Could This Be The Best Pythagoras’ Theorem Lesson Ever?
Pythagoras’ Theorem: Three Entry Points
So, you’re designing a lesson to introduce Pythagoras’ theorem to your pupils for the first time. Ultimately they’ll end up with it in algebraic form but this is a once in a lifetime opportunity to explore and reveal some deep lasting connections within shape and space and geometry. Simply telling or teaching them the equation is a far less richer route to learning than guiding them on a journey of discovery as if they were finding it for the very first time themselves… just like the great man himself did way back when in the late 6th Century BC. After all, the things we see or find out for ourselves are the things we remember most and will always stay longer in the memory than things we hear or read or get told by someone else. So here are three of my own favourite entry points which will have great visual impact, lasting mathematical value and hopefully deliver deeper understanding… before we sink our teeth into finding unknown sides of a right-angled triangle and ultimately solve problems using it in two and three dimensions. And anyway… what teacher or pupil doesn’t cherish the opportunity to indulge in a little bit of Higher Tier colouring-in once in a blue moon?
There are numerous geometric demonstrations and this one is near fool-proof since it begins with the primitive 3, 4, 5 Pythagorean Triple and can be accomplished using nothing more complicated than cm² paper, a ruler and scissors. The vertices of all the initial and divided shapes connect at easy to locate lattice-points and the lines of dissection are identified using parallel and perpendicular simple vectors so make sure you prompt your pupils to identify these essential features of the diagram for greater accuracy and also make some illuminating connections to other topics. The shapes numbered 1, 2, 3, 4, 5 don’t even need to be rotated to fully cover the green square… but no need to reveal that to them. The activity finishes with the theorem in algebraic and numerical form pupils have explored in a very tactile, creative and kinaesthetic way. I almost feel sympathy for Henry Perigal whose original dissection this is an improvement upon. His own dissection is engraved on his tombstone but is difficult to replicate and the intersection of his lines of dissection do not begin at easily identified lattice points on the precise and accurate cm² paper we enjoy and use daily in our classrooms. Evidently this is a modern invention unavailable to mathematicians of days long gone.
Reasoning features more prominently within the new National Curriculum and constructing a chain or process of mathematical arguments leading to an algebraic proof will be something pupils have to become well versed in if they are targeting high grades. This easy to follow algebraic proof again begins by constructing the primitive 3, 4, 5 right-angled triangle. Four congruent copies are produced which enclose a 5×5 square all of which lie within a 7×7 grid on cm² paper. Again parallel and perpendicular vectors of equal length appear which enable us to verify that the red shape is indeed a perfect square. Naturally we assume a number of prior skills as expanding brackets and evaluating the area of an algebraic triangle must be correctly performed before the final expression is simplified and the final equation arises. As with the first activity it’s not a demanding task to verify numerically the equation works since 5² = 3² + 4² and also for the five shapes within the large square 7² = 5² + 4(½ × 3 × 4)
The third exercise is the Spiral of Theodorus often simply referred to as the Pythagorean Spiral. Whoever claims ownership they are simply replicating a process Mother Nature has perfected over thousands of Millennia and she will continue to demonstrate mastery of this mathematical function long after all our names are lost in time. I present this entry point last since it requires Pythagoras’ theorem to identify the length of each contiguous right-angled triangle’s hypotenuse but these lengths can be seen to progress at predictably increasing length. By presenting this task on (x, y) axes, pupils can follow each step as the model indicates the coordinates of each vertex to 1 d.p. History records that Plato, tutored by Theodorus, questioned why he stopped at √17. The reason is commonly believed to be that the √17 hypotenuse belongs to the last triangle that does not overlap the figure. But why should our pupils stop there? The interactive model continues to… a number which I won’t reveal here for fear of spoiling the surprise. You may ask pupils to guess as the activity progresses what number triangle results in the spiral a) overlapping itself and b) exceeding the boundaries of the coordinate grid. If you want to be really mean to your pupils you may wish to hold back the accurate coordinate information and let them get on with it without any further input once the first three triangles have been constructed and watch gleefully as minor errors in their construction become compounded in a case of what one might call Mathematical Chinese Whispers. As with all three activities, how you use them, what assistance you give your pupils, what commentary and exposition you provide as the tasks progress is entirely up to you. Perhaps this is an excellent case of less is more. It may lead to you reflecting if this model of introducing new teaching content is one you wish to emulate, adapt and evolve to suit your own teaching style and importantly the learning process your pupils best respond to. Enjoy!
Matt Dunbar is the author of Trinity Maths. You can download the Interactive Excel Spreadsheets and Pupil PDF Templates which demonstrate and support these three activities FREE from www.trinitymaths.com