x=a, y=b: When?

The plan for our current KS4 interim curriculum was to follow the one year Edexcel SoL (over 4 terms) and use the final term of year 11 for targeted revision. But then pandemic.

In this structure, I’ve tried to look at the sequencing in a way to seed difficult concepts as a through line in “easier” mathematics, so that when we get to the stuff that’s hard to conceptualise, pupils aren’t rudderless.

This was the rough plan for foundation Spring 2 in year 10. (I know it’s broken down into individual lessons, which I don’t believe are a thing, but as I said: Interim curriculum.)

In this post, I want to talk about the stuff in transformations (the yellow).

Rather than teaching seemingly unrelated transformations and moving on, the aim is to use this as a springboard into other areas of maths. All the vector stuff takes place in the context of translations (the higher plan spends much longer on this and covers all vectors) as a broader teaching point about movement, and all the reflection stuff takes place within plotting very simple straight line graphs.

No matter how many ways I’ve tried to teach straight line graphs, it’s the lines in the form of x=a and y=b that trip kids up. There’s loads of reasons for this. Personally I think it’s because of the (quite right) focus on gradient as a key feature of a straight line and what that looks like algebraically. This gets fuzzy to articulate in a way pupils can understand with x=2 for example.

Even when talking about perpendicular gradients having a product of negative 1, pupils really struggle to conceptualise what this means with perpendicular lines like x=2 and y=3 etc.

I don’t want to de-rail pupils with this (albeit lovely) kind of thing just yet – especially for our foundation kids.

But I do want them to plot these lines automatically.

And I do want them to be able to reason effectively and link across other areas of maths with these lines.

My compromise here is to seed this through something else, in this case reflections, where familiarity can be explored later during the straight line graphs work.

I considered putting these lines in with loci and constructions as explaining the graph x=3 as the locus of points when x is 3 has been the most effective way to prompt pupils to understand the which are horizontal and which are vertical, but given the section on graphs immediately follows this it seems better to put it here.

While the task above can be done by reading off the scales for lengths, we then present them with the same thing, but without gridlines to show how the properties of the graphs themselves is enough.

What follows are a series of questions that play with these lines and ask questions around them in a way to get pupils to become flexible in their thinking and application.

These tasks have all been designed so that they are accessible, but probing. For many of them there is also an efficient bit of reasoning to explore with multiple ways to secure a solution. I’ve intended to use minimal variation where appropriate, and been sparing in my use of fraction arithmetic and area of polygons to try and keep accessibility at the forefront of the pupil experience.

The final two tasks are designed to allow pupils practice at filtering information on the same diagram.

Although, I also couldn’t resist a bit of generalising with the final triangle area question either! The mathematics isn’t particularly challenging, but the reading and unpicking the mathematical properties from the language used in the conditions can be a huge stumbling block for pupils in KS5.

[All the activities in this post are linked at the top with solutions, and while this post outlines a suggested sequence for use, it’s no way exhaustive or considered optimal, but an ongoing reflection and sharing of materials.]

This post accidentally forms the first part of a wider set on straight line graphs and coordinate geometry:

Section 2: Gridless Gradient

Section 3: Gradients on Grids

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