Todd removes a piece of the shape shown. Give a possible example for each of the three possible outcomes stated.
a) The perimeter increased. Remember we can find the perimeter of a shape by adding together the lengths of all its sides. We can think of it a bit like the total distance travelled around the shape. Let’s begin by finding the perimeter of the shape shown.
We’ll start by counting each of the units. That’s one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, and 22. So the perimeter of the shape given is 22 units. And for part a, we’re looking to find a way to increase the perimeter. We’re trying to find a way to make the distance travelled around the shape longer.
Since we have to remove a piece, the best way to do this is to cut into the shape. By going into the shape and back out, we’re increasing the total distance travelled when we trace the outside of the shape. One way we could do this is to cut into the side of the shape as shown. We could alternatively have cut in from the top or the right-hand side of the shape.
Let’s just check that the perimeter of that shape has increased. In fact, if we repeat the process for finding the perimeter as before, we can see that the perimeter of the shape is 26 units. And the perimeter has indeed increased as required.
b) The perimeter remained the same. The trick here is to look at one of the corners of the shape. The perimeter of just this part is two units. By removing this corner, the perimeter is still two units at this point.
In fact, as long as we remove only one corner, the perimeter will stay exactly the same. We can check that the perimeter remains the same. And in fact, once again, we have a perimeter of 22 units as required.
c) The perimeter decreased. We’ve already seen that if we cut a piece out of the shape, the perimeter increases and if we cut a corner off, it remains the same. This time we’re going to need to cut an entire section off to decrease the perimeter. We could choose to do that vertically as shown or alternatively we could cut it horizontally and take the entire row of squares from the top.
Let’s check the perimeter of this shape: one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20. We have indeed reduced the perimeter by two units. The perimeter has decreased.