unit 2: rational numbers - part 2€¦ · unit 2: rational numbers - part 2. 10 9 8 7 6 5 4 3 2 1...

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Unit 2:Rational Numbers - Part 2

10 9 8 7 6 5 4 3 2 1

Eureka Math™

Grade 7, Module 2

Student File_AContains copy-ready classwork and homework

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A Story of Ratios®

Multiplication & Division Patterns & Fluency

Classwork

Analyze the patterns below:

Chart A Chart B

What observations can you make about Chart B:

What observations can you make about Chart A:

7 •2 A STORY OF RATIOS

S.39

Multiplying Integers (A)Find each product.

1×8 = 0×2 = (−5)×1 = (−9)× (−1) =

6×8 = (−1)× (−6) = (−7)×5 = (−7)× (−2) =

(−6)×5 = (−6)×8 = 3×4 = (−4)×6 =

0× (−9) = (−1)×5 = 8× (−3) = (−3)× (−8) =

5×2 = 1× (−1) = 7×8 = (−2)× (−9) =

4×7 = (−2)× (−5) = (−4)×2 = 5× (−1) =

(−9)× (−2) = (−1)×9 = 4×4 = (−1)× (−2) =

(−8)×6 = (−9)× (−9) = 3× (−1) = 2× (−7) =

4×0 = 8×1 = 3× (−2) = 5× (−9) =

0× (−5) = (−2)×6 = 3× (−9) = (−3)× (−9) =

(−6)×6 = 5×4 = 0×8 = (−5)× (−4) =

3× (−3) = (−1)× (−1) = (−2)×5 = (−8)×1 =

(−5)× (−2) = (−1)×2 = 8×9 = 9×1 =

(−5)×9 = (−1)× (−7) = (−2)×1 = 3×9 =

4× (−4) = 8×0 = (−2)× (−3) = 4× (−5) =

6×5 = 8× (−4) = (−7)× (−4) = (−4)×4 =

(−3)×9 = (−2)×9 = (−2)×8 = 1× (−8) =

6× (−3) = (−1)×0 = (−5)×2 = (−5)×6 =

(−9)×3 = (−9)×5 = (−5)× (−5) = (−7)× (−8) =

9×7 = 5× (−2) = (−8)× (−1) = 9× (−9) =

7× (−4) = 3× (−6) = 3×2 = 2×3 =

0× (−6) = 0×3 = (−3)×8 = 4×6 =

9×2 = 3× (−4) = 8× (−7) = (−3)× (−4) =

(−2)× (−4) = 6× (−5) = (−9)×2 = (−8)× (−8) =

(−6)× (−3) = 8× (−9) = 7×2 = (−8)×2 =


S.40

Column 1 Column 2 Column 3 Column 4

Multiplying Integers (B)Find each product.

