6 July 2014

Linear Graphs

One of the topics that my pupils struggle with the most, from Key Stage 3 right through to Year 12, is linear graphs.

My bright Year 11s know the rule for finding the gradient of a perpendicular line, but can't work out how to answer a GCSE question like this:
Strangely, I teach this differently at GCSE and A level.  At GCSE my pupils write down
y = mx + c
and substitute values for y, m and x, then solve for c.  Whereas at A level my pupils use the equation
y - y1 = m(x - x1)
Essentially the same method but a slightly different approach. Perhaps I should be more consistent and introduce the 'A level method' at GCSE. I'd like to hear what other teachers do.

In this post I’ll suggest some ideas and resources for teaching linear graphs.

Key words
Let's start with the basics. There's a lot of new vocabulary in coordinate geometry, some of which pupils will have encountered at primary school. This is a nice activity for discussing new words. This topic also lends itself well to a vocabulary knowledge survey as suggested in my earlier post

Here's a nice trick for remembering the words parallel and perpendicular (this could be helpful for spellings - parallel is often misspelt): 

and this activity is suitable for Year 6 or 7. 

Introducing the form y = mx + c
I'd normally start this topic in Year 7 or 8 with plotting straight line graphs (Teachit Maths provides an excellent Drawing Straight Line Graphs booklet for this purpose), before exploring the form y = mx + c.

This topic provides an excellent opportunity for pupil investigation using Desmos. Use a worksheet to guide pupils through the lesson:

Once pupils are able to interpret the values of m and c then they can attempt this exercise or this true/false activity (both from rogradymaths.blogspot.co.uk). Teachit Maths also has a nice 'Understanding y = mx+c' worksheet and two activities on horizontal and vertical lines.

My blog post 'All about gradient' features ideas and resources for teaching gradient (the post focuses mainly on real-life applications of the concept of steepness, but also features links to gradient worksheets and activities).

Understanding parallel and perpendicular gradients
So here's the key information that pupils need to understand:

If they know how to work out a gradient as 'rise over run' (or equivalent) then pupils should be able to derive the 'negative reciprocal' rule for perpendicular gradients from an activity like this one from MathsPad.

Here's a few activities for a lesson on perpendicular lines:

Bringing it all together
When students study linear graphs at AS level, they don't really learn anything new - my GCSE pupils should have the necessary skills and knowledge to answer C1 coordinate geometry questions. I gave my Year 11 pupils questions 5 - 7 from this AS worksheet last year - it took them ages to answer each question, but they were certainly capable. Essentially it's just the wording of the questions that throws them.

Here's some more recommended resources suitable for GCSE students:

Finally, fun for puzzle-lovers in linear relationships sudoku and a nice Key Stage 3 Christmas activity on plotting linear graphs from Teachit Maths - Mistletoe and Lines.

Finally, ever wondered which countries use y = mx + c notation and which countries use something different? Check out this map.


  1. Readers might be interested in this blog post about the different methods for teaching the equation of a straight line http://cavmaths.wordpress.com/2014/03/29/the-straight-lines-debate/

  2. There's a great parallel and perpendicular line activity in my 15th gems post: http://www.resourceaholic.com/2014/11/gems15.html

  3. I love your worksheet for straight line graphs on Desmos! I'm planning on using it, thanks for sharing. And thanks for all the other great resources here too, I'm glad I discovered your blog!

    1. Thank you! That's great. I really appreciate the feedback.