From Computation to Algebra Exploring linkages between computation, algebraic thinking and algebra...

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