TASMANIAN Department of Education

1. The key ideas and strategies that underpin Multiplicative Thinking Presented by Dianne Siemon Support for this project has been provided by the Australian Research Council, RMIT University, the Victorian Department of Education and Training, and the Tasmanian Department of Education. TASMANIAN Department of Education

2. KEY IDEAS AND STRATEGIES: • Early Number(counting, subitising, part-part-whole, trusting the count, composite units, place-value) • Mental strategies for addition& subtraction(count on from larger, doubles/near doubles, make-to-ten) • Concepts for multiplication and division(groups of, arrays/regions, area, Cartesian Product, rate, factor-factor-product) • Mental strategies for multiplication and division(eg, doubles and 1 more group for 3 of anything, relate to 10 for 5s and 9s facts) • Fractions and Decimals(make, name, record, rename, compare, order via partitioning)

3. COUNTING: “Jenni can count to 100 ...” To count effectively, children not only need to know the number naming sequence, they need to recognise that: • counting objects and words need to be in one-to-one correspondence; • “three” means a collection of three no matter what it looks like; • the last number counted tells ‘how many’.

4. SUBITISING & PART-PART-WHOLE: “But can Jenni read numbers without counting?” To develop a strong sense of number, children also need to be able to: • recognise collections up to five without counting subitising); and • name numbers in terms of their parts (part-part-whole knowledge). Eg, for this collection see “3” instantly but also see it as a “2 and a 1 more”

5. Eg, How many? Close your eyes. What did you see?

6. Try this:

7. …and this: What difference does this make?

8. Try this:

9. … and this: What did you notice?

10. What about this? Would colour help? How? Why?

11. But what about? How do you feel?

12. The numbers 0 to 9 are the only numbers most of us ever need to learn ... it is important to know everything there is to know about each number. For this collection, we need to know: • it can be counted by matching number names to objects: “one, two, three, four, five, six, seven, eight” and that the last one says, how many; • it can be written as eight or 8; and • it is 1 more than 7 and 1 less than 9.

13. But we also need to know 8 in terms of its parts, that is, 8 is 2 less than 10 6 and 2 more 4 and 4 double 4 3 and 3 and 2 5 and 3, 3 and 5 Differently configured ten-frames are ideal for this

14. TRUSTING THE COUNT: This recently recognised capacity* builds on a number of important early number ideas. Trusting the count has a range of meanings: • initially, children may not believe that if they counted the same collection again, they would get the same result, or that counting is a strategy to determine how many. • Ultimately, it is about having access to a range of mental objects for each of the numerals, 0 to 9, which can be used flexibly without having to make, count or see these collections physically. * See WA Department of Education, First Steps in Mathematics

15. Trusting the count is evident when children: • know that counting is an appropriate response to “How many …?” questions; • believe that counting the same collection again will always produce the same result irrespective of how the objects in the collection are arranged; • are able to subitise (ie, identify the number of objects without counting) and invoke a range of mental objects for each of the numbers 0 to ten (including part-part-whole knowledge); • work flexibly with numbers 0 to ten using part-part-whole knowledge and/or visual imagery without having to make or count the numbers; and • are able to use small collections as composite units when counting larger collections (eg, count by 2s, or 5s)

16. MENTAL STRATEGIES FOR ADDITION: Pre-requisites: • Children know their part-part-whole number relations (eg, 7 is 3 and 4, 5 and 2, 6 and 1 more, 3 less than 10 etc); • Children trust the count and can count on from hidden or given; • Children have a sense of numbers to 20 and beyond (eg, 10 and 6 more, 16)

17. 1. Count on from larger for combinations involving 1, 2 or 3 (using commutativity) For example, for 6 and 2, THINK: 6 … 7, 8 for 3 and 8, THINK: 8 … 9, 10, 11 for 1 and 6, THINK: 6 … 7 for 4 and 2, THINK: 4 … 5, 6 This strategy can be supported by ten-frames, dice and oral counting

18. For example: Cover 5, count on Cover 4, count on

19. 2. Doubles and near doubles For example, for 4 and 4, THINK: double 4, 8 for 6 and 7, THINK: 6 and 6 is 12, and 1 more, 13 for 9 and 8, THINK: double 9 is 18, 1 less, 17 for 7 and 8, THINK: double 7 is 14, 1 more, 15 This strategy can be supported by ten-frames and bead frames (to 20) can be used to build doubles facts

