Here is part of a standard number line.

I might call this a “unitised” number line, showing how many ones I have.

One use of this number line might be to assist in the teaching of addition and subtraction of whole numbers. I could also use it to teach multiplication, say by two. For this purpose, I’ve found it useful to modify my number line as follows:

Here, two number lines whose units are scaled differently have been merged. We might call this a “bi-unitised” number line, the units being ones and twos. I discovered these pictorial representations fairly recently and have found them to be very powerful in helping develop a relational understanding of number. Such bi-unitised number lines might help us begin to think about multiples and factors. From there, we might make links to the relatively more challenging concepts of division, ratio and proportion.

These three concepts are encompassed within the teaching and learning of fractions which, as I argued in my previous post (“Some thoughts on working with fractions”), are essentially just numbers that arise when one integer is divided by another. The simplest subset of fractions is the unit fractions, where 1 is divided by each of the positive integers in turn. Here is an attempt to represent some of the unit fractions on separate but aligned number lines.

One thing I like about the image above is that it allows us to compare the sizes of the unit fractions. Imagine adding more number lines showing ^{1}/_{5}, ^{1}/_{6}, … What patterns would emerge? What questions might arise?

Extending both scales on a given bi-unitised number line (let’s just call it a relational number line) gives us even more representational power. Here are some pairs that allow us to visualise, compare and contrast both division and multiplication of one by two and three.

I like the reciprocal nature of these pairs of images.

Relational number lines are particularly useful for visualising the relationship between improper fractions and mixed numbers: how many ones would appear on this relational number line and where would they be placed?

And what is the largest number represented on both scales of all the relational number lines shown above? What other numbers can you name on each scale of each line?

With a little imagination, we can investigate multiplication and division of fractions by whole numbers. Draw some relational number lines that would help you to:

Multiply ^{3}/_{5} by 4;

Divide 2 by 5;

Find out how many ^{2}/_{3} you would need to make 2^{2}/_{3}.

We can look at other equivalences:

complete the upper scale on the two relational number lines below.

How else could you use relational number lines to support teaching and learning in your context?

Thank you for joining me in thinking about learning, teaching and doing mathematics. I hope it has been time well spent.