Lesson Video: Making a Whole: Halves, Thirds, and Quarters Mathematics

Video Transcript

Making a Whole: Halves, Thirds, and Quarters

In this video, we’re going to be thinking about halves, thirds, and quarters. And we’re going to learn how to find two fractions with the same denominator which together make one whole.

Let’s start with a cheese sandwich, which is a good place to start. Now let’s imagine that a mouse wants to eat this sandwich but doesn’t fancy it all in one go. And so he decides to cut it into two equal parts: one part for now, one part for later. Now, from what we know about fractions, we know that each of these two equal parts is a half. And so if we put the whole sandwich back next to it to look at, we can see straightaway that two-halves are the same as one whole.

We could show this idea using fraction strips, too. If this strip represents one whole, then the blue strip must be worth one-half. And how many of these half strips will make one whole? As we’ve said already, there are two-halves in one whole. One-half and one-half make a whole.

Let’s think about another sandwich. This one’s strawberry jam. But it’s still one whole sandwich, isn’t it? Now what if our mouse got hold of this sandwich and thought to themselves, I’m a bit full up to eat this all in one go, so I’m going to divide it into four equal parts. Then, I can eat it over four days. And do you remember how to divide a shape into four equal parts quite quickly? We split it into half and then half again. The mouse still has a whole sandwich to eat, but it’s just in bits. And each of those bits is called one-quarter. Remember, if we split a shape or a number into four equal parts, they’re quarters.

And so what can we tell from what we’ve just done? We can tell that four-quarters are the same as one whole. And if we know that four-quarters make one whole, then we can use this to find other facts. For example, if we have one-quarter of a fraction strip, what more do we need to make one whole? We need another one-, two-, three-quarters. One-quarter and another three-quarters makes four-quarters altogether. And we know that four-quarters are the same as one whole.

Now can you remember what the title of this video was? There was another type of fraction mentioned. We’ve already looked at halves and quarters. And so finally, we need to think about thirds. So we’ll start with one whole sandwich again. The filling inside this sandwich is green and yellow. Perhaps it’s best not to ask what’s inside this one. And to cut the sandwich into thirds, we’re gonna have to divide it into three equal parts. Each of our three equal parts is worth one-third. One-third, two-thirds, three-thirds go together to make one whole.

So now we’ve looked at all three types of fractions that were mentioned in the title. We’ve thought about how to make a whole using halves, thirds, and quarters. Now there’s something interesting about the facts that we’ve found out. And to spot it, we’re going to need to remind ourselves how to write fractions. And remember our first fact. We said that one whole is the same as two-halves. Now, how are we going to write two-halves as a fraction? Now usually if we wanted to write down the value of two-halves of a sandwich, we just write one. It’s one whole sandwich, isn’t it? We don’t say I’ve got two-halves of a sandwich.

But if we did want to write this as a fraction, we’d need to think about what the top number and the bottom number in a fraction mean. Let’s quickly remind ourselves. The bottom number in a fraction, which is called the denominator, is the number of equal parts that the whole amount has been split into. And if we’re talking about halves, how many equal parts is that? As we’ve said already, it’s two equal parts. So any fraction with a two as the bottom number or the denominator is talking about halves.

Now do you remember what the top number or the numerator in a fraction represents? This is the number of those equal parts that have been selected, the number that we’re talking about. This is why when we first learn about halves, we only talk about one-half. And that’s why our numerator is one. But now we want to show two-halves. And so we’re going to need a numerator of two. One whole is equal to two-halves.

Let’s do the same with the other two fractions that we’ve learned about. We’ve also found out that one whole is the same as three-thirds, didn’t we? Let’s write this as a fraction. Don’t forget the bottom number, the denominator, shows us how many equal parts we’ve split up the whole amount into. This time there are three equal parts, so the denominator is three, and the numerator represents the number of parts that we’ve selected, the number that we’re talking about. We don’t want one-third or two-thirds. We want three thirds, don’t we? But here’s another fraction then that’s exactly the same as one whole, three-thirds.