20. For example: Ten-frames

21. For example: Count: 6 and 6 is 12, and 1 more, 13 Bead Frame (to 20) Double-decker bus scenario

22. 3. Make to ten and count on For example, for 8 and 3, THINK: 8 … 10, 11 for 6 and 8, THINK: 8 … 10, 14 for 9 and 6, THINK: 9 … 10, 15 for 7 and 8, THINK: double 7 is 14, 1 more, 15 Ten-frames and bead frames (to 20) can be used to bridge to ten, build place-value facts (eg 10 and 6 more , sixteen)

23. For example: For 8 and 6 …

24. For example: Think: 10 … and 4 more ... 14

25. MENTAL STRATEGIES FOR SUBTRACTION: For example, for 9 take 2, THINK: 9 … 8, 7(count back) for 6 take 3, THINK: 3 and 3 is 6 (think of addition) for 15 take 8, THINK: 15, 10, 7(make back to 10) Or for 16 take 9, THINK: 16 take 8 is 8, take 1 more, 7(halving) 16, 10, 7(make back to 10) 9, 10, 16 … 7 needed (think of addition) 16, 6, add 1 more, 7(place-value)

26. CONCEPTS FOR MULTIPLICATION: Establish the value of equal groups by: • exploring more efficient strategies for counting large collections using composite units; and • sharing collections equally. Explore concepts through action stories that involve naturally occurring ‘equal groups’, eg, the number of wheels on 4 toy cars, the number of fingers in the room, the number of cakes on a baker’s tray ...., and stories from Children’s Literature, eg, Counting on Frank or the Doorbell Rang See Booker et al, pp.182-201 & pp.221-233

27. 1. Groups of: 3 fours ... 4, 8, 12 4 threes ... 3, 6, 9, 12 Focus is on the group. Really only suitable for small whole numbers, eg, some sense in asking: How many threes in 12? But very little sense in asking: How many groups of 4.8 in 34.5? Strategies: make-all/count-all groups, repeated addition (or skip counting).

28. 2. Arrays: Rotate and rename 4 threes ... THINK: 6 and 6 3 fours ... THINK: 8, 12 Focus on product (see the whole, equal groups reinforced by visual image), does not rely on repeated addition, supports commutativity (eg, 3 fours SAME AS 4 threes) and leads to more efficient mental strategies Strategies: mental strategies that build on from known, eg, doubling and addition strategies

29. 3. Regions: Rotate and rename 4 threes ... THINK: 6 and 6 3 fours ... THINK: 8, 12 Continuous model. Same advantages as array idea (discrete model) – establishes basis for subsequent ‘area’ idea. Note: For whole number multiplication continuous models are introduced after discrete – this is different for fraction models!

30. 4. ‘Area’ idea: 14 3 3 by 1 ten and 4 ones 3 by 1 ten ... 3 tens 3 by 4 ones ... 12 ones Think: 30 ... 42 Supports multiplication by place-value parts and the use of extended number fact knowledge, eg, 4 tens by 2 ones is 8 tens ... Ultimately, 2-digit by 2-digit numbers and beyond

31. The ‘Area’ idea (extended): 33 24 Supports multiplication by place-value parts, eg, 2 tens by 3 tens is 6 hundreds... Ultimately, that tenths by tenths are hundredths and (2x+4)(3x+3) is 6x2+18x+12

32. 5. Cartesian Product: Eg, lunch options 4 different types of filling 2 different types of fruit 3 different types of bread 3 x 4 x 2 = 24 different options Supports ‘for each’ idea and multiplication by 1 or more factors

33. 6. Rate: Eg, 5 sweets per bag. 13 bags of sweets. How many sweets altogether? Eg, Jason bought 3.5 kg of potatoes at $2.95 per kg. How much did he spend on potatoes? These problems require thinking about the ‘unit’. In this case, 1 bag and 1 kg respectively Eg, Samantha’s snail travels 15 cm in 3 minutes. Anna’s snail travels 37 cm in 8 minutes. Which is the speedier snail? This problem involves rate but actually asks for a comparison of ratios which requires proportional reasoning. Rate builds on the ‘for each’ idea and underpins proportional reasoning

34. MENTAL STRATEGIES FOR MULTIPLICATION: The traditional ‘multiplication tables’ are not really tables at all but lists of equations which count groups, for example: This is grossly inefficient 1 x 3 = 3 2 x 3 = 6 3 x 3 = 9 4 x 3 = 12 5 x 3 = 15 6 x 3 = 18 7 x 3 = 21 8 x 3 = 24 9 x 3 = 27 10 x 3 = 30 11 x 3 = 33 12 x 3 = 36 1 x 4 = 4 2 x 4 = 8 3 x 4 = 12 4 x 4 = 16 5 x 4 = 20 6 x 4 = 24 7 x 4 = 28 8 x 4 = 32 9 x 4 = 36 10 x 4 = 40 11 x 4 = 44 12 x 4 = 48 3 fours not seen to be the same as 4 threes ... 10’s and beyond not necessary

