Area Model, Beyond Multiplication into Division

Or so I hoped, discovering the: Unexpected Boundaries of the Area Model in Division

Jun 14, 2025

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In a previous post, "A Planned and Purposeful Journey of Multiplication" we explored how the area model can serve as a consistent visual bridge from early multiplication concepts right through to expanding and factorising binomials. My ambition was to unite all multiplicative operations under this elegant framework, so naturally, my attention then turned to division. After all, in the abstract, division is simply the inverse of multiplication. What I didn't foresee, however, was the bumpy path this exploration would take, leading to a crucial new realisation about the very model I had come to trust.

Initial Inspiration: Connecting Numeric and Algebraic Division

Two key things spurred this deeper exploration. The first was the compelling connection between multiplying a binomial by a trinomial and dividing a polynomial using the area model. The structural parallels are striking.

Secondly, I witnessed a demonstration of this very concept during one of Jonathan Hall's excellent Maths Fuddles – I highly recommend these for great discussions around maths and more.

From seeing this demonstration, which involved a 2 by 3 grid used for dividing a three-digit by a two-digit number, I thought, "Brilliant! I have a way of foreshadowing algebraic division with the numeric – this is exactly the kind of connection I was looking for!"

Only, I couldn't quite remember how the division worked, or how to set it up exactly. I knew that the divisor was split into its base ten components, as this mirrored the algebraic approach.

A CPD Dive into Area Model Division

As a group of schools, we use Complete Maths for our CPD support. Not feeling entirely confident with my own memory of the method, I steered a CPD session towards division, all the way up to division of polynomials using the area model. It was a superb session led by Sam Blatherwick.

Sam's session truly highlighted an approach to division using the area model. He encouraged us to decompose the numbers and work flexibly to break them down, with one side of the model (the divisor) not being partitioned into its factors, but rather the dividend being "chunked" to fit the divisor's dimensions, as shown below.

Now, this method does have its own inefficiencies, but I believe it also develops a strong sense of number. If a student was already comfortable using 'chunking' for their division, I think this area model approach would be a fantastic way to move them forward in their conceptual understanding and also develop fluency with that chunking approach.

Sam also showed us, (as I’ve seen before from Mark McCourt) that the familiar "bus stop" division symbol is effectively two sides of a rectangle, reinforcing the connection between division and the area model's geometric foundation.

It was a great session, including some brilliant ideas around dividing fractions that I hope to write up in a future post. However, I hadn't been explicit enough in my instructions to Complete Maths and had simply hoped that Sam would read my mind and deliver the exact aspect of dividing using the area model that I was hoping for.

Unfortunately, however good Sam is (and I would highly recommend him if you’re looking for someone to deliver CPD), it turns out he’s not actually a mind reader! This, however, provided me with the impetus to work it out for myself.

The Problem and the Realisation

So I did, and this was the initial result.

I encountered a problem. I started going into decimals. Into negatives. All sorts of complexities.

First, let's consider how I create questions like this. One of my goals when creating questions is to have the numeric foreshadow the algebraic. To do this, I'll typically use the algebraic form to generate the questions.

For instance, I start with the answer from a triple bracket expansion. I used a question generator to expand (𝑥 + 6)(𝑥 + 4)(𝑥 + 3), which gave me 𝑥³ + 13𝑥² + 54𝑥 + 72.

After this, I substituted 𝑥 = 10 to convert it to base 10, giving me 2912, and I knew that 16 was a factor (since 𝑥 + 6 would become 16 when 𝑥 = 10). I then attempted to recreate the division I had previously seen, the effect of which can be seen in the placeholder above.

This is the very approach I take when trying to mirror multiplying two-digit by two-digit numbers with binomials.

But it had failed me. Where was I going wrong? Why wasn't this approach working?

Then I realised. My coefficients of 𝑥 in the algebraic expansion were often beyond 10. So, when I started working in base ten, these 'extra' values had been absorbed into the calculation in a way that didn’t neatly map onto the area model's structure for division.

Making it Work: The Key Constraint

With this in mind, I now knew that to make it work effectively, I needed to keep my coefficients of x less than 10. This ensures that when we substitute x = 10, the place value columns in the numeric representation don't "spill over" and complicate the neat partitioning required by the area model.

Let's consider an example where this constraint is met. If we start with the algebraic expression I decided to make my values small to meet this constraint. I started with (𝑥 + 1)² and then multiplied this by (𝑥 + 2). This gives me the expression, 𝑥³ + 4𝑥² + 5𝑥 + 2, all of my constants within the expression are now less than 10. Which should result in the calculation working nicely. So let’s try this again. First we change the unknown base to 10. Which gives us the calculation 1452 ÷ 12.

Great, so now we have a numeric example which foreshadows the algebra. Let’s keep the example and do the polynomial division next to it. In order to see the parallels between them.

So we now have a way of foreshadowing the algebraic with the numeric, so the last question to ask, is, is this beneficial? Does it help our students? Does it add value?

While there maybe some structural benefit, at this point the fluency of factorising and expanding binomials is probably enough to be able to make the connections and continue the model into polynomial division. So the numeric is unlikely to give enough of a gain, unless there is a better way that what I have outlined above.

Reaching the Useful End

Did the approach stop working entirely? No, it was workable, but it wasn't reliably useful. It had limitations to the point where the calculations could become too complex for the very students we are trying to support.

I had, quite by accident, reached the useful end of the approach for numeric cases. Can I set it up so that the division works cleanly? Yes, for carefully constructed examples, I can certainly foreshadow the algebraic with the numeric. But I can't reliably introduce it as a consistent, dependable approach to numeric division.

There was so much promise, but an ideological pursuit of uniting all multiplicative operations under one model cannot be pursued to the detriment of our students. After all, this pursuit is about enabling students to develop their mental models and acquire build knowledge upon their pre-existing schema, alongside their proficiency in the approach. Unless limited to special cases, the complexity that could arise would not make this useful or transferable approach as a general numeric division strategy.

Unless, of course, I've missed something in the way I've done this? I'm always open to new insights!