1. Have a go…
KS1
KS2

2. Concrete, Pictorial and Abstract Methods
My name is Paul Rowlandson, Assistant Principal at Trinity Academy…

3. Importance of CPA
In his research on the cognitive development of children (1966), Jerome Bruner proposed three modes of representation:
Enactive representation (action-based)
Iconic representation (image-based)
Symbolic representation (language-based)

4. Importance of CPA
‘Bruner's constructivist theory suggests it is effective when faced with new material to follow a progression from enactive to iconic to symbolic representation; this holds true even for adult learners. Bruner's work also suggests that a learner even of a very young age is capable of learning any material so long as the instruction is organized appropriately.’
McLeod, S. A. (2008)

5. Importance of CPA 3 + 4 = 7 Concrete – The ‘doing’ stage
Brings concepts to life by allowing children the experience of handling physical objects
Pictorial – The ‘seeing’ stage
Encouraging children to visualise concrete experiences and provides a link to the abstract
Drawing a representation of their experience
Abstract – The ‘symbolic’ stage
Using mathematical symbols to model problems
3 + 4 = 7

6. Multiple Representations
Discussion
What concrete and pictorial resources do you use in maths?

7. Using CPA methods Today we are going to focus on: Place value
Addition and Subtraction
Multiplication and Division

8. Number bonds

9. Number bonds
Children need to see that numbers can be made in a variety of ways.
Using one hand…
Show me three
Show me another way, and another…
Using two hands…
Show me five
Show me another way, and another
How is your way different to a friend?

10. Number bonds How can we build and show number bonds?
How can we use these resources for other number bonds?

11. Number bonds
Ten frames- making 5 in different ways

12. Number bonds
Ten frames- making 5 in different ways

13. Number bonds
Ten frames- making 5 in different ways

14. Number bonds
Ten frames- making 5 in different ways

15. Number bonds
How many ways can you make 6? What number facts can you show placing the counters in different ways?

16. Number bonds
Ten frames

17. Number bonds
Ten frames

18. Number bonds
Ten frames

19. Number bonds
Ten frames

20. Number bonds 6 6 How many ways can you make the pan scales balanced?
Workings 6 6

21. Number bonds
Part - whole model ? ?

22. Number bonds
Part - whole model 3 4 7 5 7 1 7 4 7 7 6

23. Number bonds
Activity How many ways can you make 10 using the part-whole model? Did you use a strategy? 10 10

24. Number bonds
Bar model 10 10 10 6 4 3 7 10 10 10 5 5 2 8 10 10 10 4 6 1 9

25. Place value columns

26. Place value columns
Use Styrofoam cups to see the value of each digit.

27. Place value columns
Introducing tens and ones Count straws up to ten. When we reach ten we can put these together to make 1 ten. How many do you have? Is this more or less than your partner?

28. Place value columns
Take one lot of 10 away. What is your new number? Which column changed? Build the number 24 Take 6 away. What difficulties did you face? What did you have to do? How did this affect the columns? I have ten bundles of ten. I call this one hundred.

29. Place value columns
Build 243 using place value counters. Add 8 to the ones column.
Model
Workings
H T 0 10 1
100 10 1
100 1
What has happened to the columns?
Key vocabulary: exchange
Can we exchange any counters?

30. Place value columns
Build 314 using place value counters. Subtract 4 from the ones column.
Model
Workings
H T 0 10 1
100 10 1
100
What has happened to the columns?
Key vocabulary: exchange
Can we exchange any counters?

31. Place value columns
Build 406 using place value counters. Subtract 10 from the tens column.
Model
Workings
H T 0
100 10 1
100 10 1
100 10 1
What has happened to the columns?
Key vocabulary: exchange
Can we exchange any counters?

32. Place value columns
We can begin to make rules by using concrete objects. Testing a rule Can you think of a number that changes four columns when you add 8 to it?

33. Place value Working walls
Use the working wall as a reminder of the work you have already done.
Encourages children to use equipment or visualise problem.

34. 29 Place value How many ways can you now represent this number?
Use other pictorial images, drawings of concrete objects that we have not used.

35. 29 Place value How many ways can you show 29? Twenty nine T O 10 + 19
20 + 9 29

36. Addition

37. Addition- Part whole model
Solve… =
Model
Calculations  ? 3 4

38. Addition Solve… 8 + 1 = Model Calculations 8 + 1 = 9 1 + 8 = 9 8 + = 9
= 9
+ 1 = 9 8 1

39. Addition – parts and wholes
Solve… =
Model
Calculations ? 8 1
8 + 1 = 9
1 + 8 = 9
= 9
+ 1 = 9 8 1 ?

40. Addition- Regrouping to make 10
Solve…
Model
Calculations = You try: = = 11

41. Addition- Regrouping to make 10
Solve…
Model
Calculations = +3 +1 11 10 3 1 11 7

42. Addition- Regrouping to make 10
Solve…
Model
Calculations = 11 ? 11 10 7

43. Addition- Column method
Solve… =
Model
Calculations 15
+34 ___ T O 4 9
Key vocabulary: exchange
Can we exchange any ones for a ten?

