Representations: The Area Model

Sharing a post I wrote a few months ago for Be Bold Maths.

Research

The EEF guidance for Improving Mathematics in Key Stages Two and Three outline eight recommendations. They are:

  1. Use assessment to build on pupils’ existing knowledge and understanding;
  2. Use manipulatives and representations;
  3. Teach pupils strategies for solving problems;
  4. Enable pupils to develop a rich network of mathematical knowledge;
  5. Develop pupils’ independence and motivation;
  6. Use tasks and resources to challenge and support pupils’ mathematics;
  7. Use structured interventions to provide additional support;
  8. Support pupils to make a successful transition between primary and secondary school.
Improving Maths Guidance Has Lessons for Us All #EEF | @LeadingLearner
Education Endowment Foundation – Improving Mathematics in Key stages Two and Three summary document

Focussing on raising attainment and understanding in the Maths classroom can be overwhelming, especially for trainee teachers. In order for students to develop a rich network of Mathematical knowledge, we need to ensure that connections are made between Mathematical facts, procedures and concepts. In practice, when presented with a scheme of work and timetabled learning hours, we sometimes fail to make these deeper connections in learning and bridge the gaps. Instead, we focus on teaching methods and procedures that may not allow links to each other. No wonder some student see Maths as learning a bunch of rules!

Using manipulatives and representations does not necessarily mean getting out Cuisenaire rods, algebra tiles and multi-link cubes every lesson. It is about ensuring students know how to use their existing knowledge to solve problems. In this blog I will talk about the use of the area model as a representation for several areas within Maths teaching.

Some ways the grid method/area model be used

  • Multiplication
  • Division
  • Expanding brackets – double and triple
  • Factorising polynomials of degree 2
  • Factorising polynomials of degree greater than 2
  • Completing the square – coefficient of 1 or not 1
  • Algebraic long division

How I Implemented It

This model is based on the concept of finding the area of a rectangle. Students see this early on in their mathematical education and are, mostly, fluent at working out area of rectangles. They also need to be able to work backwards to find missing lengths. If these two concepts are not in place then the model will not allow them to develop the deeper understanding with factorising polynomials, for example.

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I only taught students the column method for multiplication and the ‘bus stop’ method for division for a long time, until a student showed me how they had learnt these using the grid method in primary school. This method works well because in general, students tend to have a strong conceptual understanding on place value. To carry out two or three-digit multiplication student need to recall multiplication facts and be able to add. They are able to recognise the value of digits and can break them up. For this reason, when presented with a grid for multiplication, they are able to partition the numbers and carry out the multiplication using their multiplication facts.

I have only recently started using the grid method for division and I highly recommend it. It allows the students to clearly see that division is the inverse of multiplication and that the link the two together. An example slide from my lesson is shown below.

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The fluidity of this model means we can use it for expanding brackets too. Once students have completed some work on algebraic simplification, we can introduce expanding single, double and triple brackets using this method. What I really love is that, unlike FOIL (why is this even taught?), this method works every single time without needing to remember what order to multiply the terms in.

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Seeing this method for division as an inverse of multiplication allows us to apply it when factorising polynomials too. In the past, I have taught expanding and factorising as two different subtopics and stated they are inverses without explicitly showing my students. The area model shows us to visualise this inverse property.

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Top Tips

  • Practice examples yourself first to see where and how the method can be used for the several topics.
  • Practice delivery of examples – what will this look like on the whiteboard? What will it look like in student books? What will you say/emphasise during delivery?
  • Talk to your department on exploring different methods – are they/do they need to be consistent for everyone?
  • Work with feeder primary schools to see how they teach – could there be shared planning on certain topics?
  • Accept that some students may need other methods, but could you achieve department consistency by having a method that ‘all students should see’?

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