One of my happiest memories of learning mathematics at secondary school was applying Pythagoras’ Theorem to calculate the lengths of various ladders that had been leant against a wall. There was so much involved in solving the problem: sketching the diagram, manipulating the formula a2+b2=c2, substituting the known values in, using a calculator to evaluate the result and finally rounding to the required accuracy. I remember this process making me feel a sense of satisfaction and enjoyment. However, it was not until teaching it myself, many years later, did I realise what I was missing out on.
As a youngster aiming to pass an exam it was not required or even expected of me to understand the origins of the theorem. I simply had to have good recall of a2+b2=c2 with c the longest length. When I teach Pythagoras’ Theorem now I think back to my experiences and try to deepen my students understanding by showing the origins of the formula through experimentation and deriving it using algebraic proof.
I use an interactive Geogebra file firstly to illustrate that changing the lengths of the vertical and horizontal sides has a direct effect on the magnitude of the hypotenuse and secondly to provide Pythagorean Triples for our investigation. As a class we take the measurements for each side of various right-angled triangles in the hope of identifying how the three lengths maybe connected. However, we soon realise that looking at the lengths alone will not be sufficient to elicit any connection between them.
The class then works in small groups to discuss where such a connection may indeed lie. After a while one group will inevitably decide it maybe worthwhile to investigate the area of a square drawn on each side. To aid their experiment they only use Pythagorean Triples. Eventually, the students begin to recognise the sum of the two squares on the shorter two sides equals the area of the square on the hypotenuse side. We test this hypothesis using right-angled triangles with integer lengths and then proceed to any right-angled triangle. Whilst there can be an issue with accuracy the students come to the conclusion that their hypothesis is correct.
At this point the group work is used to encourage the students to come up with a clear and concise definition of Pythagoras’ Theorem using their own words. A nice YouTube video that I found illustrates their hypothesis very nicely. It is important at this stage to explain to the class the difference between a demonstration and a proof.
There are many ways to prove Pythagoras’ Theorem. My favourite is have the class cut out four congruent right-angled triangles of base ‘b’ and height ‘a’ and arrange them to create a square where the hypotenuse side, ‘c’ is the length of each side. Strangely, this simple arrangement of triangles often turns out to be more complicated than expected. The class is then challenged to derive the algebraic area of the square taking into account the four congruent triangles and the smaller inside square. Hence, Pythagoras’ Theorem is proven.
The main teaching point for this lesson is not the algebraic proof nor the experimental demonstration but the concept that this a2+b2=c2 formula does not simply exist but arises from known mathematical principals that are accessible to GCSE grade C students.
Click here to download the resources used in this lesson.
Click here to watch the proof of Pythagoras Theorem.
How do you teach Pythagoras’ Theorem?
Do you know of any other proofs or YouTube demonstrations?
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