns $10 shoveling his neighbor’s driveway. He uses all of the $10 to pay back some of his debt. Write an equation to represent what the amount of his new debt will be.
14. Describe a situation that can be represented by the equation –6 + 6 = 0.
15. Describe a situation that can be represented by the equation –10 + (–10) = –20.
16. Describe a situation that can be represented by the equation 25 + (–5) = 20.
Challenge Problem 17. For what integer values of a and b is |a + b| = |a| + |b|?
Grade 7 Unit 1: Working with Rational Numbers
Lesson 3: SUBTRACTION AS “TAKING AWAY”
Exercises
Exercises 1. 3 – 4 = A –7 B –1 C 1 D 7 2. Use this number line to show –4 – (–8).
–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0
3. Use this number line to show –1 – 6.
1
2
3
4
5
6
7
8
9 10
Grade 7 Unit 1: Working with Rational Numbers
Lesson 3: SUBTRACTION AS “TAKING AWAY”
10 9 8 7 6 5 4 3 2 1 0 –1 –2 –3 –4 –5 –6 –7 –8 –9 –10
4. 5 – 11 = 5. 9 – (–6) = 6. –7 – 6 = 7. –3 – (–3) = 8. 15 – 32 = 9. –42 – (–26) = 10. 85 – 93 =
Exercises
Grade 7 Unit 1: Working with Rational Numbers
Lesson 3: SUBTRACTION AS “TAKING AWAY”
Exercises
11. The temperature was 4°F at 11 p.m. Overnight the temperature fell by 9°F. What was the morning temperature in degrees Fahrenheit?
12. Sophie’s credit card statement said she owed $75. Then the credit card company said it make a mistake and took away $15 of this debt. a. Write a subtraction expression to represent this situation. b. How much does Sophie owe now?
13. Atoms contain particles called protons and electrons. A proton has a positive charge and an electron has a negative charge. A neutral atom has a total charge of 0. Sometimes an atom gains or loses 1 or more electrons and becomes a charged particle called an ion. a. An oxygen atom has 8 protons, with a combined charge of 8, and 8 electrons, with a combined charge of –8. Write an equation to show that the total charge for an oxygen atom is 0.
b. Suppose the oxygen atom from Exercise 13a gains 2 electrons. What would be the charge of the resulting oxygen ion? Write an equation to show your answer.
c. An aluminum atom has 13 protons and 13 electrons. Suppose an aluminum atom loses 3 of its electrons. What would be the charge of the resulting ion? Write an equation to show your answer.
14. A hiker is in a canyon, at an elevation of –14 ft (that is, 14 ft below sea level). She descends 37 more ft. What is her elevation now?
Grade 7 Unit 1: Working with Rational Numbers
Lesson 3: SUBTRACTION AS “TAKING AWAY”
Exercises
15. Describe a situation that can be represented by the equation –6 – 6 = –12.
16. Describe a situation that can be represented by the equation 10 – 15 = –5.
Challenge Problem 17. For what integer values of a and b is |a – b| = a – b?
Grade 7 Unit 1: Working with Rational Numbers
Lesson 4: SUBTRACTION AS DISTANCE
Exercises
Exercises For Exercises 1–5, use this number line. –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0
1
2
3
4
5
6
7
8
9 10
1. What is the distance between –5 and 0? 2. What is the distance between 5 and –7? 3. What is the distance between –8 and –3? 4. Explain how to use subtraction to find the distance between 6 and –4. 5. Explain how to use subtraction to find the distance between –8 and –1. 6. What is the distance between –43 and 72 on a number line? 7. What is the distance between –561 and –305 on a number line? 8. If a – b = –12, what is the value of b – a? 9. When does |a – b| = –(a – b)? 10. In the state of California, the highest point is located on Mount Whitney and the lowest point is located in Death Valley. Mount Whitney has an elevation of 14,494 ft. Death Valley has an elevation of –282 ft. What is the difference in elevation between the two locations?