(−6)×0 = 7×3 = 6× (−10) = (−3)× (−5) =

8× (−2) = (−4)× (−10) = 10× (−3) = 3×5 =

9× (−4) = 10×4 = 10× (−4) = 5×9 =

0× (−10) = 11×11 = 2×3 = (−4)× (−12) =

(−4)× (−6) = (−10)× (−2) = 3×12 = 4×7 =

2×4 = 3× (−3) = (−12)× (−12) = (−9)×5 =

9× (−7) = 9×8 = (−1)×10 = (−1)× (−2) =

4× (−12) = (−6)× (−5) = 10× (−1) = (−7)× (−9) =

7×4 = 6× (−5) = 9× (−12) = 8×1 =

(−2)×1 = (−11)×2 = 12×3 = (−4)×3 =

7× (−8) = 11×2 = 7×11 = (−9)× (−12) =

(−12)×7 = 4×10 = 8×5 = 0×3 =

11×7 = 1× (−6) = (−11)×4 = 0× (−6) =

11× (−9) = 4× (−2) = 2× (−11) = (−5)×12 =

(−3)×1 = (−1)×11 = 7× (−10) = (−7)× (−3) =

(−11)× (−11) = 8×4 = (−3)×12 = (−10)× (−6) =

2×7 = (−5)×10 = (−7)×5 = (−2)×2 =

6× (−4) = 10× (−11) = (−4)× (−3) = (−8)× (−2) =

2×12 = (−4)×1 = (−4)×7 = (−1)×5 =

4× (−8) = (−2)× (−11) = (−10)×7 = (−8)×9 =

(−1)×2 = (−9)× (−8) = 1×5 = (−6)×12 =

(−10)× (−4) = (−11)× (−10) = 1× (−12) = 3× (−7) =

(−3)× (−4) = 8×12 = 2× (−8) = 0×8 =

5× (−7) = 0×11 = (−10)×10 = (−8)×0 =

4× (−7) = 11×1 = (−3)×8 = (−2)× (−10) =


S.41


Multiplying Integers (C)Find each product.

(−1)×2 = (−7)×0 = (−7)×7 = 8× (−4) =

(−3)× (−18) = 5×13 = 2×1 = (−8)× (−19) =

(−8)× (−7) = 20× (−9) = 11× (−1) = 9× (−4) =

(−11)×12 = 3× (−2) = (−5)×9 = 16×5 =

(−10)×1 = 6× (−12) = (−6)×3 = (−6)× (−9) =

11× (−16) = 1× (−19) = 17× (−5) = (−13)× (−5) =

10× (−8) = (−10)× (−9) = 10×12 = 5× (−11) =

(−14)×1 = 17×7 = 4×2 = (−8)× (−2) =

19× (−8) = 10× (−7) = (−11)×6 = 7× (−18) =

(−6)× (−4) = (−20)×13 = 9×7 = 14× (−8) =

6×5 = 13×17 = 12×2 = 1×10 =

(−2)×13 = (−20)× (−20) = (−18)× (−20) = 20×6 =

2×8 = 20×15 = 15×14 = 17× (−1) =

(−6)×20 = (−3)× (−15) = (−20)×15 = (−9)×1 =

(−11)× (−4) = (−1)× (−19) = 11× (−6) = 9× (−17) =

18× (−1) = (−19)×13 = 9×0 = 8×9 =

4× (−3) = 10× (−1) = 13×19 = 2× (−11) =

6× (−6) = 1×2 = 6× (−9) = 0× (−4) =

19×17 = 19×1 = (−10)× (−3) = 11× (−3) =

(−17)× (−9) = 17×14 = (−8)×14 = (−7)×17 =

12× (−14) = 11× (−8) = (−9)×13 = 11× (−17) =

18× (−7) = (−19)× (−17) = (−19)× (−10) = 18×1 =

20× (−6) = 12×8 = (−3)× (−11) = (−14)×0 =

(−16)×7 = 2× (−8) = (−12)× (−9) = 16× (−14) =

(−3)× (−5) = 13× (−15) = 15×17 = 8× (−5) =


S.42


Dividing Integers (A)Find each quotient.

