Week 2

{Describe the changes and invariance achieved by combinations of rotations, reflections and translations}

This content, listed in curly brackets, is officially intended only for 'more highly attaining students'. However, it should be accessible, and I hope it will appeal, to a wider range of students. Our particular tasks involve pairs of transformations: two rotations, or a rotation and a translation, or two reflections.

MONDAY: This is a fairly straightforward task, which we build on in later tasks.

Students might well be able to predict that the resultant transformation is a rotation through 180˚, though it is unlikely that they will determine the position of the centre without first drawing the two images of the flag. [The centre is mid-way between P and Q and two units to the right.]

TUESDAY: This is quite a rich and subtle task. The tracing paper might help to convey the idea that when we perform a transformation, we are really transforming the whole plane, not just individual objects. It turns out that the point in the plane that returns to its starting position is the centre of rotation of the half turn that we found in Task 02A. The point that returns to its starting position is in effect an invariant point - there is only one such point in this task, which of course is the case for a rotation, where the only invariant point is the centre of rotation.

It will be interesting to see how students locate the desired point. Do they make lots of trials? Do they manage to get useful feedback from their trials? Do they use the experience gained from Task 02A?

Having found the desired point, it is easy to confirm that it is indeed 'invariant':

WEDNESDAY: Here we reverse the order of the rotations used in the previous two tasks. The resultant rotation will again be through 180˚, but will the centre be in the same place as before?

Student might have difficulty performing the rotations here, as the flag and the image being rotated are further from their centres than in the previous tasks. Also the second image goes off the page. Rather than rotating the flag and its image, do some students look for the centre by adopting the 'invariant point' approach of Task 02B?

THURSDAY: The previous task demonstrates that pairs of rotations are not (necessarily) commutative. The same applies to a rotation and translation, as becomes evident in this task.

 

It will interesting to see how students approach this task. Do they expect the two transformations to be commutative? Do they search for the desired translation by trial and improvement or do they use a more analytic approach by working backwards? If we apply the inverse of R to Flag G, we can see that we can get from Flag F to this intermediate image by a translation of 2 units to the left and 6 units up.

Students might notice that there seems to be a simple relation between the two translations. Can they express it in general terms and explain why it is true?

A nice extension to the task is to find the centre of the single rotation that would map F onto G. A further extension is to express the foot of each flag in coordinate form [(5, 3) and (11, 5), say] and to find a (general) rule for using these coordinates to find the coordinates of the centre.

FRIDAY: This task involves two reflections and is based on the fact that two reflections are equivalent to a rotation about their point of intersection,  through twice the angle between the mirrors. In this case the mirrors intersect at 45˚ and the reflections are equivalent to a 90˚ rotation. However, students don't need to know all this in order to tackle the task. Indeed, without this knowledge the task becomes a rich investigation.

There are infinitely many pairs of mirrors that will map F onto H, but probably the two most obvious ones are these. It will interesting to see whether students come up with these solutions, perhaps through a mixture of analysis, judicious trial and improvement, and a bit of luck.