from computation to algebra exploring linkages between computation, algebraic thinking and algebra...
DESCRIPTION
7 x 99 Recognising Structure 25 xTRANSCRIPT
![Page 1: From Computation to Algebra Exploring linkages between computation, algebraic thinking and algebra Kevin Hannah, National Coordinator, Secondary Numeracy](https://reader033.vdocuments.mx/reader033/viewer/2022051200/5a4d1b527f8b9ab0599a7ef9/html5/thumbnails/1.jpg)
From Computation to Algebra
Exploring linkages between computation,
algebraic thinking and algebra
Kevin Hannah, National Coordinator,Secondary Numeracy Project
![Page 2: From Computation to Algebra Exploring linkages between computation, algebraic thinking and algebra Kevin Hannah, National Coordinator, Secondary Numeracy](https://reader033.vdocuments.mx/reader033/viewer/2022051200/5a4d1b527f8b9ab0599a7ef9/html5/thumbnails/2.jpg)
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.
![Page 3: From Computation to Algebra Exploring linkages between computation, algebraic thinking and algebra Kevin Hannah, National Coordinator, Secondary Numeracy](https://reader033.vdocuments.mx/reader033/viewer/2022051200/5a4d1b527f8b9ab0599a7ef9/html5/thumbnails/3.jpg)
7 x 99
Recognising Structure
25 x 9997 + 56
![Page 4: From Computation to Algebra Exploring linkages between computation, algebraic thinking and algebra Kevin Hannah, National Coordinator, Secondary Numeracy](https://reader033.vdocuments.mx/reader033/viewer/2022051200/5a4d1b527f8b9ab0599a7ef9/html5/thumbnails/4.jpg)
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.
![Page 5: From Computation to Algebra Exploring linkages between computation, algebraic thinking and algebra Kevin Hannah, National Coordinator, Secondary Numeracy](https://reader033.vdocuments.mx/reader033/viewer/2022051200/5a4d1b527f8b9ab0599a7ef9/html5/thumbnails/5.jpg)
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.
![Page 6: From Computation to Algebra Exploring linkages between computation, algebraic thinking and algebra Kevin Hannah, National Coordinator, Secondary Numeracy](https://reader033.vdocuments.mx/reader033/viewer/2022051200/5a4d1b527f8b9ab0599a7ef9/html5/thumbnails/6.jpg)
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
![Page 7: From Computation to Algebra Exploring linkages between computation, algebraic thinking and algebra Kevin Hannah, National Coordinator, Secondary Numeracy](https://reader033.vdocuments.mx/reader033/viewer/2022051200/5a4d1b527f8b9ab0599a7ef9/html5/thumbnails/7.jpg)
Using Materials
46 + = 83
10 20 30 40 50 60 70 80 900 100
46 83
410 10 10
3
37
![Page 8: From Computation to Algebra Exploring linkages between computation, algebraic thinking and algebra Kevin Hannah, National Coordinator, Secondary Numeracy](https://reader033.vdocuments.mx/reader033/viewer/2022051200/5a4d1b527f8b9ab0599a7ef9/html5/thumbnails/8.jpg)
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
![Page 9: From Computation to Algebra Exploring linkages between computation, algebraic thinking and algebra Kevin Hannah, National Coordinator, Secondary Numeracy](https://reader033.vdocuments.mx/reader033/viewer/2022051200/5a4d1b527f8b9ab0599a7ef9/html5/thumbnails/9.jpg)
Using Number Properties
39 + = 93 27 + = 52 46 + = 82 55 + = 72 17 + = 64
5425361747
![Page 10: From Computation to Algebra Exploring linkages between computation, algebraic thinking and algebra Kevin Hannah, National Coordinator, Secondary Numeracy](https://reader033.vdocuments.mx/reader033/viewer/2022051200/5a4d1b527f8b9ab0599a7ef9/html5/thumbnails/10.jpg)
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.