(−28)÷7 = (−9)÷ (−9) = (−18)÷ (−6) = 28÷7 =

18÷ (−9) = (−3)÷1 = (−49)÷7 = (−14)÷7 =

15÷5 = (−12)÷ (−4) = (−25)÷ (−5) = (−18)÷ (−3) =

(−40)÷ (−8) = 4÷ (−4) = (−36)÷ (−6) = 14÷ (−2) =

(−42)÷6 = (−10)÷2 = 56÷8 = 24÷3 =

(−42)÷ (−7) = 12÷ (−2) = 25÷ (−5) = 7÷ (−1) =

(−18)÷ (−2) = 21÷3 = 42÷7 = (−5)÷5 =

(−40)÷5 = 18÷3 = 45÷5 = (−48)÷8 =

48÷8 = 18÷ (−6) = 24÷6 = (−54)÷ (−6) =

(−54)÷9 = 63÷ (−7) = 9÷9 = (−21)÷ (−7) =

56÷ (−7) = 36÷9 = 56÷ (−8) = 18÷6 =

21÷7 = 8÷1 = 25÷5 = 14÷ (−7) =

(−21)÷7 = 24÷ (−3) = 32÷ (−8) = 63÷7 =

81÷ (−9) = (−9)÷ (−3) = (−10)÷ (−2) = (−35)÷ (−7) =

(−2)÷2 = (−15)÷ (−3) = 6÷ (−3) = (−45)÷ (−5) =

(−6)÷ (−3) = (−36)÷6 = 54÷6 = (−5)÷ (−5) =

(−42)÷ (−6) = 24÷ (−4) = (−24)÷ (−4) = 16÷ (−2) =

(−12)÷6 = 56÷7 = (−36)÷ (−9) = (−48)÷6 =

(−8)÷8 = 8÷ (−4) = 64÷8 = (−20)÷4 =

3÷ (−1) = 9÷3 = 5÷ (−1) = (−7)÷7 =

(−35)÷ (−5) = (−48)÷ (−6) = (−64)÷ (−8) = (−56)÷ (−7) =

9÷ (−1) = (−9)÷3 = 14÷7 = (−15)÷3 =

49÷7 = 36÷6 = 7÷7 = 9÷1 =

64÷ (−8) = 35÷7 = 16÷8 = 12÷6 =

(−21)÷3 = (−16)÷8 = (−56)÷8 = 48÷ (−6) =


S.43


Integer Division (B)Find each quotient.

96÷ (−12) = (−40)÷ (−10) = (−55)÷5 = 63÷9 =

(−63)÷ (−9) = (−8)÷2 = (−90)÷ (−10) = 36÷ (−6) =

72÷12 = 33÷11 = 49÷7 = (−100)÷10 =

35÷5 = (−25)÷5 = 48÷ (−4) = 20÷10 =

(−24)÷ (−12) = (−96)÷8 = 60÷ (−5) = (−30)÷5 =

14÷2 = (−14)÷ (−7) = (−16)÷2 = (−110)÷10 =

(−66)÷11 = (−63)÷9 = 80÷ (−10) = (−36)÷ (−12) =

18÷9 = 18÷ (−2) = 64÷ (−8) = 4÷4 =

64÷8 = (−99)÷ (−9) = 60÷ (−10) = (−110)÷ (−11) =

84÷12 = (−25)÷ (−5) = (−22)÷2 = (−56)÷ (−8) =

(−40)÷ (−5) = 1÷ (−1) = 2÷2 = (−21)÷ (−3) =

(−6)÷ (−1) = (−24)÷12 = (−24)÷ (−4) = 33÷ (−3) =

(−70)÷ (−7) = 30÷ (−5) = 50÷10 = 3÷ (−3) =

28÷ (−7) = 66÷ (−6) = (−72)÷12 = 15÷3 =

(−48)÷ (−12) = (−14)÷7 = 72÷ (−6) = (−36)÷ (−3) =

(−120)÷10 = 70÷7 = (−56)÷8 = 120÷10 =

(−132)÷ (−12) = 7÷1 = (−70)÷ (−10) = 20÷ (−2) =

144÷ (−12) = (−28)÷7 = 14÷7 = 30÷6 =

55÷5 = 21÷ (−7) = (−27)÷9 = (−20)÷ (−4) =

(−45)÷ (−9) = 120÷ (−10) = 28÷ (−4) = 12÷ (−1) =

8÷1 = 66÷ (−11) = (−36)÷ (−9) = 24÷3 =

5÷ (−5) = (−6)÷ (−6) = 8÷ (−1) = 8÷4 =

(−15)÷3 = 50÷5 = 54÷6 = (−36)÷6 =

(−24)÷4 = 96÷8 = (−12)÷ (−6) = 60÷5 =

(−36)÷3 = 24÷4 = 28÷4 = (−88)÷11 =


S.44


Integer Division (C)Find each quotient.

