Transformation Matrices

This Geogebra page is to show where transformation matrices come from and how they work.

You can download it or run it in the browser. Find the links and basic instructions on TES.com 

You control the page by dragging points around on the coordinate axes. The vectors labelled a and b become the base vectors of the transformed coordinate space. A point (2,3) will be transformed to the point with position vector 2a + 3b.

Here are some ideas for how to this tool with a class:

1) Display axes on the board which students can write over with their answers then quickly show the correct answers.

I would base the lesson around a set of worksheets with predrawn axes and questions with matrix multiplications for matrices:

Section A:     Stretch, reflection, rotation, enlargements 

Section B:     Shears 

Section C:     Combinations of different transformations - some combinations where the order is not important and some where the order is important. 

Section D:     Exploring the determinant and the area ratio of the object and image (Section D).

Start with an easy one:

Give these questions as a starter to the group:

M_1 = {{1,0},{0,-1}}

P = {2,2}        P' = M_1P

Q = {3,2}        Q' = M_1Q

R = {3,4}         R' = M_1R

S = {3,3}         S' = M_1S 

(written out as a matrix and column vectors)

Find P', Q', R', S'.  (Its pronounced 'P prime')

They'll probably finish this quickly, so maybe you can add some extra points like T(-2,3), U(4, -2) and V(x,y).

Display the Geogebra page with calculation, image and inksplat hidden, P,Q,R,S in the wrong places. a should be at (1,0) and b at (0,-1). Don't let them see the screen when you change the vectors.

Ask them to show you where the points should be on the coordinate axes. Drag the points to the positions they tell you and allow them to make mistakes and self-correct as a group. Reveal the 'image' to check everthing is right.

If Ricky hasn't already blurted out the obvious, ask what they notice (think, pair, share) and get them to tell you what happened, and keep asking until you get a high quality description.

Then give them five minutes to finish section A of the worksheet while you walj around and check their multiplication, then quickly go through the answers using the tool and elicit high quality descriptions of the transformations.

2) Revision / lesson starter

Display a transformation matrix, with the calculation show, obscured by the inksplat (you can move it around with the mouse using the little grey triangles). Hide the image points. The class have to finish the calculation, plot the points and tell you what transformations happened in what order.