1. Multiplication represents a many to one correspondence.
Additive reasoning 2 + 3, , 4 + 1
In the move to multiplicative reasoning we need to now see this as one set/ group / or unit of 5.
One five
1 (5)
1 x 5

2. But it is also five ones (additive composition)
5 (1)
5 x 1

3. By the commutative property 1 x 5 = 5 x 1 but this image does not make both expressions clear….

4. By moving to arrays, we help learners see that multiplication is not just a count, it is a measure of space or growth
One down and 5 across

5. See the 3 fives? Can you also see five threes?
This growth or measure of space is not obvious in a one times relationship…. It is much easier to see in a 3 x 5
Arrays allow you to track the change or growth in two directions.
See the 3 fives? Can you also see five threes?
3(5) or 3 x 5 = 5(3) or 5 x 3

6. See the 3 fives? Can you also see five threes?
An equal groups model is much less flexible.
The five threes are almost impossible to see at the same time.
See the 3 fives? Can you also see five threes?
3(5) or 3 x 5 = 5(3) or 5 x 3

7. See how the space covered is growing. It is taller or wider now
See how the space covered is growing. It is taller or wider now. It is 3 times as tall or wide.

8. Much more difficult to focus on the growth or change in size because counting interferes. There are 3 times as many dots.

9. Demonstrate an understanding of multiplication to 5 × 5 by:
representing and explaining multiplication using equal grouping and arrays
creating and solving problems in context that involve multiplication
modeling multiplication using concrete and visual representations, and recording the process symbolically
relating multiplication to repeated addition
relating multiplication to division

10. Demonstrate an understanding of multiplication to 5 × 5 by:
representing and explaining multiplication using equal grouping and arrays
creating and solving problems in context that involve multiplication
modeling multiplication using concrete and visual representations, and recording the process symbolically
relating multiplication to repeated addition
relating multiplication to division

11. Demonstrate an understanding of division by:
representing & explaining
using equal sharing and equal grouping
creating & solving problems in context that involve both equal sharing and equal grouping
modelling equal sharing and equal grouping
concretely, visually then recording
relating division to repeated subtraction (I vehemently disagree)
relating division to multiplication. [C, CN, PS, R]

12. 5 3 15 ÷ 5 = 3 Demonstrate an understanding of division by:
representing & explaining
using equal sharing and equal grouping   5 3
15 ÷ 5 = 3

13. I had 15 chairs. I want to set them up at 5 tables
I had 15 chairs. I want to set them up at 5 tables. How many chairs per table? (equal shares).
5 tables 3
15 ÷ 5 = 3

14. I had 15 chairs. I need to put 5 at each table. How many tables needed
I had 15 chairs. I need to put 5 at each table. How many tables needed? (equal group)
5 chairs 3
15 ÷ 5 = 3

15. See how the space covered is growing. It is taller or wider now
See how the space covered is growing. It is taller or wider now. It is 3 times as tall or wide.

16. Demonstrate an understanding of fractions by:
explain that a fraction represents a part of a whole
describing situations in which fractions are used
comparing fractions, same whole with like denominators [C, CN, ME, R, V]
We use fractions to achieve more accurate measurements

17. 1 ÷ 3 = 1 3 Demonstrate an understanding of fractions by:
explain that a fraction represents a part of a whole
describing situations in which fractions are used
1 ÷ 3 = 1 3

18. 1 ÷ 3 = 1 3 Demonstrate an understanding of fractions by:
explain that a fraction represents a part of a set
1 ÷ 3 = 1 3

19. Arrays allow for more robust imagery of multiplication:
Learners can see: “the groups of”
“the change in space covered (area)”
“the proportional growth.
It is growing or shrinking at an equal rate
“the dimensionality”
The link to addition without reverting to ones
(the count is increasing by an equal quantity each time
The link to division, fractions and decimals

20. Represents dimensional growth Grows proportionally
By the commutative property 1 x 5 = 5 x 1 but this image does not make both expressions clear….
By moving to arrays, we help learners see that multiplication is commutative…. But also that it fills area
Represents dimensional growth
Grows proportionally
1 x 5 = 5 5 x 1 = 5

21. Multiplication represents a many to one correspondence.
is about units being repeated, There is direction involved. The units grow horizontally, vertically, or both. And they can grow in 3 Dimensions (so flat paper won’t do it all)
We use rectangles (squares) and prisms to model the growth in Grade3, 4 and 5.
If you can multiply,, you can divide. They are inverses.
Know some facts, use them to know more facts....
Look for relationships.

