6..3.2 Understand and use rotations (4)
6..3.3 Understand and use reflections (4)
6..3.4 Understand and use enlargements (1)
In this set of tasks, students are asked to complete a larger version of a wire shape that is placed in a different orientation. Thus the tasks could be said to involve rotation, reflection and enlargement, but without the need to find centres or mirror lines.
There are two broad strategies for solving the tasks, which might be termed local and global, and it is interesting to see which one the students favour.
Monday: In this task, a wire shape has been doubled in size, turned through 90˚ and flipped over. Thinking globally, we can, for example, say that each segment of the larger shape is perpendicular to the corresponding segment in the smaller shape. Thinking locally, we can, for example, say that end point A' and the other endpoint (which we might want to call F') are the same distance from B' and directly opposite.
It might be helpful mentally to label every vertex of the shape, so that it can be referred to as ABCDEF. We have not done so on the diagram itself, to avoid obscuring the shape.
Tuesday: Here the figure has been rotated through 135˚ and a line the length of the side of a grid square has become the length of the diagonal of a grid square (an enlargement with scale factor ×√2).
Wednesday: Here the side of a unit square is mapped onto the diagonal of a 1 unit by 2 units square (an enlargement of ×√5). The images of vertical and horizontal lines are composed of such diagonals, but what about the image of the slanting segment EF?
It is a nice challenge to compare the lengths of EF and E'F'. Is E'F' √5 times as long? It is also nice to find a way of confirming that angle D'E'F' is 45˚.
If we place our shapes onto the Cartesian plane, and image that, say, A and A' are both at the origin, as in the diagram below, then the original plane, as represented by the green grid lines, is transformed into the plane represented by the brown grid lines. Further, we can represent the transformation by the matrix shown beneath the diagram.
[Note: We are not suggesting that this treatment would work with students at Key Stage 3, but it might be of interest to, say, A Level students.]
We can take this further still, by breaking the transformation into two recognisable parts: a rotation of 116.6˚ anticlockwise [given that tan⁻¹(0.5) ≃ 26.6˚] and an enlargement with scale factor ×√5. In turn, these can be represented by the matrices below:
Thursday: A new and simpler shape, except that one of the segments, CD, is at a relatively strange angle - not horizontal, not vertical, not at 45˚. When CD is rotated through 45˚, what is its new orientation?!
Having found the image of the whole shape, it is again a nice challenge to compare lengths and relate them to the scale factor, and to verify that angle BCD has been conserved. [It might again be useful to put the original shape inside a 'box' - in this case a 1-by-3 rectangle.]
Friday: Here we use Thursday's relatively simple shape again, but rotated through an obscure angle. The orientation of the image of CD may come as a surprise!