20÷2 = 30÷ (−10) = (−50)÷ (−10) = 24÷ (−6) =

288÷ (−18) = (−85)÷ (−5) = (−36)÷4 = 117÷13 =

136÷ (−8) = (−171)÷19 = 240÷15 = (−64)÷16 =

168÷12 = (−200)÷20 = 14÷ (−7) = (−99)÷11 =

240÷ (−15) = (−120)÷ (−8) = 102÷ (−17) = (−130)÷ (−10) =

(−140)÷ (−10) = (−210)÷15 = (−224)÷ (−14) = (−221)÷ (−17) =

(−13)÷ (−1) = 120÷8 = 144÷ (−8) = 24÷6 =

400÷ (−20) = 39÷ (−13) = (−3)÷1 = 84÷7 =

(−200)÷10 = 224÷ (−16) = 66÷11 = (−4)÷4 =

88÷11 = (−60)÷12 = 288÷16 = 192÷12 =

288÷ (−16) = (−90)÷6 = 90÷ (−10) = (−288)÷16 =

133÷19 = 55÷5 = 128÷8 = 144÷ (−12) =

48÷ (−8) = (−306)÷17 = (−64)÷4 = (−65)÷ (−13) =

35÷5 = 34÷ (−17) = 252÷ (−14) = (−36)÷9 =

51÷3 = (−18)÷ (−9) = (−33)÷ (−3) = 221÷ (−13) =

12÷ (−12) = (−60)÷ (−6) = 72÷ (−6) = 6÷ (−1) =

34÷2 = 204÷12 = 209÷19 = 221÷ (−17) =

(−72)÷6 = 18÷3 = (−104)÷8 = 247÷ (−13) =

84÷12 = 11÷ (−11) = 33÷ (−3) = 360÷18 =

98÷ (−14) = 54÷ (−18) = 156÷ (−13) = (−187)÷ (−11) =

(−60)÷ (−5) = 15÷3 = 143÷11 = 60÷6 =

(−280)÷ (−20) = 240÷16 = 108÷ (−9) = 60÷ (−6) =

160÷10 = (−85)÷17 = 192÷ (−16) = (−361)÷19 =

(−35)÷ (−7) = (−30)÷ (−10) = 126÷ (−7) = (−144)÷18 =

90÷15 = (−153)÷9 = (−152)÷ (−8) = (−36)÷ (−2) =


S.45


30 − 6 ∙ (8) ÷ 2 + 8

30 − 48 ÷ 2 + 8

Order of Operations with Integers

Examples:

1) 𝟑𝟎 − 𝟔 ∙ 𝟓 + 𝟑 ÷ 𝟐 + 𝟐𝟑

2) (𝟒 + 𝟐)𝟐 ∙ 𝟏𝟎 − 𝟏 3) −𝟕×𝟒 + 𝟖 ÷ 𝟐

1 Parentheses: __________________________________

30 − 6 ∙ (8) ÷ 2 + 2A

30 − 24 + 8

6 + 8

14

Teacher Tips: * Work with only two numbers at a time* Think of the downward arrow as the equalsign to that one pa rt of the problem* Bring down everything that you have not used each time. That includes numbers and symbols.

2 Exponents: ___________________________________ 3 Multiply & Divide: _______________________________

4 Add & Subtract: ________________________________


S.46

4) 𝟖(𝟒 − 𝟓) + (−𝟕) 5) 𝟐 ∙ 𝟒 − (𝟓)𝟐 + 𝟑

6) −𝟖 𝟒 − (−𝟓 + 𝟑) 7) Y𝟏𝟔𝟖∙ −𝟐 𝟑 + −𝟕

8) (𝟐∙𝟒)𝟐

Y𝟖(𝟐𝟑)9) 𝟔𝟐× −𝟏 + (𝟑𝟑 − 𝟑𝟑)


S.47


Converting Decimals to Fractions Using Place Value

Converting Decimals to Fractions - READ IT - WRITE IT - REDUCE IT


Converting Fractions to Decimals By Scaling Up & Down

Example 1: Using Place Values to Write Fractions as Decimals

Write each fraction as a decimal.

Example 2: Scaling to Rewrite Fractions and Convert to Decimals

First, scale each fraction to rewrite with a place value denominator. Then write as a decimal.

Problem Set


Convert each decimal to its equivalent fraction in lowest terms.

Convert each fraction to a decimal.


Fluency in Converting Between Decimals & Fractions

Classwork

Example 1: Decimal Representations of Rational Numbers

In previous lessons we have learned common fraction to decimal equivalences. Analyze the charts below.

What do these fractions have in common? What do these fractions have in common?

