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1 Lesson Dividing with Integers

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2 Lesson Dividing 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 learn what to do when you see positive and negative integers in division questions. Key words: integer quotient positive negative

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3 Lesson Dividing with Integers You already know how to divide by positive whole numbers. Now you’re going to see how a number line can be helpful in answering division questions with positive and negative integers. This is similar to what you saw in the previous Lesson with multiplication.

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4 5 coins Division Means Breaking a Number into Equal Parts Lesson Dividing with Integers “Divide” is another way of saying “share out equally.” 10 ÷ 2 10 coins

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5 A large bag contains 12 apples. The apples are shared equally among 4 students. How many apples does each student get? Example 1 Solution follows… Lesson Dividing with Integers Solution You can see from the picture that if the apples are shared out equally, each student will get 3 apples.

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6 Lesson Dividing with Integers You can also think of division as the opposite of multiplication. Example 1 asked you to find 12 ÷ 4. You can rewrite that as Then you can use “guess and check” to work out what the question mark must represent: Try substituting 1:4 × 1 = 12 Try substituting 2:4 × 2 = 12 Try substituting 3:4 × 3 = 12 which is the same as saying 12 ÷ 4 = ? 4 × ? = 12 When you do a division, the result is called the quotient.

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7 18 ÷ 3 21 ÷ 7 12 ÷ 6 26 ÷ ÷ 4 Guided Practice Solution follows… Lesson Dividing with Integers Solve without a calculator: 3 × 6 = 18, so 18 ÷ 3 = 6 7 × 3 = 21, so 21 ÷ 7 = 3 6 × 2 = 12, so 12 ÷ 6 = 2 2 × 13 = 26, so 26 ÷ 2 = 13 4 × 25 = 100, so 100 ÷ 4 =

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8 Guided Practice Solution follows… Lesson Dividing with Integers Solve without a calculator: 54 ÷ 9 88 ÷ ÷ 5 48 ÷ 12 9 × 6 = 54, so 54 ÷ 9 = 6 11 × 8 = 88, so 88 ÷ 11 = 8 5 × 14 = 70, so 70 ÷ 5 = × 4 = 48, so 48 ÷ 12 =

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9 You can think about this line in another way: Division Can Also Be Shown on a Number Line Lesson Dividing with Integers Here is a number line showing the multiplication 4 × 6 = 24. So the number line also shows that 24 ÷ 4 = 6 It shows that the number 24 can be divided into 4 equal parts, each of size 6. It also shows that the number 6 is one-fourth of the way from 0 to 24.

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10 Example 2 Solution follows… Lesson Dividing with Integers The number line shows the line from 0 to 35 divided into 5 equal parts. You can see that 7 is one-fifth of the way from 0 to 35, so the answer must be 35 ÷ 5 = 7. Use this number line to solve 35 ÷ 5. Solution

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11 Guided Practice Solution follows… Lesson Dividing with Integers Use this number line to answer Exercises 10–12: 102 ÷ ÷ 3 85 ÷ 5 When 102 is divided into 2 equal parts, each part has a size of 51. When 102 is divided into 3 equal parts, each part has a size of 34. When 85 is divided into 5 equal parts, each part has a size of

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12 Guided Practice Solution follows… Lesson Dividing with Integers Use this number line to answer Exercises 13–18: ÷ ÷ ÷ ÷ ÷ ÷

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13 Negative Numbers Can Be Used in Division Lesson Dividing with Integers Dividing with negative integers is similar to dividing with positive integers. You still have to work out how many equal parts go into the number.

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14 Example 3 Solution follows… Lesson Dividing with Integers Calculate –42 ÷ 7 using the number line. The number –6 is one-seventh of the way from 0 to –42 on the number line. So the answer is –6. Solution You can check this by working out 7 × (–6) = –42

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–64 ÷ –64 ÷ –40 ÷ –48 ÷ 3 Guided Practice Solution follows… Lesson Dividing with Integers Use this number line to answer Exercises 19–22: –8 –32 –16

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16 Dividing with Integers Lesson Rules for dividing integers positive ÷ positive = positive positive ÷ negative = negative negative ÷ positive = negative negative ÷ negative = positive 6 ÷ 2 = 3 (–6) ÷ 2 = –3 (–6) ÷ (–2) = 3 6 ÷ (–2) = –3 For example: The rules for dividing integers are really similar to the multiplication rules you saw in the last Lesson.

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17 Example 4 Solution follows… Lesson Dividing with Integers You can use a number line to work out that 42 divides into 7 equal groups of 6, so 42 ÷ 7 = 6. Calculate (–42) ÷ (–7). Solution Then using the rules for dividing integers shown above, you can see that –42 ÷ (–7) = 6 Rules for dividing integers positive ÷ positive = positive positive ÷ negative = negative negative ÷ positive = negative negative ÷ negative = positive

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18 Example 5 Solution follows… Lesson Dividing with Integers Calculate –120 ÷ (–40). You can use the rules for dividing integers instead of the number line. Solution The answer will be the same as 120 ÷ 40: –120 ÷ (–40) = 120 ÷ 40 = 3 Check: –40 × 3 = –120 Rules for dividing integers positive ÷ positive = positive positive ÷ negative = negative negative ÷ positive = negative negative ÷ negative = positive You are dividing a negative by a negative, so the answer is positive.

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19 Guided Practice Solution follows… Lesson Dividing with Integers Say whether the following will give positive or negative answers. (You don’t need to work out the actual solutions.) 23. –18 ÷ –27 ÷ (–9) 25. –17 ÷ (–1) ÷ (–25) 27. –363 ÷ –1008 ÷ (–24) negative positive negative positive negative positive

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20 Independent Practice Solution follows… Lesson Dividing with Integers Solve: ÷ ÷ 5 3. –36 ÷ ÷ (–11) 5. –27 ÷ (–9) 6. –126 ÷ 9 8 –3 3 9 –6 –14

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÷ ÷ ÷ (–5) Independent Practice Solution follows… Lesson Dividing with Integers Use a number line to solve: 9 –8 53

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–126 ÷ ÷ (–4) 12. –64 ÷ (–4) Independent Practice Solution follows… Lesson Dividing with Integers Use a number line to solve: –47 –21 16

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23 Independent Practice Solution follows… Lesson Dividing with Integers The Garcias are going to the store. Mrs. Garcia is going to divide $42 equally among her three children so they have pocket money. How much money does each child get? A submarine is –400 m from the surface of the ocean. It rises to half of this depth. At what depth is the submarine now? Emma’s bank account is overdrawn. Her balance is –$240. She pays some money into the account so that her overdraft is reduced to one-third of the old balance. What is the new balance on Emma’s account? $14 –200m –

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24 Lesson Dividing with Integers Round Up You’ve now learned rules for adding, subtracting, multiplying, and dividing integers. In the next Lesson, you’ll see these rules applied in real-life situations.

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