![Page 11: From Computation to Algebra Exploring linkages between computation, algebraic thinking and algebra Kevin Hannah, National Coordinator, Secondary Numeracy](https://reader033.vdocuments.mx/reader033/viewer/2022051200/5a4d1b527f8b9ab0599a7ef9/html5/thumbnails/11.jpg)
Extending from Computation
Video
Extending the picture used to build the computational skills
Adjusting from a number line
to a strip diagram
![Page 12: From Computation to Algebra Exploring linkages between computation, algebraic thinking and algebra Kevin Hannah, National Coordinator, Secondary Numeracy](https://reader033.vdocuments.mx/reader033/viewer/2022051200/5a4d1b527f8b9ab0599a7ef9/html5/thumbnails/12.jpg)
Solving Equations
19 + = 43
19 43
19 43
![Page 13: From Computation to Algebra Exploring linkages between computation, algebraic thinking and algebra Kevin Hannah, National Coordinator, Secondary Numeracy](https://reader033.vdocuments.mx/reader033/viewer/2022051200/5a4d1b527f8b9ab0599a7ef9/html5/thumbnails/13.jpg)
Solving Equations
2X = 28
3n + 1 = 28
2p + 1 = p + 9
![Page 14: From Computation to Algebra Exploring linkages between computation, algebraic thinking and algebra Kevin Hannah, National Coordinator, Secondary Numeracy](https://reader033.vdocuments.mx/reader033/viewer/2022051200/5a4d1b527f8b9ab0599a7ef9/html5/thumbnails/14.jpg)
Solving Equations
2X = 28 X
28 X
![Page 15: From Computation to Algebra Exploring linkages between computation, algebraic thinking and algebra Kevin Hannah, National Coordinator, Secondary Numeracy](https://reader033.vdocuments.mx/reader033/viewer/2022051200/5a4d1b527f8b9ab0599a7ef9/html5/thumbnails/15.jpg)
Solving Equations
3n + 1 = 28
Does the 1 have to be that small?
![Page 16: From Computation to Algebra Exploring linkages between computation, algebraic thinking and algebra Kevin Hannah, National Coordinator, Secondary Numeracy](https://reader033.vdocuments.mx/reader033/viewer/2022051200/5a4d1b527f8b9ab0599a7ef9/html5/thumbnails/16.jpg)
Solving Equations
3n + 1 = 28
28 n 1 n n
![Page 17: From Computation to Algebra Exploring linkages between computation, algebraic thinking and algebra Kevin Hannah, National Coordinator, Secondary Numeracy](https://reader033.vdocuments.mx/reader033/viewer/2022051200/5a4d1b527f8b9ab0599a7ef9/html5/thumbnails/17.jpg)
Solving Equations: what students did
2p + 1 = p + 9
9 pp p 1
![Page 18: From Computation to Algebra Exploring linkages between computation, algebraic thinking and algebra Kevin Hannah, National Coordinator, Secondary Numeracy](https://reader033.vdocuments.mx/reader033/viewer/2022051200/5a4d1b527f8b9ab0599a7ef9/html5/thumbnails/18.jpg)
Solving Equations: what students did
2p + 1 = p + 9
9 pp p 1
![Page 19: From Computation to Algebra Exploring linkages between computation, algebraic thinking and algebra Kevin Hannah, National Coordinator, Secondary Numeracy](https://reader033.vdocuments.mx/reader033/viewer/2022051200/5a4d1b527f8b9ab0599a7ef9/html5/thumbnails/19.jpg)
Solving Equations: what students did
2p + 1 = p + 9
9 pp p 1
![Page 20: From Computation to Algebra Exploring linkages between computation, algebraic thinking and algebra Kevin Hannah, National Coordinator, Secondary Numeracy](https://reader033.vdocuments.mx/reader033/viewer/2022051200/5a4d1b527f8b9ab0599a7ef9/html5/thumbnails/20.jpg)
Solving Equations: what students did
2p + 1 = p + 9
5 pp p 1 4
![Page 21: From Computation to Algebra Exploring linkages between computation, algebraic thinking and algebra Kevin Hannah, National Coordinator, Secondary Numeracy](https://reader033.vdocuments.mx/reader033/viewer/2022051200/5a4d1b527f8b9ab0599a7ef9/html5/thumbnails/21.jpg)
Solving Equations: what students did
2p + 1 = p + 9
1 pp p 1 p
![Page 22: From Computation to Algebra Exploring linkages between computation, algebraic thinking and algebra Kevin Hannah, National Coordinator, Secondary Numeracy](https://reader033.vdocuments.mx/reader033/viewer/2022051200/5a4d1b527f8b9ab0599a7ef9/html5/thumbnails/22.jpg)
Solving Equations: what students did
2p + 1 = p + 9
1 pp p 1 8
![Page 23: From Computation to Algebra Exploring linkages between computation, algebraic thinking and algebra Kevin Hannah, National Coordinator, Secondary Numeracy](https://reader033.vdocuments.mx/reader033/viewer/2022051200/5a4d1b527f8b9ab0599a7ef9/html5/thumbnails/23.jpg)
Solving Equations: some more
4x = 28
6x + 2 = 44
5x + 4 = 2x + 25
![Page 24: From Computation to Algebra Exploring linkages between computation, algebraic thinking and algebra Kevin Hannah, National Coordinator, Secondary Numeracy](https://reader033.vdocuments.mx/reader033/viewer/2022051200/5a4d1b527f8b9ab0599a7ef9/html5/thumbnails/24.jpg)
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
![Page 25: From Computation to Algebra Exploring linkages between computation, algebraic thinking and algebra Kevin Hannah, National Coordinator, Secondary Numeracy](https://reader033.vdocuments.mx/reader033/viewer/2022051200/5a4d1b527f8b9ab0599a7ef9/html5/thumbnails/25.jpg)
Next: Abstract to . . .
