Division of Fractions: A Learning Trajectory Designed to Promote Conceptual Understanding

1. Division of Fractions: A Learning Trajectory Designed to Promote Conceptual Understanding Amanda Geist Centennial Middle School amanda.geist@bvsd.org

3. Do students really understand what this means? • Can they draw a picture to represent this problem? • Can they give a real life situation that could be represented by this problem?

4. Background • At the time I implemented this project I taught honors 7th grade math students • In general, many of them caught onto math “rules” quickly, but sometimes lacked the understanding behind them • These students were exposed to division of fractions the previous year in 6th grade • I began asking students questions like draw a picture or give a real life situation in class and realized they couldn’t do either task

5. Project Exploration • Can students gain conceptual understanding of dividing fractions, even if they’ve already been exposed to the tip of the iceberg (formal algorithms)? • Is it possible for students to truly remember the formal algorithms (if they don’t have floating capacity for the iceberg)? • Can the iceberg work backwards?

6. Project Implementation • Pre-Assessment • Daily number talks (around 10 minutes at the start of class) for a period of 6 weeks • Pose division situations in context, students share strategies and ideas, teacher introduces other strategies • Post-Assessment

7. Division Pre-Assessment

8. Pre-Assessment Conclusions • Students who solved problems correctly used a wide range of strategies ranging from informal to formal • Many students were unsure of what to do with the remainder in fraction division situations • Many students couldn’t remember exactly how to implement the formal division of fractions algorithm • Students did better solving fraction division problems in context rather than out of context. Many of them resorted to informal methods due to the context.

9. Common Denominator Algorithm Multiply by the Reciprocal Give a Real Life Situation that could be represented by the problem Number Line Ratio Table Drawing Pictures Unifix Cubes Exploration of Division Problems in Context

10. 20 ÷ 5 “If you equally divide 20 into 5 groups, how many will be in one group?” “How many groups of 5 fit inside of 20?” • Quotative • Grouping • Measurement • Partitive • Fair-Share

11. “If you equally divide 4 among ¾ of a group, how much will be in one group?” “How many groups of ¾ fit inside of 4?” • Quotative • Grouping • Measurement • Partitive • Fair-Share

12. “If you equally divide 20 into 5 groups, how many will be in one group?” “How many groups of 5 fit inside of 20?” “How many groups of ¾ fit inside of 4?” “If you equally divide 4 among ¾ of a group, how much will be in one group?”

13. Quotative Pathway (Also known as grouping or measurement) Eventually leads to the common denominator algorithm 20 ÷ 5 = “How many groups of 5 fit inside of 20?” = “How many groups of ¾ fit inside of 6?”

14. You have three bars of cheese. If it takes 2/3 bar of cheese to make one pizza, how many pizzas can you make? You have three bars of cheese. If it takes 4/3 bar of cheese to make one pizza, how many pizzas can you make?

15. Drawing a picture. I have 3 bars of cheese. It takes 2/3 bar to make one pizza. How many pizzas can I make?

16. Drawing a picture. I have 3 bars of cheese. It takes 4/3 bar to make one pizza. How many pizzas can I make?

17. Underlying Question “How many groups of ____ bars of cheese can be made from ____ bars of cheese?” “How many times does ____ fit inside of ____?”

18. Number Line How many times does 2/3 fit inside of 3?

19. Number Line How many times does 4/3 fit inside of 3?

20. Ratio Table How many times does 2/3 fit inside of 3?

21. Ratio Table How many times does 4/3 fit inside of 3?

22. These are Examples of Division You have 5 bars of cheese. It takes 2/3 bar of cheese to make a pizza. How many pizzas can you make? • How many groups of 2/3 fit inside of 5?

23. Bows require ¾ yard of ribbon. How many bows can you make from 2 yards of ribbon? • How many groups of 3/4 fit inside of 2? • Division number sentence_______________ (Use unifix cubes)

24. A serving size is 2/3 cookies. How many servings can be made from 4 cookies? • How many groups of 2/3 fit inside of 4? • Division number sentence_______________ (Draw a Picture)

25. How many groups of ____ fit inside of ____? (Number Line) Ask students to give a real life context

26. How many groups of ____ fit inside of ____? (Ratio Table) Ask students to give a real life context

27. A serving size of rice is 1/6 cup. How many servings can be made from 1/2 of a cup? • In other words, how many times does 1/6 fit inside of 1/2?

28. A serving size of rice is 1/2 cup. How many servings can be made from 1/6 of a cup? • In other words, how many times does 1/2 fit inside of 1/6?

29. A serving size of cereal is 3/8 cup. How many servings can be made from 1/2 of a cup? • In other words, how many times does 3/8 fit inside of 1/2?

30. A serving size of cereal is 1/2 cup. How many servings can be made from 3/8 of a cup? • In other words, how many times does 1/2 fit inside of 3/8?

31. A serving size of cereal is 1/3 cup. How many servings can be made from 3/4 of a cup? • In other words, how many times does 1/3 fit inside of 3/4?

32. If we split the bars into the same size pieces, we’re asking how many times does 4/12 fit inside of 9/12?

33. If we split the bars into the same size pieces, we’re asking how many times does 4/12 fit inside of 9/12?

34. Tip of the Iceberg: Common Denominator Algorithm • A serving size of rice is 2/3 cup. How many servings can you make from 5/6 cup? • How many times does 2/3 fit inside of 5/6?

35. Tip of the Iceberg: Common Denominator Algorithm • A serving size of rice is 2/7 cup. How many servings can you make from 4/5 cup? • How many times does 2/7 fit inside of 4/5?

36. Partitive Model Pathway (Also known as fair sharing) Eventually leads to multiplying by the reciprocal 20 ÷ 5 = “If you equally divide 20 into 5 groups, how many will be in one group?” = “If you equally divide 6 among ¾ of a group, how much will be in one group?”

37. 6 strawberries (equally sized) fill ¾ of a cup. How many strawberries will fill one cup? • If you equally share 6 among ¾ of a group, how much will be in one group?

38. 4 cookies fill 2/3 of a cookie jar (a very small cookie jar!) How many cookies will fill one jar? • If you equally share 4 among 2/3 of a group, how much will be in one group?

39. 5 kiwis weigh 4/5 of a pound. How many kiwis will weigh one pound? • If you equally share 5 among 4/5 of a group, how much will be in one group?

40. If you equally share 1/3 cake among 3 people, how much cake does each person get? • If you equally share 1/3 among 3 groups, how much is in one group?

41. If you equally share 2/3 cake among 5 people, how much cake does each person get? • If you equally share 2/3 among 5 groups, how much is in one group?

42. 1/2 of a cake fills 2/3 of a container. How much cake will fill one container? • If you equally share 1/2 among 2/3 of a group, how much is in one group?

43. 1/2 of a cake fills 2/3 of a container. How much cake will fill one container?

44. 1/2 of a cake and fills 2/3 of a container. How much cake will fill one container?

45. Tip of the Iceberg: Multiply by the Reciprocal • 3/4 of a pizza fills 2/3 of a container. How much pizzas will fill one container?

46. If it takes 5/8 hours to pain 2/3 of a room, how many hours will it take to paint 1 room?

47. “If you equally divide 6 among ¾ of a group, how much will be in one group?”

48. If you equally share 4 among 2/3 of a group, how much will be in one group?

49. If you equally share 5 among 4/5 of a group, how much will be in one group?

50. Summative Exit Ticket (Post Test) 1) 2) Draw a picture to represent this problem 3) Give a real life situation that could be represented by this problem