1. Beyond Invert and Multiply: Making Sense of Fraction Computation Julie McNamara November 6 and 7, 2014

2. Have you ever heard…. Yours is not to reason why, Just invert and multiply!

3. In contrast to….. “Children who are successful at making sense of mathematics are those who believe that mathematics makes sense.” -Lauren Resnick

4. CCSS Standards for Mathematical Practice 1.Make sense of problems and persevere in solving the 2.Reason abstractly and quantitatively. 3.Construct viable arguments and critique the reasoning of others. 4.Model with mathematics. 5.Use appropriate tools strategically. 6.Attend to precision. 7.Look for and make use of structure. 8.Look for and express regularity in repeated reasoning.

5. NCTM Mathematics Teaching Practices 1.Establish mathematics goals to focus learning. 2.Implement tasks that promote reasoning and problem solving. 3.Use and connect mathematical representations. 4.Facilitate meaningful mathematical discourse. 5.Pose purposeful questions. 6.Build procedural fluency from conceptual understanding. 7.Support productive struggle in learning mathematics. 8.Elicit and use evidence of student thinking.

6. Brendan, Grade 4

7. Fractions as numbers… “ In mathematics, do whatever it takes to help you learn something, provided you do not lose sight of what you are supposed to learn. In the case of fractions, it means you may use any pictorial image you want to process your thoughts on fractions, but at the end, you should be able to formulate logical arguments in terms of the original definition of a fraction as a point on the number line.” -Wu, 2002, p. 13

8. Rod Relations Using your Cuisenaire rods, find as many fractional relationships as you can. For example: 1 orange = 2 yellows, so 1 yellow = ½ orange

9. Making Sense of Fraction Addition

10. Whole Number Addition and Subtraction Strategies Decomposing/recomposing Associative property Commutative property Renaming

11. Use the Cuisenaire Rods to solve: 1212 1212 brown rod + brown rod 1414 1414 brown rod + brown rod 1212 1414 brown rod + brown rod

12. Addition with Cuisenaire Rods, V1 and V2 Version 1: All problems use brown rod as the whole May need to rename one addend Version 2: Problems use different rods as the whole May need to rename both addends

13. Get to the Whole! Decomposing and recomposing fractions to “get to the whole” when adding and subtracting. 3434 3434 +

14. : Will’s Strategy Video from Beyond Invert & Multiply (in press) 3434 3434 +

15. : Belen’s Strategy 3434 3434 + Video from Beyond Invert & Multiply (in press)

16. : Malia’s Strategy 3535 4545 + Video from Beyond Invert & Multiply (in press)

17. Student work

20. Making Sense of Fraction Multiplication

21. Connecting Multiplication to Addition

22. Using repeated addition to solve 1212 x 6 Video from Beyond Invert & Multiply (in press)

23. Connecting to Whole Number Multiplication

24. Farmer Liz FruitVegetable s

25. Farmer Liz FruitVegetable s

26. Farmer Liz Farmer Liz wants to further partition the two halves of her field such that -- half of the fruit section will be planted with fruit trees and half with fruit bushes half of the vegetable section will be planted with vegetables that grow above ground and half with vegetables that grow below ground

27. Farmer Liz Fruit trees Fruit bushes Above ground vegetables Below ground vegetables

28. Farmer Liz What fraction of Farmer Liz’s field is planted with fruit trees? What fraction of Farmer Liz’s field is planted with fruit bushes? What fraction of Farmer Liz’s field is planted with above ground vegetables? What fraction of Farmer Liz’s field is planted with below ground vegetables?

29. Multiplying fractions How does the “field” show that = 1212 1212 x 1414 ?

30. Farmer Bruce Farmer Bruce is going to plant half of his field with flowers and leave the other half unplanted to use as a pasture. He plans to use 1/3 of the flower half for roses, 1/3 for daisies, and 1/3 for geraniums. What fraction of the field will be planted with roses? What fraction of the field will be planted with daisies? What fraction of the field will be planted with geraniums?

31. Farmer Bruce Farmer Bruce decided that he really didn’t like geraniums and instead planted 2/3 of the flower section with daisies and kept 1/3 for roses, how much of the field is planted with daisies?

32. Tell Me All You Can Before coming up with an exact answer, consider what you know about the answer as a means of getting a sense of the “neighborhood” of the answer.

33. Tell Me All You Can The answer will be less than ________ because _________. The answer will be greater than ________ because _________. The answer will be between ________ and ________ because _________.

34. What do you know about ? 1212 x 2 6 Video from Beyond Invert & Multiply (in press)

35. What do you know about ? 1212 x 4 5 Video from Beyond Invert & Multiply (in press)

36. Making Sense of Fraction Division

37. Two types of division situations: Quotative (also called measurement division): Size of group is known; number of groups is unknown 6 ÷ 2: How many 2 ’s are in 6? Partitive : Number of groups is known; how many in each group is unknown 6 ÷ 2: Split 6 into 2 groups  6 is 2 of what?

38. Quotative Division 6 ÷ 2 : How many 2 ’s are in 6

39. How Long? How Far? Part 1

40. Video from Beyond Invert & Multiply (in press)

41. Quotative Division

42. Reasoning about 1 ÷ 1616

43. Reasoning about 2 ÷ 1616

44. Reasoning about 10 ÷ 1313

45. Reasoning about 6 ÷ 3434

46. Partitive Division 6 ÷ 3: Split 6 into 3 groups  6 is 3 of what?

47. How Long? How Far? Part 2 Beach Clean-Up (2 people) DistanceEach person cleans 8 miles4 miles 2 miles 1 mile ½ mile ?

48. How Long? How Far? Part 2 1212 ÷ 3 1616 ÷ 2 3434 ÷ 3

49. Do you always have to invert and multiply? Your friend tells you she doesn’t understand why your teacher makes you invert and multiply to divide fractions. She says you can just divide across the numerators and denominators to get your answer. She shows you the two examples below to prove her point: What do you think of her idea?  Is she right?  If so, why? If not, why not? 49

50. What about in this case?

51. Write a word problem that can be solved by the expression below:

52. CCSS Number and Operations - Fractions 3.NF: Develop understanding of fractions as numbers. 4.NF: Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. 5.NF: Use equivalent fractions as a strategy to add and subtract fractions. 5.NF: Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

53. Fractions as numbers… “ In mathematics, do whatever it takes to help you learn something, provided you do not lose sight of what you are supposed to learn. In the case of fractions, it means you may use any pictorial image you want to process your thoughts on fractions, but at the end, you should be able to formulate logical arguments in terms of the original definition of a fraction as a point on the number line.” -Wu, 2002, p. 13

54. Remember…. “Children who are successful at making sense of mathematics are those who believe that mathematics makes sense.” -Lauren Resnick

55. Coming Soon!

56. Thank you!!! juliemcmath@gmail.com