Learning Targets: Use models to subtract one integer with absolute value less than 10 from another. Subtract integers. Solve real-world problems by subtracting integers. SUGGESTED LEARNING STRATEGIES: Summarizing, Sharing and Respondiing, Create Representations, Use Manipulatives To track the winter temperatures, the students also have to subtract integers to find temperature differences. One way to subtract integers is to use models. Example A Use counters to subtract 5 ( 3). Step 1: Use counters to represent 5. 5: Step 2: Subtract 3 negative counters. Step 3: The difference is the number of counters that remain. Solution: 5 ( 3) = 2 Try These A Use counters to find each difference. You may draw the counters or use counters from your teacher. a. 6 ( 1) b. 4 ( 2) c. 5 ( 5) 1. Make Use of Structure. Consider the subtraction method used in Example A. a. You can use the method to subtract positive integers when the number being subtracted is less than the beginning number. Explain why. b. To subtract negative integers using the method in Example A, the number being subtracted must be greater than the beginning number. Explain why. 100 SpringBoard Mathematics Course 1, Unit 2 Integers
ACTIVITY 8 Zero pairs can be used to subtract other combinations of integers. Example B Use counters to subtract 6 ( 2). Step 1: Start with 6 positive counters. Step 2: You need to subtract 2 negative counters. There are no negative counters in Step 1. So insert 2 zero pairs. MATH TIP Remember, zero pairs have a sum of zero ( 1 + 1 = 0), so adding a zero pair is like adding zero to an expression. It does not change the value of the expression. Step 3: Subtract the 2 negative counters. Step 4: The difference is the number of counters that remain. Solution: 6 ( 2) = 8 Try These B Use counters to find each difference. You may draw the counters or use counters from your teacher. Check students models. a. 2 ( 3) b. 7 ( 1) c. 6 ( 4) 2. Look at each subtraction sentence and its corresponding addition sentence. Subtraction Sentence Addition Sentence 5 ( 3) = 2 5 + (+3) = 2 6 ( 2) = 8 6 (+2) = 8 8 (+5) = 13 8 + ( 5) = 13 7 (+4) = 3 7 + ( 4) = 3 a. How did the operation symbol change? CONNECT TO ENGLISH Subtracting a negative number is somewhat similar to stating a double negative in English. In English, the two negatives result in a positive to form an affirmative, or nonnegative, statement. Example: I do not disagree. b. How did the sign of the integer being subtracted change? c. What do you notice about the answer to each subtraction sentence and its corresponding addition sentence? Activity 8 Adding and 101
3. Write a corresponding addition sentence for the following subtraction sentence. 4 (+6) = 10 You have used models to subtract integers. You have used addition to subtract integers. Rule for : Change the subtraction sign to an addition sign and change the integer being subtracted to its opposite. Then use the rule for adding integers. 4. Use the rule for subtracting integers. Show your work. a. 5 7 b. 4 ( 9) c. 2 ( 4) d. 8 ( 6) e. 3 (+8) f. 6 ( 2) g. 6 ( 6) h. 3 ( 3) i. 4 (+9) j. 5 (+8) k. 3 (+5) l. 7 9 m. 7 ( 2) n. 6 (+3) 102 SpringBoard Mathematics Course 1, Unit 2 Integers
ACTIVITY 8 5. At the ski resort, the outside temperature is 7 F. The wind chill factor makes it feel like it is 12 F. a. Model with mathematics. Write an expression to model the difference between the actual temperature and what the temperature feels like with the wind chill. b. What is the difference between the actual temperature and what the temperature feels like with the wind chill? 6. At the state capital, the outside temperature is 18 F. The wind chill factor makes it feel like it is 3 F. What is the difference between the actual temperature and what the temperature feels like with the wind chill? Explain your reasoning. 7. The temperature at 6 a.m. was 15 F. At 6 p.m. the temperature was 20 F. What was the change in temperature? 8. a. Evaluate 8 3 and 3 8. b. Does reversing the order of the numbers in subtraction change the difference? Explain your thinking. Activity 8 Adding and 103
Check Your Understanding 9. Show how to use counters to subtract 5 ( 7). 10. How is the expression 8 2 related to the expression 8 + ( 2)? 11. Use the rules for subtracting integers to find each difference. a. 2 ( 3) b. 1 6 Write a subtraction number sentence for the models shown. 12. 13. Use counters to find each difference. 14. 1 ( 8) 15. 6 ( 2) 16. 3 9 17. 7 ( 6) ACADEMIC VOCABULARY An elevation is a distance above or below a point of reference, such as ground level or sea level. Subtract. 18. 4 ( 4) 19. 5 ( 12) 20. 8 ( 17) 21. 10 ( 8) 22. 34 ( 3) 23. 9 ( 9) 24. 20 ( 4) 25. 7 16 26. Reason quantitatively. Is 5 ( 5) the same as 5 5? Explain your reasoning. 27. Make sense of problems. A diver jumps from a platform 16 feet above the water to a depth of 6 feet below the surface of the water. What is the distance he dives? 28. A skyscraper is 1,150 feet above ground level. It extends 35 feet below ground level. What is the total length of the skyscraper from top to bottom? 29. A submarine at an elevation of 48 meters descends 12 meters. What is its elevation after it descends? 30. From Monday to Tuesday, the temperature fell from 23 F to 6 F. How many degrees is this temperature change? 104 SpringBoard Mathematics Course 1, Unit 2 Integers