The Numeracy Professional Development Project in Secondary Schools

1
The Numeracy Professional Development Project in
Secondary Schools
Addition and Subtraction Strategies

• Kevin Hannah
• National Coordinator,
• Secondary Numeracy Project

2
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

3
Strategy Framework

• 0 Emergent
• 1 One-to-one counting
• 2 Counting from one with materials
• 3 Counting from one by imaging
• 4 Counting on
• 5 Early Part-Whole
• 6 Advanced Part-Whole
• 7 Advanced Multiplicative
• 8 Advanced Proportional

4
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.

5
Answers Only Please

• 46 19
• 61 27
• 14 ? 101
• 78 124
• 9000 8985
• 403 - 98

• 7. 47 y 83
• 8. 53 - m 27
• 9. 2x 1 x 7
• 10. 2x - 1 8 - x
• 11. 26 7 ? 12
• 12. 88 x 120 ?

6
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

7
Using Materials
37
10
10
10
4
3
8
Using Materials
33
46
9
Encouraging Imaging
24
10
Encouraging Imaging
26
57
11
Using Number Properties
44
From 18 add 2 to get to 20 add 40 to get to
60 add 2 to get to 62 Total add 44
12
Using Number Properties
13
Where to from here?

• A subtraction strategy
• Other strategies on number line
• Link to Algebra - solving equations

14
Using Materials- subtraction
81 - 47
34
10
10
10
3
1
15
Imaging - subtraction
64 - 37
27
16
Imaging - subtraction
83 - 26
57
17
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
18
Other strategies problems

• 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.

19
Other subtraction strategies
81 - 47
34
20
Other subtraction strategies
81 - 47
34
21
Other subtraction problems
92 - 65
27
22
Other subtraction problems
92 - 65
27
23
Other subtraction problems
- 18 43
61
24
Other subtraction problems
- 18 43
61
25
Other addition problems
27 72
45
26
Other addition problems
27 72
45
27
Other addition problems
57 26
83
28
What is Numeracy?

• Its about making sense of numbers.
• Its about problem solving with numbers.
• Its about understanding base 10.
• Its about learning with meaning.
• Its about preparing for algebra

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Solving Equations
47 83
30
Solving Equations
53 - x 27
53 - 53 - x 27 - 53
- x -26
x -26 -1
x 26
31
Solving Equations
53 - x 27
53 - x x 27 x
53 27 x
53 - 27 27 -27x
26 x
32
Solving Equations
53 - 27
53
27

33
Solving Equations
2X 1 X 7
X
X
1
7
X
34
Solving Equations
2X 1 7
7
35
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

36
Solving Equations
2X - 1 X 7
X
X
7
1
X
37
Solving Equations
X - 1 2X - 7
7
X
1
X
X
38
Solving Equations
X - 1 2X - 7
7
X
1
X
X
7 X 1
X 6
39
Solving Equations
2X - 1 8 - X
X
X
X
1
8
40
Solving Equations
2(X 1) 18
X
X
1
1
18
41
Solving Equations
2(X 1) 18
X
X
1
1
42
Solving Equations
2(X 1) 18
X
X
1
1
18
43
Solving Equations
X 3 2
X
3
2
44
Solving Equations
X
4
10
45
Solving Equations
X
4
46
What is Numeracy?

• Its about making sense of numbers.
• Its about problem solving with numbers.
• Its about understanding base 10.
• Its about learning with meaning.
• Its about preparing for algebra