Video: Related Subtraction Equations

In this video, we will learn how to write a family of subtraction facts linking three numbers up to 10 and model them with part–whole models.

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Video Transcript

Related Subtraction Equations

In this video, we’re going to learn how to write a family of subtraction facts. And these facts are going to link three numbers up to 10. We’re also going to learn how to model these subtraction facts using part–whole models.

Here are seven pictures hanging on the wall. Unfortunately, they haven’t been hung very well because some of them start to fall off. There were seven pictures to begin with. One, two, three, four pictures fell down. And there are one, two, three pictures left. We could write a subtraction fact to show what’s happened here. Seven take away four equals three.

Another way we could write exactly the same thing is by starting with the answer. Three pictures is what we get if we have seven and we take away four. Three equals seven subtract four. Can you see how these subtraction facts are related? They’re linked, aren’t they? They contain exactly the same numbers. And in a way, they’re saying exactly the same thing. In this example, we’ve looked at two facts that are related, but we can actually find more than two.

Let’s try another example. Let’s imagine that we’re fed up of our pictures falling off the wall. And so we decide to put them on a shelf. Now, this shelf isn’t big enough to fit all seven pictures on; we can only fit six. There are four pictures that have red frames and two that have green frames. Let’s draw a part–whole model to show this. Four red frames and two green frames make six altogether. Four and two go together to make six.

Now what if we take away one of these parts? Should we take away the red picture frames to begin with? There we go. Now, let’s use number cards to make a subtraction. We started with six, we’ve taken away four red picture frames, and we’re left with two green picture frames. Six subtract four equals two. And do you remember how we said we could write this another way if we start with the answer? Two green picture frames are what we get if we start with six and we take away four red picture frames. Two equals six subtract four. Can you see we’ve used exactly the same number cards, haven’t we, for both facts? We’ve just moved them around.

Now, I wonder, “are there any other subtraction facts in this family?” If we know that four and two are two parts that go together to make six, what else do we know? Perhaps we could start with our six picture frames again. And instead of taking away four red ones, we could take away the two green ones. Now what subtraction facts can we see? We started with six frames. This time we’ve taken away two green picture frames, and we’re left with the four red ones. Six subtract two equals four. And once again, we could juggle these number cards around to make the same subtraction fact, but one where we start with the answer. Four is what we’re left with if we start with six and we take away two.

So, these four subtraction facts are all part of the same family. They contain the same numbers. And we can see how to find the facts in the part–whole model. If we start with a whole amount then take away one of the parts, we’ll be left with the other part. Let’s say that again whilst we look at the other subtraction. If we start with a whole and we subtract one of the parts, we’ll be left with the other part.

How good do you think you are at spotting these families of subtraction facts? Let’s try some questions where we have to practice this skill.

There are five carrots. A rabbit eats three of them. Which of the following gives the number of carrots left? Three subtract two equals five, five subtract two equals three, five equals three subtract two, five subtract three equals two, or three equals five subtract two. Which of the following is another subtraction equation for the number of carrots left? Three equals five subtract two, two equals five subtract three, five subtract two equals two, or five subtract two equals three.

We’re given lots of subtraction facts to choose from in this question. And did you notice they all sound very similar. That’s because the numbers five, three, and two keep cropping up, don’t they? It could be very easy to get confused with this question, so we’re going to have to read it really carefully. To begin with, we’re told that there are five carrots. Can you see them in the picture? One, two, three, four, five. But these five carrots aren’t around for long because we’re told that a rabbit eats three of them. And we can see these in the picture too, can’t we? These are the carrots that have been crossed out, one, two, three. So, there were five carrots to begin with and a rabbit has eaten three of them.

Our first question asks us, which of the following gives the number of carrots left? Well, first of all, can you see how many carrots are left, one, two. Now, we’re given the five different subtraction facts to choose from, but only one of them is a way of showing what’s happened to these carrots. Let’s try and write down what’s happened as a subtraction fact and then look for a fact that matches what we’ve written.

We were told we had five carrots to start with, so let’s write down the number five. And we also know that a rabbit eats three of them. This is the same as taking away three of them, isn’t it? So, we’re going to write the subtraction symbol and then the number three, five take away three. Now, as we’ve just said, we’ve counted how many carrots are left, and there are two. So, the subtraction that shows us the number of carrots left is five take away three equals two. Now can you see this subtraction fact in the ones that we’re given? Yes, we can. Five take away three equals two.

But there’s another part to this question. We’re asked which of the following is another subtraction equation for the number of carrots left. And we’re asked this question because although we can write five subtract three equals two, there’s another subtraction fact that we can write. This time, let’s go through each subtraction really carefully and try to understand what it’s telling us.

Firstly, three equals five take away two. Now, this is interesting because this subtraction starts with the answer and then we have the subtraction. But what it means is three is what we have left over if we start with five and we take away two. So, we could model this using counters. Here are five counters. And it is correct; three is what’s left over if we start with five and take away two. But it’s not what’s happened to our carrots, is it? The rabbit ate or took away three of them, not two. So, the subtraction we’re looking for is not this one.

The next subtraction says two equals five take away three. This is another calculation that starts with the answer. But if we think about what it means, it’s telling us that two is what’s left if we start with five and take away three. And if we model this with counters, we can see that this is exactly what happens to our carrots. Two carrots are what’s left over if we start with five and take away three. And if we write the subtraction underneath the one we’ve already got at the top, we can see that it’s almost exactly the same. The parts of it are just switched around a little bit. Five take away three equals two. Two equals five take away three.

If there are five carrots and a rabbit eats three of them, we can find out the number of carrots left by working out five take away three equals two. And another way of saying exactly the same thing is two equals five take away three.

