Surds is a topic which always raises a titter on Twitter and some clamour in the classroom; I’m sure my pupils are chuckling because it rhymes with nerd and not for reason of any other aural similarity! Building pupils’ knowledge and conceptual understanding as they make their journey through square numbers and square roots, estimation and accuracy and the efficiency of surd form, and on to rational and irrational numbers, requires that they follow a route that needs to be logical and meaningful; otherwise they stumble around this odd topic and at best learn some procedural rules, but are left still questioning “why am I working in surds?” and “what even is a surd?”.
Knowledge of square numbers and square roots, identifying square factors and use of a calculator are all pre-requisite skills, but how should we develop deeper understanding and not just reasonably confident procedural manipulation of surd form?
One starting point is to investigate Pythagoras’ Theorem and the understanding of square root form. Using the NRICH Tilted Squares session is one way to deepen understanding of squares and square roots. This is part of a series called “Dotty Grids - an opportunity for exploration” from NRICH which does exactly what is says on the html tin. A recent find which looks to be a very rich resource area is Geogebra Tube. Search for “Pythagoras Theorem” and you will be deluged with resources and activities to try out.
A favourite activity of mine has to be the Spiral of Theodorus, which builds up a seashell-like shape using square roots. This shape can then be used in various art works and thereby could help pupils develop a love of and understanding of the maths in nature. It’s well worth spending time on a search for Spiral of Theodorus and incorporating this into some classroom display. The artwork that your pupils create could be very original and clever in design.
Locating square numbers on a number line is a worthwhile activity: by first identifying where the squares for lie on a number line, pupils can then work forwards and backwards to estimate where lies, lies, etc. With a calculator, pupils can compare their estimates of the squares with the true values, and also vice versa: ask them what they notice when they work to one decimal place, comparing e.g. and both rounded to one decimal place. An investigation of this kind will make clear the value – and efficiency – of the surd form,
The manipulation of surd form can be very well practised with Tarsia puzzles. There are many good lesson ideas at resourceaholic.com too, and within the NCETM Departmental Workshop on surds there are suggested activities from the NCETM, including the history of surd use, and suggested activities.
Once your pupils become more proficient at manipulating surds, they could respond to this Inquiry Prompt from Rachel Horsman. The mathematical notes on the website provide plenty of rich and challenging surd questions to the point where your pupils will be doing a triple back somersault with confidence and grace!
And if ever there's a moan that surds are dull, show your pupils the highly addictive Angry Surds (love the pun!) game.
Page header by Alan Levine (adapted), some rights reserved
Theodorus Spiral in the public domain, courtesy of Wikimedia Commons