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### 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.