Combined Transformations and Invariance: Probing Questions
Whether you are looking for a question to stimulate discussion in lesson, or a challenge at the end of a homework, then hopefully you will find these useful.
Contents
Convince Me That... keyboard_arrow_up
Back to Top
I use Convince Me That questions lots in my lessons and homeworks. Providing students with a statement and challenging them to come up with as many different ways of convincing you as possible can lead to some fascinating discussions. The different ways of seeing the same thing can also help improve the depth of students’ understanding. Thanks so much to the Thornleigh Maths Department, in particular Erica Richards, Anton Lewis and Gareth Fairclough for helping me put these together, and we will endeavour to keep adding more!
Given a shape on a co-ordinate grid, you can carry out different single transformations that result in 0, 1, 2 or 4 points being invariant (but not 3???)
All types of transformations apart from translations can leave at least one point invariant
The order you carry out combinations of different pairs of transformations may matter to the final result
You can always produce the same result of two reflections with one rotation
VI3 Treatment keyboard_arrow_up
Back to Top
We devised VI3 Treatment as a versatile way of giving students meaningful follow-up work at once we have marked their homework. The idea is that students are challenged to come up with 3 things with certain constraints. These are ideal to use as an extension for students who have got everything correct, and also as further purposeful practise for students who have got a particular question wrong. Use the ideas below and adapt them accordingly, using different numbers where appropriate. Either mark them yourself or better still, get other students to do it. Thanks so much to the Thornleigh Maths Department, in particular Erica Richards, Anton Lewis and Gareth Fairclough for helping me put these together, and we will endeavour to keep adding more!
Draw a square in the top-right quadrant. Now describe 3 different transformations that result in 0, 1 and 2 vertices of the square becoming invariant
Describe 3 pairs of combinations of transformations (each pair must contain 2 different types of transformation) that result in an object returning to its original position
Describe 3 combinations of transformations that would transform the point (2,3) to (-1, 1)