22. These multiple meanings are why we must not leave students believing multiplication is “just addition”.
There is a dimensional and proportional growth involved.
Times as many, times greater or less than
Multiplication involves growth or shrinkage.

23. These multiple meanings are why we must not leave students believing multiplication is “just addition”.
There is a dimensional and proportional growth involved.
Times as many, times greater or less than
Multiplication involves growth or shrinkage.
We use rectangles (squares) and prisms to model the growth in Grade3, 4 and 5.
If you can multiply,, you can divide. They are inverses.
Know some facts, use them to know more facts....
Look for relationships.

24. BASIC FACTS 1.) Multiplication IS NOT “JUST” REPEATED ADDITION.
2.) EQUAL GROUPS arranged into ARRAYS allow learners to “see” the proportional change that is occurring when we multiply.
area, volume, capacity, rate of change, ratios, place value for both whole and decimal numbers , cartesian products, all represent multiplicative functions.
3.) Division is not a different operation. It is the inverse.
4.) Zero and one represent special cases of multiplication/division
5.) Multiplication is COMMUTATIVE.
6.) Zero and one represent special cases.
7.) THE DISTRIBUTIVE PROPERTY allows us to compute complex problems with ease

25. x2 xy y2
x y x + y
(x + y)(x + y)= x2 + 2xy + y2

26. x2 xy y2
x y x + y
(x + y)(x + y)= x2 + 2xy + y2

27. 20 x 30=
600
4 x 30=
120
4 x 6 = 24
20 x 6 = 120 30 + 6
24 x 36 = (20+4) (30+6)

28. 20 x 30=
600
4 x 30=
120
4 x 6 = 24
20 x 6 = 120 30 + 6
x x 36
600
864
I see or 864.
The order you add in does not matter. Use your mental math strategies.

29. 60 x 6
360
7 x 6 42 6
For 7 x 6 I use mathematical reasoning:
5 x 6 = 30 plus 2 x 6 =12.
I know therefore 7 x 6 = 42
Mental strategies for adding include place value re grouping: = 100 now add 302
Include stringing: =
There would be no need to compensate on these numbers... too easy. Answer is 402

30. Demonstrate an understanding of multiplication to 5 × 5 by:
representing and explaining multiplication using equal grouping and arrays
creating and solving problems in context that involve multiplication
modelling multiplication using concrete and visual representations, and recording the process symbolically
relating multiplication to repeated addition
relating multiplication to division

34. Our students should be tested on: Their willingness to engage and persevere in challenging and extending their learning
Their ability to BUILD, EXPLAIN, REPRESENT and COMPARE multiplication facts as equal groups and arrays.
Their ability to BUILD, EXPLAIN, REPRESENT and COMPARE division facts as equal groups or fair (equal) shares.
Their ability to BUILD, EXPLAIN, REPRESENT and COMPARE multiplication and division facts.
Their ability to recall facts to 5 x 5.

37. Our students should be tested on: Their willingness to engage and persevere in challenging and extending their learning
Their ability to BUILD, EXPLAIN, REPRESENT and COMPARE multiplication facts as equal groups and arrays.
Their ability to BUILD, EXPLAIN, REPRESENT and COMPARE division facts as equal groups or fair (equal) shares.
Their ability to BUILD, EXPLAIN, REPRESENT and COMPARE multiplication and division facts.
Their ability to BUILD, EXPLAIN, REPRESENT and COMPARE examples of the commutative property, the distributive property .(extended to why we can generalize for multiplying with tens and hundreds) and the special cases of 1 and 0.
Their ability to recall at minimum 5 known facts to generate without stopping to count individual items, at minimum 3 facts related to each (this will become automatic by Grade 5) to interpret remainders, to multiply with tens, hundreds, thousands
Their ability to communicate mathematically: commutative property, distributive property, product, factor, area, quotient, divisor, times as many, times as much, area, dimensions
Their ability to critically analyze a solution and determine if it is correct, if there is a different way to arrive at it, if one way is superior to another and if so why.
ALL WITHIN PROBLEM SOLVING CONTEXTS??? CREATE AND SOLVE PROBLEMS

38. Apply the properties of 0 & 1 for multiplication, & the property of 1 for division
Describe & apply mental strategies, to determine * & related ÷ facts to 9 x 9. Build recall to 7x7.
Demonstrate an understanding of multiplication (2-or 3-digit by 1-digit) to solve problems by:
personal strategies with & without concrete materials, using arrays
connecting concrete representations to symbolic representations
estimating products
applying the distributive property
Demonstrate an understanding of area regular & irregular shapes by
recognizing that area is measured in square units
selecting and justifying referents for the units cm2
estimating area, using referents for cm2 or m2
determining and recording area (cm2 or m2)
constructing different rectangles for a given area

39. Apply mental strategies & number properties,
to determine with fluency, answers for basic multiplication facts to 81 and related division facts
Apply mental strategies for multiplication, such as:
annexing then adding zero
halving and doubling
using the distributive property.
Demonstrate, with and without concrete materials, an understanding of multiplication
(2 digit by 2 digit) to solve problems.

