adding and subtracting integers changing elevations 1 · 11/6/2013 · outside temperature at...

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Unit 1 • Integers and Rational Numbers 11

My Notes

ACTIVITY

1.2SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Think/Pair/Share, Interactive Word Wall

Ms. Flowers, a math teacher at Rachel Carson Middle School, plans a fi eld trip for her students every year. On the fi rst day of school, each student recommends a place where the class should go on the annual trip.

1. If your teacher were to plan a fi eld trip for your class this year, where would you like to go? Write your choice below and explain why this would be a good choice for your class trip.

When the students report to math class, many have ideas about where they would like to go. Marcus wants the class to go to Death Valley this year. Th e other students think that Death Valley would not be a pleasant place to visit so they tell Marcus to convince them this is a good choice. In his research, Marcus fi nds that Death Valley is very interesting and could be called “a land of extremes.”

He was surprised to learn that Death Valley has some high mountains and some very low canyons. He noticed that some elevations in Death Valley are above sea level and are listed as positive numbers, while some are below sea level and are listed as negative numbers. He also noticed that the online atlas listed sea level as 0 feet. For a moment he was confused about what these numbers meant, but then remembered that last year he learned that these numbers are called integers.

2. In the chart below, list what you already know about integers and what more you would like to learn about them. When you are fi nished, your class will make a chart showing everyone’s information.

What I know about integers:

What I want to learn about integers:

Changing ElevationsAdding and Subtracting Integers

11-22_SB_MS2_1-2_SE.indd 1111-22_SB_MS2_1-2_SE.indd 11 12/23/09 4:55:16 PM12/23/09 4:55:16 PM

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12 SpringBoard® Mathematics with MeaningTM Level 2

My Notes

ACTIVITY 1.2continued

Adding and Subtracting Integers Changing Elevations Changing Elevations

3. Marcus found that one of the lowest elevation points on Earth is in Death Valley. Some of Earth’s lowest locations and their elevations are listed below.

Lake Assal (Africa): -509 ft .

Qattara Depression (Egypt): -435 ft .

Laguna del Carbón (Argentina): -344 ft .

Death Valley (United States): -282 ft .

Valdes Peninsula (Argentina): -131 ft .

Graph the integer for each elevation on this number line.

4. Death Valley also has some higher points of elevation. Th ey include:

• Shoshone, with an elevation of 1585 feet• Dante’s View, with an elevation of 5500 feet• Butte Valley, with an elevation of 4400 feet

Graph the integer for each elevation on this number line.

5. Look back at the number lines in Questions 3 and 4. What do you notice about the order of the integers on the number line?

6. Using your generalization from Question 5, order the integers in each set from least to greatest.

a. -1360, -282, -435

b. -410, -505, -509, -180

c. -266, -227, -410, -433

SUGGESTED LEARNING STRATEGIES: Visualize, Create Representations, Quickwrite, Look for a Pattern, Group Presentation, Think/Pair/Share

–525 –500 –475 –450 –425 –400 –375 –350 –325 –300 –275 –250 –225 –200 –175 –150 –125 –100 –75 –50 –25 0

0 500 1000 20001500 2500 3000 3500 4000 4500 5000 5500 6000


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My Notes

Th e students began thinking about elevation. Th ey reasoned that there must be places as far above sea level as there are below sea level.

7. For example, the elevation of the Salton Sea in California is -227 feet and Jamestown, Louisiana is approximately 227 feet above sea level. Graph the integer for each location.

8. Describe the position of these two integers in relationship to zero.

9. Telescope Peak is the highest point in Death Valley. Its elevation is 11,049 feet.

a. If a student begins a hike at sea level, and hikes to the top of Telescope Peak, how far above sea level will he be?

b. If a diving bell begins a dive at sea level, and dives to a point below sea level that is as far from sea level as Telescope Peak, how far below sea level would it be?

c. What are the two distances in questions 8 and 9 called?

10. Why are the distances of the hike and the dive with their changes in elevation an example of two integers with the same absolute value?

11. Write the absolute value of each integer.

SUGGESTED LEARNING STRATEGIES: Quickwrite, Look for a Pattern, Interactive Word Wall, KWL, Think/Pair/Share

Adding and Subtracting IntegersChanging Elevations Changing Elevations


Integer 23 -46 1659 0 -79,865Absolute Value

–300 –250 –200 –150 –100 –50 0 50 100 150 200 300250


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My NotesMy Notes



12. Evaluate each expression.

a. |-128| b. |56| c. |0|

Th e students in Ms. Flower’s class were starting to get excited about their fi eld trip, but Lindsay was concerned that they would not be able to stay warm at night. When camping, even when it is very hot during the day, it can be very cold at night. Lindsay decided to do some research on sleeping bags.

