Similarity

I saw this great monster GCSE question on Twitter (thanks @ReviseJustMaths) - it inspired me to write a post about similarity!

Taken from Edexcel Higher Paper 1 November 2013

In this post I'll focus on resources and methods for teaching similar triangles.

Two triangles are similar if they're 'equiangular' (a bit of a mouthful so I teach 'AAA' instead). GCSE questions often require us to first match up equal angles before we start calculating unknown sides using ratio/proportional methods.  This 'matching up' normally involves using angle rules - see my post on FUN angles. The key point to emphasise in teaching is that it really really helps to draw the two triangles separately, and in the same orientation. Here's three examples to illustrate this:

Example 1 - Separate Triangles

Example 1 is straightforward - it's clear to see from the angle labels that the 8cm side in the first triangle is proportional to the side labelled x in the second triangle.  I'd still encourage my pupils to redraw the two triangles in the same orientation though.

Example 2 - Joined Triangles

Example 2 is less obvious.  My pupils often make the mistake of thinking that side BC is proportional to side CE.  I teach them to label all equal angles in the diagram, then draw triangles ABC and EDC separately. They'd then see that because <ABC = <CDE (alternate angles) and <ACB = <ECD (vertically opposite angles), the side BC is actually proportional to the side CD. 

Example 3 - Combined Triangles

In Example 3 we can use the same approach as in Example 2. Label all the equal angles in the diagram ie <ABD = <ECD (corresponding angles), then draw the triangles ABD and ECD separately. In this question it's likely we'll be asked to work out the length BC, which just requires a bit of thinking (ie calculate the length BD first, then subtract the length CD).  

Resources

• I love these resources from the Mathematics Assessment Project. In particular, the sorting cards on pages 15 - 18 provide a challenging and engaging activity.  I asked my pupils to sort the cards into three categories (similar/not similar/can't tell), stick them onto A3 paper and annotate with reasons.  
• Similar pairs is a nice matching activity from Don Steward.
• This activity from Teach Mathematics has been tried by Mr Collins and reviewed in this blog post. 
• This is a great booklet of practice questions. There's also some challenging questions in this worksheet from mathsmalakiss.com.
• CIMT has lots of excellent similarity resources.
• This is a lovely rich task from Shell Centre

For resources for area and volume of similar figures, see my resource library.