It Stands to Reason
In this article we explore a set of activities through which pupils can deepen their conceptual understanding of, and increase the fluency of their reasoning about, similarity. Similarity is one of the five key geometrical ideas (the others are symmetry, invariance, transformation and congruence) that should be used to support pupils to build sound spatial and geometrical reasoning skills (ref. Key Ideas in Teaching Mathematics, Anne Watson, Keith Jones, Dave Pratt). As Tony Gardiner explains in Teaching Mathematics at Secondary Level, deductive reasoning in geometry is based on three organising principles: the congruence criterion, the parallel criterion and the similarity criterion, the latter of which allows pupils to deal with ratios, scaling and enlargement. It is best taught after the basic consequences of congruence and parallelism have been explored, and once pupils are confident when working with ratios and reasoning multiplicatively.
A pupil draws a 2-by-1 rectangle …
… and ‘grows’ it, stage by stage - click the image above to view the full sequence (PDF)
To proceed from one stage to the next, a new rectangle is created from the current stage, drawn such that the shorter side of the “new rectangle” is the same length as the longer side of the current whole rectangle, and the side-lengths of the new rectangle are in the ratio 1 : 2. For example, at Stage 2 above the longer side-length of the current whole rectangle is 8. To create the Stage 3 diagram the pupil now visualises one of those two sides of length 8 as the shorter side of the next rectangle that she is about to create. Because she wants the side-lengths to be in the ratio 1 : 2, while she is drawing the longer sides she has in mind that she has to keep going until they are of length ‘16’ (2 × 8). She draws the longer sides perpendicular to the side of length 8 in the direction that will make the new rectangle ‘cover’, or include, the whole existing rectangle …
The selection, out of the two possibilities, of the longer side of the previous whole rectangle to become the shorter side of the new rectangle (and the direction of the ‘extension’) at each stage proceeds clockwise. Pupils will probably realise that at each stage they can create the whole new rectangle by merely appending (‘sticking-on’) a new rectangle to the other longer side of the current whole rectangle. It is hoped that pupils will also notice for themselves that the ratio of the side-lengths of every appended rectangle is always the same, although not this time 1 : 2. It is important that pupils both see, and endeavour to explain, the following: how does a rectangle with sides in the ratio 1 : 2 plus a rectangle with sides in the ratio 2 : 3 combine to make another rectangle with sides in the ratio 1 : 2?
This recursive process is fairly straightforward for pupils to carry out if the ratio 1 : 2 is changed to any other ratio 1 : n where n is a whole number, but becomes more tricky with a ratio such as 2 : 3, e.g.
Whatever ratio is chosen, and whichever stage you stop at, the figure is always an interesting combination of overlapping rectangles that are all similar to the outside rectangle, and distinct rectangles that are again similar to each other, and your pupils can reason about the relationship between the two ratios
In devising, and then carrying out, this process, the pupil is experimenting with a recursive rule and the idea of ‘same shape’. The pupil is using the spatial aspects of geometry (spatial thinking and visualisation) which, although distinct from, are very closely entwined with the deductive aspects of geometry, in particular deductive reasoning from geometrical axioms. Pupils need to be clear about when they are experimenting and conjecturing, and when they are working deductively. The recursive rules explored in this article lead to pupils creating mathematically similar shapes, which then provide contexts and opportunities for both spatial thinking and deductive reasoning. Whether the recursive rules are given to them or made-up by them, your pupils will encounter and use, again and again, the similarity criterion, which they should become confident to express conventionally as
two polygons, … and …. are similar if
- corresponding angles are equal, , etc
- and corresponding sides are proportional, : : etc.
The following examples show how the spatial thinking involved in creating the images goes ‘hand-in-hand’ with deductive reasoning: engaging successfully in one supports successful engagement in the other.
Starting point: an isosceles triangle.
Recursive procedure: from the centre-point of the side between the equal angles of any existing isosceles triangle draw line segments parallel to the other two sides to meet the other two sides at points that you then join with another line segment.
The next five images (from left to right) show the first stage, and part of the second stage, of this procedure.
Pupils can decide for themselves how many times they will apply the procedure. Invite them to describe relationships between shapes in their image – for example they may pick out ‘same shapes’ that are not the same size (similar, non-congruent, shapes). They should realise for themselves that labelling points makes it easier to identify particular shapes - although they could also use colours, for example:
It is important to challenge pupils to use mathematical facts that they know to justify their assertions that particular shapes are the ‘same shape’. As usual, encourage discussion and constructive challenge: does pupil B agree with pupil A’s assertion? Can pupil C refine and improve pupil A’s justification. Does pupil D think that A, B or C has given the clearest reasoning? And so on.
What happens if the lines parallel to sides are drawn from the midpoint of one of the equal sides? Why?
Does the same procedure when applied to ANY (scalene) triangle produce similar shapes?
This question provides many opportunities for pupils to reason deductively.
Pupils who struggle to create sufficiently accurate diagrams could work on triangle dotty paper.
Starting point: a right-angled triangle.
Recursive procedure: from the right angle of a right-angled triangle in the image draw a perpendicular line segment to meet the opposite side of that triangle:
There are two possible second stages …
If the left-hand option is taken, a pupil could continue to create an interesting image in this way …
… but there are many other equally effective systematic ways to continue!
There are lots of opportunities, in any of the pleasing images that can be created, for pupils to use reasoning, in particular to prove that ALL the triangles are similar!
In examples 2 and 3 the application of the recursive rule repeatedly cuts-up shapes to create smaller and smaller similar shapes: the whole image stays the same size. In the next three examples, as in example 1, the whole image grows, as larger and larger similar shapes are created. In both kinds of example continuation of the process for ever – ‘to infinity’ – can be imagined and discussed by your pupils. The next two examples are relatively easy to create because they can be drawn on a triangle-dotty grid.
Starting point: a regular hexagon
The recursive procedure is simple to follow and describe – it is the kind of recursive rule that pupils might think of for themselves.
Starting point: an isosceles trapezium
The image on the left shows the effect of using a recursive procedure that produces a separate new similar trapezium at each step.
The image on the right, however, shows the kind of recursive procedure that pupils may think will create a new separate similar shape at each stage, but actually doesn’t – it produces overlapping similar shapes, but not at every step. (Shapes similar to the starting trapezium shown in Step 1 are outlined in red.)
This kind of ‘mistaken prediction’ can be a rich subject for discussion and learning: asking why the separate trapezia in the image on the right are not similar will prompt lots of worthwhile debate. If pupils use rules that generate shapes that aren’t similar, they will get a much deeper understanding than if they only use rules that do: seeing “what it’s not as well as what it is” is one of the key principles of conceptual variation.
Starting point: two touching identical circles fitting exactly in another circle …
Pupils will understand the recursive procedure from their own, or given, diagrams, as follows …
A powerful benefit of creating and investigating recursive procedures that produce similar shapes is that, depending on the shapes involved, pupils have opportunities to draw on their prior knowledge of a wide range of mathematical procedures and results. For example, in this circle situation they can use their knowledge of the relationship between the radius and the circumference of any circle to prove that in every ‘row’ of identical circles the sum of the circumferences equals the circumference of the largest single surrounding circle, perhaps starting with a diagram like this that shows the relationships between the diameters of the nested circles.
Your pupils should be able to explain why ALL circles are similar, and this would be a good opportunity to discuss it. This could lead to a rich discussion about length and area ratios: given two similar shapes, what’s the same and what’s different?
Page header by Brian Hillegas (adapted), some rights reserved