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LESSON 5: All Mixed Up • M2-69

LEARNING GOALS• Interpret and determine sums and differences of rational

numbers in real-world contexts.• Represent and apply the additive inverse in

real-world contexts.• Determine distance as the absolute value of the

difference between two signed numbers in real-world contexts.

• Solve real-world and mathematical problems involving operations with rational numbers.

You have learned how to add and subtract with signed numbers using models and rules. How can you solve real-world problems by adding and subtracting signed numbers?

WARM UPDetermine each sum.

1. 1 __ 6 1 1 __ 3

2. 2 __ 7 1 2 __ 5

3. 1 __ 2 1 3 __ 5

4. 1 __ 3 1 4 __ 5

All Mixed UpAdding and Subtracting Rational

Numbers

5

C02_SE_M02_T01_L05.indd 69C02_SE_M02_T01_L05.indd 69 4/7/17 10:54 AM4/7/17 10:54 AM

M2-70 • TOPIC 1: Adding and Subtracting Rational Numbers

Getting Started

No Matter the Number

Consider this addition problem: 23 3 __ 4 1 4 1 __ 4 5 ?

1. Draw a model to add the two numbers and determine the sum. Explain how your model represents the addition problem.

2. How would your model be different if the addends were integers? How would the model be the same?

3. Describe how you can apply the rules you have learned about adding signed numbers to determine the sum of 23 3 __ 4 and 4 1 __ 4 .

Remember, addends are the numbers being added.



You can use what you know about adding and subtracting positive and negative integers to solve problems with positive and negative fractions and decimals.

Yesterday, Katrina was just $23.75 below her fundraising goal. She got a check today for $12.33 to put toward the fundraiser. Describe Katrina’s progress toward the goal.

Sum of Rational Numbers

Problems

ACTIVIT Y

5.1

WORKED EXAMPLE

You can model this situation using addition:

–$23.75

currently belowthe goal

+ $12.33

got a check forthis amount

• Estimate.

Katrina will still be below her goal, because 223.75 1 12.33 , 0.

–23.75

+ 12.33

0

• Determine the sum.

223.75 1 12.33 5 211.42

Are we sure this is the right answer? How can I check?

1. Explain how the final sum was calculated.



2. What does the sum mean in terms of the problem situation?

3. Explain how you can know that addition is the correct operation to use to solve this problem.

Sketch a model to estimate each sum or difference. Then determine each solution and write an equation.

4. The table shows the freezing points of some of the elements in the periodic table.

a. Patricia and Elliott are trying to figure out how much the temperature would have to increase from the freezing point of hydrogen to reach the freezing point of phosphorus. Patricia says the temperature would have to increase 545.7°, and Elliott says the temperature would have to increase 322.3°. Who is correct?

b. Francine and Lisa are trying to figure out how much the temperature would have to increase from the freezing point of nitrogen to reach the freezing point of mercury. Francine says the temperature would have to increase 308 1 ___ 20 °, and Lisa says the temperature would have to increase 383 9 ___ 20 °. Who is correct?

ElementFreezing

Point (°F)

Helium 2458

Hydrogen 2434

Oxygen 2368.77

Nitrogen 2345 3 __ 4

Chlorine 2149.51

Mercury 237 7 ___ 10

Phosphorus 111.7



5. A drilling crew dug to a height of 245 1 __ 4 feet during their first day of drilling. On the second day, the crew dug down 9 1 __ 3 feet more than on the first day. Describe the height of the bottom of the hole after the second day.

6. The ancient Babylonians were writing fractions in 1800 BCE. But they did not have a concept of zero until about 1489 years later. In what year did the Babylonians develop the concept of zero?

7. Ben purchased lunch today at the school cafeteria for $2.75. Before today, Ben owed $9.15 on his lunch account. What is the status of his lunch account after today?

8. The highest mountain in the world is Mt. Everest, whose peak is 29,029 feet above sea level. But the tallest mountain is Mauna Kea. The base of Mauna Kea is 19,669 feet below sea level, and its peak is 33,465 feet above its base. How much higher above sea level is Mt. Everest than Mauna Kea?

The abbreviation bce means “before the common era,” or before Year 0.



Rational Number Difference

Problems

ACTIVIT Y

5.2

The freezing point of chlorine is –149.51° Fahrenheit. The element zinc freezes at much higher temperatures. The freezing point of zinc is 787.51° Fahrenheit. How many more degrees is the freezing point of zinc than the freezing point of chlorine?