7•2

Example 2: Converting Rational Numbers to Decimals Using Long Division

Use the long division algorithm to find the decimal value of − 34 .

Exercise 1

Convert each rational number to its decimal form using long division.

a. − 78 =

b. 316

=

Lesson 14: Converting Rational Numbers to Decimals Using Long Division

A STORY OF RATIOS

This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka- math.orgG7-M2-SE-1.3.0-05.2015

S.72

http://creativecommons.org/licenses/by-nc-sa/3.0/deed.en_US



7•2

Example 3: Converting Rational Numbers to Decimals Using Long Division

Use long division to find the decimal representation of13.

Exercise 2

Calculate the decimal values of the fraction below using long division. Express your answers using bars over the shortest sequence of repeating digits.

a. − 49 b. − 1

11

c. 17

d. − 56


A STORY OF RATIOS


S.73




7•2

Example 4: Fractions Represent Terminating or Repeating Decimals

How do we determine whether the decimal representation of a quotient of two integers, with the divisor not equal to zero, will terminate or repeat?

Example 5: Using Rational Number Conversions in Problem Solving

a. Eric and four of his friends are taking a trip across the New York State Thruway. They decide to split the cost oftolls equally. If the total cost of tolls is $8, how much will each person have to pay?

b. Just before leaving on the trip, two of Eric’s friends have a family emergency and cannot go. What is eachperson’s share of the $8 tolls now?


A STORY OF RATIOS


S.74




7•2

Problem Set

1. Convert each rational number into its decimal form.19

= _______________ 16

= _______________ 29

= _______________

13

= _______________ 26

= _______________ 39

= _______________

49

= _______________ 36

= _______________ 59

= _______________

23

= _______________ 46

= _______________ 69

= _______________

79

= _______________ 56

= _______________ 89

= _______________

One of these decimal representations is not like the others. Why?

Lesson Summary

The real world requires that we represent rational numbers in different ways depending on the context of a situation. All rational numbers can be represented as either terminating decimals or repeating decimals using the long division algorithm. We represent repeating decimals by placing a bar over the shortest sequence of repeating digits.


A STORY OF RATIOS


S.75




7•2

Enrichment:

2. Chandler tells Aubrey that the decimal value of − 117 is not a repeating decimal. Should Aubrey believe him?

Explain.

3. Complete the quotients below without using a calculator, and answer the questions that follow.

a. Convert each rational number in the table to its decimal equivalent.

111

= 2

11=

311

= 4

11=

511

=

611

= 7

11=

811

= 9

11=

1011

=

Do you see a pattern? Explain.

b. Convert each rational number in the table to its decimal equivalent.

099

= 1099

= 2099

= 3099

= 4599

=

5899

= 6299

= 7799

= 8199

= 9899

=

Do you see a pattern? Explain. c. Can you find other rational numbers that follow similar patterns?


A STORY OF RATIOS


S.76




7•2

Lesson 16: Applying the Properties of Operations to Multiply and Divide Rational Numbers

A STORY OF RATIOS


S.77

Commutative Property

of Addition

Commutative Property of

Multiplication

Associative Property of Addition

Associative Property of

Multiplication

Identity Property

of Addition

Identity Property of

Multiplication

Distributive Property

RED ORANGE YELLOW GREEN BLUE PURPLE WHITE Directions: Determine which property is represented by the definitions or examples in the

boxes below. Color the box to match the color listed under the mathematical properties in

the color key above.

(a+b)+(c+d) = (a+d)+ (b+c) Adding zero to any number(n) results in

the number(n) remaining unchanged

3(4)=4(3) 5 = 0 + 5

1m=m 3 + 5 + 8 = 8 + 3 + 5 A number (n)

multiplied by 1, will always equal the

number (n).

(x+y) + z=z + (x+y)

The order in which you multiply numbers together

can change without changing the product.

-4(2x + 6) = -8x – 24 3+(4+7) = (3+4)+7 3y · 1 = 3y

(x + y) + z = x + (z + y) 4·6·8·9 = 9·6·4·8 50(1) = 50

The grouping of numbers in an addition

problem can change without changing the

sum.

