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2/5

1. How would you explain meaningfully to 5 th graders so that they can understand why the answer for 1 ÷ 2/5 is 2½. “ 1 ÷ 2/5 = ?” means. “how many two-fifths are in one? ”. 1. 2/5. 1 ÷ 2/5. There are 2½ two-fifths in 1. 2½  2/5 = 1. ↔ 1 ÷ 2/5 = 2½.

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  1. 1. How would you explain meaningfully to 5th graders so that they can understand why the answer for 1 ÷ 2/5 is 2½. “1 ÷ 2/5 = ?” means “how many two-fifths are in one?” 1 2/5 1 ÷ 2/5 There are 2½ two-fifths in 1. 2½ 2/5 = 1 ↔ 1 ÷ 2/5 = 2½

  2. 2. Without computing, predict in a meaningful manner the answer in each case (a) (b) (c) (d) (e)

  3. Suppose that you have 2/3 of a large pizza. If each child eats 1/6 of a pizza, how many children can the 2/3 pizza feed? (a) Write an equation using division to represent the problem (let n be the answer). 2/3 ÷ 1/6 = n Repeated-subtraction view of division (b) Write another equation using division to represent the problem (let n be the answer). 2/3 ÷ n = 1/6 Sharing-equally view of division (c) Write an equation using multiplication to represent the problem (let n be the answer). n  1/6 = 2/3 Repeated-addition view of multiplication (d) How are the three equations conceptually different?

  4. 4. 1/3 of a cake makes 1 serving. How many servings can we get from 4/5 of a cake? (a) Write an equation to represent the problem.Let x be the answer. x 4/5 ÷ 1/3 =

  5. 4. 1/3 of a cake makes 1 serving. How many servings can we get from 4/5 of a cake? (b) Solve this problem in a way such that it is meaningful for a 5th grader. 4/5 1/3 ? 4/5 ÷ 1/3

  6. 4. 1/3 of a cake makes 1 serving. How many servings can we get from 4/5 of a cake? (b) Solve this problem in a way such that it is meaningful for a 5th grader. 4/5 1/3 What does 4/5 ÷ 1/3 actually mean? How many 1/3 are in 4/5? 4/5 ÷ 1/3

  7. 4. 1/3 of a cake makes 1 serving. How many servings can we get from 4/5 of a cake? (b) Solve this problem in a way such that it is meaningful for a 5th grader. 4/5 1/3 How many 1/3 are in 4/5? 4/5 ÷ 1/3 More than 2 but less than 3.

  8. 4. 1/3 of a cake makes 1 serving. How many servings can we get from 4/5 of a cake? (b) Solve this problem in a way such that it is meaningful for a 5th grader. 4/5 How do you find the exact answer? 1/3 How many 1/3 are in 4/5? 4/5 ÷ 1/3 2 5 Answer: 2

  9. We have seen why the answer is 2 2/5. Can you see 12/5 in these pictures? 4/5 1/3 How many 1/3 are in 4/5? 4/5 ÷ 1/3 12 5 2 5 Answer: 2 or

  10. We have seen why the answer is 2 2/5. Can you see 12/5 in these pictures? 4/5 1/3 How many 1/3 are in 4/5? 4/5 ÷ 1/3 12 5 2 5 Answer: 2 or

  11. 4. 1/3 of a cake makes 1 serving. How many servings can we get from 4/5 of a cake? (c) What is the referent whole for the dividend 4/5? One cake 4/5 of a cake 1/3 How many 1/3 are in 4/5? 4/5 ÷ 1/3 12 5 2 5 Answer: 2 or

  12. 4. 1/3 of a cake makes 1 serving. How many servings can we get from 4/5 of a cake? (d) What is the referent whole for the divisor 1/3? One cake How many 1/3of a cake are in 4/5 of a cake? 4/5 of a cake Can you see 12  5? ofa cake 1/3 How many 1/3 are in 4/5? 4/5 ÷ 1/3 12 5 2 5 Answer: 2 or

  13. 4. 1/3 of a cake makes 1 serving. How many servings can we get from 4/5 of a cake? (d) What is the referent whole for the divisor 1/3? One cake How many 5/15of a cake are in 12/15 of a cake? 4/5 of a cake How many 5 pieces are in 12 pieces? ofa cake 1/3 4/5 ÷ 1/3 12 5 12 5 = Answer:

  14. 4. 1/3 of a cake makes 1 serving. How many servings can we get from 4/5 of a cake? (e) What is the referent whole for the quotient 2 ? 2 5 1/3 of a cake of 1/3 cake is equal to 4/5 cake. 4/5 of a cake 2 14 5 3 5 41 2 535 2  = ÷ = 2 ofa cake 1/3 How many 1/3 are in 4/5? 4/5 ÷ 1/3 12 5 2 5 2 5 or Answer: 2 2

  15. In summary, what is the referent whole for each of these numbers? The referent whole for the quotient is the divisor. 2 14 5 3 5 41 2 535 2  = ÷ = 2 In repeated-subtraction view of division, the referent whole for the dividend and divisor are by default 1 unit. In fractional part of a quantity view of multiplication, the referent whole for the multiplicand and the product is by default 1 unit. 2 2/5 is a fraction of another fraction. Its referent whole is the multiplicand.

  16. Use this Virtual Manipulative at this link to reason about Division involving Fractions http://nlvm.usu.edu/en/nav/frames_asid_265_g_3_t_1.html?open=activities&from=category_g_3_t_1.html

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