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Fitting In

The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest equilateral triangle which fits into a circle is LMN and PQR is an equilateral triangle with P and Q on the line LM and R on the circumference of the circle. Show that LM = 3PQ

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Look Before You Leap

The diagonals of a square meet at O. The bisector of angle OAB meets BO and BC at N and P respectively. The length of NO is 24. How long is PC?

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Two Ladders

Two ladders are propped up against facing walls. The end of the first ladder is 10 metres above the foot of the first wall. The end of the second ladder is 5 metres above the foot of the second wall. At what height do the ladders cross?

Folding Squares

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Why do this problem?

This problem begins with a simple paper fold, which leads to an elegant result. Learners need to estimate lengths and work accurately to construct and measure lines. Once they have made a conjecture it can be justified with geometrical arguments using angles and ratio. To prove the results knowledge of similar triangles is needed, but the problem could equally well be used as an introduction to the idea of similarity.

Possible approach

Start by giving everyone a square of paper which they can fold according to the diagrams. Encourage them to measure as accurately as possible the two sections of the diagonal which are formed. Collect together some measurements and ask what the learners notice. What do they think would happen with squares of different sizes? In order to collect a lot of data in a short space of time, small groups could create different sized squares and construct the lines and measure them, sharing the results within their group.

Once a pattern has emerged bring the class together and ask what they have found. Some learners will be convinced that the relationship will always hold because they have tried it with lots of examples, so there is a good opportunity to discuss the distinction between demonstration and mathematical proof.

In order to justify formally what they have noticed, learners might find it useful to discuss what they know to be true, and what a convincing argument shows to be true. This might be for example that certain angles are equal.
Once pairs of equal angles are identified a route to a proof becomes clearer.

The last part of the question asks about other quadrilaterals. Some learners will be secure enough in the proof for squares to attempt to prove or disprove the cases for other quadrilaterals straight away without the need for accurate diagrams or folding.

Key questions

What does it look like the folds are doing to the lines?
How could you verify what you think is true?
What other mathematics do you know that might be useful here?
Does this work for other quadrilaterals?

Possible extension

The problem Take a Square II builds on these ideas and extends them to different fractions of the line.

Possible support

The problem Two Ladders could be used to investigate similar triangles.