Warm Up Problem of the Day Lesson Presentation Lesson Quizzes

Warm Up Evaluate each expression. 1. 17 · 5 2. 8 · 34 3. 4 · 86 4. 20 · 850 5. 275 ÷ 5 6. 112 ÷ 4 85 272 344 17,000 55 28

Problem of the Day A teacher assigns 5 points for a correct answer, –2 points for an incorrect answer, and 0 points for leaving the questioned unanswered. What is the score for a student who had 22 correct answers, 15 incorrect answers, and 7 unanswered questions? 80

Sunshine State Standards MA.7.A.3.1 Use and justify the rules for…multiplying and dividing…integers. Also MA.7.A.3.2, MA.7.A.5.2

You can think of multiplication as repeated addition. 3 · 2 = 2 + 2 + 2 = 6 and 3 · (–2) = (–2) + (–2) + (–2) = –6

Additional Example 1A: Multiplying Integers Using Repeated Addition Use a number line to find each product. 2 · (–6) Use the Commutative Property. + (–6) + (–6) -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 Think: Add -6 two times. 2 · (–6) = –12

Additional Example 1B: Multiplying Integers Using Repeated Addition Use a number line to find each product. –8 · 3 –8 · 3 = 3 · (–8) Use the Commutative Property. + (–8) + (–8) + (–8) –24–23–22–21–20–19–18–17-16-15-14-13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 Think: Add –8 three times. –8 · 3 = –24

Use a number line to find each product. Check It Out: Example 1A Use a number line to find each product. 3 · –4 +(–4) +(–4) -16 -14 -12 -10 -8 -6 -4 -2 0 2 4 –12 +(–4)

Use a number line to find each product. Check It Out: Example 1B Use a number line to find each product. –5 · 3 –15 –30 –25 –20 –15 –10 –5 0 5 –5 · 3

Multiplication and division are inverse operations Multiplication and division are inverse operations. They “undo” each other. Notice how these operations undo each other in the patterns shown. Remember!

The patterns below suggest that when the signs of integers are different, their product or quotient is negative. The patterns also suggest that the product or quotient of two negative integers is positive.

If the signs are: Your answer will be: the same positive different MULTIPLYING AND DIVIDING TWO INTEGERS If the signs are: Your answer will be: the same positive different negative

Additional Example 2: Multiplying Integers Find each product. A. –6 · (–5) –6 · (–5) Both signs are negative, so the product is positive. 30 B. –4 · 7 –4 · 7 The signs are different, so the product is negative. -28

Check It Out: Example 2 Find each product. A. –2 · (–8) 16 B. 5 · (–5) –25

Check It Out: Example 2 Find each product. C. –11 · 11 –121 D. –100 · (–28) 2,800

Additional Example 3: Dividing Integers Find each quotient. A. 35 ÷ (–5) 35 ÷ (–5) Think: 35 ÷ 5 = 7. –7 The signs are different, so the quotient is negative. B. –32 ÷ (–8) –32 ÷ (–8) Think: 32 ÷ 8 = 4. The signs are the same, so the quotient is positive. 4

Additional Example 3: Dividing Integers Find the quotient. C. –48 ÷ 6 –48 ÷ 6 Think: 48 ÷ 6 = 8. –8 The signs are different, so the quotient is negative.

Check It Out: Example 3 Find the quotient. A. –12 ÷ 3 –4 B. –24 ÷ (–6) 4

Check It Out: Example 3 Find the quotient. C. 18 ÷ (–9) –2

Zero divided by any number is zero, but you cannot find an answer for division by zero. For example –6 ÷ 0 ≠ 0, because 0 · 0 ≠ –6. We say that division by zero is undefined.

Additional Example 4: Diving Application Roxy went scuba diving. She descended 14 feet from her original position. She then rose, going up in 2 increments of 4 feet each. Her final position was 16 feet below sea level. Find Roxy's original position. Work backward to find Roxy's original position. Use negative integers to represent positions below sea level. Start with Roxy's final position. Because Roxy went up in 2 increments of 4 feet each, subtract to find her position before rising. 21

Additional Example 4 Continued Roxy went scuba diving. She descended 14 feet from her original position. She then rose, going up in 2 increments of 4 feet each. Her final position was 16 feet below sea level. Find Roxy's original position. Use the order of operations. –16 – (2 · 4) = –16 – 8 Add the opposite of 8. = –16 + (–8) = –24 ft Then add the distance she descended. –24 + 14 = –10 ft Roxy's original position was 10 feet below sea level. 22

Find position before descending: –20 + (3 5) = –20 + 15 = –5 ft Check It Out: Example 4 Sid went scuba diving. He rose 12 feet from his original position. He then descended, going down in 3 increments of 5 feet each. His final position was 20 feet below sea level. Find Sid’s original position. Find position before descending: –20 + (3 5) = –20 + 15 = –5 ft Find position before rising: –5 – 12 = –17 ft Sid’s original position was 17 ft below sea level. 23

Lesson Quizzes Standard Lesson Quiz Lesson Quiz for Student Response Systems

Lesson Quiz: Part I Use a number line to find the product. 1. –8 · 2 Find each product or quotient. 2. –3 · 5 · (–2) 3. –75 ÷ 5 4. –110 ÷ (–2) –16 30 –15 55

Lesson Quiz: Part II 5. The temperature in Bar Harbor, Maine, was –3 °F. It then dropped during the night to be four times as cold. What was the temperature then? –12°F

Lesson Quiz for Student Response Systems 1. Identify the number line that helps to find the product. –7 · 3 A. –21 B. –10

Lesson Quiz for Student Response Systems 2. Identify the product. –4 · 6 · (–4) A. 96 B. 2 C. –2 D. –96

Lesson Quiz for Student Response Systems 3. Identify the quotient. –66 ÷ 6 A. 10 B. 11 C. –10 D. –11

Lesson Quiz for Student Response Systems 4. Identify the quotient. –122 ÷ (–2) A. –62 B. –61 C. 61 D. 62

Lesson Quiz for Student Response Systems 5. The level of water in a tank is 16 feet. The level decreases steadily, and the tank is emptied in 8 hours. What was the change in water level in the tank during the first hour? A. 3 ft B. 2 ft C. –2 ft D. –3 ft