13p + 6 = 10p + 423p = 36p = 12
m + 41 = 7m + 56m = 36m = 6
![Page 26: From Computation to Algebra Exploring linkages between computation, algebraic thinking and algebra Kevin Hannah, National Coordinator, Secondary Numeracy](https://reader033.vdocuments.mx/reader033/viewer/2022051200/5a4d1b527f8b9ab0599a7ef9/html5/thumbnails/26.jpg)
Solving Equations:brackets
2(X + 1) = 18 X X 1 1
9 9
![Page 27: From Computation to Algebra Exploring linkages between computation, algebraic thinking and algebra Kevin Hannah, National Coordinator, Secondary Numeracy](https://reader033.vdocuments.mx/reader033/viewer/2022051200/5a4d1b527f8b9ab0599a7ef9/html5/thumbnails/27.jpg)
Solving Equations: brackets
2(X + 1) = 18 X
18 X 1 1
![Page 28: From Computation to Algebra Exploring linkages between computation, algebraic thinking and algebra Kevin Hannah, National Coordinator, Secondary Numeracy](https://reader033.vdocuments.mx/reader033/viewer/2022051200/5a4d1b527f8b9ab0599a7ef9/html5/thumbnails/28.jpg)
Solving Equations: subtraction
19 + = 43
19 43
19 43
![Page 29: From Computation to Algebra Exploring linkages between computation, algebraic thinking and algebra Kevin Hannah, National Coordinator, Secondary Numeracy](https://reader033.vdocuments.mx/reader033/viewer/2022051200/5a4d1b527f8b9ab0599a7ef9/html5/thumbnails/29.jpg)
Solving Equations: subtraction
n - 17 = 64
n64 17
64 ? 17
![Page 30: From Computation to Algebra Exploring linkages between computation, algebraic thinking and algebra Kevin Hannah, National Coordinator, Secondary Numeracy](https://reader033.vdocuments.mx/reader033/viewer/2022051200/5a4d1b527f8b9ab0599a7ef9/html5/thumbnails/30.jpg)
Solving Equations:subtraction
2X - 1 = X + 7
7 X
1 X
X
![Page 31: From Computation to Algebra Exploring linkages between computation, algebraic thinking and algebra Kevin Hannah, National Coordinator, Secondary Numeracy](https://reader033.vdocuments.mx/reader033/viewer/2022051200/5a4d1b527f8b9ab0599a7ef9/html5/thumbnails/31.jpg)
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
![Page 32: From Computation to Algebra Exploring linkages between computation, algebraic thinking and algebra Kevin Hannah, National Coordinator, Secondary Numeracy](https://reader033.vdocuments.mx/reader033/viewer/2022051200/5a4d1b527f8b9ab0599a7ef9/html5/thumbnails/32.jpg)
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.
![Page 33: From Computation to Algebra Exploring linkages between computation, algebraic thinking and algebra Kevin Hannah, National Coordinator, Secondary Numeracy](https://reader033.vdocuments.mx/reader033/viewer/2022051200/5a4d1b527f8b9ab0599a7ef9/html5/thumbnails/33.jpg)
Solving Equations
2X - 1 = 8 - X X
1 8 X X
![Page 34: From Computation to Algebra Exploring linkages between computation, algebraic thinking and algebra Kevin Hannah, National Coordinator, Secondary Numeracy](https://reader033.vdocuments.mx/reader033/viewer/2022051200/5a4d1b527f8b9ab0599a7ef9/html5/thumbnails/34.jpg)
Solving Equations
X - 1 = 2X - 7 7
1 X
X X
![Page 35: From Computation to Algebra Exploring linkages between computation, algebraic thinking and algebra Kevin Hannah, National Coordinator, Secondary Numeracy](https://reader033.vdocuments.mx/reader033/viewer/2022051200/5a4d1b527f8b9ab0599a7ef9/html5/thumbnails/35.jpg)
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.
