Getting back to basics: area

Thanks to Headteacher, Nick Hart, for this month's maths blog post focusing on the topic of area and strategies for getting to the basics right!

Looking for a free activity on area? Download this Year 5 activity which includes activities to project on the whiteboard, additional practice questions and Teacher Notes. Download now. 


I remember teaching area a good few years back and decided to use the context of carpeting a room as a starting point.  Square carpet tiles, it seemed, despite being part of my own general knowledge, was an entirely mind blowing concept for my year 4 class who knew carpet to come only in rolls.  Of course, this odd context completely derailed what I had planned for them.  

Teaching area no doubt has to start with a concrete experience but for novices, there is too much distraction here that will overload working memory.  Once children are comfortable with the basic principle, they can apply it to contexts with trickier language later on.  A better starting point is for children to see how many square tiles fit into a rectangle:
maths blog area
This shouldn’t cause too many problems but is the foundation for everything that follows.  Children needn’t stick with rectangles - the same idea applies to other irregular shapes.  It’s just that placing square tiles in, say, a pentagon will yield an approximate area:
maths blog area 2 
Children need sufficient time to practise finding the area in this way, with different shapes to internalise the idea that area is the space inside a 2D shape.  

A pictorial representation is simply a rectangle with a grid and to find the area, children count the squares.  Again, doing this with irregular shapes and therefore needing to count half squares and quarter squares is too important to miss out.  It’s at this stage that we need to push for efficiency, not just for efficiency’s sake but to make a strong conceptual link to the formula that we all know for calculating the area of rectangles.  Children need to be counting in rows and columns:
maths area
 maths blog area
It’s this efficient counting of squares inside a rectangle that provides proof for the length multiplied by breadth formula.

When children fully understand what area is and can calculate efficiently, they can move on to trickier ideas such as compound shapes.  For children that struggle to visualise the standard strategy of splitting a compound shape into two or more rectangles, an initial step is needed.  We need to show children that joining two rectangles creates a compound shape and how the area is made of the original rectangles:
maths blog area
 
Children could simply practise combining rectangles to see the different compound shapes that could be made and how the area changes with each combination.  With that concept solidified, we can show children how it works in reverse where we start with a compound shape and need to break it up into rectangles to make working out the area a much simpler process:
maths blog area 
I’ve been very careful in which sides I’ve labelled here.  For the children that might struggle the most, designing questions and tasks with too much going on will not set them up to succeed.  In the example above, although not all the sides are labelled, there is no need to work out any of the unknown side lengths because each rectangle has adjacent sides which are known.  The important thing is that children are practising is to visualise and annotate where compound shapes need to be split.  It is important that over time, as their confidence builds, they are set questions where the compound shapes are in different orientations, questions where the splitting line is both horizontal and vertical and questions where multiples splits are necessary.  Clarity of thinking on the teacher’s part is crucial here – getting children to focus on the most efficient way of breaking up the compound shape is an often neglected conceptual step.

For a truly deep conceptual understanding of area, and to arm children with a range of strategies to solve problems like this, they need to see other ways of doing it.  A subtraction model for finding the area of a compound shape is more difficult, but as long as we clearly model the process and the mathematical thinking children can succeed.  You simply look at the area of two rectangles, overlay the larger one with the smaller one and figure out the area of the larger rectangle that can be seen:
 maths blog area
In time, this overlay of shapes will turn into cutting out a section of a larger rectangle to reinforce the idea that a particular section is not part of the shape concerned.  We can then show children questions like this as a scaffold to help them to visualise what the area of the larger shape is, what the area of the smaller shape is and how to calculate the area of the compound shape:
maths blog area
 
The thinking process, and abstract representation here is (12 x 10) – (6 x 7).  A great task is to give children lots of images like the one above, lots of calculations, brackets and all, and match them up.
Beyond working on basic fluency of finding the area of compound shapes, there are ample opportunities to reason too.  Tasks that require a deeper level of thinking ensure that all children are challenged while giving children who have been struggling sufficient time to master the basics.  Here’s a particular favourite:
maths blog area
Because the yellow rectangles are not labelled, children can only estimate their area in comparison to what is known – the larger rectangle.  This estimation should then be used to reason whether each rectangle would yield the solution to the problem.  For example, rectangle C is approximately slightly less than half of the larger rectangle, perhaps 182.  The question requires that the yellow rectangle’s area must be approximately 62 therefore rectangle C doesn’t work.  Looking at rectangle B, roughly 6 of them would fit into the larger rectangle, therefore the area of the yellow rectangle is approximately 40 divided by 6 and using times tables knowledge, that’s about the same as 42 divided by 6 which is 7 – the area of rectangle B must be about 72.  This kind of works and in comparison to rectangle A, which is less than half the size of rectangle B, it is now very obvious that the best solution is rectangle B.

I’ve deliberately kept the labelling of sides consistently simple.  Children often have difficulty at first figuring out unknown and perhaps too often, we race ahead to that style of question before children have fully understand what area is and therefore resulting in overload.  Working out unknowns is tricky enough in its own right and children need a solid understanding of additive reasoning in order to do that.

With the pressures of curriculum coverage, topics such as area can sometimes be raced through. However, lingering with these topics and squeezing out as much mathematical thinking as possible can help children to make links to other areas of maths and experience success which then breeds motivation to master other aspects of maths.

Thanks to Headteacher, Nick Hart, for this month's maths blog post focusing on the topic of area and strategies for getting to the basics right!

Looking for a free activity on area? Download this Year 5 activity which includes activities to project on the whiteboard, additional practice questions and Teacher Notes. Download now. 

Tags

mathematics, maths, Rising Stars Mathematics

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