Transcript Add_Sub

The Numeracy Professional
Development Project
in Secondary Schools
Addition and Subtraction
Strategies
Kevin Hannah
National Coordinator,
Secondary Numeracy Project
Addition and Subtraction strategies
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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 Framework
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1
2
3
4
5
6
7
8
Emergent
One-to-one counting
Counting from one with materials
Counting from one by imaging
Counting on
Early Part-Whole
Advanced Part-Whole
Advanced Multiplicative
Advanced Proportional
Objective
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To explore links between numeracy and algebra
In particular:
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To show that images used to help solve number problems also
develop understanding for solving complicated linear
equations.
Answers Only Please
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2.
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6.
46 + 19
61 – 27
14 + ? = 101
78 + 124
9000 – 8985
403 - 98
7.
8.
9.
10.
11.
12.
47 + y = 83
53 - m = 27
2x + 1 = x + 7
2x - 1 = 8 - x
26 + 7 = ? + 12
88 + x = 120 + ?
A Teaching Progression
Start by:
 Using materials, diagrams to illustrate and
solve the problem
Progress to:
 Developing mental images to help solve
the problem
Extend to:
 Working abstractly with the number
property
Using Materials
46 + 37 = 83
10
10
10
4
0
10
20
30
40
3
50
46
60
70
80
83
90
100
Using Materials
48 + 33 = 81
10
10
10
2
0
10
20
30
40
48
1
50
60
70
80
90
100
80
90
100
81
29 + 46 = 75
40
1
0
10
20
30
29
40
50
5
60
70
75
Encouraging Imaging
39 + 24 = 63
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30
40
39
10
10
50
3
60
70
63
Encouraging Imaging
28 + 26 = 54
2
20
28 30
10
10
40
4
50 54
16 + 57 = 73
4
16 20
50
3
70 73
60
Using Number Properties
18 + 44 = 62
From 18:
add 2 to get to 20
add 40 to get to 60
add 2 to get to 62
Total:
add 44
Using Number Properties
39 + 54 = 93
27 + 25 = 52
46 + 36 = 82
55 + 17 = 72
17 + 47 = 64
Where to from here?
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A subtraction strategy
Other strategies on number line
Link to Algebra - solving equations
Using Materials- subtraction
81 - 47 = 34
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10
10
3
0
10
20
30
40
1
50
47
60
70
80
81
90
100
Imaging - subtraction
64 - 37 = 27
3
30
40
37
10
10
50
4
60
70
64
Imaging - subtraction
83 - 26 = 57
4
26 30
50
3
80 83
Using Number Properties
82 - 17 = 65
From 17:
add 3 to get to 20
add 60 to get to 80
add 2 to get to 82
Total:
add 65
Other strategies & problems
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20
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60
70
80
90
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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 = 34
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0
10
20
30
34
1
40
41
10
10
50
10
60
10
70
80
81
90
100
Other subtraction strategies
81 - 47 = 34
10
0
10
20
30 3
31
10
40
34
10
50
10
60
10
70
80
81
90
100
Other subtraction problems
92 - 27 = 65
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10
20
30
10
40
10
10
50
27
92 - 65 =
60
10
70
10
10
80
90
92
100
Other subtraction problems
92 - 27 = 65
5
0
10
20
30
40
50
60
70
65
65+
= 92
10
10
80
2
90
92
100
Other subtraction problems
61 - 18 = 43
10
0
10
20
30
40
43
61
50
8
60
70
61
= 43 + 18
80
90
100
Other subtraction problems
61 - 18 = 43
10
0
10
20
18
10
30
10
40
10
50
3
60
61
18 + 43 = 61
70
80
90
100
Other addition problems
45
10
0
10
20
30
27
+ 27 = 72
10
10
10
40
50
60
5
70
72
+ 27 = 27 +
80
90
100
Other addition problems
45
+ 27 = 72
7
0
10
20
30
40
50
45
45
10
10
60
70
72
= 72 - 27
80
90
100
Other addition problems
57 + 26 =
83
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0
10
20
30
40
50
60
57
10
70
6
80
83
90
100
What is Numeracy?
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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
47
83
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
X
X
X
1
7
Solving Equations
2X + 1 = 7
X
X
7
1
A Teaching Progression
Start by:
 Using materials, diagrams to illustrate and
solve the problem
Progress to:
 Developing mental images to help solve
the problem
Extend to:
 Working abstractly with the property
Solving Equations
2X - 1 = X + 7
X
X
X
7
1
Solving Equations
X - 1 = 2X - 7
7
X
X
X
1
Solving Equations
X - 1 = 2X - 7
7
X
X
X
7 = X+1
X = 6
1
Solving Equations
2X - 1 = 8 - X
X
X
8
X
1
Solving Equations
2(X + 1) = 18
X
1
X
18
1
Solving Equations
2(X + 1) = 18
X
1
9
X
1
9
Solving Equations
2(X + 1) = 18
X
X
18
1
1
Solving Equations
X+3=2
X
3
2
Solving Equations
x+4
3
= 10
X

x
3
x
3
x
3
4
10
Solving Equations
x+4
3
= 10
X
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10
10
4
10
What is Numeracy?
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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