So let’s end with quarters then. We know that a quarter is when we split the whole amount into four equal parts. So we know our denominator is going to be four. And we said one whole is the same as four-quarters, didn’t we? And so we need the numerator to be four, too.

Now, can you see anything interesting about these fractions? In particular, what do you notice if you look at each fraction on their own? In the fraction two-halves, the top number and the bottom number are the same. That’s also true for three-thirds and four-quarters. So we can say that when a fraction’s numerator and denominator are the same, the fraction is worth one whole. If you did a maths test with four questions in it and you got four questions right, you’d have got four out of four or one whole test right. If you had a cake and you cut it into three equal pieces and you ate all three of them, you’d have eaten three out of a possible three or one whole cake. So whenever we see both the top number and the bottom number exactly the same in a fraction, we know we’re talking about one whole.

Now, we’ve gone through quite a lot there. Let’s see how much we remember. Let’s try some questions where we have to put into practice what we’ve learned about making a whole.

This whole shape is a circle. Choose two shapes that can combine to make the whole.

To start within this question, we’re shown a picture of a circle. But you know this question isn’t about shapes, not really anyway. The key word in our first sentence is the word “whole.” And we can see that this word also crops up later on, where we’re told what to do. We need to try and make the whole.

Now when we talk about one whole in maths, we’re not talking about a hole in the ground. This word is spelled with a W. We’re talking about all of the amount or all of the shape. And so if we want to make the entire circle, all of the circle, we’re given some parts of a circle to help us do this. We could say these are fractions of a circle. This question isn’t really about shapes. It’s about fractions. And of course, we can’t just make a whole circle using one of these shapes. We’re told we need to choose two of them.

To help us do this, let’s look at each of these shapes one by one and decide what fraction we can see. As we’ve said already, each one shows part of a circle. So do you think it might help if we draw in the rest of the circle each time? Let’s look at our first fraction for a moment. Don’t forget when we’re talking about fractions, we need to split the whole amount into equal parts. And at the moment, we haven’t done that. That’s better. Now we can see that the whole circle has been split into four equal parts, but only one of those parts is shaded. This fraction is one-quarter.

Now let’s take a look at our second shape. So we’ll complete the circle again. And make sure that all our parts are the same size. It looks like we’ve split the circle into four equal parts again, doesn’t it? This fraction shows a number of quarters, too. But instead of one part being shaded, this time we can see two. The fraction that we can see in our second picture is two-quarters. So it looks like maybe this question isn’t just about fractions. It’s about quarters.

Now let’s step back for a moment and go back to that whole shape that we have at the top. Let’s split up this shape into four equal parts, too. So how many quarters could we see that make up the whole shape? One, two, three, four. So we can say that four- quarters make one whole. This is helpful for us. We’re looking to make four-quarters. Alright, let’s look at our last shape and work out which fraction we can see here. Then we’ll try to answer the question. This time we can see three-quarters have been shaded, so the shapes that we’re given show one-quarter, two-quarters, and also three-quarters.

Now, don’t forget we’re looking to find four-quarters somehow because we know four- quarters make one whole. So which two shapes can we choose if we want to have four-quarters altogether? Can you see? We know that two and two make four, but we haven’t got two lots of two-quarters. We’ve only got one shape that’s worth two-quarters.

What else makes four? One and three make four, don’t they? And if we look carefully at our final shape, can you see that all that’s missing is one-quarter? So if we took that first one-quarter and turned it slightly, it would fit just like a jigsaw in the missing gap. We know that four-quarters make one whole. And so we knew if we wanted to make the whole circle, we needed to find four-quarters somehow. The two shapes that can combine to make the whole circle are the one that shows one-quarter and the one that shows three-quarters.

Complete the following sentence: Two-thirds and what make a whole.

In this question, we’re given a sentence that we need to complete. This is a little bit like an addition sentence. It’s about putting two parts together to make a total amount. Now one of the parts that we’re given is this fraction here, two-thirds. We don’t know what the second part is. This is the part we need to find. So we put something together with two-thirds and we make a whole. What goes together with two-thirds to make one whole?