11. In Marcus’s town, the school, the library, and the park are all on the same street. The school is 13 blocks west of the park and the library is 5 blocks west of the park. What is the distance between the library and the school?
Grade 7 Unit 1: Working with Rational Numbers
Lesson 4: SUBTRACTION AS DISTANCE
Exercises
12. Karen’s house, the fire station, and the grocery store are all on the same street. The fire station is 8 blocks north of Karen’s house. The grocery store is 12 blocks south of Karen’s house. How far apart are the fire station and the grocery store?
13. The highest and lowest recorded temperatures in the state of Iowa are 118°F and –47°F. What is the difference between these two temperatures?
Challenge Problem 14. A rectangle has vertices (–2, –6), (4, –6), (–2, –9), and (4, –9). What is the area of the rectangle?
Grade 7 Unit 1: Working with Rational Numbers
Lesson 5: ADDING AND SUBTRACTING
Exercises
Exercises 1. 3
1 1 + –3 = 2 2
2. –0.64 + 1.7 = 3. 4
2 3 –7 = 3 5
4. –5.03 – (–2.92) = 5.
4 3 + – = 7 4
6. –0.5 – 0.8 = 7. –
5 5 + – = 8 6
8. Maya owes her mother $26.75. a. Maya borrows another $5.50 from her mother to buy a ticket to the school play. Write an equation to represent Maya’s total debt to her mother.
b. Maya earned $12 babysitting and used all of that money to repay some of her debt to her mother. Write an equation to represent Maya’s new debt.
9. The city of Chicago has a biking and walking path along its lakefront. a. Jack is on the path at the Oak Street Beach. The north end of the path is 7.55 mi from him and the south end of the path is 10.76 mi from him. What is the total length of the path?
b. Doggie Beach is 3.21 mi north of Jack. Foster Beach is 6.55 mi north of him. How far is it from Doggie Beach to Foster Beach?
c. The Shedd Aquarium is 3.29 mi south of Daniel. The Theater on the Lake is 1.81 mi north of him. What is the distance between the aquarium and the theater?
Grade 7 Unit 1: Working with Rational Numbers
Lesson 5: ADDING AND SUBTRACTING
Exercises
10. In December at the Amundsen-Scott South Pole Station, the mean daily low temperature is –29.3°C and the mean daily high temperature is –26.5°C. What is the difference between these two temperatures?
Challenge Problem 11. Write a word problem that requires adding a negative number and a positive number.
Grade 7 Unit 1: Working with Rational Numbers
Lesson 6: PROPERTIES OF OPERATIONS
Exercises
Exercises 1. What property tells us that 6
1 1 +0=6 ? 2 2
A Additive identity property B Inverse property of addition C Commutative property of addition D Associative property of addition
2. What is the additive inverse of
7 ? 9
3. What is the additive inverse of –0.053?
4. What is the additive inverse of –a?
In Exercises 5 and 6, choose the property that justifies the step. Next to each step, there should be a drop-down menu from which students choose from the following: Additive Identity Property, Additive Inverse Property, Commutative Property of Addition, Associative Property of Addition 2 1 5 2 1 5 5. 4 + 3 + 6 = 4 + 3 + 6 8 3 3 8 3 3 5 = 4 + 10 8 = 14
5 8
6. (3.8 – 4.5) – 3.8 = (3.8 + (–4.5)) + (–3.8) = (–4.5 + 3.8) + (–3.8) = –4.5 + (3.8 + (–3.8)) = –4.5 + 0 = –4.5
Grade 7 Unit 1: Working with Rational Numbers
Lesson 6: PROPERTIES OF OPERATIONS
Exercises
7. Find the error(s) in Marcus’s calculation. Calculate the correct value. 5 + 4 – (–5) = 5 + (–5) – 4 = 0 – 4 = 4 – 0 =4
8. Find the error(s) in Lucy’s calculation. Calculate the correct value. 12 – (12 + 17) = (12 – 12) + 17 = 0 + 17 = 17
9. Find the error(s) in Jack’s calculation. Calculate the correct value. 20 – 12.5 + 7.5 = 20 – (12.5 + 7.5) = 20 – 20 =0 In Exercises 10–12, simplify the expression. 4 3 2 10. 3 + 1 + 2 − 5 9 5
4 9
11. 16 – 9 – 7 + 20 – 9
12. (–0.09 + 0.73) + (–1 + 0.27)
13. Circle all equivalent expressions. 5 – 12 + 8 – 3
3 – 8 + 12 – 5
8 – 12 + 5 – 3
Challenge Problem 14. Use properties to prove that the expressions you selected in Exercise 12 are equivalent.
