Developing Multiplicative Reasoning

1. Developing Multiplicative Reasoning Mike Askew King’s College London

2. Comparing

3. Smarties • There are two jars. • One contains 25 smarties • The other contains 75 smarties • The smarties are poured into one jar • How many smarties in the jar?

4. Water • There are two jars. • One contains water at 25o C • The other contains water at 75o C • The water is poured into one jar • What is the temperature of water in the jar now?

5. What fraction (1)? • 2/5 • 3/5 • 2 1/2 1 1/2 2/3 1 2/3

6. What fraction (2)?

7. MaTCH - School culture • Lack of belief in children’s ability to think and be creative • ‘Empty vessels’ model of teaching • Low expectations, especially about children’s language

8. Development emphases • Teaching methods:Use of discussion and cooperative workChildren explaining methods • Teaching approaches:Developing tools for thinking • Addition through empty number line • Multiplication through arrays and areas

9. Artifacts and tools • Artifacts: symbols, concrete materials, diagrams, words • Tools:artifacts linked to mental representation

10. Interpreting children’s thinking • Artifacts – empty number line, arrays, areas – introduced by teacher • Tool use – evidence for this from children’s talk and recording?

11. Artifacts into tools Multiplicative reasoning

12. Multiplication arises from • Repeated sets (3 tables, 4 at each) • Scale factors (3 times as long as) • Rectangular areas (3 rows of 4 chairs) • Cartesian product (3 breads, 4 fillings)

13. Division arises from: • Measurement/grouping (quotition) • 12 children at tables of 4 • Sharing (partition) • 12 children at 4 tables

14. Teaching approach • ‘Ban’ on teaching multiplication as repeated addition • Use of arrays and areas line to model strategies • Later use of ratio tables for linking with division

15. Early work: describing arrays • • • • • • • • • • • • • • • 

16. Why arrays? • Links with 10 x 10 multiplication grid • Less obvious link with addition • Cartesian product can be modelled on an array (bread x fillings) • Extension of one dimensional number line into two dimensions • Tool for thinking with

17. Torn grids (1)

18. Torn grids (2)

19. 35 x 23

20. 35 x 23

21. 89 x 75

22. 7 x 32

23. 26 x 16

24. 42 x 36

25. 32 x 69

26. Reduced scale

27. 15 x 14 - child’s extension

28. 9 x 22

29. 34 x 9

30. Orientation

31. Left waiting • 650 people want to travel into Athens by train • There are 11 carriages • Each carriage carries 45 people • How many people are left waiting for the next train?

32. Left waiting 1

33. Left waiting 2

34. Left waiting 3

35. Turnstiles • There are 42 turnstiles at the North East end of the stadium • 285 (210) people go through each turnstile. • How many people entered the stadium by the North East entrance?

36. Turnstiles 1

37. Turnstiles 2

38. Grandmother’s sweets • Grandmother arrives with a bag of 24 sweets. • There are three children at home who decide to share the sweets equally. • Just about to eat them when another child arrives. • And then the twins arrive.

39. 24 shared between 3

40. 24 shared between 4

41. 24 shared between 6

42. 24 shared between 3 and 4

43. 24 shared between 6

44. How many numbers? • Have have 6 apples in one bag and four in another. If I put them together, how many apples do I have? • I have 4 bags each containing 6 apples. How apples do I have?

45. Ratio tables • I am putting apples into bags. There are six apples in each bag. I fill seven bags. How many apples is that? • Multiplication

46. Ratio tables • I am putting apples into bags. There are six apples in each bag. I have 42 apples. How many bags can I fill? • Division as repeated subtraction/grouping

47. Ratio tables • I am putting apples into bags. There are seven bags. I have 42 apples. If I put the same number of apples in each bag, how many is that? • Division as sharing

48. Marathon • Sam is running the London marathon • Every two miles she drinks 100 ml of water • How much does she drink over the 26 miles?