44. Addition- Column method
Solve… =
Model
Calculations 15
+34 ___ T O 4 9
Key vocabulary: exchange
Can we exchange any ones for a ten?

45. Addition- Column method
Hafsa is counting cars. She counts 15 blue and 34 red. How many cars does Hafsa count?
Model
Calculations = = = = = 15 ? 15 34

46. Addition- Column method
Solve… =
Model
Calculations 35
+17 ___ T O 10 10 10 1 1 1 1 1 10 1 5 2 1 10
Key vocabulary: exchange
Can we exchange any counters?

47. Addition- Column method
Patryk earns £17. He adds this to the £35 he has already saved. How much money does Patryk have now?
Model
Calculations = = = = = = 35 ? 35 17

48. Addition- Column method
Solve… =
Model
Calculations
243
+368 ____ 10 1
100
100
100
100 10 1 10 10 1 6 1 1 1 1
100 10 1
You try: = =
Key vocabulary: exchange
Can we exchange any counters?

49. Addition – column method= =
Remember
Build both numbers in the correct columns
Exchange counters if needed
Complete abstract method alongside
Draw bar model
Write all calculations bar model shows

50. Subtraction

51. Subtraction- Part whole model
Solve… – 6 =
Model
Calculations
 10 6 ?

52. Subtraction- find the difference
Solve… – 11 =
Model
Calculations 12 – 11 = 1

53. Subtraction - make 10
Solve… – 5 =
Model
Calculations 14 – 5 =

54. Subtraction - make 10
Solve… – 5 =
Model
Calculations 14 – 5 =

55. Subtraction - make 10
Solve… – 5 =
Model
Calculations 14 – 5 = 9

56. Subtraction- column method
Solve… – 27 =
Model
Calculations 49
-27 _ T O 2 2

57. Subtraction- Column method
Solve… =
Model
Calculations 35
-17 ___ T O 1 10 10 10 1 2 1 1 1 1 1 1 1 1 1 1 1 8
Key vocabulary: exchange
Can we exchange any counters?

58. Subtraction- column method
Solve… – 88=
Model
Calculations
234
____ 10 1
100 1 1 2 1
100
100 10 10 10 1 1 1 1 10 10 10 10 1 1 1 1 14 6 10 10 10 10 1 1 1 1 10 10 1 1
Key vocabulary: exchange
Can we exchange any counters?
You try:
345 – 167=
402 – 58=

59. Multiplication and division
Importance of multiplication and division facts
How can we help children learn these facts?
Make sure children understand the link between multiplication and division by using the inverse.
Ensure that children practice their times tables every day but in slightly different formats.
Teach it in a variety of ways – missing numbers, concrete with numicon and cubes, solve problems.

60. Multiplication
Muffins come in boxes of 4. Peter buys 6 boxes of muffins. How many muffins does Peter buy all altogether?
Model
Calculations ?
6 x 4= 24 4 4 4 4 4 4

61. Multiplication 23 × 6 _ 13 8 Solve… 6 x 23 = Model Calculations
100 10 1
100 10 1 1 10
100 1 10 1 10 23
× 6 _ 1 10 1 1 10 13 8 1 10 1 10
You try:
5 x 16 =
4 x 34 = 1 1 1 10

62. What’s the same? What’s different?
Tom has 5 packets of sweets. Each packet contains 42 sweets. How many sweets does Tom have altogether? Tom has 42 packets of sweets. Each packet contains 5 sweets. How many sweets does Tom have altogether?

63. Division

64. Division – 2 ways Partitive (sharing)
I have 42 sweets and I want to give them to 3 people. Each person will receive an equal amount.

65. Division (sharing)
Jane has 30 cakes. She wants to share them equally between five boxes. How many should go in each box?
Model
Calculations 30 30
÷ 5
= 6
? ? ? ? ?
Number of cakes in each box = 6 ?
In this version, we are splitting 30 into 5 equal groups.

66. Division
Solve… ÷ 3 =
Model
Calculations 10 10 10 10 1 1
42 ÷ 3 =

67. Division 42 ÷ 3 = Solve… 42 ÷ 3 = Model Calculations 10 1 1 1 1 1 1 1

68. Division 42 ÷ 3 = 14 Solve… 42 ÷ 3 = Model Calculations 65 ÷ 5 =10
You try:
65 ÷ 5 = 10 10

69. Division – 2 ways Quotative (grouping)
I have 42 sweets and I want to give them out 3 at a time. How many people will receive 3 sweets?

