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Question Video: Finding the Number of Objects in a Given Pattern When Extending Mathematics • 4th Grade

Predict how many hexagons will be present if the given pattern is extended to have a total of 38 polygons.


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.

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