1. Negotiating the Multiplicative Reasoning Map

4. Key words:
Units
Units of units
Partitioning vs Sharing out
Groups of become units of
Areas/ Volumes: Directional
Division/ Fractions/Decimal Fractions
Number Properties

5. Proportional Growth in 1, 2 or 3 directions
Multiplicative reasoning is not captured by a single image. The Alberta Curriculum supports several multiplication images.

6. Growth in 1,2,3 directions
Multiplication can represent a proportional growth or shrinkage that moves ‘directionally’. (Lorway, 2017). Numberlines capture this movement in one direction, area models in two directions and volume models in three directions.

7. I see 10 + 10 + 10 + 10 but the growth was from 10 to 40, that is 4 times as much
Multiplicative growth on a number line can easily be mislabeled as “just repeated addition.” But you have to focus on the units inside the new unit. The 4 tens are now one 40. But each ten is one ten 4 (1x10) or 4 tens or 40 or 1x 40
While addition is definitely related, this growth results in an overall expansion from 10 to 4 times as many as ten or 40.

8. Fold

9. Build arrays

10. Number Properties

11. There is no division without multiplication.....

12. Students who struggle with division do not have a strong enough understanding of multiplication....

15. Arrays or area models are introduced in Grade 3 in the outcomes.
Grade 2 teachers can lay a foundation for this kind of thinking by building two digit numbers into the hundred grid.
Students see the decades as

16. Kinder and Grade one build collections for numbers to 10.
Build with repeating equal units. Arrange into rectangular arrays.
See how they can come apart in equal groups in two directions.
Blocks on blocks also demonstrates this. This student noticed that the dot collection could be translated to to 3 sets of two blocks with one leftover. But she also saw two sets of 3 blocks. Still one leftover. The threes fit on the twos because they both represent 6.

17. Task: Grab handfuls. Use 2 or 3 fingers to slide together units of 2 or 3. Push to create a “line up” that creates a rectangular array.
Use ChunkitZ to fill rectangles or squares in one direction or to create your own rectangle or square. Place Chunkitz on ChunkitZ. See the unit in each direction.
Teachers can support the development of this “directional” thinking in Kindergarten and grade one by including arrays and attention to arrays in the work students do with images to develop understandings of number to 10 and to 20.

18. Partitioning divides a whole into segments or parts.
This is also a “directional” change that signals a multiplicative relationship.

19. Paper folding is one of the tasks that prepares students for partitioning. One paper is folded into equal sections. Accordian folding, back and forth like a “fan” mimics partitioning of a measuring unit.

20. Grade 3 is the first time the word multiplication appears in an outcome and the imagery of multiplication as a growth in 2 directions is quite evident.
The role of area models and array thinking in supporting the development of multiplicative reasoning as well as as an effective tool for promoting and sustaining fact recall and retention is well documented in research. (Jacob & Mulligan, 2014; Jacob & Willis, 2003; Kinzer & Stanford, 2014; Outhred, L., & Mitchelmore. M. (2004). Siemon et al., 2011; YoungLoveridge, 2005)

21. Area models make visible: the commutative property
The role of area models and array thinking in supporting the development of multiplicative reasoning as well as as an effective tool for promoting and sustaining fact recall and retention is well documented in research. (Jacob & Mulligan, 2014; Jacob & Willis, 2003; Kinzer & Stanford, 2014; Outhred, L., & Mitchelmore. M. (2004). Siemon et al., 2011; YoungLoveridge, 2005)

22. In one image you can see that 4 x 3 = 3 x 4
That’s the commutative property. 3 4

23. I see threes horizontally: pull them apart
I could say 4 groups or sets of 3 = 12.

24. But I also see units of 4 running vertical. Pull them apart.
I could say 3 groups or sets of 4 = 12

25. If you turn the original array you can still pull threes one way and fours the other.
4 x 3 equals 3 x 4 4 4 3 3

26. Area models make visible:
that multiplication is related to repeated addition
The role of area models and array thinking in supporting the development of multiplicative reasoning as well as as an effective tool for promoting and sustaining fact recall and retention is well documented in research. (Jacob & Mulligan, 2014; Jacob & Willis, 2003; Kinzer & Stanford, 2014; Outhred, L., & Mitchelmore. M. (2004). Siemon et al., 2011; YoungLoveridge, 2005)