WORKED EXAMPLE

You can model this situation using subtraction:

787.51

freezing pointof zinc

– (–149.51)

freezing pointof chlorine

• Estimate.

The answer is greater than 787.51, because 787.512(2149.51) 5 787.51 1 149.51.

787.51

0–(–149.51)

• Determine the difference. Write an equation.

787.512(2149.51) 5 937.02

1. What does the difference mean in terms of the problem situation?

2. Explain how you can check the answer.


Sketch a model to estimate. Then determine each solution and write an equation.

3. The temperature in Wichita, Kansas, is 23°C. The temperature in Ryan’s hometown is 18° colder than that. What is the temperature in Ryan’s hometown?

4. To qualify to compete in the high jump finals, athletes must jump a certain height in the semi-finals. Clarissa jumped 2 3 __ 8 inches below the qualifying height, but her friend Anika made it to 1 5 __ 6 inches over the qualifying height. How much higher was Anika’s semi-final jump compared with Clarissa’s?

5. The Down Under roller coaster rises up to 65.8 feet above the ground before dropping 90 feet into an underground cavern. Describe the height of the roller coaster at the bottom of the cavern.

NOTES


Adding and Subtracting

Rational Numbers

Determine each sum or difference.

1. 4.7 1 (23.65)

3. 3.95 1 (26.792)

5. 2 3 __ 4 1 5 __ 8

7. 2 3 __ 4 2 5 __ 8

9. 2 7 ___ 12 2 5 __ 6

2. 2 2 __ 3 1 5 __ 8

4. 2 5 __ 7 1 ( 21 1 __ 3 )

6. 27.38 2 (26.2)

8. 22 5 __ 6 1 1 3 __ 8

10. 237.27 1 (213.2)

ACTIVIT Y

5.3


NOTES


11. 20.8 2 (20.6)

13. 0.67 1 (20.33)

15. 27300 1 2100

17. 24.7 1 3.16

19. 2325 1 (2775)

12. 2 3 __ 7 1 (21 3 __ 4 )

14. 242.65 2 (216.3)

16. 23 5 __ 8 2 (22 1 __ 3 )

18. 26.9 2 (23.1)

20. 22 1 __ 5 2 1 3 ___ 10


NOTES


TALK the TALK

Mixing Up the Sums

Represent each number as the sum of two rational numbers. Use a number line to explain your answer.

1. 22.1

2. 25 2 __ 3

3. 4 7 __ 9

4. 5.8

5. 21 4 __ 7



Assignment

PracticeConsider the subtraction expression 21.3 2 (22.4).

1. Use a number line to solve the problem.

2. Use a two-color counter model to solve the problem.

Calculate each sum. Be sure to estimate first.

3. 12 2 __ 5 1 (23 1 __ 4 ) 4. 5.3 1 (27.45)

5. 2 5 __ 8 1 8 3 __ 8

Calculate each difference. Estimate before calculating.

6. 28.38 2 11.29

7. 7 2 __ 3 2 (24 1 __ 4 ) 8. 24 5 __ 6 2 6 2 __ 3

RememberThe opposite of a number is called the additive

inverse of the number. The absolute value of the

difference between two numbers is a measure of

the distance between the numbers.

WriteExplain in your own words how adding and

subtracting positive and negative numbers with

fractions and decimals is different from and similar

to adding and subtracting with whole numbers.

StretchDetermine each solution. Let 23 5 x, 25 5 y, and 24 5 k.

1. |x 2 y 2 y 2 k 1 y 1 x|2. k 2 y 2 k 1 x 1 k

3. 2 k 2 y 1 x 2 y



Review1. Determine each difference. Then, write a sentence that describes the movement on the number line that

you could use to solve the problem.

a. 7 2 (26)

b. 25 2 13

2. Shilo is riding her bicycle across the state of Georgia to raise money for her favorite charity. The distance

in miles that she can travel varies directly with the length of time in hours she spends riding. Assume that

her constant of proportionality is 18. What does the constant of proportionality represent in this problem?

3. The constant of proportionality between the number of children on a field trip and the number of

teachers on the trip is 14 ___ 3 . There are 70 children on a field trip. How many teachers are on the trip?

4. Determine two unit rates for each given rate.

a. 12 students ate 4.5 pizzas.

b. Shae painted 1 __ 3 of the wall in 1 __ 4 hour.


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