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Developing Multiplicative Reasoning

Developing Multiplicative Reasoning. Mike Askew King’s College London. Comparing. 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?. Water. There are two jars.

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Developing Multiplicative Reasoning

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  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?

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