2 4 Multiplying and Dividing Integers Additional Example

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

2 -4 Multiplying and Dividing Integers Additional Example 1 B: Multiplying Integers Using Repeated Addition Use a number line to find each product. – 8 · 3 = 3 · (– 8) + (– 8) Use the Commutative Property. + (– 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

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

2 -4 Multiplying and Dividing Integers Check It Out: Example 1 B Use a number line to find each product. – 5 · 3 – 15 – 5 · 3 – 30 – 25 – 20 – 15 – 10 – 5 0 5

2 -4 Multiplying and Dividing Integers Remember! Multiplication and division are inverse operations. They “undo” each other. Notice how these operations undo each other in the patterns shown.

2 -4 Multiplying and Dividing Integers 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.

2 -4 Multiplying and Dividing Integers MULTIPLYING AND DIVIDING TWO INTEGERS If the signs are: Your answer will be: the same positive different negative

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

2 -4 Multiplying and Dividing Integers Check It Out: Example 2 Find each product. A. – 2 · (– 8) 16 B. 5 · (– 5) – 25

2 -4 Multiplying and Dividing Integers Check It Out: Example 2 Find each product. C. – 11 · 11 – 121 D. – 100 · (– 28) 2, 800

2 -4 Multiplying and Dividing Integers Additional Example 3: Dividing Integers Find each quotient. A. 35 ÷ (– 5) – 7 B. – 32 ÷ (– 8) 4 Think: 35 ÷ 5 = 7. The signs are different, so the quotient is negative. Think: 32 ÷ 8 = 4. The signs are the same, so the quotient is positive.

2 -4 Multiplying and Dividing Integers Additional Example 3: Dividing Integers Find the quotient. C. – 48 ÷ 6 – 8 Think: 48 ÷ 6 = 8. The signs are different, so the quotient is negative.

2 -4 Multiplying and Dividing Integers Check It Out: Example 3 Find the quotient. A. – 12 ÷ 3 – 4 B. – 24 ÷ (– 6) 4

2 -4 Multiplying and Dividing Integers Check It Out: Example 3 Find the quotient. C. 18 ÷ (– 9) – 2

2 -4 Multiplying and Dividing Integers 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.

2 -4 Multiplying and Dividing Integers 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.

2 -4 Multiplying and Dividing Integers 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 – 16 – (2 · 4) = – 16 – 8 operations. Add the opposite = – 16 + (– 8) = – 24 ft of 8. Then add the distance she descended. – 24 + 14 = – 10 ft Roxy's original position was 10 feet below sea level.

2 -4 Multiplying and Dividing Integers 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.

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

2 -4 Multiplying and Dividing Integers 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