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Some local pupils lost a geometric opportunity recently as they surveyed the cars in the car park. Did you know that car tyres, and the wheels that they on, are a rich source of geometry?

Mirror, Mirror...

Stage: 3 Challenge Level: Challenge Level:1

Why do this problem?

This problem could be used as an extension task once students have learnt to draw reflections accurately. It can provide a valuable mathematical challenge for students who are ready to move on while others in the class need more practice with their basic drawing skills.

Alternatively, this problem and the two related problems ...on the Wall , and Who Is the Fairest of Them All? could form a unit of work on combined transformations. All three problems ask students to consider the effect of combining two transformations, and then challenge them to describe the single transformations that produce the same results.

Possible approach

As an extension task, all that is needed is to provide the problem as a worksheet to a pair of students who make sense of it together. When they have established the combined transformation for one specific example, a teacher intervention may be appropriate, to move the focus to the general case - asking the key questions below.

With a full class, encourage different students to start with slightly different spacings of parallel lines and flag positions. The teacher intervention above could become a full class discussion. Students could regroup according to similarities/differences in the final combined transformations in order to acquire more information without needing to do a lot more drawing.

Key questions

What if the flag was in a different place?
What if you reflected in the other line first?
What if the lines were both at 45 degrees to the horizontal?
What if the lines were both at 60 degrees to the horizontal? (use isometric paper )
What, precisely, does the final position of the flag depend on?
Can you prove it?

Possible extension

Ask students to summarise their findings in exactly 20 words (!) - then ask what would happen if the lines were not parallel (ie. Move them on to ...on the Wall)
Ask (suitably experienced) pupils to create a dynamic geometry file that demonstrates their findings.

Possible support

Do some introductory reflection work - filling in missing halves of shapes, reflecting patterns in vertical, horizontal and diagonal lines, etc.
Try Hidden Meaning and Reflecting Squarely
Ensure students use square paper and encourage them to draw their reflections accurately.