3(6·5) = 5(3·6) 0 + (-100)= -100 The order in which you add numbers together

can change without changing the sum.

ef + eg = e(f + g)

9 + 1 = 1 + 9

The grouping of numbers in a multiplication

problem can change without changing

the product.

12 + 28 = 4(3 + 7) -2(8·1)= 8(-2·1)

x(yz)=z(xy) y + 0 = y Multiply a sum by multiplying each

addend separately and then add the products.

abc=bca

Identifying & Applying Mathematical Properties




Directions: For questions 1 through 6, write the property that is represented by the given equation.

1. 0 + $%&= $

%&

2. 5.4 + 2.9 = 2.9 + 5.4

3. &

9 % + 3 ./+ 4 %

/= 9 %

&+ (3 .

/+ )

4. −6.1 + 0 = −6.1

5. − 5%6+ %

&= %

&+ (− 5

%6)

8.2 + 4.94.9 + 3.86. 8.2 + = + 3.8

Directions: For questions 7 through 10, complete each equation to make it a true statement.

7. 5.5 + 7.1 + 6.5 = 5.5 + (7.1 + ______________)

8. −21 + _________ = −21

9. . +:

$: := __________ + .

10. 2 .%6+ 7 %

;+ 9 $

%6= 2 .

%6+ ( ____________ + 9 $

%6)

11. Use the commutative property of addition to complete the equation. Then simplify to showthat the two sides of the equation are equal.

$%&

%%%&

+ =

12. Use the associative property of addition to complete the equation. Then simplify to showthat the two sides of the equation are equal.

3.7 + 6.5 + 4.5 =

13. What is the value of p in this equation?

𝑝 + 3 %&

/;

%;; &

/ + = + + 3 %

%;

/;

A. B. C. 1 D. 3 %&

14. Which statement is true?A. -7 + (-11) = -11 + 7 B. 0 + ;

= == %

3 $:

C. + 5 %&+ 8 %

&= 3 $

:+ (5 %

&+ 8 %

&) D. 9.4 + 1 = 9.4

7 •2 Lesson 16 A STORY OF RATIOS

S.78

/4 %

7 •2 Lesson 16 A STORY OF RATIOS

; ;

Directions: For questions 15 – 20, write the property that is represented by the given equation.

15. 1 • / = /

16. 3.4 • 6.9 = 6.9 • 3.4

17. &

2 % • 3 .:• 5 %

&= 2 %

&• (3 .

:• 5 %

.)

18. −/;• ./= .

/• (− /

;)

19. 2.5 • (2 • 6.3) = (2.5 • 2) • 6.3

20. %/

;=

%&= %

/;=

%/

• + • + • %&

Directions: For questions 21 – 25, complete each equation to make it a true statement.

21. (3.5 • 6.4) • 5.2 = 3.5 • (6.4 • __________)

22. -17 • ________ = -17

23. 1 .%6• 2 /

; = ________ • 1 .

%6

24. &.• %;• 15 = &

.• (__________ • 15)

25. 2.8 • (5.6 + __________) = 2.8 • 5.6 + 2.8 • 8.4

26. Which statement is true?A. -8 • (-12) = -12 • 8 B. 2 &

.• • &

5= 2 &

.• (4 %

&• &5)

C. 7.4 • (2.5 + 3.9) = 7.4 • 2.5 • 2.9 D. 1 • −

4 %&./= .

/• 0

27. Use the associative property of multiplication to complete the equation. Then show that thetwo sides of the equation are equal.

(2.7 • 4.1) • 3.2 =

28. Use the distributive property of multiplication over addition to complete the equation. Thenshow that the two sides of the equation are equal.

/;

; .: %6• + =

29. Use the commutative property of multiplication to complete the equation. Then show thatthe two sides of the equation are equal.

1.5 • 25.4 =

S.79

7 •2 Lesson 17

Lesson 17: Addition and Subtraction of Rational Numbers

Classwork

Exercise 1: Real-World Connection to Adding and Subtracting Rational Numbers

Suppose a seventh grader’s birthday is today, and she is 12 years old. How old was she 3 12 years ago? Write anequation, and use a number line to model your answer.

