WEEK 17

 

6.3 Transforming shapes

6..3.2 Understand and use rotations (3)
6..3.3 Understand and use reflections (3)

This week's tasks involve reflection and rotation, but not of everyday objects; rather, the tasks are based on items from the CSMS Reflection and Rotation test that I developed.

Monday: This task looks rather unprepossessing, but it has proved to be quite challenging for some students. In part, this might be because students are rarely confronted with slanting mirror lines, other than the occasional line at 45˚ to the horizontal and vertical.

This task appeared on the CSMS Reflection and Rotation test, which was given to a representative sample of 449 Year 9 students back in the 1970s. Their responses, which were sketched freehand, were coded according to the scheme below. We devised some clearly defined rules for deciding whether, for example, a response was deemed to be correct or merely adequate. Of course, one could have come up with different rules. Suffice to say, about two-thirds of the sample (2% + 21% + 26% + 16% = 65%) had at least a fairly good sense of where the image should go and what kind of slope it would have. However, a sizeable minority of students (about one quarter of the sample) thought that the vertical flag's image would also be vertical, with some (10% of the sample) placing the image in roughly the right place (code 5) but others (14% of the sample) placing it horizontally (or vertically) across from the object (code 7).

Tuesday: In this task we ask students to assess a typical code 7 and code 5 response (see above). It is interesting to see whether students can identify the features captured by these codes and, if so, how they evaluate them.

Wednesday: This is another item from the CSMS Reflection and Rotation test. We found that 20% of the Year 9 sample answered this correctly, with roughly another 40% drawing the image with the correct slope but in the wrong position. It was rare for students to draw the image with one or other end-point in the correct position but with a wrong slope.

Thursday: This shows some responses to Wednesday's task. We found that it was quite common for students to place the image such that the base-points of the object and image were symmetrical about the centre of rotation, as in these three examples. Students who did this, usually drew an image with the correct slope (22% of the total sample), as in the case of Cleo and Dev, with a few (3% of the total sample) getting the slope wrong too, as with Eze.

Friday: This task is based on another item from the CSMS test. The original item was answered correctly by 14% of the Year 9 sample (using the criterion for 'correct' that we had devised), with another 25% giving a point roughly equidistant from the flags (like Hui, below) but not one that would give a 90˚ rotation. The point drawn by Gina, below, was particularly popular (21% of the sample) as was the 'symmetrical' point drawn by Flo (10%).

Note: It is interesting to consider the role that tracing paper can play in rotation tasks. Tracing paper could be useful in Task 17C for giving a sense of how the flag moves, although students would still need to determine where the movement stops. Its usefulness in 17E, say, to help locate a centre, is far less clear cut. It would provide an effective way of testing whether a chosen point is correct; however, it won't necessarily give useful feedback beyond that: if a chosen point turns out to be wrong, it can be difficult to see where an improved choice might be - so more a case of trial and error upon error than trial and improvement.

Extra: Here are two further tasks that might be interesting to try with students. They are designed to assess whether students have some sense of the idea that when an object is rotated through a given angle, the whole plane of which it is a part is rotated through that angle. In the case of the disc stuck to the wheel of a scooter (Task 17F), when the wheel turns through 90˚, so does the disc, even if we don't know its precise location. Similarly, if a piece of card is rotated (Task 17G) so that an arrow drawn on the card becomes horizontal, a second arrow on the card parallel to the first will also become horizontal.