Well, underneath our sentence, did you notice we’ve got quite a helpful little diagram to help us? It’s a long, thin rectangle, little bit like a strip of paper. In fact, shall we imagine that it’s a strip of paper. We could say that it’s worth one whole strip. Let’s label it with the number one. But we can also see something else about this strip of paper. It’s been divided or split up into three equal parts. Each part is worth one-third. And so we can see just by looking at this picture one very important piece of information that’s gonna help us answer the question. How many thirds make up one whole? We can see that three-thirds make a whole, can’t we? And we can use this fact to help us.

Now we said that the first fraction in our sentence was two-thirds. But we also said that one whole is the same as three-thirds. So if we’ve already got two-thirds and we want to get three-thirds, how many thirds do we need to add? Two-thirds plus one more third make three-thirds, doesn’t it? So we can say that two-thirds and one-third make a whole. And so the missing part of our sentence is a fraction. The answer is one- third.

What number is missing? What-quarters and a three-quarters make one.

Can you see where the number is missing from our sentence? It’s the top number or the numerator in a fraction. So by finding this missing numerator, we’re going to turn this sentence into an addition. We can put two fractions together to make one. And underneath our sentence, we’re given a fraction strip to help us. Now we know that with any fraction strip, when we look at the whole amount, it’s worth one, one whole. And if we look for a moment at this particular fraction strip, we can see that it’s been split up into four equal parts or quarters. Each part is worth one-quarter, and there are one-, two-, three-, four-quarters altogether. So we could write four-quarters equals one. They’re both worth exactly the same.

If this was a chocolate bar and you broke up each of the four pieces and ate them all, you’d have eaten one whole chocolate bar or four-quarters. Now if we want to make one whole just like this sentence tells us, we’re going to need to make four-quarters. The second fraction in our addition sentence is three-quarters. So, how many quarters are we going to need to add to three-quarters to make those four-quarters that we need? Well, if we already have three-quarters, we only need one more quarter, don’t we? We know that four-quarters are the same as one whole. And so if we want to make one, we’re really looking to make four-quarters. And so we know we can say one- quarter and three-quarters make one. The missing number or the numerator in our fraction was one.

What number is missing? Three-what equals one.

Now when we say three-what as we read that question out loud, it might sound a bit strange, but we can see that what we’ve got here is a fraction. There’s the numerator or the top number. And then we’ve got a line, but we can see that the missing number is the bottom number in this fraction. So we don’t know if we’re talking about halves or thirds or quarters. That’s why we have to read the question as three-what. This is the number that’s missing.

How do we know what this missing number’s going to be? Well, the first thing that we can say is the fraction that we’re given is equal to one, one whole. And there’s something we know about fractions that are worth the same as one whole. Do you remember what it is? Let’s think about halves for a moment. How many halves equal one whole? Two-halves. What about quarters? How many quarters equal one? Four-quarters are the same as one whole, aren’t they?

And these two examples should remind us something really important about fractions. If the numerator, which is the top number, and the denominator, or the bottom number, are the same, the fraction is worth one whole. And we could have looked at lots and lots of different types of fractions. But in the two examples we’ve just seen, two-halves and four-quarters, can you see that the numerator and the denominator are exactly the same? Both of these fractions are worth one whole.

Now that we’ve reminded ourselves about this, let’s come back to the question because we know we’re looking for another fraction that equals one whole. And the only thing we know about our fraction is the numerator. And that’s three. And as we’ve just said, if this fraction is going to be worth one, we need the numerator and the denominator to be the same. So if the top number’s three, the bottom number must also be three. Three-thirds equal one. And we know this is true, don’t we? We know that any fraction where the numerator and the denominator are the same is worth one whole. So if the numerator is three, we also know the denominator must be three too. The missing number in this fraction is three. Three-thirds equals one.

We’ve gone through quite a lot in this video. What have we learned? We’ve learned how to use models of halves, thirds, and quarters to find fractions that make one whole. We also noticed that if the numerator and the denominator of the fraction are the same, that fraction is worth one.