Copyright © University of Cambridge. All rights reserved.
'Isosceles Triangles' printed from http://nrich.maths.org/
Why do this problem?
Paul Andrews, a respected
mathematics educator based at Cambridge University, explains why he
"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."
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.
- 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
Repeat for isosceles triangles of areas $50$ and $15$. Which
starting areas lead to more/fewer solutions? Is the number of
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?
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.