Video Transcript
Predict how many hexagons will be
present if the given pattern is extended to have a total of 38 polygons.
Now, the word polygon in our
question simply means a 2D shape which has straight sides. We can see that our pattern
contains two types of 2D shapes that have straight sides or two types of
polygon. There are six-sided shapes or
hexagons, and there are three-sided shapes or triangles. And these make an alternating
pattern: hexagon, triangle, hexagon, triangle, and so on. Where does our repeating pattern
start and end?
If we look carefully, we can see
that the part of the pattern that’s repeated is the first four shapes. We start with a large hexagon. Then we have a small triangle with
its base level with the base of the hexagon. Then we have another large
hexagon. And then we have a triangle. But this time, it’s in a different
position to the first. At this point, we start the pattern
all over again, large hexagon, small triangle with the base on the bottom, and so
on.
Now that we know which part of the
pattern is repeated, one way to find the answer could be to sketch the shapes again
and again until we get a total of 38 polygons. And then, we could just count up
how many hexagons we’ve drawn. The trouble is, this might take
quite a long time, and there is a quicker way to find the answer. The question actually wants us to
choose a quick method because it asks us to predict how many hexagons there will be
and not calculate the answer. So let’s try to predict how many
there will be.
We’ve already noticed that there’s
an alternating pattern of a hexagon, then a triangle and so on. Even though two triangles are in
different positions, it’s still one type of shape and then the other, all the way
along through the pattern. And because we know which part of
the pattern is repeated, we’re able to look at how the number of hexagons change as
this pattern grows. Let’s just look at the first two
shapes to start with. We can see that there are two
polygons and one of them is a hexagon. Let’s extend this to look at the
part that’s repeated, the first four shapes in the pattern.
This time, there are four polygons
and two of them are hexagons. And as we increase the number of
shapes in the pattern, the number of hexagons is always half the total number of
polygons. There are six polygons we can see,
and three of them are hexagons. And so, without having to draw
every single shape, we know that if we extend our pattern to have a total of 38
polygons, all we need to do is to find half of 38. 38 divided by two equals 19. And so, we can predict that there
will be 19 hexagons. What if we did sketch every single
polygon up to 38? We can see that our prediction of
finding half of 38 gave us the correct answer. There are 19 hexagons.