Example 1: Representing Sums of Rational Numbers on a Number Line

a. Place the tail of the arrow on 12.

b. The length of the arrow is the absolute value of −3 12, �−3 1

2� = 3 12.

c. The direction of the arrow is to the left since you are adding a negative number to 12.

Draw the number line model in the space below.


A STORY OF RATIOS

This work is deriv e d from Eureka Math ™ and licensed by Great Minds. ©2015 -Grea t Min ds. eure ka math.org G7-M2-SE-1.3.0-05.2015

S.80

7 •2 Lesson 17

Exercise 2

Find the following sum using a number line diagram: −2 12 + 5.

Example 2: Representing Differences of Rational Numbers on a Number Line

Find the following difference, and represent it on a number line: 1 − 2 14.

a.

Now follow the steps to represent the sum:

b.

c.

d.

Draw the number line model in the space below.


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7 •2 Lesson 17

Exercise 3

Find the following difference, and represent it on a number line: −5 12 − (−8).

Exercise 4

Find the following sums and differences using a number line model.

a. −6 + 5 14

b. 7 − (−0.9)

c. 2.5 + �− 12�

d. − 14 + 4


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7 •2 Lesson 17

e. 12− (−3)

Exercise 5

Create an equation and number line diagram to model each answer.

a. Samantha owes her father $7. She just got paid $5.50 for babysitting. If she gives that money to her dad, howmuch will she still owe him?

b. At the start of a trip, a car’s gas tank contains 12 gallons of gasoline. During the trip, the car consumes 10 18

gallons of gasoline. How much gasoline is left in the tank?

c. A fish was swimming 3 12 feet below the water’s surface at 7:00 a.m. Four hours later, the fish was at a depth

that is 5 14 feet below where it was at 7:00 a.m. What rational number represents the position of the fish with

respect to the water’s surface at 11:00 a.m.?


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7 •2 Lesson 17

Problem Set

Represent each of the following problems using both a number line diagram and an equation.

1. A bird that was perched atop a 15 12 -foot tree dives down six feet to a branch below. How far above the ground is

the bird’s new location?

2. Mariah owed her grandfather $2.25 but was recently able to pay him back $1.50. How much does Mariah currentlyowe her grandfather?

3. Jake is hiking a trail that leads to the top of a canyon. The trail is 4.2 miles long, and Jake plans to stop for lunchafter he completes 1.6 miles. How far from the top of the canyon will Jake be when he stops for lunch?

4. Sonji and her friend Rachel are competing in a running race. When Sonji is 0.4 miles from the finish line, she noticesthat her friend Rachel has fallen. If Sonji runs one-tenth of a mile back to help her friend, how far will she be fromthe finish line?

5. Mr. Henderson did not realize his checking account had a balance of $200 when he used his debit card for a$317.25 purchase. What is his checking account balance after the purchase?

6. If the temperature is −3℉ at 10: 00 p.m., and the temperature falls four degrees overnight, what is the resultingtemperature?

Lesson Summary

The rules for adding and subtracting integers apply to all rational numbers.

The sum of two rational numbers (e.g., −1 + 4.3) can be found on the number line by placing the tail of an arrow at −1 and locating the head of the arrow 4.3 units to the right to arrive at the sum, which is 3.3.

To model the difference of two rational numbers on a number line (e.g., −5.7 − 3), first rewrite the difference as a sum, −5.7 + (−3), and then follow the steps for locating a sum. Place a single arrow with its tail at −5.7 and the head of the arrow 3 units to the left to arrive at −8.7.


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7 •2 Lesson 18

Lesson 18: Applying the Properties of Operations to Add

and Subtract Rational Numbers

Classwork

Example 1: The Opposite of a Sum is the Sum of its Opposites

Explain the meaning of “The opposite of a sum is the sum of its opposites.” Use a specific math example.

Rational Number

Rational Number

Sum Opposite of

the Sum

Opposite Rational Number

Opposite Rational Number

Sum

Exercise 1

Represent the following expression with a single rational number.