27. 3
The role of area models and array thinking in supporting the development of multiplicative reasoning as well as as an effective tool for promoting and sustaining fact recall and retention is well documented in research. (Jacob & Mulligan, 2014; Jacob & Willis, 2003; Kinzer & Stanford, 2014; Outhred, L., & Mitchelmore. M. (2004). Siemon et al., 2011; YoungLoveridge, 2005)
I see 3, 6, 9, 12

28. 3
The role of area models and array thinking in supporting the development of multiplicative reasoning as well as as an effective tool for promoting and sustaining fact recall and retention is well documented in research. (Jacob & Mulligan, 2014; Jacob & Willis, 2003; Kinzer & Stanford, 2014; Outhred, L., & Mitchelmore. M. (2004). Siemon et al., 2011; YoungLoveridge, 2005)
Or 3 plus 3 then double it

29. When I look at the fours I see 4, 8, 12.
The role of area models and array thinking in supporting the development of multiplicative reasoning as well as as an effective tool for promoting and sustaining fact recall and retention is well documented in research. (Jacob & Mulligan, 2014; Jacob & Willis, 2003; Kinzer & Stanford, 2014; Outhred, L., & Mitchelmore. M. (2004). Siemon et al., 2011; YoungLoveridge, 2005)
When I look at the fours I see 4, 8, 12.

30. OR HOW ABOUT
5/6 * 7/8 =
The role of area models and array thinking in supporting the development of multiplicative reasoning as well as as an effective tool for promoting and sustaining fact recall and retention is well documented in research. (Jacob & Mulligan, 2014; Jacob & Willis, 2003; Kinzer & Stanford, 2014; Outhred, L., & Mitchelmore. M. (2004). Siemon et al., 2011; YoungLoveridge, 2005)

31. How do you add 6 and 43 hundredths 7 and six tenths times?
But all multiplications cannot be explained with repeated addition: Try x 7.6
How do you add 6 and 43 hundredths 7 and six tenths times?
The role of area models and array thinking in supporting the development of multiplicative reasoning as well as as an effective tool for promoting and sustaining fact recall and retention is well documented in research. (Jacob & Mulligan, 2014; Jacob & Willis, 2003; Kinzer & Stanford, 2014; Outhred, L., & Mitchelmore. M. (2004). Siemon et al., 2011; YoungLoveridge, 2005)

32. Area models explain the timestable grid .

33. Look familiar? 1 2 3 4 5 20 25

34. Area models make visible: that multiplication is related to division
The role of area models and array thinking in supporting the development of multiplicative reasoning as well as as an effective tool for promoting and sustaining fact recall and retention is well documented in research. (Jacob & Mulligan, 2014; Jacob & Willis, 2003; Kinzer & Stanford, 2014; Outhred, L., & Mitchelmore. M. (2004). Siemon et al., 2011; YoungLoveridge, 2005)

35. 3
3 x 4 = 12 4 12

36. 12 ÷ 4 3 4
12 divided into equal sets of 4s

37. 12 ÷ 4 ? 12 4
Think what times 4 = 12
? X 4 = 12

38. 12 ÷ 3 3 4
12 divided into equal sets of 3s

39. 12 ÷ 3 4 3
Turn it, it still works
12 divided into equal sets of 3s

40. Area models allow students to:
Represent division using equal sharing and equal grouping

41. Area models to 5 x 5 are small enough to hold in visual memory…
Area models to 5 x 5 are small enough to hold in visual memory….. Students can practice with them as ”flashcards” until they have automatic recall. (Lorway, 2017).
tiles Stress moving in sets or groups of, not counting by ones.
folding The relationships that emerge are related to multiplication relationships. When thirds cross fourths, twelfths emerge
3, 4, and 12 are related.
COMPARE
grids Stress seeing the “groups of” or “units” in rows crossing columns.
rods 3 four rods cover the same area as 4 three rods. Cuisenaire rods remove the distraction of seeing individual ones. the focus is on thinking in units.
With grids

42. and the total amount of things, objects, space that is now created
In order to understand multiplication as a way of thinking learners must understand and connect three elements:
groups of equal size
numbers of groups
and the total amount of things, objects, space that is now created
Students who are able to construct and coordinate these elements in both multiplication and division problems before they carry out a count are thinking multiplicatively. When learners are able to think multiplicatively they can apply the commutative property, the distributive property and inverse relations to solve problems.
Kouba (1989), Steffe (1992), Mulligan and Mitchelmore (1997) and Mulligan & Watson (1998).