Grade 7 Unit 1: Working with Rational Numbers
Lesson 7: Putting It Together 1
Exercises
•
Read through your work on the Self Check task and think about your other work in this lesson.
•
Write what you have learned.
•
What would you do differently if you were starting the Self Check task now?
•
Record your ideas. Keep track of any strategies you have learned.
•
Complete any exercises that you have not finished from this unit.
Grade 7 Unit 1: Working with Rational Numbers
Lesson 11: MULTIPLYING INTEGERS
Exercises
Exercises 1. 2(– 7) = A –14 B –7 C 7 D 14 2. Use this number line to show 4(–2).
–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0
1
2
3
4
5
6
7
8
9 10
3. Find each of these products. a. 4 • –9 = b. 3 • –9 = c. 2 • –9 = d. 1 • –9 = e. 0 • –9 = 4. Describe the pattern you see in the equations in Exercise 3.
5. Extend the pattern you described in Exercise 4 to find these products: a. –1 • –9= b. –2 • –9= c. –3 • –9= 6. Use a pattern like the one used in Exercise 5 to show that –4 • –4 = 16.
7. –4 • 11 = 8. –9(–8) = 9. 8(–3) =
Grade 7 Unit 1: Working with Rational Numbers
Lesson 11: MULTIPLYING INTEGERS
Exercises
10. –15(–20) = 11. 33(3) = 12. (–2)(–2)(–2) = 13. (–3)(3)(–3) = 14. The temperature at midnight is 0°F. The temperature drops 2°F every hour for the next 7 hours. Write an equation for the temperature at 7 a.m.
15. Sophie saves $5 of her allowance each week. a. What does the expression 20 • 5 represent in this situation? b. What does the expression -6 dot symbol 5 represent in this situation?
16. Karen spends $9 of her savings each week to go to the movies. a. What does the expression 6(–9) represent in this situation? b. What does the expression –2(–9) represent in this situation?
Challenge Problem 17. Consider the expression a(b + 5), where a and b are integers. a. When will the value of the expression be positive? b. When will the value of the expression be negative?
Grade 7 Unit 1: Working with Rational Numbers
Lesson 12: PROVING RULES FOR MULTIPLYING
Exercises
Exercises 1 1. –1 • = 2 2. –1 • (–4.09) = In Problem 3, a drop-down menu should be included next to each step. The menu should contain these choices: Multiplicative Identity Property, Multiplication Property of –1, Associative Property of Multiplication, and Commutative Property of Multiplication. 1 1 1 • 3, which is . Select a property These steps show that – • –3 equals 3 9 that justifies each step. 9 1 1 − • −3 = −1• − 11• (•−−1• 3) −1• 11 • ( −1• 3) 911 • −33 = −1• 911 • ( −1• 3) = 9 9− − − 99 •• − −33 = = − −1• −1• 1• 9 •• (( − 1• 33)) 9 1 99 1 = −1• • −1 • 3= −1• 1 • −1 • 3 9 = −1• 911 • −1 • 3 = =− −1• −11 •• 33 1• 99 •• − 9 1 1 = −1• −1• • 3= −1• −1• 1 • 3 = −1• 1• −1• 1• 911 • 3 9 = =− −1• − −1• 99 •• 33 9 1 = ( −1• −1) • • 3=( −1• −1) • 11 • 3 9 =( −1• −1) • 911 • 3 = (( − −1• 1• − −11)) •• 9 •• 33 = 9 1 1 1 9 = 1• • 3 = 1• 9 = 1• 911 ••• 333 = 1• = 1• 99 • 3 1 11 9 = •3 = 1 •3 = 9 91 • 3 = = 99 1•• 33 =9 • 3 9 3. Which property tells you that –1(–3 + 7) = (–1)(–3) + (–1)(7)? A Associative property of multiplication B Additive identity property C Multiplication property of –1 D Distributive property 4. 18 • –0.1 = 5. –
3 2 •– = 4 3
Grade 7 Unit 1: Working with Rational Numbers
Lesson 12: PROVING RULES FOR MULTIPLYING
Exercises
6. –1.2 • 0.09 = 7. 7 • –45.98 = 8.
