1.2. Adding and Subtracting Integers Changing Elevations. My Notes ACTIVITY


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1 Adding and Subtracting Integers SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Think/Pair/Share, Interactive Word Wall Ms. Flowers, a math teacher at Rachel Carson Middle School, plans a field trip for her students every year. On the first 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 field 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. ACTIVITY 1.2 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. The 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 finds 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 finished, your class will make a chart showing everyone s information. What I know about integers: What I want to learn about integers: Unit 1 Integers and Rational Numbers 11
2 ACTIVITY 1.2 Adding and Subtracting Integers SUGGESTED LEARNING STRATEGIES: Visualize, Create Representations, Quickwrite, Look for a Pattern, Group Presentation, Think/Pair/Share 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 Death Valley also has some higher points of elevation. They 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 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 , 282, 435 b. 410, 505, 509, 180 c. 266, 227, 410, SpringBoard Mathematics with Meaning TM Level 2
3 Adding and Subtracting Integers ACTIVITY 1.2 SUGGESTED LEARNING STRATEGIES: Quickwrite, Look for a Pattern, Interactive Word Wall, KWL, Think/Pair/Share The students began thinking about elevation. They 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 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. Integer ,865 Absolute Value Unit 1 Integers and Rational Numbers 13
4 ACTIVITY 1.2 Adding and Subtracting Integers MATH TERMS 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. SUGGESTED LEARNING STRATEGIES: Think Aloud, Summarize/Paraphrase/Retell, Think/Pair/Share, Quickwrite 12. Evaluate each expression. a b. 56 c. 0 The students in Ms. Flower s class were starting to get excited about their field 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. The 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. Location Alamosa, Colorado Yellowstone National Park St. Cloud, Minnesota Bartlett, New Hampshire Fairbanks, Alaska Sleeping Bag Rating Outside Temperature º 010º 103º º +154º Will sleeper be warm? 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. 14 SpringBoard Mathematics with Meaning TM Level 2
5 Adding and Subtracting Integers ACTIVITY 1.2 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, 60 0, , 3 15, , 4 Inequalities can also be graphed on a number line. 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. EXAMPLE 1 Graph the inequality: x < 2. Step 1: Step 2: 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 Think 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: The number line above is the graph of x < 2. Unit 1 Integers and Rational Numbers 15
6 ACTIVITY 1.2 Adding and Subtracting Integers TRY THESE A SUGGESTED LEARNING STRATEGIES: KWL, Group Presentation, Think/Pair/Share, Close Reading Graph each inequality. a. y < 6 b. s > 5 c. b 2 d. a 7 Integers without signs are considered to be positive. Some of the students were excited about going hiking on their field 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. The first number in a problem is where you start your hike. The operation sign in the middle of the problem tells you whether you are going to face in the positive direction or the negative direction. The sign of the second number tells you if you are going to step backward () or forward (+). The 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: The answer is SpringBoard Mathematics with Meaning TM Level 2
7 Adding and Subtracting Integers ACTIVITY 1.2 SUGGESTED LEARNING STRATEGIES: Create Representations, Use Manipulatives TRY THESE B Illustrate the move for each problem. Then record your answer: a (3) = b. 6 + (6) = c. 5 + (3) = d = 17. Work with a partner. For each problem, roll a number cube twice. Use the numbers you roll to make a problem. Then use the number line to solve the problem. Look for patterns while you work. a.  + ( ) = WRITING MATH Parentheses are placed around a signed number to make a problem easier to read. b.  + = c. + ( ) = d. + = 18. What patterns did you notice about integer addition as you worked? Unit 1 Integers and Rational Numbers 17
8 ACTIVITY 1.2 Adding and Subtracting Integers SUGGESTED LEARNING STRATEGIES: Self Revision/Peer Revision, Look for a Pattern, KWL, Group Presentation 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 difference: 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: The answer is SpringBoard Mathematics with Meaning TM Level 2
9 Adding and Subtracting Integers ACTIVITY 1.2 SUGGESTED LEARNING STRATEGIES: Create Representations, Use Manipulatives TRY THESE C Illustrate the move for each problem. Then record your answer. a (3) = b = c. 5  (3) = d. 62 = 20. For each move that you illustrated in Try These C, show a different move that starts at the same integer and has the same answer as the original problem. The first one is done for you. a (3) = 5 + (+3) = 2 b = c. 5  (3) = d. 62 = Unit 1 Integers and Rational Numbers 19
10 ACTIVITY 1.2 Adding and Subtracting Integers SUGGESTED LEARNING STRATEGIES: Self Revision/Peer Revision, Group Presentation, Look for a Pattern, Use Manipulatives 21. How did you find 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. 20 SpringBoard Mathematics with Meaning TM Level 2
11 Adding and Subtracting Integers ACTIVITY 1.2 SUGGESTED LEARNING STRATEGIES: Use Manipulatives, Group Presentation, Think/Pair/Share 23. Work with a partner. For each problem, roll a number cube twice. Use the numbers you roll to make a problem. Then 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) = b (6) = c (2)  2 = 25. Describe a situation that matches each equation. a = 0 b. $75 + ($75) = $0 Unit 1 Integers and Rational Numbers 21
12 ACTIVITY 1.2 Adding and Subtracting Integers CHECK YOUR UNDERSTANDING Write your answers on notebook paper. Show your work. 1. Give a real world situation for the integer Order these integers from least to greatest: 3. Evaluate 36. 0, 132, 458, 789, Give two values that make these inequalities true. a. 0 < d b. z < 37 c. 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 (8) b c. 6  (6) d a. Use a number line to illustrate this problem and find its solution. b. Write a problem for a different move that begins at the same integer and has the same answer. 9. Describe a situation that matches this equation = MATHEMATICAL What relationship exists REFLECTION between addition and subtraction? Explain using integer models. 22 SpringBoard Mathematics with Meaning TM Level 2
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