Th e temperature rating of a sleeping bag describes the lowest outside temperature at which the bag will keep a sleeper warm.

13. Look at each situation below and decide if the sleeping bag is the correct bag for the temperature of the camping location based on the rating of the bag.

LocationSleeping Bag

RatingOutside

TemperatureWill sleeper

be warm?Alamosa, Colorado +35 60º

Yellowstone National Park 0 -10º

St. Cloud, Minnesota -10 -3º

Bartlett, New Hampshire -20 15º

Fairbanks, Alaska +15 -4º

14. List the locations for which the sleeping bags are not rated for the temperature and explain why.

15. How did being able to compare integers help you know which sleeping bag would keep a sleeper warm? Explain below.

SUGGESTED LEARNING STRATEGIES: Think Aloud, Summarize/Paraphrase/Retell, Think/Pair/Share, Quickwrite

The absolute value of a number is its distance from zero on a number line. Distance or absolute value is always positive, so |-6| = 6 and |6| = 6.

MATH TERMS


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My Notes

SUGGESTED LEARNING STRATEGIES: Close Reading, Visualize, Group Presentation, Create Representations



You can use inequalities to compare integers.

16. For each pair of integers, write two inequalities comparing the integers using the given inequality symbol.

Integers < >

35, 600, -10

-10, -315, -2015, -4

Inequalities can also be graphed on a number line.

EXAMPLE 1

Graph the inequality: x < 2.

Step 1: Graph the integer in the inequality on the number line. Since 2 is the integer, locate 2 on the number line.

• If the inequality sign is < or >, graph the point as an open circle because the integer is not part of the solution.

• If the inequality sign is ≤ or ≥, graph the point as a closed circle because the integer is part of the solution.

Step 2: Th ink about the group of numbers the inequality is describing and draw an arrow on the number line to represent them. Since the numbers described in the inequality are less than 2, the arrow goes to the left because that is where all numbers less than 2 are.

Solution: Th e number line above is the graph of x < 2.

READING MATH

Read the symbol > as “greater than.”

Read the symbol < as “less than.”

Read the symbol ≥ as “greater than or equal to.”

Read the symbol ≤ as “less than or equal to.”

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 2 3 4 5 6 7 8 9 101

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 2 3 4 5 6 7 8 9 101


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My Notes




TRY THESE A

Graph each inequality.

a. y < 6

b. s > -5

c. b ≥ -2

d. a ≤ 7

Some of the students were excited about going hiking on their fi eld trip. Ms. Flowers knew that thinking about hiking could help the students understand adding and subtracting integers. She explained adding and subtracting integers using ideas related to hiking.

• Th e fi rst number in a problem is where you start your hike.• Th e operation sign in the middle of the problem tells you

whether you are going to face in the positive direction or the negative direction.

• Th e sign of the second number tells you if you are going to step backward (-) or forward (+).

• Th e answer to the integer problem is where your hike ends.

EXAMPLE 2

Find the sum: -4 + (-2)

Use this number line to follow along with this example.

Step 1: Start at -4.Step 2: Face the positive direction (right).Step 3: Since the 2 is negative, take 2 steps backward (to the left ).Solution: Th e answer is -6.

SUGGESTED LEARNING STRATEGIES: KWL, Group Presentation, Think/Pair/Share, Close Reading

Integers without signs are considered to be positive.

–8 –6 –4 –2 0 2 4 6 8

–8 –6 –4 –2 0 2 4 6 8

–8 –6 –4 –2 0 2 4 6 8

–8 –6 –4 –2 0 2 4 6 8

–8 –6 –4 –2 0 2 4 6 8


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My Notes




TRY THESE B

Illustrate the move for each problem. Th en record your answer:

a. -5 + (-3) =

b. 6 + (-6) =

c. 5 + (-3) =

d. 6 + 2 =

17. Work with a partner. For each problem, roll a number cube twice. Use the numbers you roll to make a problem. Th en use the number line to solve the problem. Look for patterns while you work.

a. - + (- ) =

b. - + =

c. + (- ) =

d. + =

18. What patterns did you notice about integer addition as you worked?

SUGGESTED LEARNING STRATEGIES: Create Representations, Use Manipulatives

WRITING MATH

Parentheses are placed around a signed number to make a problem easier to read.