70. Division (grouping)
Jane has 30 cakes. She wants to pack them into boxes with 5 cakes in each box. How many boxes will she need?
Model
Calculations 30 30
÷ 5
= 6 5 5 5 5 5 5
Number of boxes needed = 6 ?
In this version, we are counting how many fives go into thirty.

71. Division
Solve… ÷ 5 =
Model
Calculations
5 615 H T O

72. Division
Solve… ÷ 5 =
Model
Calculations
5 615 H T O 1

73. Division
Solve… ÷ 5 =
Model
Calculations
5 615 H T O 1 1

74. Division
Solve… ÷ 5 =
Model
Calculations
5 615 H T O 1 2 1

75. Division
Solve… ÷ 5 =
Model
Calculations
5 615 H T O 12 1 1

76. 12 3 Division 5 615 Solve… 615 ÷ 5 = Model Calculations H T O
5 615 H T O 12 3 1 1
You try:
936 ÷ 4 =

77. Multiple representations
‘Used well, manipulatives can enable pupils to inquire themselves- becoming independent learners and thinkers. They can also provide a common language with which to communicate cognitive models for abstract ideas.’
Drury, H. (2015)
“If we do not use concrete manipulations, then we can not understand mathematics. If we only use concrete manipulations, then we are not doing mathematics.”
Gu (2015)

78. Fractions and Decimals

79. Introducing fractions
How many ways can you show 1 4 ?
0.25
2 8
25%

80. Comparing fractions
How can you use the strips of paper to compare fractions?
Calculations
1 2 > 1 3
2 4 = 4 8
1 5 < 2 5

81. Finding fractions of amounts
Find of 20
Calculations
20 ÷ 5 =
3 x 4 = 4 12

82. Finding fractions of amounts
Solve… of 18 =
Calculations
18 ÷ 6 =
3 x 5 = 3 15

83. Finding fractions of amounts
On your white boards use the counters to work out: 1 3
of 15 2 3
of 15 2 3
of 21 2 7
of 21
What are the links between the questions?
What question could you ask next?

84. Finding fractions of amounts
Design a tower which follows one of the following rules:
Each colour represents 1 3
One in every four is blue
There are of one colour

85. Adding Fractions = 1 5 3 5
Model
Calculations 1 5 3 + 4 5 =

86. Adding fractions Solve… Model Calculations + = You try: 2 + = 6 2 + =3 5 + 1 = 5 9 + 2 =

87. Adding Fractions = 3 4 1 4
Model
Calculations 3 4 1 + 4 =
= 1

88. Adding fractions Solve… Model Calculations You try: 3 5 1 2 5 3 5 1 2+ 1 = 2 5 +
Model
Calculations
You try: 3 5 + 1 = 2 5 + 3 5 - 1 = 4 2 8 + = 6 -

89. Adding fractions 3 6 + 5 6 Solve: 2 5 + 4 5 Model Calculations=
Model
Calculations =
You try:
1 1 5 2 1

90. Subtracting fractions
Solve… 5 7 - 2 =
Model
Calculations
You try: 5 7 - 2 =
3 7 7 9 - 3 =

91. Subtracting fractions
Solve… 5 6 - 2 =
Model
Calculations
You try:
3 6 5 6 - 2 = 7 9 - 3 =

92. Converting fractions Convert 8 6 into a mixed number fraction Model
Calculations
You try:
8 6
2 6 = 1
7 5 1
2 6
9 3

93. Fractions and Decimals
Convert into a decimal
Model
Calculations
2 ÷ 10 = 1
0.1 1
0.1
0.2

94. Fractions and Decimals
Convert into a decimal
Model
Calculations
1 ÷ 4 = 1
0.1
0.01
0.1
0.1
0.01 1
0.25

95. Fractions and Decimals
Prove… is 0.3
Model
Calculations
3 ÷ 10 = 1
0.1 1
0.1
0.3

96. Fractions and Decimals
Prove… is 0.2
Model
Calculations
1 ÷ 5 = 1
0.1
0.1 1
0.2

97. Multiple representations
‘Used well, manipulatives can enable pupils to inquire themselves- becoming independent learners and thinkers. They can also provide a common language with which to communicate cognitive models for abstract ideas.’
Drury, H. (2015)
“If we do not use concrete manipulations, then we can not understand mathematics. If we only use concrete manipulations, then we are not doing mathematics.”
Gu (2015)