WEEK 16

 

6.3 Transforming shapes

6..3.3 Understand and use reflections (2)

This week's tasks mainly involve reflection, mostly of everyday objects, but they require fairly careful analysis.

Monday: This task could be said to involve enlargement as well as reflection. We can think of the drawing on the mirror's surface as an enlargement, scale factor ×0.5, of the image seen in the mirror, with the centre of enlargement at the viewer's eye. This means the drawing is the same size, regardless of the viewer's distance from the mirror (as long as they can reach the mirror to draw...).

 

You might want to try this, or a similar activity, in the classroom, if you can find a suitable reflective surface. Rather than a person's face, you could use a more easily measurable object placed next to the viewer's face, such as a book or a 30 cm ruler.

Tuesday: The clock shows half-past-two. Some students might over-compensate for the reflection and decide on half-past-three.

In 10 minutes' time, the clock will look something like this (note that the hour hand has moved slightly too, in the same direction as the minute hand):

Wednesday: The story in this task suggests that the word MATHS is written on the outside of the door. If that is the case, the word will look like mirror writing when viewed directly from inside the room, but not if it is viewed through some kind of mirror.

Thursday: This task takes quite a lot of coordinating! First, let us assume the bus is moving so smoothly it conveys no physical sensation of movement to Vera (if she shuts here eyes). Vera is facing forwards but the images she sees on here phone's screen seem to be receding into the distance in front of her, so it looks to her as if she is moving backwards at the same speed as the bus is moving forwards.

Things get more complicated if Vera can also see objects directly, out of the corner of her eye. These objects seem to be moving backwards at the same speed relative to Vera as she thinks she is moving. So if, say, her bus is moving forwards at 30 mph and her screen tells her she is moving backwards at 30 mph, then these objects would appear to be overtaking her by going backwards 30 mph faster. This might make sense if all she was seeing was stationary buses pointing the other way, or a line of railway carriages, but a bit unsettling if they were stationary objects like houses.

Friday: This is an example of the classic Heron's Problem. It is similar to task 08E but more general in that points A and B are not equally distant from the blue line.

Some students may feel intuitively that the desired point P will be towards the left-hand end of the segment of the blue line shown in the diagram. Others may well choose the midpoint of the segment. An elegant way of solving the task is to consider the shortest distance between point A and a point B', where B' is the reflection of B in the blue line.