22 September 2018

Vectors for Enlargement

I'm going to do a series of short posts on approaches or methods that teachers might not have seen before. I'm very aware that there will be many people who already use these approaches, but I figure that not everyone will have seen them so they are worth sharing.

Today, vectors for enlargements. I first saw this on Twitter years ago but I can't remember who tweeted it (apologies!). Later I saw my former colleague Lizzie Stokes (@misstokesmaths) using this approach in her teaching.

How I used to teach enlargement
I'll illustrate my previous method with a negative enlargement question:
I'd tell my students to draw a line to any vertex from the centre of enlargement. Then I'd tell them to count how many squares they've moved to get to the first vertex. I always modelled this slowly on the board. I showed the process of counting carefully ('count the corners, not the spaces!') because I find they often miscount.

Then I circled the scale factor and said that because the scale factor is negative two, to find the image they have to double the distance from the centre of enlargement. But because it's negative, they have to go in the opposite direction from the centre. So in this case, we started by going two units right and one unit up so now we have to go four units left and two units down. I emphasised that we're going the opposite direction because it's a negative enlargement. We then do the same thing for each of the other vertices in turn. At the end, all the corresponding vertices should be joined with straight lines going through the centre of enlargement. If not, you've miscounted!

I'd show another example with a different scale factor, then get them to practise a lot of these on printed worksheets. It was normally a relatively quiet lesson as they all had to do a lot of counting! I'd go round berating them for using pen when I'd told them to use pencil ten thousand times.

In hindsight I realise that this method could be better. It works fine for super smart students (most things do!), but others struggled with it. One problem is that you have to hold lots of details in your head while you work though the question. Another problem is that the method is different to the method for positive enlargement - there's an extra thing to remember.

How I now teach enlargement
They start by labelling each vertex with a letter and then finding the vector that takes them from the centre of enlargement to each vertex. They are already fluent in using vectors from our work on translation so this should be straightforward.

They write down the vectors rather than trying to hold the information in their head like they did before. Then all they have to do is multiply each of their vectors by the scale factor, and this gives them the vector for each of the vertices of the image. So they know exactly where to draw the image - it's all calculated and recorded before they start drawing.
I've used 'Year 11' vector notation in the picture above for clarity, though it's not essential at this stage.

This method is identical regardless of whether we are enlarging with a positive or negative integer or a fraction. So suddenly negative enlargements are no harder than positive enlargements (assuming students know how to multiply negatives). A clear and consistent method all round - I don't know why I didn't always do it like this.

Any others?
I hope that was helpful if you'd not seen it before!

I've blogged a number of times before about alternative methods for various topics (see my posts tagged 'methods'). If there are any approaches or methods that you use that you think other teachers might not use, I'd love to hear them! Please comment below or email me or tweet me.


  1. I do like methods that apply no matter the level of difficulty. Thanks for this.

  2. Never taught it like this but I'm teaching it tomorrow so I'm going to try it - thanks!