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Square Areas

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Isosceles Triangles

Stage: 3 Challenge Level: Challenge Level:1

Why do this problem?

Paul Andrews, a respected mathematics educator based at Cambridge University, explains why he likes this problem :
"I first saw it used as a homework question more than ten years ago with a year seven class (11 year old students) in Budapest and have used it with every group I have taught since - initial teacher training, mathematics education research students, in-service; everyone gets it. I love it because it forces an acknowledgement of so many different topics in mathematics and is sufficiently challenging to keep almost any group meaningfully occupied within a framework of Key Stage Three mathematics content. This, for me, is the key to a good problem; Key Stage Three content alongside non-standard and unexpected outcomes. For example, notwithstanding the obvious problem solving skills necessary for managing such a non-standard problem, I think it requires an understanding of coordinates, isosceles triangles and the area of a triangle. It requires an awareness of the different factors of 18 and which are likely to yield productive solutions. It requires, also, an understanding not only of basic transformations like reflection and rotation but also an awareness of their symmetries. Moreover, the solution, which is numerically quite small, is attainable without being trivial. In short, I love this problem because of the wealth of basic ideas it encapsulates and the sheer joy it brings to problem solvers, of whatever age, when they see why the answer has to be as it is. It is truly the best problem ever and can provoke some interesting extensions."

Possible approach

This printable worksheet may be useful: Isosceles Triangles.

The richness of this task might best be exposed by working on the problem in small groups. The groups should be encouraged to keep asking the questions: are our solutions valid? have we found all of the solutions?

Once groups feel that they have finished, they could try to explain clearly to the class why they have found all of the solutions. Those who missed out solutions could be encouraged to think about why these solutions were overlooked.

Key questions

  • How do you know that your triangles have the correct area?
  • Are there any more like that?
  • Can you explain why you are certain that there are no more solutions?

Possible extension

Repeat for isosceles triangles of areas $50$ and $15$. Which starting areas lead to more/fewer solutions? Is the number of solutions predictable?
The problem doesn't insist that one edge is horizontal/vertical - what new triangles can be found, with all sides sloping? Can you prove that you have got them all?

Possible support

The problem can be used as a context for practising: drawing isosceles triangles, using co-ordinates accurately, calculating areas, communicating results, working with others, etc.

When the students have worked out the basic possibilities for the isosceles triangles, they could cut them out to help to search for congruent solutions.