1. Addition & Subtraction Conceptual Progress Sort Cards
Subject Knowledge Enhancement Programme
Day 1: Session 2
Adwick Conference Centre 16 Jan 2015

2. Organise the cards into a progressive order.
The illustrations are only ONE example – other illustrations may be more appropriate Your may decide cards need to appear in more than one place OR add cards of your own ! What concepts need to be grasped to progress from card to card?

3. Early number development
Appreciation of size
Alternative representations
Emerging number sense
Comparing
Ordering
Estimation
Rounding
Calculations
Operations
Equivalence
within & across
systems
Patterns
Sequences

4. “One, two, three…”
“There are
three bears”

5. “One, two, three…”

6. Early number development
Appreciation of size
Alternative representations
Emerging number sense
Comparing
Equivalence
Calculations
Estimation
Patterns

7. 4
HTU , HTU , HTU

9. Early number development
Appreciation of size
Alternative representations
Emerging number sense
Comparing
Equivalence
Calculations
Estimation
Patterns

12. Early number development
Appreciation of size
Alternative representations
Emerging number sense
Comparing
Equivalence
Calculations
Estimation
Patterns

13. Concrete
Pictorial
Abstract
3 + 2 = 5

14. 3 + 2 = 5
4 + 1 = 5
2 + 3 = 5
= 5

16. Stuff + stuff = more stuff OR Stuff grows?

17. Big stuff “take away” smaller stuff?1 1
Pay 2 1 ?

18. Subtraction: Is it “take away” or “and” a negative number?
To subtract or take away suggests an operation whereas negative is a sign.
Unfortunately minus seems to mean both – should we make a distinction ?
Big stuff “take away” small stuff : what does the”-” mean in 7 – 3 ?

19. Taking away more than you start with1 1
Pay 5 ? 1
Taking away more than you start with

21. Multiplication 1 1 1 1 1 1 1 1 1 1 1 1
Going back again to basics

22. “How would you say......?”
5 x 3
What different ways could this be expressed verbally?
Do they all mean the same thing?

23. is a derivative of which one?
Is it important to make the distinction between 3x5 & 5x3?
Five “times” three
is a derivative of which one?

24. “5 multiplied three times”
5 x 3
“5 multiplied three times”
“Five lots of 3” x3
Do we actually teach the concept of an operation , when ? OR does it just happen?

26. Division
The BAR symbolism comes into its own, and the link between “divided into” and “shared by” is easily demonstrated. Is there any meaning to “divided by” or is it a derivative of these two?
20 divided into 4 equal pieces
20 shared equally between 4
(or 20 shared by 4)

27. “Goes into”
Similar to “times”, “goes into” reveals an ambiguity. Previously we have seen 20 ÷ 4 but this is not the same as “how many 4’s go into 20”? 5 4 4 4 4 4 5
...because the question was
“how many times does 4 go into 20?” 5 5

28. Choose an aspect of addition that relates to the level of your students.
What specific concepts can you target for development?
Try planning some activities to explore this with your students
Consider activities that integrate CPA
How about extending the more able?

29. Count number of objects
“One, two, three cars.”

30. Subitizing small amounts
“Three cars.”

31. Developing mental “counting” skills through addition of small “subitized” amounts
e.g. 3 + 4 = 7

32. Line them up to count & compare

33. Put boxes around pictures

34. Use one to one representation of objects eg boxes & tally

35. Use different coloured boxes to represent the sum of different amounts5 3

36. Using precise language in combination diagrams
addend
addend
sum

37. Use different coloured boxes to compare different amounts5 3

38. Using precise language in comparison diagrams
minuend
difference
subtrahend

39. Use different sized boxes to represent different amounts5 3

40. Use numerical abstract representation5

41. Use language to compare e. g
Use language to compare e.g. more, less, longer, shorter, taller, shorter, bigger, smaller

42. Use numerical comparisons, e.g. …….. more than, ………… less than

43. Write numerical sentences to describe a situation
3 + 2 = 5
47 – 35 = 12

44. Use precise language to refer to parts of a number sentence

45. Group number sentences together that relate a similar or equivalent fact
2 + 3 = 5
5 = 2 + 3
3 + 2 = 5
5 = 3 + 2
5 - 3 = 2
3 = 5 - 2
5 - 2 = 3
2 = 5 - 3

46. Build up a set of easily recalled facts that relate to specific number sentences
1 + 9
2 + 8
3 + 7
4 + 6
5 + 5
11 + 9
12 + 8
13 + 7
14 + 6
15 + 5
= 10
= 20

47. Find missing values in a number sentence
7 =  + 3
 + 3 = 8
10 =  - 3
12 -  = 8

48. Illustrate a number sentence on a number line
3 + 5 = 8 2 4 6 8 10

49. Illustrate a number problem as a bar diagram
Rulers
How much would you need to spend in order to buy three pencils that cost 5 pence each and two rulers that cost 11 pence each?
Pencils 5 11

50. Illustrate number sentences in a “Maths story”
3 + 5 = 8 8 = Three friends in the park are joined by five more so there are eight altogether. They decide to play a game of football, there are four players in each team.

51. Develop an understanding that a single fact can reveal associated facts
8 + 9 = 17
= 25
8 + 7 = 15
= 150
= 515

52. Use “make & break” to create equivalent number sentences
4 + 4 = 8
(2 + 2) + 4 = 8
2 + (2 + 4) = 8
2 + 6 = 8

53. Say & read two digit numbers as N tens, M AND know the common number “name”23 17
“Two tens, three ; Twenty-three
“One ten and seven ; Seventeen

54. Appreciate the size of a number.
123
12 3

55. Know the equivalence of place value in tens and use “make & break” to support addition and subtraction
(20 + 4) + (30 + 7) = 61
54 – 28
(50 + 4) – (20 + 8)
( ) – (20 + 8)
( ) – (20 + 8)
= 26

56. Develop column addition using manipulatives to scale24
+ 17 41 1

57. Develop column subtraction using manipulatives to scale1 2 32
- 14

58. Counting up and down in tens…..
172
182
192
202
212
….. … 87 97
107
117 ….

59. Consolidate column addition through use of “scaled” manipulatives24
+ 17 U U T U U U U U U U

60. Consolidate column subtraction through use of “scaled” manipulatives1 T U T U U 32
- 14 2 T U T U U T U U U U U U U U

61. Use boxes to represent more than one object15 9

62. Use repeated addition groups to introduce “lots of”15

63. “Break” quantities into “lots of” and “remainders”
14 = four lots of 3 with 2 left over

64. Develop easily recalled facts based on knowledge of “lots of” tens
Three lots of 10
is equal to 30 T T

65. Develop easily recalled facts based on knowledge of “lots of” fives15

66. Develop easily recalled facts based on knowledge of “lots of” twos2 4 6 8 10

67. Extend knowledge and skills in two columns to include hundreds
234
+ 175
584
- 325

68. Know that numbers expressed with different units of measure mean different “sizes”
11 mm cm

69. Use knowledge of unit equivalence to convert amounts
3 ft
4 yd

70. Understand “negative” in context of a directed number

71. Know the distinction between subtraction and the addition of negative integers