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Reflecting Squarely

Stage: 3 Challenge Level: Challenge Level:1

Why do this problem?

This problem makes students think about the symmetries of many different shapes. There will be opportunities for instant feedback, and a challenge that will engage the whole class in an activity that offers choice and room for creativity. The students have to take responsibility for checking each other's work, since the teacher clearly can't be expected to anticipate all the possible combinations that might be generated.

Possible approach

This problem could easily be used as it stands, as one of many activities on reflection symmetry. It can also be expanded, leading into a richer task:

As students enter, have copies of the three pieces available (on an OHP/blu-tacked to the board/on desks - card shapes or multilink (interlocking cubes)). Ask the class to find a few arrangements that are symmetrical. Keep a record of correct solutions as they are suggested.

Give students time to work individually/in pairs trying to find all the others. After a while students could be invited to draw new solutions on the board. Say that students have found only six arrangements, suggest that the set of three shapes is called a '6-ways-set'.

"Now it is time to 'beat the problem'. You can design your own three shapes, like the original, all made from squares on a square grid, with a total area of 10. Do you think we can find 3 shapes which can be put together symmetrically in more ways than the original problem? i.e. we're looking for a 7-ways-set, an 8-ways-set... (If anyone finds more symmetrical combinations for the original problem, then this task becomes even more challenging!)

"Between us, can we find a complete collection: a 0-ways-set, a 1-ways-set, a 2-ways-set... ?"
"When you have designed a set, and think you have found all the symmetrical arrangements, draw them clearly and stick your work to the board, for others to check." The board could be prepared with headings: 0-ways-sets, 1-ways-sets, 2-ways-sets... Ask all students to take responsibility for checking at least one displayed solution and confirm that it is in the right category.

Keep the work on display at the end of the lesson, so that students (from this class or another) can add to it over the next couple of weeks.

Key questions

Where can the mirror lines be?
Is there a systematic way of checking that you've found all the arrangements?


Possible extension

Students might like to look at Andrei's solution and try to understand the logic behind his approach.

Encourage students to find three shapes that have very few possible arrangements (or none), and/or more than anything found so far.

Possible support

Provide mirrors, and/or scissors so students can cut out their arrangement of shapes and fold them to check potential mirror lines.