1 1 1 1 •– • •– = 2 3 4 5
9. Maya begins a hike at an elevation of 0 ft. Each half hour, her elevation decreases by 13 ft. What is her elevation after 2 hr?
Challenge Problem 10. Explain why the product of a positive number and a negative number is negative.
Grade 7 Unit 1: Working with Rational Numbers
Lesson 13: DIVIDING
Exercises
Exercises 1. –1 ÷ 1 = 2. Write a division expression to represent this situation: Lucy’s book is due to the library today, but she has not finished reading it. She will owe the library 12¢ for every day her book is late. If she does not return the book, after how many days will she owe the library 72¢? 3. Explain how you can use the rules for finding the sign of a product to find the sign of a quotient.
4. 34 ÷ –17 = 5. –42 ÷ –6 = 6. –9 ÷ 18 = 7. –
5 5 ÷ = 12 6
8. –0.4 ÷ –0.02 = 9. 55 ÷ –0.5 = 10. Is it possible for two numbers to have a product that is positive and a quotient that is negative? Explain.
11. Sophie hiked from an elevation of –8 ft to an elevation of –72 ft in 16 min. On average, how much did her elevation change each minute?
Grade 7 Unit 1: Working with Rational Numbers
Lesson 13: DIVIDING
Exercises
Challenge Problem 12. Find numbers a and b that satisfy the given criteria. If it is impossible to find numbers a and b for the given criteria, explain why. a. b. c. d. e. f.
a • b > a ÷ b and a and b are both positive. a • b < a ÷ b and a and b are both positive. a • b > a ÷ b and a and b are both negative. a • b < a ÷ b and a and b are both negative. a • b > a ÷ b, a is negative, and b is positive. a • b < a ÷ b, a is negative, and b is positive.
Grade 7 Unit 1: Working with Rational Numbers
Lesson 14: THE DISTRIBUTIVE PROPERTY
Exercises
Exercises 1. Which expression is equivalent to 3 • 5.7 + 3 • 4.3? A 3 • 5.7 • 4.3 B 3 • 5.7 + 4.3 C 5.7(3 + 4.3) D 3(5.7 + 4.3) 2. A popsicle costs $0.88. Jack wants to buy a popsicle for himself and his 4 cousins. Explain how he can use the distributive property to calculate the total cost.
For Exercises 3–5, find the product. Use the distributive property to simplify the calculation. 3. 11($7.95) 2 4. 4 5 9 5. 3.09 • 12 3 1 1 4 • •4 + 5•− • 4 592 12 5 For Exercises 9 6–9, simplify each expression. Use the properties of addition and multiplication to simplify the calculations. 6.
3 1 1 4 • •4 + 5•− • 4 9 12 5
7. 24 • 137 + 63 • 24 8. (5.4 – 6.3) ÷ 9 9. –0.007 • 0.4 • –0.02 • 10 • 1,000 • 100
Grade 7 Unit 1: Working with Rational Numbers
Lesson 14: THE DISTRIBUTIVE PROPERTY 10. Which expression has the same value as 12 – (6 – 9)? A 12 – 6 – 9 B 12 – 6 + 9 C 12 + 6 – 9 D 12 – (9 – 6)
Challenge Problem 11. Solve each equation. Use the properties of operations to help you. 2 • x • 12 = –32 3 b. –5x + 11x = 54
a. –
c. –2(x + 3) = 10 d. –8x + 64 = 0
Exercises
Grade 7 Unit 1: Working with Rational Numbers
Lesson 15: RATIONAL NUMBERS
Exercises
Exercises 1. Show that 2 is a rational number by writing it as a ratio of integers.
