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Inside Seven Squares

Stage: 2 Challenge Level: Challenge Level:2 Challenge Level:2

Inside Seven Squares

Seven squares are set inside each other. The centre points of each side of the outer square are joined to make a smaller square inside it and so on.
seven squares set inside each other
The centre square has the area of $1$ (one) square unit.
What is the total area of the four outside triangles which are outlined in red?

This problem is also available in French: Les 7 carrés

Why do this problem?

This problem provides a challenge when you are focusing on drawing lines and shapes accurately, or finding areas either by counting squares or calculation. The problem calls on learners' understanding of squares and right-angled triangles, and requires working systematically, as well as visualisation and reasonably accurate drawing.

Possible approach

You could introduce this problem by making the beginnings of the design using paper folding. Each learner will need a square of paper. (This could be made in class by folding the end of the sheet diagonally, thus forming a square and cutting off the excess.)

Take the square of paper and fold it both ways diagonally so finding the centre. Fold each corner to the centre and crease it. Thus forming the second square (in red).

After this, fold each side to the centre and crease, forming the third square (in green).

Learners could be challenged to continue this process which becomes increasingly difficult and prone to inaccuracy. This introduction will show learners where the problem "is coming from".

Provide $1$ cm$^2$ paper, or even better, square dotty paper, sharp pencils and rulers for working on the actual problem. Learners would benefit from working in pairs so that they are able to talk through their ideas with a partner but should do their own drawing. This sheet, showing the diagram, could prove useful.

Learners could be left on their own to discover the best place to start the drawing and the best size to choose, or the whole group could discuss this together before they begin working individually.

At the end of the lesson learners can discuss both their methods for tackling the drawing and for finding the area of the four triangles. There are several different ways of answering the problem and it can be done without any drawing at all, so there should be plenty to talk about. It might be useful, if no one has suggested it, to cut out the four triangles and put them together as a square.

Key questions

What would be a good way to start, with the first, smallest square or the largest one?
How big should you draw the centre square to make it easier to draw the others?
Is that a $45^o$ angle?
Now you have done the drawing, how are you going to find the area of the four triangles?

Possible extension

Learners who find this problem straightforward could find the area of each of the successive squares in the diagram and predict the size of the next squares, or try the problem Fitted, and/or Baravelle.

Possible support

Children could start with a $4 \times 4$ square and then draw a second square at $45^o$ to this with each side centred on a corner of the first square. Each side of this second square will be placed diagonally across two squares. A third and successive squares can be drawn in the same way.