35. More efficient mental strategies build on experiences with arrays and regions: Eg, 3 sixes? ... THINK: double 6 ... 12, and 1 more 6 ... 18 And the commutative principle: 3 6 Eg, 6 threes? ... THINK: 3 sixes ... double 6, 12, and 1 more 6 ... 18 6 3

36. This involves a shift in focus: From a focus on the number IN the group A critical step in the development of multiplicative thinking appears to be the shift from counting groups, for example, 1 three, 2 threes, 3 threes, 4 threes, ... to seeing the number of groups as a factor, For example, 3 ones, 3 twos, 3 threes, 3 fours, ... and generalising, for example, “3 of anything is double the group and 1 more group”. To a focus on the number OF groups

37. Mental strategies for the multiplication facts from 0x0 to 9x9 • Doubles and doubles ‘reversed’ (twos facts) • Doubles and 1 more group ... (threes facts) • Double, doubles ... (fours facts) • Same as (ones and zero facts) • Relate to ten (fives and nines facts) • Rename number of groups (remaining facts)

38. An alternative ‘multiplication table’: This actually represents the region idea and supports efficient, mental strategies (read across the row), eg, 6 ones, 6 twos, 6 threes, 6 fours, 6 fives, 6 sixes, 6 sevens, 6 eights, 6 nines

39. The region model implicit in the alternative table also supports the commutative idea: Eg, 6 threes? THINK: ….

40. The region model implicit in the alternative table also supports the commutative idea: Eg, 6 threes? THINK: 3 sixes This halves the amount of learning

41. Doubles Strategy (twos) : 2 ones, 2 twos, 2 threes, 2 fours, 2 fives ... 2 fours ... THINK: double 4 ... 8 2 sevens ... THINK: double 7 ... 14 7 twos ... THINK: double 7 ... 14

42. Doubles and 1 more group strategy (threes): 3 ones, 3 twos, 3 threes, 3 fours, 3 fives ... 3 eights THINK: double 8 and 1 more 8 16 , 20, 24 9 threes ... THINK? 3 twenty-threes THINK?

43. Doubles doubles strategy (fours): 4 ones, 4 twos, 4 threes, 4 fours, 4 fives ... 4 sixes THINK: double 4 ... 8 double again, 16 8 fours ... THINK? 4 forty-sevens THINK?

44. ‘Same as’ strategy (ones and zeros): 1 one, 1 two, 1 three, 1 four, 1 five, ... 1 of anything is itself ... 8 ones, same as 1 eight Cannot show zero facts on table ... 0 of anything is 0 ... 7 zeros, same as 0 sevens

45. Relate to tens strategy (fives and nines): 5 ones, 5 twos, 5 threes, 5 fours, 5 fives ... 9 ones, 9 twos, 9 threes, 9 fours, 9 fives ... 5 sevens THINK: half of 10 sevens, 35 8 fives ... THINK? 9 eights THINK: less than 10 eights, 1 eight less, 72

46. Rename number of groups (remaining facts): 6 sixes, 6 sevens, 6 eights ... 7 sixes, 7 sevens, 7 eights ... 8 sixes, 8 sevens, 8 eights ... 6 sevens THINK: 3 sevens and 3 sevens, 42 ... OR 5 sevens and 1 more 7 8 sevens THINK: 7 sevens is 49, and 1 more 7, 56

47. CONCEPTS FOR DIVISION: 1. How many groups in (quotition): How many fours in 12? 1 four, 2 fours, 3 fours 12 counters Really only suitable for small collections of small whole numbers, eg, some sense in asking: How many fours in 12? But very little sense in asking: How many groups of 4.8 in 34.5? Strategies: make-all/count-all groups, repeated addition

48. Quotition (guzinta) Action Stories: 24 tennis balls need to be packed into cans that hold 3 tennis balls each. How many cans will be needed? Sam has 48 marbles. He wants to give his friends 6 marbles each. How many friends will play marbles? How many threes? How many sixes? Total and number in each group known – Question relates to how many groups.

49. 2. Sharing (partition): 18 sweets shared among 6. How many each? 3 in each group 18 counters More powerful notion of division which relates to array and regions models for multiplication and extends to fractions and algebra Strategy: ‘Think of Multiplication’ eg, 6 what’s are 18? ... 6 threes

50. Partition Action Stories: 42 tennis balls are shared equally among 7 friends. How many tennis balls each? Sam has 36 marbles. He packs them equally into 9 bags. How many marbles in each bag? THINK: 7 what’s are 42? THINK: 9 what’s are 36? Total and number of groups known – Question relates to number in each group.