It Stands to Reason
In this regular feature, an element of the mathematics curriculum is chosen and we collate for you some teaching ideas and resources that we think will help your pupils develop their reasoning skills. If you’d like to suggest a future topic, please do so to email@example.com or @NCETMsecondary.
How do you introduce Pythagoras’ Theorem to your pupils? Most adults remember something about a square and that word beginning with h, but few can demonstrate complete mastery of the concept. Other adults know that having a loop of rope with 12 equally spaced knots can be used to find a right angle if it is arranged to make a triangle with sides 3, 4 and 5 … though I’ve never met anyone who has such a carefully knotted rope to hand when it’s needed …
You may like to introduce the ideas of Pythagoras using the NRICH task Tilted Squares. Pupils are encouraged to work systematically to find the areas of squares as they progressively ‘tilt’ on a square grid:
By considering how to describe the ‘tilt’ mathematically, the right angle triangle of which these squares are the ‘square on the hypotenuse’ can be made evident, and this helps pupils to suggest an appropriate connection. To complement this task, you could show a tiling pattern that is based on the theorem:
There are a number of dissections that can demonstrate Pythagoras’ Theorem; that of Perigal is on his gravestone! For some pupils, this video clip reaches the parts of their brain that other demonstrations fail to do.
However, pupils need to have deep conceptual understanding of the generality of the theorem: dissections and tiling patterns show a specific example each time. Dynamic Geometry is a good tool for demonstrations which reinforce generality. From this, pupils can develop for themselves a proof of Pythagoras’ Theorem. Pythagoras Proofs suggests alternative ways of deriving a proof, which can be scaffolded appropriately so that all pupils can access them.
- Triples and quadruples: from Pythagoras to Fermat enables pupils to experiment with generating Pythagorean triples and quadruples;
- Focus on Pythagoras draws together ideas from research and lesson observations;
- The Napkin Problem will liven up the school canteen;
- and, finally, you can display what your pupils have created on a Pinterest board similar to this one.
Page header by Juanedc (adapted), some rights reserved