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1 Lesson Multiplying with Integers

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2 Lesson Multiplying with Integers California Standard: Number Sense 2.3 Solve addition, subtraction, multiplication, and division problems, including those arising in concrete situations, that use positive and negative integers and combinations of these operations. What it means for you: You’ll see what happens when you multiply positive and negative whole numbers. Key words: integer product factor

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3 Multiplying with Integers You’ve seen how to use a number line to show what happens when you add or subtract positive and negative integers. Lesson In this Lesson you’ll see how it can be useful for doing multiplication problems too. –4……plus 9……equals 5.

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4 Multiplying with Integers Multiplication Is All About Grouping Things Multiplication is really a way of adding together groups of objects. Lesson You can do the same kind of grouping and counting on the number line. For instance, 2 × 3 just means “2 groups of 3.” There are 6 blocks in total, so 2 × 3 = 6. =+ Doing “3 groups of 2” gives the same result. There are still 6 blocks, so 3 × 2 = 6. +=+

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5 Multiplying with Integers Example 1 Solution follows… Lesson Show the answer to 2 × 3 using a number line.

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6 Multiplying with Integers Example 1 Lesson You can show the answer with 2 arrows, each of length 3: Solution You could show the same answer with 3 arrows of length 2: 3 × 2 is three times as far from 0 as 2 is. 2 × 3 is twice as far from 0 as 3 is. Show the answer to 2 × 3 using a number line

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Multiplying with Integers Guided Practice Solution follows… Lesson What multiplication is shown on each number line? 3 × 7 10 × 2 4 × 4

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8 Multiplying with Integers Multiplying by a Negative Changes the Direction Even if you’re multiplying by a negative, you’re still dealing with groups. Lesson So 3 × (–2) still means “3 groups of –2.” Just like in Example 1, there are 3 arrows of length 2 on the number line, but this time the negative sign means they’re pointing left. You can see from this number line that 3 × (–2) = –6. 222

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9 Multiplying with Integers Example 2 Calculate 4 × (–1). Solution follows… Lesson 1.2.1

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10 Multiplying with Integers Example 2 Lesson You can see from this number line that 4 × (–1) = –4. Solution –1 Calculate 4 × (–1).

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11 Multiplying with Integers Example 3 The outside temperature at midnight was 0 °F. Every hour after that, the temperature dropped by 3 °F. What was the temperature at 5 a.m.? Solution follows… Lesson The change in temperature is –3 °F each hour for five hours. So you need to solve 5 × (–3). Solution This shows that 5 × (–3) = –15, so at 5 a.m. the temperature was –15 °F. –3

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12 Multiplying with Integers Guided Practice Solution follows… Lesson What multiplication is shown on each number line? 4 × (–10) What conclusion can you make from Exercises 4 and 5? 5. 4 × (–10) is the same as 10 × (–4). It doesn’t matter which number the negative sign belongs to — the answer is the same. 10 × (–4) 6.

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13 Multiplying with Integers Guided Practice Solution follows… Lesson × (–4) –7 × 3 30 × (–2) –2 × 30 Calculate the following multiplications: –28 –21 –

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The amount of fuel in a racing car changes by –6 gallons per lap. What is the change in its fuel load over 7 laps? Multiplying with Integers Guided Practice Solution follows… Lesson A submarine changes its depth in the water by –25 feet per minute. What is its total change in depth in four minutes? –25 × 4 = –100 feet –16 × 5 = –80 feet –6 × 7 = –42 gallons 12. A bird is flying toward the ground. Its height changes by –16 feet per second. What is the bird’s total change in height in 5 seconds?