For Exercises 2–7, write each number as a ratio of integers. 2. 0.03 3. –9 4. 28% 5. 4.7 6. 0.4444 7. 0.3
For Exercises 8–11, write each number as a terminating or repeating decimal. 8. – 9.
2 9
10. 1 11.
7 8
3 20
5 12
Grade 7 Unit 1: Working with Rational Numbers
Lesson 15: RATIONAL NUMBERS
Exercises
12. a. Write each of these fractions as a decimal. 1 = 9 1 = 99 1 = 999 1 9, 999 = b. Describe the pattern you see in your results for Exercise 12a.
c. Predict the decimal form of
1 . Check your prediction by dividing. 99, 999
Challenge Problem 13. a. Use a calculator to find the decimal equivalents of all the unit fractions from 1 1 to . 2 12
b. What do you think determines whether the decimal form of a unit fraction will terminate or repeat? Test your idea on other unit fractions. (Hint: consider the factors of the denominators)
Grade 7 Unit 1: Working with Rational Numbers
Lesson 16: PUTTING IT TOGETHER 2
Exercises
•
Read through your work on the Self Check task and think about your other work in this lesson.
•
Write what you have learned.
•
What would you do differently if you were starting the Self Check task now?
•
Record your ideas. Keep track of any strategies you have learned.
•
Complete any exercises that you have not finished from this unit.
Math Grade 7 Unit 1
Working with Rational Numbers
ANSWERS Exercises FOR EXERCISES
Grade 7 Unit 1: Working with Rational Numbers
Lesson 2: MODELING INTEGER ADDITION
answers
ANSWERS 1. C 0 2. –4
–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0
3. –6 10 9 8 7 6 5 4 3 2 1 0 –1 –2 –3 –4 –5 –6 –7 –8 –9 –10
4. 7 5. –25
1
2
3
4
5
6
7
8
9 10
Grade 7 Unit 1: Working with Rational Numbers
Lesson 2: MODELING INTEGER ADDITION
answers
6. –5 7. 19 8. 17 9. –94 10. 0 11. At noon, the temperature was 3°F. –7 + 10 = 3 12. The distance from the ground to the top of the pole is 34 ft. 40 + –6 = 34 13. a. An equation that represents his total debt is –7 + (–5) = –12. b. An equation that represents his new debt is –12 + 10 = –2. 14. Answers will vary. Ask a classmate to read your situation. Possible answer: The temperature was –6°F. Then it rose by 6°F. Now the temperature is 0°F. 15. Answers will vary. Ask a classmate to read your situation. Possible answer: Lucy has a balance of –$10 on her credit card (that is, her debt is $10). Then she charges another item that costs $10. Lucy now has a balance of –$20 on her credit card. That is, her debt is $20. 16. Answers will vary. Ask a classmate to read your situation. Possible answer: Jack’s checking account balance was $25. Then the bank charged him a $5 fee. Now his balance is $20.
Challenge Problem 17. For the equation |a + b| = |a| + |b|, the integer values for a and b are both positive or both negative, or at least one of the integers is 0.