43. Area models make visible: place value as a multiplicative function
Cite research on place value as a multiplicative function.

44. Area models make visible:
the multiplicative nature of fractions

45. The role of area models and array thinking in supporting the development of multiplicative reasoning as well as as an effective tool for promoting and sustaining fact recall and retention is well documented in research. (Jacob & Mulligan, 2014; Jacob & Willis, 2003; Kinzer & Stanford, 2014; Outhred, L., & Mitchelmore. M. (2004). Siemon et al., 2011; YoungLoveridge, 2005)

46. Mathematics is used to describe and explain relationships
Mathematics is used to describe and explain relationships. Students look for relationships among numbers, sets, shapes, objects and concepts. The search for possible relationships involves collecting and analyzing data and describing relationships visually, symbolically, orally or in written form. POS p 8
“Because the learner is constantly searching for connections on many levels, educators need to orchestrate the experiences from which learners extract understanding.... Brain research establishes and confirms that multiple complex and concrete experiences are essential for meaningful learning and teaching” POS p 11
Learning through problem solving should be the focus of mathematics at all grade levels. Students develop their own problem-solving strategies by listening to, discussing and trying different strategies.
If students have already been given ways to solve the problem, it is not a problem, but practice.
Problem solving is a powerful teaching tool that fosters multiple, creative and innovative solutions. The use of visualization in the study of mathematics provides students with opportunities to understand mathematical concepts and make connections among them. Visual images and visual reasoning are important components of number, spatial and measurement sense. reasoning involves:

47. It is highly visual:
“The use of visualization in the study of mathematics provides students with opportunities to understand mathematical concepts and make connections among them.
Visual images and visual reasoning are important components of number, spatial and measurement sense. Number visualization occurs when students create mental representations of numbers.”
Mathematics Kindergarten to Grade 9 Program of Studies , Update 2016, p.2

48. Relationships Matter:
“Mathematics is used to describe and explain relationships. As part of the study of mathematics, students look for relationships among numbers, sets, shapes, objects and concepts. The search for possible relationships involves collecting and analyzing data and describing relationships visually, symbolically, orally or in written form.
Mathematics Kindergarten to Grade 9 Program of Studies , Update 2016, p.8

49. It is presented spatially:
“Spatial sense involves visualization, mental imagery and spatial reasoning. These skills are central to the understanding of mathematics.”
Mathematics Kindergarten to Grade 9 Program of Studies , Update 2016, p.8

50. Specific imagery connects across the grades:
When mathematical ideas are connected to each other or to real-world phenomena, students begin to view mathematics as useful, relevant and integrated.
“Because the learner is constantly searching for connections on many levels, educators need to orchestrate the experiences from which learners extract understanding.... Brain research establishes and confirms that multiple complex and concrete experiences are essential for meaningful learning and teaching”
(Caine and Caine, 1991, p. 5).

51. Here is my diagram. Can you explain the labels?3 4

52. 12 ÷ 4
I have 12 cans of soup and want to give 4 to each customer. How many customers will get soup? 4 12
1 person
1 person
1 person

53. 12 ÷ 4
I have 12 cans of soup, organized into 4 stacks. How many are in each stack. 4
4 stacks 12
3 in each stack

54. 3 12 12 ÷ 3 I have 12 cookies and want to put 3 in each box.
How many boxes do I need? 3 12
1 box
1 box
1 box
1 box

55. 12 ÷ 3
I have 12 cookies and 3 boxes. If I share them equally, how many cookies go in each box? 3
3 boxes 12
4 in each box