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15 Multiplying with Integers A Negative Times a Negative Equals a Positive You’ve already seen that multiplying a positive integer by a negative integer results in a negative solution. But if you multiply one negative number by another, their “–” signs cancel each other out. Lesson –4 × 2 = –8

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16 Multiplying with Integers Example 4 Calculate –3 × (–2). Solution follows… Lesson You know that 3 × (–2) means “3 groups of –2”, and 3 × (–2) = –6. Solution The extra negative sign in –3 × (–2) changes the sign again. The answer must be positive: –3 × –2 = 6

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17 Multiplying with Integers If you’ve got several negative integers to multiply, you can do it bit by bit. Lesson 1.2.1

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18 Multiplying with Integers Example 5 Calculate –3 × (–2) × (–5). Solution follows… Lesson [–3 × (–2)] × –5 Solution = 6 × (–5) = –30 Work it out in smaller parts Now, positive × negative = negative First multiply two of the numbers: –3 × (–2) = 6

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19 Multiplying with Integers You can multiply any two integers using the following rules: Lesson Rules for multiplying integers positive × positive = positive positive × negative = negative negative × positive = negative negative × negative = positive 2 × 3 = 6 (–2) × 3 = –6 (–2) × (–3) = 6 2 × (–3) = –6 For example:

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20 If there’s an odd number of negative factors, you’ll end up with one that doesn’t cancel out, so the final answer will be negative. Multiplying with Integers Lesson You can use these rules even if you’re multiplying more than two numbers together. Just count the number of “–” signs in the question. If there’s an even number of negative factors, they’ll cancel out in pairs, and the answer will be positive. Rules for multiplying integers positive × positive = positive positive × negative = negative negative × positive = negative negative × negative = positive

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21 Solution continues… Multiplying with Integers Example 6 Solve –2 × 5 × (–4) × (–10). Solution follows… Lesson Work out the “size” of the number by finding: 2 × 5 × 4 × 10 = 400 This is an odd number, so the answer will be negative. –2 × 5 × (–4) × (–10) has three minus signs. Solution So the answer must be –400.

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22 Multiplying with Integers Example 6 Lesson Solution ( continued ) = –400 Now, positive × negative = negative = 40 × (–10) Negative × negative = positive –2 × 5 × (–4) × (–10) = –10 × (–4) × –10 Negative × positive = negative To prove that the answer is –400, you can break the question down into smaller parts:

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23 Multiplying with Integers Guided Practice Solution follows… Lesson –8 × (–3) –2 × 9 2 × (–3) × (–5) –27 × (–13) × (–7) × (–17) Say whether each of the following questions will have positive or negative answers. (You don’t need to work out the actual solutions.) Negative (odd number of negative factors) Positive (even number of negative factors)

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24 Multiplying with Integers Guided Practice Solution follows… Lesson –6 × 11 × (–19) × (–83) –1 × 2 × (–3) × 4 × (–5) 225 × (–311) × (–277) × (–1008) × 47 × (–119) Say whether each of the following questions will have positive or negative answers. (You don’t need to work out the actual solutions.) Negative (odd number of negative factors) Positive (even number of negative factors) Negative (odd number of negative factors)

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× 7 2. –3 × × 4 Multiplying with Integers Independent Practice Solution follows… Lesson In Exercises 1–3, use a number line to solve the multiplication. 35 –36 44

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× (–6) × (–2) 6. –8 × (–3) Multiplying with Integers Independent Practice Solution follows… Lesson In Exercises 4–6, use a number line to solve the multiplication. –36 21 × (–2) = –21 × 2 = –42 –8 × (–3) = 8 × 3 = 24

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27 Ms. Ross is overdrawn on her bank account. Her balance is –$30. Mr. Banks is overdrawn on his bank account by 5 times the amount Ms. Ross is overdrawn. What is Mr. Banks’s account balance? Sara multiplied two negative integers together. She then multiplied her answer by another negative number. Is her final result positive or negative? Pablo multiplied two integers together. The answer that he got was –28. What integers might he have multiplied together? Multiplying with Integers Independent Practice Solution follows… Lesson –$150 negative possible answers: 1 × (–28); –1 × 28; 2 × (–14); –2 × 14; 4 × (–7); –4 ×

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28 Round Up It’s important to know what happens when you multiply by negative integers, because they appear in lots of math topics. You’ll need the rules for multiplying again when you learn about dividing with negative integers in the next Lesson. Lesson Multiplying with Integers

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