Grade 7 Unit 1: Working with Rational Numbers
Lesson 3: SUBTRACTION AS “TAKING AWAY”
answers
ANSWERS 1. B –1 2. 4
–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0
3. –7 10 9 8 7 6 5 4 3 2 1 0 –1 –2 –3 –4 –5 –6 –7 –8 –9 –10
1
2
3
4
5
6
7
8
9 10
Grade 7 Unit 1: Working with Rational Numbers
Lesson 3: SUBTRACTION AS “TAKING AWAY”
answers
4. –6 5. 15 6. –13 7. 0 8. –17 9. –16 10. –8 11. The morning temperature is –5°F. 4 – 9 = 4 + (–9) = –5 12. a. A subtraction expression that represents this situation is –75 – (–15). b. Sophie now owes $60. –75 – (–15) = –75 + 15 = –60 13. a. An equation that shows the total charge is 8 + (–8) = 0.° b. The charge of the resulting oxygen ion would be –2. 0 + (–2) = –2. c. The charge of the resulting aluminum ion would be 3. 13 + (–13) – (–3) = 0 + 3 = 3 14. Her elevation is now –51 ft. –14 – 37 = –14 + (–37) = –51 15. Answers will vary. Ask a classmate to read your situation. Possible answer: The temperature at midnight was –6°F. By 2:00 a.m., the temperature had dropped another 6°F to –12°F. 16. Answers will vary. Ask a classmate to read your situation. Possible answer: Jack had $10 in his checking account. Then he wrote a check for $15. Now his balance is –$5.
Challenge Problem 17. |a – b| = a – b when a ≥ b.
Grade 7 Unit 1: Working with Rational Numbers
Lesson 4: SUBTRACTION AS DISTANCE
answers
ANSWERS 1. 5 units 2. 12 units 3. 5 units 4. To find the distance between 6 units and –4 units, find the difference. 6 – (–4) = 6 + 4 = 10 5. To find the distance between –8 units and –1 unit, find the absolute value of the difference. |–8 – (–1)| = |–8 + 1| = |–7| = 7 6. The distance on a number line is 115 units. 7. The distance on a number line is 256 units. 8. The value of b – a is 12. 9. The equation |a – b| = –(a – b) is true when a ≤ b. 10. The difference is 14,776 ft.
Grade 7 Unit 1: Working with Rational Numbers
Lesson 4: SUBTRACTION AS DISTANCE
answers
11. The distance is 8 blocks. The school, library, and park have the coordinates of –13, –5, and 0, respectively. The distance from the school to the library in blocks is shown with this equation: |–13 – (–5)| = |–13 + 5| = |–8| = 8 12. The distance is 20 blocks. Lana’s house, the fire station, and the grocery store have the coordinates of 0, 8, and –12, respectively. The distance from the fire station to the grocery store is shown with this equation: 8 – (–12) = 8 + 12 = 20 13. The difference is 165°F.
Challenge Problem 14. The area is 18 square units. The vertices (–2, –6) and (4, –6) are on the same horizontal line. To find the length of the side connecting them, find the difference of their x-coordinates: 4 – (–2) = 4 + 2 = 6 The vertices (–2, –6) and (–2, –9) are on the same vertical line. To find the length of the side between them, find the difference of their y-coordinates: |–9 – (–6)| = |–9 + 6| = |–3| = 3 The area is then 3 • 6 = 18 square units.
Grade 7 Unit 1: Working with Rational Numbers
Lesson 5: ADDING AND SUBTRACTING
answers
ANSWERS 1. 0
2. 1.06
3. –2
14 15
4. –2.11
5. –
5 28
6. –1.3
35 24 35 11 or −1 (as mixed number) 7. − 24 24 11 −1 24 8. a. Maya’s total debt is represented by the equation –$26.75 + (–$5.50) = –$32.25. −
b. Maya’s new debt is represented by the equation –$32.25 + $12 = –$20.25.
9. a. The total length of the path is 18.31 mi. b. It is 3.34 mi from Doggie Beach to Foster Beach. c. The distance between the aquarium and the theater is 5.1 mi.
10. The difference is 2.8°C.
Challenge Problem 11. Answers will vary. Ask a classmate to check your work.
