1. 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 1 ÷ 2/5 “1 ÷ 2/5 = ?” means“how many two-fifths are in one?” There are 2½ two-fifths in 1. ↔ 1 ÷ 2/5 = 2½ 2½  2/5 = 1

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

3. 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). (b)Write another equation using division to represent the problem (let n be the answer). (c)Write an equation using multiplication to represent the problem (let n be the answer). (d)How are the three equations conceptually different? 2/3 ÷ 1/6 = n 2/3 ÷ n = 1/6 n  1/6 = 2/3 Repeated-subtraction view of division Sharing-equally view of division Repeated-addition view of multiplication

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. 4/5 ÷ 1/3 = x

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 5 th grader. 1/3 4/5 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 5 th grader. 1/3 4/5 4/5 ÷ 1/3 What does 4/5 ÷ 1/3 actually mean? How many 1/3 are in 4/5?

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 5 th grader. 1/3 4/5 4/5 ÷ 1/3 How many 1/3 are in 4/5? 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 5 th grader. 1/3 4/5 4/5 ÷ 1/3 How many 1/3 are in 4/5? How do you find the exact answer? Answer: 2 2525

9. 1/3 4/5 4/5 ÷ 1/3 How many 1/3 are in 4/5? We have seen why the answer is 2 2 / 5. Answer: 2 2525 or 12 5 Can you see 12/5 in these pictures?

10. How many 1/3 are in 4/5? We have seen why the answer is 2 2 / 5. Answer: 2 2525 or 12 5 Can you see 12/5 in these pictures? 1/3 4/5 4/5 ÷ 1/3

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? 1/3 4/5 4/5 ÷ 1/3 How many 1/3 are in 4/5? Answer: 2 2525 One cake of a cake or 12 5

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? 1/3 4/5 4/5 ÷ 1/3 How many 1/3 are in 4/5? Answer: 2 2525 One cake of a cake or 12 5 How many 1/3 of a cake are in 4/5 of a cake? Can you see 12  5?

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? 1/3 4/5 4/5 ÷ 1/3 Answer: One cake of a cake 12 5 How many 5/15 of a cake are in 12/15 of a cake? How many 5 pieces are in 12 pieces? 12  5 =

14. 4. 1/3 of a cake makes 1 serving. How many servings can we get from 4/5 of a cake? 1/3 4/5 4/5 ÷ 1/3 How many 1/3 are in 4/5? Answer: 2 2525 1/3 of a cake of a cake (e)What is the referent whole for the quotient 2 ? 2525 of 1/3 cake is equal to 4/5 cake. 2 2525 2  = 2 1 4 5 3 5 or 12 5 ÷ = 2 4 1 2 5 3 5

15. 2  = 2 1 4 5 3 5 ÷ = 2 4 1 2 5 3 5 2 2 / 5 is a fraction of another fraction. Its referent whole is the multiplicand. The referent whole for the quotient is the divisor. 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. In summary, what is the referent whole for each of these numbers?

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