What I've Learned About Teaching... Area

This is a topic that has fast become a favourite of mine to teach! I've always had an "all area is based on the area of a rectangle" approach to teaching area and showed students how to adapt the formula for area of a rectangle for other shapes, but I've found students often forget because I've not done it in enough depth, and I have to remind them every time it comes up. I was getting fed up of students forgetting to half, or halving when they shouldn't and making silly errors. I realised I wasn't teaching it as well as I thought I was. So when I collaboratively planned a lesson on area of a triangle with someone else from the Maths Hub and was given the suggestion of allowing students to cut shapes up and generate the formula themselves, I was nervous as I've had nightmare 'hands on' lessons in the past and hate the tidying up after cutting out tasks so try to avoid them. But after trying these activities with my year 8s last year, I can safely say a) it wasn't as bad as I imagined it would be in terms of mess and potential off task behaviour, students were well involved and it wasn't too messy and b) it was completely worth it for the long term benefits it brought to the students- they had a deeper understanding which meant they learned it better, more flexibly and the students remembered it longer. I also learned different ways of seeing the area - bonus as I love learning new methods!

It does need to be carefully scaffolded enabling students to eventually generate a formula for themselves (generally this happens collectively as a class rather than each student independently depending on how competent they are with algebra). Before even exploring any formula, students need a solid understanding of two things. Firstly the concept of area and secondly the units we use to measure area, ie square centimetres/square metres and what they actually are. The basic concept of area and understanding the units for area are linked and important foundational knowledge needed before looking at any formula.

The key to these activities is providing students with sufficient time to explore themselves, giving timely hints and nudges but also knowing when to stop and draw the students back together. If all else fails, you can throw the method under the visualiser yourself and ask questions to get the students to pull apart what has happened themselves, pretending if you wish that you have seen a student doing it. These tasks are perfect opportunities to celebrate different methods, and also to question for understanding and get students explaining someone else's method. Show one solution under the visualiser without telling the students anything. “How has this student shown us the area? Tell the person next to you” and provide short sharp bursts of opportunities to discuss, then clarify the method as a class, “Why are we using this length, why not that one?”. Make sure you have an idea of the different methods (detailed below) that work beforehand and dedicate enough time to discuss the different methods (depending on your group - the more methods the better ideally as it improves flexibility using formula, however you also don't want to cognitively overload your students). I then tend to generalise with words first before students practice, and one of the practice questions will have the algebraic lengths so they generalise the formula themselves eventually.

Be ready for the misconceptions - trying to shoehorn pieces of shapes which don't fit, overlapping pieces, leaving gaps and some students might cut out every single little centimetre square and part squares. If needed, you can show some of these under the visualiser to discourage them away from this, "Why do we think it might not be a great idea to do..." or "Why won't... work?".

The impact of these activities? Students learn how the formula work and where they come from, some will generate the formula themselves, and they remember how to find the area of all the following shapes with relative ease. They rarely ask me how to do it in retrieval tasks.

1.      Area of a Rectangle

Most students will have an awareness of how to find the area of a rectangle, but if they do you can always ask them to persuade you to really see if they understand. Rectangles are important as they have the square centimetres all nicely lined up and the formula is easiest to generate ("do I need to count every single square centimetre? Is there a more efficient way?"). All the other shapes have areas which can always be based on the area of a rectangle so it is worth spending some time on, even with secondary students. To support the work with perpendicular heights later on, it may be worth giving rectangles with multiple dimensions within the rectangle and asking them to identify which they need to use and why, bringing in the terminology of perpendicular height in this familiar context.

2.      Area of a Triangle

Give students a triangle on centimetre squared paper. I normally draw a couple of the same triangle on squared paper and photocopy, giving each student at least two triangles to work with.

