Compare the areas of the shaded parts in the figure below using the symbol is less than, is greater than, or is equal to.
In the figure, we can see two circles. And part of both circles has been shaded. Now, the question asks us to compare the areas of these shaded parts. Another way of saying this is which is bigger. Is the shaded area in the first shape less than the shaded area in the second shape? Is it a large area? Or are they both the same size? We need to choose the correct symbol to write in between the two shapes. Now, we could try and turn our heads and look at these two areas and think which one looks the largest. And it is possible to answer the question that way. But more importantly, this question is really getting us to think about fractions and what it means to split a shape into different numbers of parts. What can we see about these two shapes?
Well, the first thing that we can see is that both circles are exactly the same size. This is useful to know because if one of the circles was larger than the other, then perhaps the shaded part would be larger too. But they’re both exactly the same size. And what’s different about them is the number of parts that they’ve been split into. It’s important to see that the parts that each circle has been split into are equal. This matters because it really helps us to compare the shaded parts. The first shape has been split into four equal parts. And we can see that one out of those four parts is shaded. That’s the same as one-quarter. We could say one-quarter of the shape has been shaded. Our second shape has been split into less equal parts, only three. And so we can say that one out of three equal parts is shaded. That’s the same as saying one-third.
Now, we can use this information to help us compare the areas. The first circle has been split into more equal parts. But because it’s been split into more equal parts than the one on the right, each part is smaller. One-quarter is less than one-third. And so the correct symbol to use in between these two shaded areas is the one that represents is less than.
If two shapes are exactly the same size but one of those shapes is split into more equal parts, the size of each part in the first shape will be less than the size of each part in the second shape. One-quarter is less than one-third. We’ve compared the areas of the two shaded parts. And we’ve chosen the correct symbol to write in between them. The area of the shaded part in the first shape is less than the area of the shaded part in the second shape.
The correct symbol to use in between the two circles is the one that represents is less than.