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The Numeracy Professional Development Project in Secondary Schools
Kevin Hannah
National Coordinator,
Secondary Numeracy Project
Addition and Subtraction Strategies
Addition and Subtraction strategies The Number Strategy Framework A Model for a Teaching Progression Some Questions Using materials Encouraging Imaging Towards number properties Subtraction strategies Subtraction and addition problems From Number to Algebra - solving equations
Strategy Framework0 Emergent1 One-to-one counting2 Counting from one with materials3 Counting from one by imaging4 Counting on5 Early Part-Whole6 Advanced Part-Whole7 Advanced Multiplicative8 Advanced Proportional
Objective To explore links between numeracy and algebra
In particular: To show that images used to help solve number problems
also develop understanding for solving complicated linear equations.
Answers Only Please
1. 46 + 19
2. 61 – 27
3. 14 + ? = 101
4. 78 + 124
5. 9000 – 8985
6. 403 - 98
7. 47 + y = 838. 53 - m = 279. 2x + 1 = x +
710. 2x - 1 = 8 - x11. 26 + 7 = ? +
1212. 88 + x = 120
+ ?
A Teaching ProgressionStart by: Using materials, diagrams to illustrate and
solve the problemProgress to: Developing mental images to help solve
the problemExtend to: Working abstractly with the number
property
Using Materials
46 + = 83
10 20 30 40 50 60 70 80 900 100
46 83
410 10 10
3
37
Using Materials
48 + = 81 33
10 20 30 40 50 60 70 80 900 10048
210 10 10
1
81
29 + = 75 46
10 20 30 40 50 60 70 80 900 10029
140 5
75
Encouraging Imaging
39 + = 63
39 63
110 3
24
30 40 50 60 70
10
Encouraging Imaging
28 + = 54 26
20 40 6028
210 4
30 50
10
54
16 + = 73 57
16
450 3
20 70 73
Using Number Properties
18 + = 62 44
From 18: add 2 to get to 20add 40 to get to 60add 2 to get to 62
Total: add 44
Using Number Properties
39 + = 93
27 + = 52
46 + = 82
55 + = 72
17 + = 64
54
25
36
17
47
Where to from here? A subtraction strategy Other strategies on number line Link to Algebra - solving equations
Using Materials- subtraction
81 - 47 =
10 20 30 40 50 60 70 80 900 100
47 81
310 10 10
1
34
Imaging - subtraction
37 64
310 4
30 40 50 60 70
10
64 - 37 =27
Imaging - subtraction
83 - 26 =57
26
450 3
30 80 83
Using Number Properties
From 17: add 3 to get to 20add 60 to get to 80add 2 to get to 82
Total: add 65
82 - 17 =65
Other strategies & problems
10 20 30 40 50 60 70 80 900 100
The number line is a versatile tool and image. It can be used to support and explain a
variety of strategies. It can be used to solve a wide range of
problems. It can prepare students for algebra.
Other subtraction strategies
81 - 47 =
10 20 30 40 50 60 70 80 900 100
41 81
6 10 10 101
34
10
34
Other subtraction strategies
81 - 47 =
10 20 30 40 50 60 70 80 900 100
31 81
3
10 10 10
34
10
34
10
Other subtraction problems
92 - = 65
10 20 30 40 50 60 70 80 900 100
27 92
10 10
27
5 10101010
92 - 65 =
Other subtraction problems
92 - = 65
10 20 30 40 50 60 70 80 900 100
65 92
10 10
27
5 2
65+ = 92
Other subtraction problems
- 18 = 43
10 20 30 40 50 60 70 80 900 100
43 61
8
61
10
= 43 + 18 61
Other subtraction problems
- 18 = 43
10 20 30 40 50 60 70 80 900 100
18 61
3
61
10
18 + 43 = 61
10 10 10
Other addition problems
+ 27 = 72
10 20 30 40 50 60 70 80 900 100
27 72
10 10
45
51010
+ 27 = 27 +
Other addition problems
+ 27 = 72
10 20 30 40 50 60 70 80 900 100
45 72
10 10
45
7
= 72 - 27 45
Other addition problems
57 + 26 =
10 20 30 40 50 60 70 80 900 100
57 83
83
1010 6
What is Numeracy? It’s about making sense of numbers. It’s about problem solving with numbers. It’s about understanding base 10. It’s about learning with meaning. It’s about preparing for algebra
Solving Equations
47 + = 83
47
83
47 83
Solving Equations
53 - x = 27
53 - 53 - x = 27 - 53 - x = -26 x = -26 ÷ -1 x = 26
Solving Equations
53 - x = 27
53 - x +x = 27 +x 53 = 27 +x 53 - 27 = 27 -27+x 26 = x
Solving Equations
53 - = 27
27
53
27 53
Solving Equations
2X + 1 = X + 7
7 X
X X 1
Solving Equations
2X + 1 = 7
7
X X 1
A Teaching ProgressionStart by: Using materials, diagrams to illustrate and
solve the problemProgress to: Developing mental images to help solve
the problemExtend to: Working abstractly with the property
Solving Equations
2X - 1 = X + 7
7
X
1
X
X
Solving Equations
X - 1 = 2X - 7
7
1
X
X X
Solving Equations
X - 1 = 2X - 7
7
1
X
X X
7 = X + 1
X = 6
Solving Equations
2X - 1 = 8 - X
X
1 8
X X
Solving Equations
2(X + 1) = 18
X
18
X 1 1
Solving Equations
2(X + 1) = 18
X X 1 1
9 9
Solving Equations
2(X + 1) = 18
X
18
X 1 1
Solving Equations
X + 3 = 2
X
2
3
10
X
4
€
x3+ 4 =10
€
x3
€
x3
€
x3
Solving Equations
X 4
€
x+43
=10
10 10 10
Solving Equations
What is Numeracy? It’s about making sense of numbers. It’s about problem solving with numbers. It’s about understanding base 10. It’s about learning with meaning. It’s about preparing for algebra