The options are to ask them to make a rectangle or to show how the area of a triangle formula can be derived. There are multiple ways to do it. Students can cut the triangle in two pieces at half of the height, then chop the top into two triangles and use them to fill the gaps. This creates a rectangle with the same area as they have used all the paper, not lost or added any paper. The rectangle has a height which is half the original perpendicular height of the triangle and the length was the same as the triangle. See diagram below - thick blue lines are where to cut the triangle.

They could instead use two triangles to make a rectangle. They cut one of the triangles along the perpendicular height into two separate triangles, which fill the gaps on the second triangle to make a rectangle. However, the rectangle we have created has used two of the original triangles. The height of the new rectangle is the same as the perpendicular height of the original triangle, and the length is the same as the original triangle. But as we have used two triangles to create the rectangle, the area of the rectangle will be twice the area of the triangle.

I'd do mini whiteboard work at this point, to check students have understood the formula and can identify the correct sides. Sometimes I will split this into two separate checks, first which are the perpendicular height and length, then finding the area, depending on the class.

Make sure you spend time highlighting perpendicular heights, showing triangles in different orientations and types, providing too much information so students are solid in choosing the correct pair of perpendicular height and length. You can also show how there are three pairs of lengths and perpendicular heights which give the same answer and explore why that is.

I really like this question from NCETM to explore which information they need for both area and perimeter of a rectangle:

3.      Area of a Parallelogram

Again give students a parallelogram on centimetre squared paper and ask them to create a rectangle with the same area.

The easiest way to see it is if they cut a right angled triangle off one end and slide it into the gap at the other side, but realistically they can do it anywhere to get the same result. The height of the rectangle is the perpendicular height of the parallelogram. The length of the rectangle is the same as the original parallelogram. So the area of the parallelogram is 

Make sure to include plenty of practice with perpendicular heights and lengths, ensuring they look different.

I'd do a quiz again like the below on mini whiteboards:

4.      Area of a Trapezium

Give the students a trapezium on centimetre squared paper. Again, I would draw this on square centimetre paper and put several of the same on one sheet and photocopy, handing a couple to each student.

Personally I’d ask them again to make a rectangle with the same area as the original trapezium, however they can make a parallelogram relatively easily, which they should already be competent with. There are again several ways of making a rectangle from the trapezium.

The students will use half the perpendicular height whichever method they use. My preferred method is to find half the height and cut across to get two trapeziums, then cut a triangle off the smaller trapezium, rotate to fit into the gap and then rotate the remains into the gap at the other side. Then students can see the new rectangle has a height which is half the perpendicular height of the original trapezium, and the length of the rectangle is the combined lengths of the parallel sides of the original trapezium. This method also works if they don’t cut the top trapezium into a triangle and smaller trapezium, they can cut into two trapeziums instead but it is then trickier to see what the new length of the trapezium is. So the method ends up being half the perpendicular height, then multiply by the sum of the parallel sides:  

A slightly different method is to identify half the perpendicular height, then cut the right angled triangles from the perpendicular height down from each side. The triangles then rotates to fill the gaps at the top of the trapezium. This is essentially finding the average length of the parallel sides, which is where they add the sides and half, then multiply by the height:  

Another method is to use two trapezia and rotate one 180 degrees, cut a right angled triangle off one side and fill the gap to make a bigger rectangle. This time the height is the perpendicular height of the original trapezium, the length of the new rectangle is the sum of the parallel sides of the original trapezium, so the area of the bigger rectangle is height x (sum of parallel sides)

HOWEVER, we used two trapezia to make the rectangle, so one trapezium would then be half the big rectangle: 

Mini whiteboard quiz to check for understanding, again this can be broken down depending on class- which are the parallel sides? Which is the perpendicular height? Depending on the class.

During practice, ensure you show different types of trapeziums in different orientations – imagine they are estimating the area under a curve using trapezia at KS4! They need to be confident spotting trapezia however they may come across them.

I also think showing all three methods and providing questions where one method is easier to use than the other two is a great exercise to develop some fluency and flexibility in how students use this more complex formula. Which method did we use for this trapezium? Why?