Ensuring that pupils see alternative views of the same family of objects is an important feature of good teaching. We need to ensure that our learners experience what is sometimes called a ‘rational’ set of examples that highlight nonprototypical aspects (Ryan and Williams, 2007). In other words, examples we consider and problems that we meet need directly to challenge misconceptions and develop a deep understanding of geometrical forms.
In considering characteristics of a rectangle, this set would challenge some students’ thinking:
Other useful ideas include encouraging pupils to rotate their work (or themselves), imagining a final product, or including additional components in a diagram to search for known objects that otherwise may be hidden.
Now try this: How would one find the area of each square in the diagram below?
Some pupils say it’s impossible because no measurements are included. Some choose a numerical value for a dimension of a shape and work through the problem.
Others label the radius of the circle r and work through the squares separately.
The large square has a width (and therefore height) equal to the diameter of the circle. Hence its area is 2r x 2r = 4r^{2}
The smaller square …?
This is where these pupils often start to scratch their heads. One pupil in particular went through the following argument:
The small square has diagonal 2r, which is also the hypotenuse of the isosceles right angled triangle.
So, applying Pythagoras’ theorem, the side length of the square, s can be found by solving;
(2r)^{2} = 2s^{2}
4r^{2} = 2s^{2}
√2 r = s
The area of a square is the square of its side length so the area of the square is 2r^{2}
2r is also the base of an isosceles right angled triangle whose height is r, so the area is a half of 2r^{2}. Since we have two of these the answer is 2r^{2}
I was genuinely impressed, but the pupils in question thought that the answer was too simple to warrant this train of thought.
Another then suggested simply rotating the central square 45º. Is it all a lot clearer now?
You may now realise that you really don’t need to think much beyond the fact that the small square is half the area of the large square. Not only is life so much simpler, but I would argue that this gives us a much more general answer that doesn’t require including measurements and variables; in fact the relationship between all three shapes can be derived no matter which shape has a dimension included.
Allowing pupils to adapt mathematical objects in a way that doesn’t affect important characteristics encourages careful consideration – this pupil knew that rotation left area invariant – and can result in some much simpler problems to solve.
SOMETHING TO TRY:
KS1: Which shapes are triangles? Why or why not:
(D Clements and J Sarama, Young Children's Ideas About Geometric Shapes, Teaching Children Mathematics, Vol. 6, 2000)
KS2: Is more of the rectangle blue or yellow? Why?
KS3: Which of the following are parallelograms?
KS4: Explain the Bride's Chair proof (Euclid's Proof) of Pythagoras' theorem:
http://www.cuttheknot.org/pythagoras/morey.shtml
KS5: How is a curve of a constant width constructed? E.g. A Reuleaux triangle
https://www.ics.uci.edu/~eppstein/junkyard/reuleaux.html
