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Can you work out the area of the inner square and give an explanation of how you did it?


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2001 Spatial Oddity

With one cut a piece of card 16 cm by 9 cm can be made into two pieces which can be rearranged to form a square 12 cm by 12 cm. Explain how this can be done.

Hidden Squares

Age 11 to 14
Challenge Level

Why do this problem?

Students often get hung up on shapes being oriented in a particular way. This problem involves squares arranged in all sorts of orientations, so students will need a secure understanding of the properties of squares. There is also the opportunity to practise working with coordinates in all four quadrants.


Possible approach

This printable worksheet may be useful: Hidden Squares

It might be useful to start by playing a few games of Square It so that students are challenged to think about squares which are not in the usual orientation.

Hand out the worksheet and explain the problem. Encourage students to work in pairs to find some of the hidden squares, and while they work, circulate to listen to discussions and see what strategies are emerging.

After students have had a chance to find a couple of squares, bring the class together to share strategies. You may wish to ask:
"If these two points are corners of the same square, where would the other two corners be?"
Or choose a student who you know has found a square that not many others have found yet:
"Tell me the coordinates of two adjacent corners of your square. Now, can everyone else work out where the other two corners must be?"

Next, give students some time to finish the first task and then apply their strategies to the second task using coordinates in all four quadrants.

A nice way to finish would be by going back to Square It to see if they are more successful now that they have had more experience looking for tilted squares. Further lessons might look at the problems Square Coordinates, Opposite Vertices, or Tilted Squares.


Possible support

Suggest that students start by looking for the three squares (four in the second problem) whose sides are parallel to the coordinate axes.

Possible extension

Square Coordinates encourages exploration of the relationship between coordinates of the vertices of squares.
Opposite Vertices challenges students to find squares given two opposite vertices.
Tilted Squares encourages exploration of the area of squares and leads to Pythagoras' Theorem.