–8 –6 –4 –2 0 2 4 6 8

–8 –6 –4 –2 0 2 4 6 8

–8 –6 –4 –2 0 2 4 6 8

–8 –6 –4 –2 0 2 4 6 8

–12 –11 –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 4321 5 6 7 8 9 10 11 12

–12 –11 –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 4321 5 6 7 8 9 10 11 12

–12 –11 –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 4321 5 6 7 8 9 10 11 12

–12 –11 –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 4321 5 6 7 8 9 10 11 12

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My Notes




19. Use the patterns that you noticed to write rules about adding integers. Include an example in your explanation.

a. A positive integer plus a positive integer.

b. A negative integer plus a negative integer.

c. A negative integer plus a positive integer.

d. A positive integer plus a negative integer.

e. An integer and its opposite

Now consider subtracting integers.

EXAMPLE 3

Find the diff erence: -4 - (-2)

Use this number line to follow along with this example.

Step 1: Start at -4.Step 2: Face the negative direction (left ).Step 3: Since the 2 is negative, take 2 steps backward (to the right).Solution: Th e answer is -2.

SUGGESTED LEARNING STRATEGIES: Self Revision/Peer Revision, Look for a Pattern, KWL, Group Presentation

–8 –6 –4 –2 0 2 4 6 8

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My Notes




TRY THESE C

Illustrate the move for each problem. Th en record your answer.

a. -5 - (-3) =

b. -6 - 2 =

c. 5 - (-3) =

d. 6 - 2 =

20. For each move that you illustrated in Try Th ese C, show a diff erent move that starts at the same integer and has the same answer as the original problem. Th e fi rst one is done for you.

a. -5 - (-3) = -5 + (+3) = -2

b. -6 - 2 =

c. 5 - (-3) =

d. 6 - 2 =

SUGGESTED LEARNING STRATEGIES: Create Representations, Use Manipulatives

–8 –6 –4 –2 0 2 4 6 8

–8 –6 –4 –2 0 2 4 6 8

–8 –6 –4 –2 0 2 4 6 8

–8 –6 –4 –2 0 2 4 6 8

–8 –6 –4 –2 0 2 4 6 8

–8 –6 –4 –2 0 2 4 6 8

–8 –6 –4 –2 0 2 4 6 8

–8 –6 –4 –2 0 2 4 6 8


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My Notes

21. How did you fi nd each move in Question 20?

22. Use the patterns that you noticed to write rules about subtracting integers. Include an example in your explanation.

a. A positive integer minus a positive integer.

b. A negative integer minus a positive integer.

c. A negative integer minus a negative integer.

d. A positive integer minus a negative integer.



SUGGESTED LEARNING STRATEGIES: Self Revision/Peer Revision, Group Presentation, Look for a Pattern, Use Manipulatives


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My Notes

23. Work with a partner. For each problem, roll a number cube twice. Use the numbers you roll to make a problem. Th en use the number line to solve the problem.

a. - - (- ) =

b. - - =

c. - (- ) =

d. - =

24. For each problem, you will need to add and subtract. You can use the rules you found or number lines to help you add and subtract the integers.

a. -3 + 6 + (-3) =

b. 4 - 6 + (-6) =

c. -5 + (-2) - 2 =

25. Describe a situation that matches each equation.

a. -6° + 6° = 0°

b. $75 + (-$75) = $0



SUGGESTED LEARNING STRATEGIES: Use Manipulatives, Group Presentation, Think/Pair/Share

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–12 –11 –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 4321 5 6 7 8 9 10 11 12

–12 –11 –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 4321 5 6 7 8 9 10 11 12

–12 –11 –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 4321 5 6 7 8 9 10 11 12


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CHECK YOUR UNDERSTANDING

Write your answers on notebook paper. Show your work.

1. Give a real world situation for the integer -100.

2. Order these integers from least to greatest: 0, -132, 458, -789, 984 3. Evaluate |36|. 4. Give two values that make these

inequalities true. a. 0 < db. z < 37c. -12 > f

5. If p stands for any positive integer and t stands for any negative integer, write two inequalities comparing p and t.

6. Graph the inequality y < 5 on a number line.

7. Simplify each problem. Use a number line if necessary.a. -3 - (-8) b. -4 - 6 c. 6 - (-6) d. 7 - 3

8. -8 - 2a. Use a number line to illustrate this

problem and fi nd its solution. b. Write a problem for a diff erent move

that begins at the same integer and has the same answer.

9. Describe a situation that matches this equation. -7 + 7 = 0

10. MATHEMATICAL R E F L E C T I O N

What relationship exists between addition and

subtraction? Explain using integer models.




adding and subtracting integers changing elevations 1 · 11/6/2013 · outside temperature at...

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