A first lesson:
(The teacher's voice is italicised.)
Draw a co-ordinate grid in your books from 0 to 12 on x and y ...
Draw one on the board as you say this, or have one already drawn.
... and draw this shape at exactly the same points.
What is it? A church, yes!
We are going to do a process involving these numbers that will change the shape.
We will do it point by point. But first, so we know what we are talking about we need to label each point. Call this point A, what are its co-ordinates? Okay, in this project we write co-ordinates vertically. We are going to work together until everyone can do this process. Copy this into your books as we do it.
1st row, 1st column, multiply in pairs and add
2nd row, 1st column, multiply in pairs and add
A A’
Repeat these instructions for every point. Invite volunteers to complete.
Label and draw each of the new points:
What has happened to the shape?
You may want to introduce the language of stretches or shears and write up things like ‘What happens to the area?’ if students mention that kind of thing.
Try out your own numbers instead of
... and see what happens to your own shape.
You may want to discuss sensible shapes to try.
Challenge: In 5 lessons time I will come in and write a set of four numbers (a matrix) on the board and your task will be to tell me what effect it will have just by looking at it.
Where this can go
You may need to share ideas about how to explore this challenge in an organised way. With less confident students the investigation can be limited to matrices of the form
Students may find several types of matrix that transform a 2D shape to a line. These are 'crushing' matrices, and it can be interesting to try and characterise them (e.g., conjecture that all matrices of the form ... will be crushes).
If students work on paper this can be pinned to a board. They must write what the matrix has done to their shape before they pin the paper to the board. Questions may arise from looking at the matrices others have tried. Someone could organise the class results and feedback how they did it. What do other people notice? So, what questions could we ask? Can you generalise or predict? What happens to the areas of your shapes?
At some point focus the class on the basic transformations: reflection, rotation, enlargement, stretch, identity, crush (ie when the shape becomes a line or point), shear. Ask if anyone has a matrix that does one of these things only. Create a space on the board for students to write up matrices that do these basic transformations, see if anyone can generalise, eg an enlargement scale factor n is:
Someone could organise the class results under these headings.
Some students may even want to extend such generalisations, eg to 3D.
Get students to try and find the inverse matrix eg of an enlargement.
Guidelines for notes at the end of the topic
A reflection has a line of symmetry.
A rotation has a centre of rotation, a number of degrees and a direction (clockwise or anti-clockwise).
An enlargement has a centre of enlargement and a scale factor. If a shape is enlarged by a scale factor of n, each side becomes n times longer and the area becomes n2 times bigger.
[You may want to draw illustrations of these points.]
Vocabulary
Reflection, rotation, enlargement, stretch, shear, crush, identity, inverse.
Further Reading
This activity & lesson start was taken from an original idea by Laurinda Brown, for a fuller write up see:
Brown, L. (1991) Stewing in your own juice. In D. Pimm and E. Love (eds) Teaching and Learning School Mathematics. London: Hodder and Stoughton.
A similar activity is also written up by Don Cohen, see:
Cohen, D. (1995) Changing shapes with matrices. Champaign, Illinois: Don Cohen – The Mathman. (Also see: http://www.mathman.biz/)
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