Lesson Video: Two-Step Problems with Bar Models: Numbers up to 10,000 | Nagwa Lesson Video: Two-Step Problems with Bar Models: Numbers up to 10,000 | Nagwa

# Lesson Video: Two-Step Problems with Bar Models: Numbers up to 10,000 Mathematics

In this video, we will learn how to solve two-step addition and subtraction problems by modeling them with bar models and writing equations.

16:31

### Video Transcript

Two-Step Problems with Bar Models: Numbers up to 10,000

In this video, we’re going to learn how to solve two-step addition and subtraction problems. And we’re going to do this by modeling the problems using both bar models and also by writing equations. Here’s an example of a two-step problem. Let’s read it carefully to find out why we might describe it as a two-step problem.

Dalia posts a gift to her friend in the United Kingdom. The parcel travels 5,619 kilometers from Cairo to London in the back of a delivery lorry. It is then flown another 662 kilometers from London to Glasgow. Finally, a mail van takes it 18 kilometers to her friend’s house. How far has the parcel traveled altogether?

Perhaps you’re still not sure why it’s called a two-step problem. Let’s go through what we need to do to find the answer. If we skip to the final question, we can see that what we need to do is to find how far the parcel’s traveled altogether. And this word “altogether” is a key word. It shows us we’re going to need to add to find the answer.

Now, there are three important numbers in this question. And they stand for the three parts to the journey that this parcel makes. The longest distance is this 5,619 kilometers it takes to drive from Cairo to London. Then there’s a flight from London up to Glasgow, which is another 662 kilometers. And then we’ve got that local mail van, which carries it the last 18 kilometers to her friend’s house. So to find out how far the parcel’s traveled altogether, we’re going to need to find the total of these three distances.

Now, one way we could show what we need to do is by using a bar model. The whole amount is unknown at the moment. This is the amount we need to find. But we do know three parts that make up the whole amount: 5,619, 662, and 18. Now, although we can use different methods to find the answer here, we’re still going to have to add these three numbers together. And we’re going to have to do it in two steps.

Now, we know that two and eight are a pair that make 10. So we could start by adding together 662 and 18. And then once we’ve done that, we could add the total to 5,619 to find the overall distance. So in this question, there are two things we need to do to find the answer. There are two steps. In this question, both of the steps are addition. But we’re also gonna be working with subtraction in this video. And sometimes a problem might need us to subtract first and then add or add then subtract. That’s why using a bar model and also writing out what we need to do in equation form is really helpful to us.

We can then see what it is we need to do to get to the final answer. Now, as this was just an example, we won’t work out the answer to this particular problem. But at least we know how we could if we wanted to. Now, probably the best way to learn how to solve two-step problems like this is just to practice doing them. So let’s go through some practice problems now. Each one is going to have two steps. And to find out what we need to do to get to the answer, we’re going to try sketching some bar models and also write equations to help.

Noah, Mason, and Benjamin decided to buy a new laptop for their father. The laptop costs 1,750 dollars. Noah decided to pay 650 dollars. Mason decided to pay 500 dollars. How much should Benjamin pay so that they can buy the laptop?

Here we’ve got what we call a word problem. It’s the sort of problem where we can’t see any symbols to tell us whether we need to add, subtract, multiply, or divide to help us solve it. Instead, we need to think carefully about the story behind the words, what do they describe. And there are a couple of things we can do to help ourselves here. A really useful tool for working out what we need to do to solve a problem is to draw a bar model. And in this question, we’ve been given a bar model already. Let’s re-read the question again right from the start, and we’ll look at how the bar model models it.

Our story begins with three brothers. We’ve got Noah, Mason, and Benjamin. And we’re told that they want to buy a new laptop for their father. The first really important piece of information we’re given is the cost of that laptop. It costs 1,750 dollars. Can you see this on the bar model? It’s written at the top here. And the way that our bar model’s been labeled, we can see that this represents the total amount. The whole length of the bar model represents 1,750 dollars. But how are our three brothers going to pay for this present for their father?

We’re told that Noah decided to pay 650 dollars. And we can see this labeled in the first part of our bar model. Next, we’re told that Mason decided to pay 500 dollars. And the second part of our bar model is labeled to show this amount. We’re finally asked how much should Benjamin pay. And we can see that this relates to the third part of our bar model. It’s the part that hasn’t been labeled, and it’s the part we need to find out.

