Let Us Reflect
1 out of 3
Where can you put the mirror across the square so that you can still "see" the whole square? How many different positions are possible?
Problem
Let Us Reflect printable sheet
You will need a mirror for this activity.
Here is a square:
Where can you put the mirror across the square so that you can still 'see' the whole square?
How many different positions are possible?
How many lines of symmetry does a square have?
Can you reflect part of the square so that you can see a smaller square?
A rectangle? A kite? A hexagon? An octagon?
What do all the shapes have in common?
This problem has been adapted from the book "Starting from Mirrors" by David Fielker, published by BEAM Education, which is now out of print.
Getting Started
You might find it useful to print off this sheet of squares.
Try experimenting by placing the mirror across the middle of the square to start with then move it gradually in one direction to see what happens.
Try a different starting position now - perhaps still across the middle, but at a different angle.
Student Solutions
Ruth has sent in some ideas for the first part of the question. She says you can put a mirror across the middle or diagonally from corner to corner to see the whole square. She sent these diagrams:
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Thank you for writing to us Siobhan, Ruth and Gemma. Well done!
Teachers' Resources
Using NRICH Tasks Richly describes ways in which teachers and learners can work with NRICH tasks in the classroom.
Why do this problem?
This problem enables children to explore the line symmetry of 2D shapes. The practical nature of the activity means that experimentation is possible and therefore it is readily accessible.
Possible approach
Ideally, children should each have a mirror for this activity. Being able to move the mirror yourself and seeing the effect is much more powerful than watching someone else doing it.
You could introduce the first part of the problem orally, giving children a chance to experiment with their mirrors. Printing off sheets of squares may be useful. Once one way is shared between the whole group, challenge pairs of children to come up with all the other ways of seeing the whole square. Record them on the board as lines on the square and ask the children what they notice (the lines all correspond with the square's lines of symmetry). The rest of the problem can be tackled by the children at their own pace.
You could have some square templates on the board, one labelled rectangle, one kite etc and learners could come to the front to mark on a line which works. Encourage them to make generalisations about where the mirror must be placed in each case.
Key questions
Where are the lines of symmetry on a rectangle/kite/hexagon/octagon?
How does this help you think about where to place the mirror on the square?
Possible extension
Learners could investigate which other 2D shapes they can 'see' using their mirror. Which are not possible and why? What would happen if the shape used was different? For example, if it was a triangle rather than a square?
Possible support
Having plastic/wooden shapes available to remind children of their properties might be helpful. Other learners might appreciate having some rough paper or a mini-whiteboard to sketch on as they tackle this problem.