25

14

−2 + 3 −35

Lesson 8: Applying the Properties of Operations to Add and Subtract Rational Numbers

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7 •2 Lesson 18

Exercise 2

Rewrite each mixed number as the sum of two signed numbers.

a. −9 58 b. −2 1

2 c. 8 1112

Exercise 3

Represent each sum as a mixed number.

a. −1 + �− 152� b. 30 + 1

8 c. −17 + �− 19�


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Exercise 4

Mr. Mitchell lost 10 pounds over the summer by jogging each week. By winter, he had gained 5 18 pounds. Represent

this situation with an expression involving signed numbers. What is the overall change in Mr. Mitchell’s weight?

Exercise 5

Jamal is completing a math problem and represents the expression −5 57 + 8 − 3 2

7 with a single rational number asshown in the steps below. Justify each of Jamal’s steps. Then, show another way to solve the problem.

= −557

+ 8 + �−327�

= −557

+ �−327� + 8

= −5 + �−57� + (−3) + �−

27� + 8

75

= −5 + �− � + �−2� + (−3) + 8

7= −5 + (−1) + (−3) + 8

= −6 + (−3) + 8

= (−9) + 8

= −1

7 •2 Lesson 18

Problem Set

1. Represent each sum as a single rational number.

a. −14 + �− 89� b. 7 + 1

9 c. −3 + �− 16�

Rewrite each of the following to show that the opposite of a sum is the sum of the opposites. Problem 2 has been completed as an example.

2. −(9 + 8) = −9 + (−8)−17 = −17

3. −�14 + 6�

4. −�10 + (−6)�

5. −�(−55) + 12�

Use your knowledge of rational numbers to answer the following questions. 6. Meghan said the opposite of the sum of −12 and 4 is 8. Do you agree? Why or why not?

7. Jolene lost her wallet at the mall. It had $10 in it. When she got home, her brother felt sorry for her and gave her$5.75. Represent this situation with an expression involving rational numbers. What is the overall change in theamount of money Jolene has?

8. Isaiah is completing a math problem and is at the last step: 25 − 28 15 . What is the answer? Show your work.

Lesson Summary

Use the properties of operations to add and subtract rational numbers more efficiently. For instance,

−529

+ 3.7 + 529

= �−529

+ 529� + 3.7 = 0 + 3.7 = 3.7.

The opposite of a sum is the sum of its opposites as shown in the examples that follow:47

47

−4 = −4 + �− �

−(5 + 3) = −5 + (−3)


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7 •2 Lesson 18

9. A number added to its opposite equals zero. What do you suppose is true about a sum added to its opposite?

Use the following examples to reach a conclusion. Express the answer to each example as a single rational number.a. (3 + 4) + (−3 + −4)b. (−8 + 1) + (8 + (− 1))

c. �− 12 + �− 1

4��+ �12 + 1

4�


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7 •2 Lesson 19

Lesson 19: Applying the Properties of Operations to Add and

Subtract Rational Numbers


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Exercise 2: Team Work!

a. −5.2 − (−3.1) + 5.2 b. 32 + �−12 78�

c. 3 16 + 20.3 − �−5 5

6� d. 1620

− (−1.8) − 45

Exercise 3

Explain, step by step, how to arrive at a single rational number to represent the following expression. Show both a written explanation and the related math work for each step.

−24 − �−12� − 12.5

7 •2 Lesson 19

Problem Set

Show all steps taken to rewrite each of the following as a single rational number.

1. 80 + �−22 415�

2. 10 + �−3 38�

3. 15

+ 20.3 − �−535�

4. 1112

− (−10) − 56

5. Explain, step by step, how to arrive at a single rational number to represent the following expression. Show both awritten explanation and the related math work for each step.

1 −34

+ �−1214�

Lesson Summary

47

Use the properties of operations to add and subtract rational numbers more efficiently. For instance,

−5 29 + 3.7 + 5 2

9 = �−5 29 + 5 2

9� + 3.7 = 0 + 3.7 = 3.7.

The opposite of a sum is the sum of its opposites as shown in the examples that follow:

−4 47 = −4 + �− �.

−(5 + 3) = −5 + (−3).


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unit 2: rational numbers - part 2€¦ · unit 2: rational numbers - part 2. 10 9 8 7 6 5 4 3 2 1...

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