[Note: GERMAN versions of the Week 10 tasks are available HERE. I welcome suggestions on how to improve the translations!]
This week we look at the Stretch transformation and the effect it has on lengths and angles. The stretch is like an enlargement of the plane, except that the plane is enlarged in only 1 direction. The length of a line-segment aligned with the direction of the stretch is enlarged by the stretch's scale factor. The length of a line-segment perpendicular to the direction of the stretch remains unchanged.
You might not want to pursue the properties of a stretch in too much depth, but the way it differs from an enlargement might help students' to develop a better awareness of the latter transformation's properties.
Monday: Here we apply a stretch, with a simple scale factor of ×2, to a zig-zag line. Unlike an enlargement, the effect is to change some of the angles of the zig-zag.
It is worth noting that the effect of a stretch is uniform. For this particular stretch, with a horizontal scale factor of ×2, the horizontal distance between any two points in the plane (or, in our case, on the elastic strip) is doubled, while the vertical distance remains unchanged. It is possible that some students will think additively. For example, they might notice that the vertex forming angle a has moved 4 units to the right and so will want to move the vertex forming angle b the same distance.
It turns out that angles a′ and b′ are both right angles. In Tuesday’s task, we ask students to explain how we can tell that this is the case.
Tuesday: Here we see the full result of Monday's stretch. It turns out that angles a and b have both become right angles. However, this doesn't necessarily mean that a = b.
We can make use of the grid lines to show that a′ = b′ = 90˚. Unfortunately, the grid lines are not of much help when it comes to comparing angles a and b, though of course we could use a protractor to show that they are not equal. The task of proving why these particular angles need not be equal when their images a′ and b′ are, is much more difficult.
A key idea that you might want to touch on is that a stretch affects angles differently and that this depends not just on their size but their 'orientation'. You could point out, for example, that the two 45˚ angles in the top left corner of the strip have been transformed in different ways - one has got larger, one has got smaller. Might there be an orientation where a 45˚ angle stays (at least roughly) the same?? One angle that can clearly stay the same is the right angle - for what orientation does this happen?
Wednesday: It might be quite challenging for some students to visualise the desired stretch here. The scale factor is not ×2 this time, though drawing a ×2 stretch might help students see that this takes things too far.
A good strategy here is to focus on angle a. Its arms have a slope of 2 horizontal, 3 vertical. Can we change this to 3 horizontal, 3 vertical?
Thursday: This shows the solution to Wednesday's task. It might be worth reflecting on the result. Notice, for example, that angles b and c have both got larger - which may not be that surprising. But angles a and d have both got smaller, which may be less obvious. Notice also that the parallel lines in the original diagram stay parallel, and so angles c and d still sum to 180˚ after the stretch. What other transformations have this property?
Angles a', c' and d' are right angles. Can students use the squared background to explain why?
The two transformed purple line-segments happen now to be equal. However, they have different-sized horizontal components. So if they were to be transformed back to what they were (using a horizontal scale factor of ×⅔), their lengths would be transformed by different amounts. [It might be easier to visualise this if one thinks of each purple line as the hypotenuse of a right-angled triangle whose other sides are horizontal and vertical - what happens to each triangle?]
Friday: In the diagram, we can think of the slanting lines as forming five isosceles triangles, each with a horizontal base sitting on the lower horizontal edge of the elastic strip. Each triangle has a vertical line of symmetry and so will remain isosceles after a horizontal stretch. The angle at the top vertex of any of these triangles will become a right angle when the base angles are 45˚, which will happen when the base is 12 units long. The horizontal scale factor needed to do this will be simpler for some of the triangles than for others....
Students are asked to change one of the angles a, b, c, d, and e, into a right angle. Do they choose at random, or are they aware that some are easier to transform than others, and choose accordingly?
The scale factor for the five angles are, respectively, ×6, ×3, ×2, ×1.5 and ×1.2.
The answer to Q2 is No! However, it is challenging to explain why. One way to do this is to establish that the marked angles are all different, for example by considering alternate agles formed by a transversal and a pair of non-parallel lines. (It turns out that the angles can be put in order, from smallest to largest, like this: a, f ,b, g, c, h, d, i, e.) When the strip is stretched horizontally (with a positive scale factor), each angle will increase but they will still all be different, so if one angle becomes a right angle, none of the others will.