40. For this series of simulations, I am going to name the lime green as 3 cm.

41. The lime one 3 or three ones.... Which in multiplication is 1 x 3 or 3 x 1...

42. I can use the rods to “prove it” or I can use cm grid paper.

43. This is a 3 x 4 because it is built with the three rod, repeated 4 times.
You build it.

44. I know 3 x 4 = 12.
How do I know that?

45. 3, 6, 9, 12
3 and 3 is 6 and double it, so 12.

46. Diagram your array for 3 x 4 and label it.12

47. If I turn the array, I change the labels to match.
I still see the four 3’s but now I label it 4 x 3.
So I can think four sets or groups of three. .
It still covers 12 squares. 4 4 3 12 3

48. When I turn it I really notice the fours more..
The purple rod is four...
Can you see how 4 fits in this array? 3 4 4 3

49. The fours are moving across the array.
Or they are moving down the array. 4 3 4

50. 3 x 4 and 4 x 3 both cover 12 squares.
IF 3 x 4 = 12 and 4 x 3 = 12 then 3 x 4 = 4 x 3
This is the commutative property
3 x 4 = 4 x 3 3 3 4 4

51. Transformations 3 4 4 3
Turns or rotations are an important ideas in mathematics.
The shape is transformed but the quantity is not.
You just change how you label it when you rotate it.

52. 3 4
Build your 3 x 4 into this grid. Label its side lengths 3 and 4. So when you remove it the labels will remind you where it sat.

53. It covers 12 so write 12 in the bottom right hand corner.3 12 4
It covers 12 so write 12 in the bottom right hand corner.

54. 3 x 4 = 12 Make an equation strip to remember 3 x 4 = 12.
Remove the rods. 12 4
3 x 4 = 12

55. What if I double the 3. So (3 x 5) + (3 x 1) = 3 x 6 Did you see this?
The 6 rod will fit. 3 3 4

56. 3 x 6 = 18 6 x 3 = 18 Turn it, then use the six rods to build it.
Turn it, then use the six rods to build it.
Make the new equation strip for 6 x 3 = 18 18 3 4 5 6 18
3 x 6 = 18
6 x 3 = 18

57. 1 2 3 4 5 6 12 18
3 x 4 = 12
4 x 3 = 12
3 x 6 = 18
6 x 3 = 18

58. 1 2 3 4 5 6 12 18
3 x 4 = 12
4 x 3 = 12
3 x 6 = 18
6 x 3 = 18

59. 1 2 3 4 5 6 12 18
3 x 4 = 12
4 x 3 = 12
3 x 6 = 18
6 x 3 = 18

60. Student have visual arrays, file cards with equations and a grid to use to practice these facts.
Invite them to talk about ways they could practice with a partner...
Suggest they create a plan for practicing or a game....
Mine follows, it is quite simple.

61. This is what it will look like if we graph it… (3, 4)
If we built from the bottom, Grade 5 and 6 could connect to the first quadrant of the Cartesian plane.
This is what it will look like if we graph it… (3, 4)
On a graph you read across the horizontal (x) axis first.
Why don’t we? Because traditionally math has been taught as though it were not a connected set of ideas. 5 4 3 2 1

62. The co-ordinates for the rotation are (4,3)5 4 3 2 1

63. Teach multiplication facts from a puzzling perspective
3 x 3
2 x 3
5 x 3
How are these 3 facts related?

64. The human brain loves to puzzle
6x5 = 30
Why did I put these together?
What do they have in common?
How are they different?

65. You want to engage students in learning
4 x 7 = 28
Puzzle, puzzle, puzzle. If you know one fact, you actually know at least 3.

66. 6 x 5 = 30
(6 x 5) + (6 x 2) = 6 x 7
6 (5+2) = 6 x 7
(6 x 7) + (6 x 2) = 6 x 9
6 (7+2) = 6 x 9

67. 6 x 5 = 30
(6 x 5) + (6 x 2) = 6 x 7
= 42
(6 x 7) + (6 x 2) = 6 x 9
= 54
(3x7)+(3x7)

68. Look what else I can figure out and practice....
6x9= 54
6x5 = 30
6x7 = 42
6 x x x 18 .

69. Connect, Compare and Explain sets of FingerFolds that go together