Exploring Area Models and Grid Method

This post contains a variety of tasks I’ve made to use at different points with nine mixed attainment year 7 classes while looking at multiplication. Eight of these classes I share with another teacher, so classes have used some/most/very few etc. of these tasks as appropriate and in different sequences.

The image above focuses on an area model of multiplication that links back to partitioning work from addition. It can’t be underestimated how confusing it is for kids to think about multiplication in terms of additive partitioning, so explicitly demonstrating this is valuable regardless of mathematical stage.

The gif below demonstrates this animation as it appears in the slides linked at the top of the post.

Where pupils were familiar with multiplying in a grid, the task below is designed to explicitly link the concrete representation to the pictorial one.

To further emphasise this link, we explored the multiplication grid using a scaled version.

Even here, pupils struggled to see how I knew the larger square in the diagram below was showing us 18 squared.

The task below asks pupils to develop fluency with the different arrangements of rectangles within a scaled grid to show combinations of multiplications.

Interestingly, pupils really struggled to articulate the following relationships in the picture and subsequent gif below. I think this is because of the cognitive load of thinking between the scaled diagrams and grids meant pupils weren’t consciously attending to commutativity.

Having spent a lot of time reiterating that a construction differs from a sketch in angles and geometrical figures, I had the language to explicitly narrate the difference between the grid diagram as a sketch as opposed to the measurable and countable arrays.

This led nicely on to discussions of ‘valid’ partitions vs. ‘efficient’ partitions.

Partitioning 9 x 10 in the ways above are all valid and work, but you just wouldn’t bother.

The task below breaks down grid multiplication into just setting up the appropriate partitions (using place value) as the focus. The example uses the commutativity explicitly too.

For pupils still struggling with ‘seeing’ the relationship between written words and numbers, the alternative answer below is colour-coded to explore this.

The gif below shows the animations to complete the first six multiplications fully if this is appropriate at this time for classes. The task has been sequenced in a way to provoke discussion around the suitability of this method for all of these questions.

The slides and their animations shown below link the difference between a scaled times table grid and multiplications as partitioning with a focus on how the changing of the size of the box in a grid won’t change the calculation. (It’s always worth explicitly explaining to kids that the reason all the boxes are the same size isn’t mathematical… It’s because I copy paste when making questions!)

The three reasoning tasks below look at partitioning with grid method in more detail and provide a high level of challenge for kids to be getting stuck into while pupils who would benefit from more procedural practice can get it.

And (finally!) the last bit of stuff in these resources focus on using arrays to begin to reason with simplifying calculations.

Unitising one of the multiplicands helped some pupils conceptualise what was happening here rather than arrays, and we had an interesting discussion how if additive operations can be represented by lengths, why is the adding of 3 an area? (We love any chance to talk about the multiplicative identity in a lesson to be honest.)

The task below was to check they could access the questions beyond that, and as such steadily removes the arrays each time.

The task below puts this idea into practice. I used rectangles around the multiplications as we haven’t explicitly come to the order of operations yet and I didn’t want to overload pupils with an extra layer of thinking when this task is for them to consider the structure of multiplication as a repeated addition. (I have included a version in the resources that doesn’t have the boxes on, as I’ll be using these questions again with classes in other year groups or as reasoning practice with KS4 at different points.

[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 is the first of a series on multiplication
Part 2: Reasoning with Subtraction and Multiplication

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