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I’m thinking about how regularly pupils get perimeter and area mixed up.
Even when thinking about perimeter they can get a little confused – they might count the squares around the outside of a shape, count a diagonal as 1cm rather than √2cm or (yes, you guessed it) only sum two sides of a rectangle to find its perimeter yet all four to find its area.
So how can we help set the foundations?
Firstly let’s just concentrate on perimeter, let’s leave area a long way away. Let’s not even measure things with a ruler or count edges, or squares at all.
Let’s think about belts. We wrap a belt around our waist and it does up, in some cases it holds up your trousers, but as fashion dictates not always. A good belt will fit snugly, not squeeze so tight that you’re bulging over the edges and not be so loose that it can slide off. In the US the term ‘shelterbelt’ is used for a line of trees or hedges around the perimeter of a field to protect against wind or soil erosion.
So which shapes will a particular belt fit around? How will you check?
Don’t forget, a perfectly shaped belt will wrap around a shape exactly with no gap and no overlap
So go back to some familiar shapes and make a perfect shaped belt for each using string. These perfect shaped belts are called perimeters. Each shape has its own perimeter and we can compare perimeters by laying our belts out in order of length.
Now choose one shape and its matching perimeter. How many other shapes have the same perimeter? Draw some. Do they all have straight edges? Curved? Wobbly?
This could go on for some time; predicting perimeters, matching perimeters to shapes, deciding whether you can find the perimeter of 3D objects around the classroom. Lots of activity and wrapping string around all sorts of objects will result in a clear understanding of what the perimeter of a shape is (yes, even circles!). I envisage some use of technology and interactive displays here too.
Now let’s revisit…but restrict our shapes to polygons.
Instead of just string let’s use straws that cleverly match the lengths of the sides of pre-drawn shapes, colour co-ordinating (so the same length is always a particular colour).
We could lay them on the shape and then lay them out in a long line to compare perimeters or thread them onto string. Pupils can record the perimeters for different shapes using initials for the colour of the straw - are there any good short cuts?
Once you know which straws make the shape does their order matter if you want the perimeter?
An informal recording strategy here could lead into some class established rules on how to record. What if I give you my record of the perimeter – can you build the shape? How many possible shapes are there? Build some!
Now concentrate on rectangles. What shortcuts are there? What about squares?
Pupils can generalise here and find the rules for the perimeter of a rectangle and square. You could go on to look at other shapes e.g. parallelograms, regular hexagons, ‘staircases’ etc.
All this without touching a ruler, nor mentioning any counting. By physically finding and comparing perimeters, pupils have a firm grasp of the concept, know that shapes with the same perimeter aren’t necessarily the same shape, and can establish some initial rules about the perimeter of a rectangle and square. They can also have plenty of fun trying problems backwards – building a shape from perimeter, as well as the more traditional finding the perimeter of a given shape.
SOMETHING TO TRY:
KS1: Wrap string around the cross-sections of different rectangular boxes to find the longest perimeter.
KS2: Find five rectangles with a perimeter of 28cm. How many rectangles are there that have a perimeter of 28cm?
KS3: Which shape has the longest perimeter? Why?
KS4: Use this diagram to find an upper and lower bound for π by comparing perimeters of the three shapes.
KS5: If you fix the perimeter of a triangle how can you maximise its area?