Grade 7 Unit 1: Working with Rational Numbers
Lesson 6: PROPERTIES OF OPERATIONS
answers
ANSWERS 1. A Additive identity property 2. –
7 9
3. 0.053 4. a 5. Associative Property of Addition 6. Commutative property of addition Associative Property of Addition Inverse property of addition Additive identity property 7. Marcus switched the order of 4 and –5 in 4 – (–5), and he switched the order of 4 and 0 in 0 – 4. Switching the order is incorrect because subtraction is not commutative. Here is a correct calculation: 5 + 4 – (–5)
= 5 + 4 + 5 = 14
8. Lucy tried to apply the associative property to group 12 – 12 together. However, the associative property works only with addition. Here is a correct calculation: 12 – (12 + 17) = 12 – 29 = 12 + (–29) = –17 9. Jack tried to apply the associative property to group 12.5 + 7.5 together. However, the associative property only works with addition. Here is a correct calculation: 20 – 12.5 + 7.5 = 20 + (–12.5) + 7.5 = 7.5 + 7.5 = 15 10. 7 11. 11
Grade 7 Unit 1: Working with Rational Numbers
Lesson 6: PROPERTIES OF OPERATIONS
answers
12. –0.09 13. Equivalent expressions are 5 – 12 + 8 – 3 and 8 – 12 + 5 – 3.
Challenge Problem 14. Possible answer: 5 – 12 + 8 – 3 = 5 + (–12) + 8 – 3 = 5 + 8 + (–12) – 3
Commutative property of addition
= 5 + (8 + (–12)) – 3
Associative property of addition
= (8 + (–12)) + 5 – 3
Commutative property of addition
= 8 – 12 + 5 – 3
Grade 7 Unit 1: Working with Rational Numbers
Lesson 11: MULTIPLYING INTEGERS
answers
ANSWERS 1. C –14 2. 4(–2) = 8 –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0
1
2
3
4
5
6
7
8
9 10
3. a. –36 b. –27 c. –18 d. –9 e. 0 4. Each time the first factor (the number –9 is multiplied by) decreases by 1, the product increases by 9. 5. a. 9 b. 18 c. 27 6. Here is a possible way to use the pattern to find the answer: 3 • –4 = –12 2 • –4 = –8 1 • –4 = –4 0 • –4 = 0 –1 • –4 = 4 –2 • –4 = 8 –3 • –4 = 12 –4 • –4 = 16 7. –44 8. 72 9. –24 10. 300 11. 99
Grade 7 Unit 1: Working with Rational Numbers
Lesson 11: MULTIPLYING INTEGERS
answers
12. –8 13. 27 14. The temperature will be –14°F. 15. a. The expression represents the amount Sophie saves in 20 weeks. b. The expression represents the total amount of changes that Sophie makes to her savings by spending 6 weeks worth of allowance. 16. a. The expression represents the total amount her savings changes in 6 weeks. b. The expression represents the total amount she saves by not going to the movies for 2 weeks.
Challenge Problem 17. a. The expression a(b + 5) will be positive when a is negative and b is less than –5 or when a is positive and b is greater than –5. b. The expression a(b + 5) will be negative when a is negative and b is greater than –5, or when a is positive and b is less than –5.
Grade 7 Unit 1: Working with Rational Numbers
Lesson 12: PROVING RULES FOR MULTIPLYING
answers
ANSWERS 1 1 1. –1 • = – 2 2 2. –1 • (–4.09) = 4.09
11 1 1 3. − −33 = = − −1• −1• 1• 9 •• (( − 1• 33)) − 9 •• − 9 9 11 =− −1• −11 •• 33 = 1• 9 •• − 9 11 =− −1• −1• = 1• − 1• 9 •• 33 9 11 = = (( − −1• −11)) •• •• 33 1• − 99 1 = 1• 1 •• 33 = 1• 99 1 == 11 ••33 = 99 • 3 9
[Multiplication property of –1] [Associative property of multiplication] [Commutative property of multiplication] [Associative property of multiplication] [Multiplication property of –1] [Multiplicative identity property]
4. D Distributive property 5. –1.8 6.
1 2
7. –0.108 8. –321.86 9.
1 120
10. Her elevation will be –52 ft (or 52 ft below sea level).
Challenge Problem 11. Possible answer: The negative number can be written as –1 times a positive number, so the product is the same as the product of –1 and two positive numbers, which is negative.