Write a subtraction sentence that matches the given model. Three equals five take away two. Five equals eight take away four. Three equals eight take away five. Five take away three equals eight. Or eight equals three take away five. Which of the following is another subtraction equation for the model? Eight take away five equals four. Eight take away four equals four. Five equals eight take away four. Or eight take away three equals five.

This question is all about the model that we can see in the picture. So, let’s spend some time looking at it and seeing what’s going on. To begin with, we’ve got a line of cubes. Let’s count how many we’ve got altogether. One, two, three, four, five, six, seven, eight. So, the whole line of cubes contains eight cubes, but we can see that it’s then split up. Can you see that it’s been broken off? So now we’ve got a group of green cubes and a group of red cubes. Let’s count how much we’ve got in each part.

We’ve got one, two, three green cubes and one, two, three, four, five red cubes. You know, we could draw a part–whole model here to show what’s going on. The whole line of cubes, as we’ve said already, is eight. But then, when it’s broken into two parts, the part of green cubes is worth three and the part of the red cubes is worth five. Five and three are two parts that go together to make eight.

Now we’re asked to write a subtraction sentence that matches this model. Now before we do that, what do you think is being taken away? Do you think the green cubes are being taken away and we’re left with the red cubes? Or do you think the red cubes are being taken away and we’re left with the green cubes? It might be hard to see just by looking at the picture, but there are some things we do know. Firstly, we know that number of cubes that there are in the whole amount is eight. So, our subtraction is definitely going to include the number eight, isn’t it?

Can you see any subtractions that don’t include the number eight? Well, this first one doesn’t, does it? We’ve got a three, a five, and a two. Let’s cross off this subtraction; it can’t be right. Something else we know about our model is that it contains three green cubes and five red cubes. So, we know our subtraction is going to include the numbers eight, three, and five. Are there any answers that don’t include these numbers? Well, if we look at the second one, we can see that there is a five and an eight, but there isn’t a three. So, again, we know straightaway this isn’t going to be the correct answer.

So, we’ve got three possible choices. Let’s have a look at them. The first subtraction says three equals eight take away five. This is interesting. It starts with the answer, doesn’t it? Often, we’ll have a subtraction where we’ll say something take away something equals something. Well, this subtraction is the other way round. Something equals something take away something. Let’s read it carefully. Three equals eight take away five. In other words, three is what’s left if we start with eight and we take away five. It looks like this matches the model, doesn’t it? Three green cubes are what’s left over if we start with eight cubes and we take away the five red cubes.

And if we quickly look at the other two subtractions, we can see they don’t actually make sense. If we start with five and then take away three, how can we have eight? That’s a bigger number. And the last subtraction doesn’t make any sense at all. Eight is what’s left over if we start with three and then take away five. I think we can cross this one through, don’t you? So, the subtraction that matches our model is three equals eight take away five.

In the second part of the question, we’re asked which of the following is another subtraction equation for the model. Let’s do exactly what we did last time. Remember, we’re looking for a subtraction that includes the numbers eight, three, and five because this is the whole and the two parts that we’re talking about. The first subtraction contains an eight and a five, but not a three. The second one we don’t have either of the two parts that we’re looking for. We’ve just got two fours. In the next subtraction, we’ve got a five and an eight again, but not a three. And so, the only subtraction that matches our model, which shows that three and five go together to make eight, is the last one.

This time, it’s as if we’ve taken away the green cubes, isn’t it? Eight cubes subtract three green cubes leaves us with five red cubes. In the model, it wasn’t very clear whether we were taking away the red or the green cubes. But one thing we did know is that we had eight cubes to start with, and we split it into three and five. And that’s how we know that the subtraction sentence that matches the model is three equals eight take away five. But we can also say eight take away three equals five.

Find the missing numbers in the subtraction sentences matching the given part–whole model. Five subtract four equals what. What subtract one equals four. One equals what subtract four. And four equals five subtract what.

In this question, we’re given four subtraction sentences and we need to find the missing numbers in them. To help us, we’re given a part–whole model. Let’s take a moment to look at it. The top number here is the whole amount, five. Let’s draw five counters to represent this whole amount. There we go. At the bottom of our part–whole model, we can see two parts that we can split the whole amount into, and they are four and one. Let’s color our counters to show that four and one go together to make five. There we are. Four and one make five.

Now let’s try to use this part–whole model and our counters to work out our missing numbers. Five subtract four equals what? Well, if we start with five counters and we take away four counters, we’re going to be left with the part that goes together with four to make five. Watch what happens when we point to the numbers on our part–whole model. Five take away four leaves us with one.

In the next subtraction, it’s the first number that’s missing. This is the number that we’re taking away from. What take away one leaves us with four? Well, as we’ve said already, we know that one and four go together to make five. And so, if we start off with five and then subtract one, we’ll be left with four. Can you see how our subtractions so far are all to do with this part–whole model? They’re part of a family of subtractions. They’re all linked.

Do you think the next two are going to be part of the same family? One equals what take away four? This is interesting because we start with the answer in this sentence. One is what we’re left with if we start with something and take away four. Can you see this is exactly the same as the subtraction above it? If we know that five take away four leaves us with one, then we know one is what we get if we start with five and take away four. The missing number in this calculation is five.

And for our final subtraction, the family four is what we get when we start with five and take away what. Well, we know the missing part here is one. Four equals five take away one. In this question, we used a part–whole model to help us find a whole family of subtractions that are all linked. They all contain the same numbers. Five take away four equals one. Five take away one equals four. One equals five take away four. And four equals five take away one. Our missing numbers were one, five, five, and one.

What have we learned in this video? We’ve learned how to write a family of subtraction facts linking three numbers up to 10.

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