From Computation to Algebra Exploring linkages between computation, algebraic thinking and algebra...
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From Computation to Algebra
Exploring linkages between computation,
algebraic thinking and algebra
Kevin Hannah, National Coordinator,Secondary Numeracy Project
Algebraic Thinking
When students demonstrate they can use principles that are generally true and do not relate only to particular numbers or patterns they may be said to be using algebraic thinking.
7 x 99
Recognising Structure
25 x 9997 + 56
Algebraic Thinking
A student who is using smart computational strategies implicitly understands the structure of our number system - place value, base 10, associative, commutative and distributive properties. They are thinking algebraically.
Extending from Computation
Step 1: Build the computational strategies of
students using visual images.Step 2: Exploit what the students are using
computationally to develop algebra.
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
Encouraging Imaging
28 + = 54 26
20 40 6028
2 10 4
30 50
10
54
16 + = 73 57
16
4 50 3
20 70 73
Using Number Properties
39 + = 93 27 + = 52 46 + = 82 55 + = 72 17 + = 64
5425361747
Extending from Computation
Step 1: Build the computational strategies of
students using visual images.Step 2: Exploit what the students are using
computationally to develop algebra.
Extending from Computation
Video
Extending the picture used to build the computational skills
Adjusting from a number line
to a strip diagram
Solving Equations
19 + = 43
19 43
19 43
Solving Equations
2X = 28
3n + 1 = 28
2p + 1 = p + 9
Solving Equations
2X = 28 X
28 X
Solving Equations
3n + 1 = 28
Does the 1 have to be that small?
Solving Equations
3n + 1 = 28
28 n 1 n n
Solving Equations: what students did
2p + 1 = p + 9
9 pp p 1
Solving Equations: what students did
2p + 1 = p + 9
9 pp p 1
Solving Equations: what students did
2p + 1 = p + 9
9 pp p 1
Solving Equations: what students did
2p + 1 = p + 9
5 pp p 1 4
Solving Equations: what students did
2p + 1 = p + 9
1 pp p 1 p
Solving Equations: what students did
2p + 1 = p + 9
1 pp p 1 8
Solving Equations: some more
4x = 28
6x + 2 = 44
5x + 4 = 2x + 25
The students’ equations
7x + 3 = 8x 9x + 7 = 10x + 3 4x + 3 = 3x + 13 3x + 5 = 29
5x + 5 = 2x + 38 5y + 7 = y + 23 11x + 2 = 10x + 5
Next: Abstract to . . .
13p + 6 = 10p + 423p = 36p = 12
m + 41 = 7m + 56m = 36m = 6
Solving Equations:brackets
2(X + 1) = 18 X X 1 1
9 9
Solving Equations: brackets
2(X + 1) = 18 X
18 X 1 1
Solving Equations: subtraction
19 + = 43
19 43
19 43
Solving Equations: subtraction
n - 17 = 64
n64 17
64 ? 17
Solving Equations:subtraction
2X - 1 = X + 7
7 X
1 X
X
Reminder: A 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:subtraction The goal is for students to work abstractly
with algebraic equations. The strips are a prop on the way to doing this.
There will come a point when the mental load of trying to construct a diagram is higher than the load required to manipulate an algebraic expression. The strips will have been dispensed with by this time.
The following two examples probably come into this category.
Solving Equations
2X - 1 = 8 - X X
1 8 X X
Solving Equations
X - 1 = 2X - 7 7
1 X
X X
Solving Equations: Integers Remember the goal is for students to work
abstractly with algebraic equations. The strips are a prop on the way to doing this.
And when it comes to integers, this visual representation doesn’t support them. So students need to have explored the structure of the equations and abstracted the principles before integers are introduced.
Solving Equations: Integers
X + 3 = 2 X
2 3
Extending from Computation
Step 1: Build the computational strategies of
students using visual images.Step 2: Exploit what the students are using
computationally to develop algebra.
97 + 56
Recognising Structure
Recognising Structure
97 + 56 = 100 +
Recognising Structure
97 + 78 = 100 +
Recognising Structure
97 + = 100 +
Recognising Structure
97 + x = 100 +
Recognising Structure
97 + = 100 + y
Recognising Structure
97 + = 100 +
Recognising Structure
88 + = 100 +
Recognising Structure
88 + = 120 +
Recognising Structure
88 + x = 120 + yx is 32 more than
y y is 32 less than x