1. Teaching for Understanding Division with Fractions Dr. DeAnn Huinker, Dr. Kevin McLeod, Dr. Henry Kepner University of Wisconsin-Milwaukee Math Teacher Leader (MTL) Seminar Milwaukee Public Schools June 2005 This material is based upon work supported by the National Science Foundation Grant No. EHR-0314898.

2. Session Goals  Deepen conceptual understanding of division with fractions.  Examine “why” we can invert and multiply.

3. Popcorn Party Serving Size: 2 cups How many servings from:  8 cups of popcorn  5 cups of popcorn

4. Summarize  Write the equation for each situation (8 cups; 5 cups).  Identify the meaning of each number in the equations.

5. Summarize How many 2–cup servings? 8 ÷ 2 = 4 cups cup servings serving 5 ÷ 2 = 2 cups cup serving 1212 servings

6. What does the represent?  One-half of a cup of popcorn?  One-half of a serving?  Are you sure? 1212

7. Popcorn Party Serving Size: cup How many servings from:  1 cup of popcorn  4 cups  2 cups 1212 1212

8. Summarize How many 2–cup servings? 1 ÷ = 2 cup servings 4 ÷ = 8 cups servings 1212 cup serving 1212 cup serving

9. Task Popcorn Parties # 1 & # 2 æFacilitator æPaper Strip Demonstrator

10. Algorithm

11. Task Popcorn Parties # 1 & # 2  Facilitator poses one problem at a time.  Each individual silently solves it.  On facilitator’s cue: State answer.  Demonstrator models with paper strips.  Take turns to justify your reasoning.

12. Popcorn Party #1 Serving Size: cup of popcorn How many servings can be made from: 1 cup of popcorn 2 cups of popcorn 3 1/2 cups of popcorn 1414

13. Popcorn Party #2 Serving Size: cup of popcorn How many servings can be made from: 3/4 cup of popcorn 6 cups of popcorn 2 1/4 cups of popcorn 4 1/2 cups of popcorn 3434

14. Popcorn Party Serving Size: cup How many servings from 4 cups? 3434

15. Summarize How many 2–cup servings? 4 ÷ = cups 3434 cup serving

16. Discussion In reviewing these division situations and your solutions, what are you noticing?

17. Fraction Division Algorithm Photo Sequence from Session

18. 7 X = 1 ?

19. a X = 1 ? Multiply by the inverse, which is also called the reciprocal.

20. a X = 1 a ≠ 0 Is this true for any number? What about zero? 1a1a

21. 3434 X = 1 1 ?

22. 7 ÷ 8 = How else can we write this?

23. 12 24 = ? ? 1212

24. = 1 12 1212

26. 5 ÷ = 3434 Estimate Greater than 5? Equal to 5? Less than 5?

27. Summarize How many 2–cup servings? 5 ÷ = 3434 How else can we write this?

28. 5 = 3434 1

30. = 3434 1 9292

32. = cdcd 1 abab