Proportionality

If 'proportional' or 'multiplicative' thinking/reasoning is the central idea in age 11-14 mathematics, then how might we do a better job of making approaches to this consistent across the curriculum, so that students develop a deeper understanding?

At Loughborough University, with colleagues Tom Francome and Chris Shore, we are currently working on putting together a complete set of free, editable teaching resources for mathematics at Key Stage 3 (ages 11-14), and we are particularly trying to do so in a 'coherent' fashion, meaning that everything is connected together into a story that makes sense (Note 1; Foster, Francome, Hewitt, & Shore, 2021). We hope it will be ready to release later in 2023.

In this blogpost, I'm going to reflect on some of the thinking that has gone in the 'Multiplicative Relationships' Unit in Year 7 (ages 11-12). This Unit isn't yet finished, but we have a structure and a bit of the detail, which I wanted to share here.

It seems to me that 'proportionality' is the central idea in 11-14 mathematics that so much else is related to, and so we make a big deal within the 'story of the curriculum' of $y=mx$. Moving too quickly on to $y=mx+c$ just muddies the main point, so we save the '$+c$' for later on (Foster, 2022) (Note 2). Initially, we want students to become very comfortable working with $y=mx$. Think of all the things that come under the umbrella of $y=mx$ and can be viewed as instances of this:

• straight-line graphs through the origin
• ratio and proportion
• similar triangles
• gradient
• multiplication and division as inverses
• speed, density and other 'rates'
• rearranging formulae like $V=IR$ and $F=ma$
• the basis for trigonometry (see Foster, 2021)

Often each of these content areas comes with a different notation, and this is something that we want to address in the name of 'coherence'. For example, we might write $y=mx$ for line graphs but $y=kx$ for 'proportionality', and students may miss that the 'constant of proportionality' $k$ is precisely the gradient of the corresponding $x$-$y$ graph. There is much to gain by seeing all multipliers and rates as gradients, so we try to use consistent notation to highlight this.

In addition to the different notations across topics, multiple different representations are commonly used, even just within the topic of ratio and proportion, such as ratio tables, double number lines, etc. Perhaps because everyone agrees that proportional reasoning is hard, there is a temptation to throw everything at it, piling lots of different ideas on students, hoping that something will make sense and stick. Instead, in our curriculum design work, we have tried to avoid this, and instead choose one powerful approach and then use it consistently and probe into it deeply.

Because $x$-$y$ graphs are not just a representation but part of the content of the curriculum (unlike more 'optional' representations, like ratio tables, that some teachers use and others might not), we focus throughout the LUMEN Curriculum on number lines and Cartesian graphs (which we see as two number lines coming together at right angles, intersecting at the origin). So, before we tackle 'proportionality' as such, we spend a lot of time making sense of multiplication through the family of graphs of $y=mx$.

We begin by taking two identical number lines and stretching one of them and (eventually) rotating it by 90°.

This develops into the idea of a rule, linking two number lines via points in the Cartesian plane.

We go on to stress multipliers, like the gradient, $m$, as the key number that takes you, by multiplication, from any (non-zero) number to any other number:

We use lots of examples like this to practise finding multipliers and missing numbers:

We then tackle a mixture of proportion problems with 'nice' numbers as well as ones with 'hard' numbers. Below, we begin with a situation that uses 'nice' numbers. First we raise and discuss the 'additive' error:

Then, we work multiplicatively, first 'between variables' (i.e., from one variable to the other), using the same arrow notation for multipliers as we used earlier:

Then we find 'within variables' multipliers:

We contrast multipliers 'between variables' (which we call rates) with multipliers 'within variables', which we call scale factors. Scale factors are always dimensionless, whereas rates sometimes have units (e.g., here Rosie's multiplier was £/km). 

Whether a rate or a scale factor is more convenient depends on the numbers; hence, tasks like this:

This eventually builds up to being able to use any of the elements from this kind of diagram:

(The diagram looks overwhelming with everything included. But in any situation you would only use 2 of these arrows at once, always the same colour as each other.)

This is all very much work in progress, and we'd be very glad of any thoughts or criticisms of what we're doing!

Questions to reflect on 

1. Do you agree about the centrality of proportionality in the lower secondary mathematics curriculum?

2. What do you like and dislike about the approach outlined here? 

3. In what ways is it similar to or different from what you typically do?

Notes 

1. To find out more about the LUMEN Curriculum, go to https://www.lboro.ac.uk/services/lumen/curriculum/. 

2. And, when it comes, I think that $y=mx+c$ might be better encountered as $y-c=mx$. This way, rather than seeing $y=mx+c$ as a non-example of proportionality, we see it as another example of proportionality, but we just 'have the wrong origin'. So, the shift to $y-c$ as our variable, rather than $y$, makes it understandable as another instance of a proportional relationship. 

References 

Foster, C. (2021). On hating formula triangles. Mathematics in School, 50(1), 31–32. https://www.foster77.co.uk/Foster,%20Mathematics%20in%20School,%20On%20hating%20formula%20triangles.pdf

Foster, C., Francome, T., Hewitt, D., & Shore, C. (2021). Principles for the design of a fully-resourced, coherent, research-informed school mathematics curriculum. Journal of Curriculum Studies, 53(5), 621–641. https://doi.org/10.1080/00220272.2021.1902569

Foster, C. (2022). Using coherent representations of number in the school mathematics curriculum. For the Learning of Mathematics, 42(3), 21–27. https://www.foster77.co.uk/Foster,%20Using%20coherent%20representations%20of%20number%20in%20the%20school%20mathematics%20curriculum.pdf