Grade 7 Unit 1: Working with Rational Numbers
Lesson 13: DIVIDING
answers
ANSWERS 1. –1 2. The division expression 72 ÷ 12 represents 72¢ ÷ 12¢ per day. 3. Possible answer:You can write the quotient as a product of the dividend and the 1 reciprocal of the divisor (that is, you can write a ÷ b as a • ). A reciprocal has the b same sign as the original number. So, the sign of the quotient will be the same as the sign of the product. 4. –2 5. 7 6.
−1 2
7. –
1 2
8. 20 9. –110 10. No, if two numbers have the same sign, their product and their quotient will both be positive. If they have different signs, their product and their quotient will both be negative. 11. Her elevation changed at a rate of –4 ft per minute: elevation change = –72 ft – (–8 ft) = –64 ft change per minute = –64 ft ÷ 16 min = –4 ft per minute
Challenge Problem 12. a. b. c. d. e. f.
Possible answer: a = 1 and b = 2 Possible answer: a = 1 and b = 0.5 Possible answer: a = –1 and b = –2 Possible answer: a = –1 and b = –0.5 Possible answer: a = –1 and b = 0.5 Possible answer: a = –1 and b = 2
Grade 7 Unit 1: Working with Rational Numbers
Lesson 14: THE DISTRIBUTIVE PROPERTY
answers
ANSWERS 1. D 3(5.7 + 4.3) 2. Possible answer: He can think of $0.88 as $1 minus $0.12. So, using the distributive property the total cost is 5 • $1 – 5 • $0.12, which is $5 – $0.60, or $4.40. 3. = 11($8 – $0.05) = $88 – $0.55 = $87.45
2 4. = 4 5 + 9 8 = 20 + 91 4 3 1 • 48 • + 5 • • − 4 9 5 12 = 20 9 5. = 12(3 + 0.09) = 36 + 1.08 2 = 37.08 45 + 9 3 1 4 1 •4 • + 5• •− 4 9 5 12 1 1 = + (– ) 3 3 =0
6. =
7. = 24(137 + 63) = 24 • 200 = 4,800
8. = (5.4 – 6.3) • =
5.4 6.3 – 9 9
= 0.6 – 0.7 = –0.1
1 9
Grade 7 Unit 1: Working with Rational Numbers
Lesson 14: THE DISTRIBUTIVE PROPERTY 9. = (–0.007 • 1,000) • (0.4 • 10) • (–0.02 • 100) = –7 • 4 • –2 = 56 10. B 12 – 6 + 9
Challenge Problem 11. Answers will vary. Ask a classmate to check your work. a. b. c. d.
x=4 x=9 x = –8 x=8
answers
Grade 7 Unit 1: Working with Rational Numbers
Lesson 15: RATIONAL NUMBERS
answers
ANSWERS 1.
2 1
2.
3 100
3.
–9 1
4.
28 7 , or 100 25
47 10 4, 444 1,111 , or 6. 10, 000 2, 500 5.
7.
1 3
8. –0.875 9. 0.2 10. 1.15 11. 0.416 12. a. 0.1 0.01 0.001 0.0001 b. Possible answer: Each decimal form is a repeating decimal. The repeating part is a number of 0s followed by 1. The number of 0s is 1 less than the number of 9s in the denominator.
c.
0.00001
Grade 7 Unit 1: Working with Rational Numbers
Lesson 15: RATIONAL NUMBERS
answers
Challenge Problem
13. a.
1 = 0.5 2 1 = 0.3 3 1 = 0.25 4 1 = 0.2 5 1 = 0.16 6 1 = 0.142857 7 1 = 0.125 8 1 = 0.1 9 1 = 0.1 10 1 = 0.09 11 1 = 0.083 12
b. Answers will vary. Unit fractions that have denominators of 2 and/or 5 as the only prime factors have decimal forms that terminate. All other unit fractions have decimal forms that repeat.