![Page 36: From Computation to Algebra Exploring linkages between computation, algebraic thinking and algebra Kevin Hannah, National Coordinator, Secondary Numeracy](https://reader033.vdocuments.mx/reader033/viewer/2022051200/5a4d1b527f8b9ab0599a7ef9/html5/thumbnails/36.jpg)
Solving Equations: Integers
X + 3 = 2 X
2 3
![Page 37: From Computation to Algebra Exploring linkages between computation, algebraic thinking and algebra Kevin Hannah, National Coordinator, Secondary Numeracy](https://reader033.vdocuments.mx/reader033/viewer/2022051200/5a4d1b527f8b9ab0599a7ef9/html5/thumbnails/37.jpg)
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.
![Page 38: From Computation to Algebra Exploring linkages between computation, algebraic thinking and algebra Kevin Hannah, National Coordinator, Secondary Numeracy](https://reader033.vdocuments.mx/reader033/viewer/2022051200/5a4d1b527f8b9ab0599a7ef9/html5/thumbnails/38.jpg)
97 + 56
Recognising Structure
![Page 39: From Computation to Algebra Exploring linkages between computation, algebraic thinking and algebra Kevin Hannah, National Coordinator, Secondary Numeracy](https://reader033.vdocuments.mx/reader033/viewer/2022051200/5a4d1b527f8b9ab0599a7ef9/html5/thumbnails/39.jpg)
Recognising Structure
97 + 56 = 100 +
![Page 40: From Computation to Algebra Exploring linkages between computation, algebraic thinking and algebra Kevin Hannah, National Coordinator, Secondary Numeracy](https://reader033.vdocuments.mx/reader033/viewer/2022051200/5a4d1b527f8b9ab0599a7ef9/html5/thumbnails/40.jpg)
Recognising Structure
97 + 78 = 100 +
![Page 41: From Computation to Algebra Exploring linkages between computation, algebraic thinking and algebra Kevin Hannah, National Coordinator, Secondary Numeracy](https://reader033.vdocuments.mx/reader033/viewer/2022051200/5a4d1b527f8b9ab0599a7ef9/html5/thumbnails/41.jpg)
Recognising Structure
97 + = 100 +
![Page 42: From Computation to Algebra Exploring linkages between computation, algebraic thinking and algebra Kevin Hannah, National Coordinator, Secondary Numeracy](https://reader033.vdocuments.mx/reader033/viewer/2022051200/5a4d1b527f8b9ab0599a7ef9/html5/thumbnails/42.jpg)
Recognising Structure
97 + x = 100 +
![Page 43: From Computation to Algebra Exploring linkages between computation, algebraic thinking and algebra Kevin Hannah, National Coordinator, Secondary Numeracy](https://reader033.vdocuments.mx/reader033/viewer/2022051200/5a4d1b527f8b9ab0599a7ef9/html5/thumbnails/43.jpg)
Recognising Structure
97 + = 100 + y
![Page 44: From Computation to Algebra Exploring linkages between computation, algebraic thinking and algebra Kevin Hannah, National Coordinator, Secondary Numeracy](https://reader033.vdocuments.mx/reader033/viewer/2022051200/5a4d1b527f8b9ab0599a7ef9/html5/thumbnails/44.jpg)
Recognising Structure
97 + = 100 +
![Page 45: From Computation to Algebra Exploring linkages between computation, algebraic thinking and algebra Kevin Hannah, National Coordinator, Secondary Numeracy](https://reader033.vdocuments.mx/reader033/viewer/2022051200/5a4d1b527f8b9ab0599a7ef9/html5/thumbnails/45.jpg)
Recognising Structure
88 + = 100 +
![Page 46: From Computation to Algebra Exploring linkages between computation, algebraic thinking and algebra Kevin Hannah, National Coordinator, Secondary Numeracy](https://reader033.vdocuments.mx/reader033/viewer/2022051200/5a4d1b527f8b9ab0599a7ef9/html5/thumbnails/46.jpg)
Recognising Structure
88 + = 120 +
![Page 47: From Computation to Algebra Exploring linkages between computation, algebraic thinking and algebra Kevin Hannah, National Coordinator, Secondary Numeracy](https://reader033.vdocuments.mx/reader033/viewer/2022051200/5a4d1b527f8b9ab0599a7ef9/html5/thumbnails/47.jpg)
Recognising Structure
88 + x = 120 + yx is 32 more than
y y is 32 less than x