So we’ve seen that the whole length of the bar model represents the whole cost of the laptop. And then its three parts represent the three parts of this whole amount that Noah, Mason, and Benjamin are going to pay. So what do we need to do to find the answer? As we’ve said already, there aren’t any symbols in the problem to tell us. But thankfully, we can use our bar model to help us. To find the amount that Benjamin needs to pay, we simply need to start with the whole amount and then subtract the two amounts that his brothers have already paid. Then we’ll know what’s left.

Now, there are two ways we could do this. One way to find the answer might be to start with the whole amount and subtract Noah’s payment and then whatever we’re left with to subtract Mason’s payment from it. Then we’d be left with the amount that Benjamin needs to pay. Or we could start with an addition. We could find out how much Noah and Mason have paid already. Then we’d need to subtract this from the whole amount. Now, it doesn’t matter which one of these two methods we use. Both of them are going to bring us to the right answer. Let’s use the first one, and then maybe we’ll show the second one just as a check.

So to begin with, we could subtract 650 from 1,750. This is the sort of calculation we could do in our heads because 650 is 100 less than 750. So we’ll be left with 1,100. And then if we take 1,100 and subtract the amount that Mason decides to pay, again, we could do this in our heads, 1,100 subtract 500. We know that if we had 1,000 and we subtracted 500, we’d have an answer of 500. So if we start off with 100 more than 1,000, but we still take away 500, our answer is going to be 100 more than 500. It looks like Benjamin’s going to need to pay 600 dollars, doesn’t it?

Let’s see whether that second method of ours would’ve found us the same answer. First, we’d need to add 650 and 500. This gives us a total of 1,150. This is the amount that those two brothers have paid altogether. And then if we subtract this from the whole amount, 1,750 take away 1,150, we’re left with 600.

We’ve used a bar model here to help us solve this two-step problem. And we’ve also talked about how we could solve it in two different ways. If a laptop costs 1,750 dollars, Noah pays 650 dollars of that, and Mason pays 500 dollars, then we know the amount that Benjamin needs to pay is 600 dollars.

Ethan and Anthony are competing in a video game. Ethan scores 3,700 points. Anthony scores 300 points less than Ethan. How many points does Anthony score? How many points do they score altogether?

This problem describes Ethan and Anthony, who are playing a video game. The first piece of information we’re given is Ethan’s score. We’re told that he has 3,700 points. And then we’re told Anthony’s score, or rather we’re told something about Anthony’s score. We’re told that Anthony scores 300 points less than Ethan.

Now, we’re given a bar model, and this helps us to see the relationship between these two boys’ scores. The bar that represents Ethan’s score is labeled 3,700. And although the bar representing Anthony’s score isn’t filled in yet — we don’t know it — we can see that it’s 300 less than Ethan’s. And we’ve got two questions to answer. And the first one asks us, how many points does Antony score?

Now, as we’ve said already, his score is 300 less than 3,700. We’re going to need to subtract to find the answer. But it’s the sort of calculation we could do quickly in our heads. We know that 700 take away 300 would leave us with 400. So 3,700 take away 300 is going to leave us with 3,400.

In the final part of our problem, we’re asked, “How many points do they score altogether?” What’s their overall total? Now, if we’d have been asked this right at the start, we might have said to ourselves, well, we can’t find the overall score. Because to do that, we’d need to know what Anthony’s score was, and we don’t know that. In other words, we’d have recognized this as being a two-step problem. First, we’d have needed to find out Anthony’s score, and then we’d be able to add the two together. But because we’ve already calculated Anthony’s score, we’ve done the first part of this two-step problem. We just now need to add the two boys’ scores together.

If we were to sketch another bar model just to show what we need to do here, it might look like this. We need to add together 3,700 and 3,400. Both our numbers have a three in the thousands place. And so the total of our thousands is 6,000. Then we just need to add the hundreds. Now, we know that 700 plus 300 would be the same as 1,000. So 700 plus 400 will be 100 more than this. It’s going to equal 1,100. And if we add together our 6,000 with this 1,100, we can see that the boys’ total score is 7,100.

We’ve solved the two parts of this problem by using bar models to help us. They showed us we needed to subtract to find the first part and then add for the second. The number of points that Anthony scores is 3,400 points, and the number of points they score altogether is 7,100 points.

In a library, there are a total of 3,152 science books, English novels, and mathematics books. The number of science books is 1,023, and the number of English novels is 1,210. How many mathematics books are there in the library?

In this problem, we need to find the number of maths books that there are in a library. And we’re given some information to help us. Firstly, we’re told the overall total of science, English, and the maths books altogether. And this is 3,152. You know, if we were to represent this problem using a bar model, we’d start off by drawing a bar and labeling it with this number. This is the whole amount.

Now, we know there are three parts to this whole amount. There’s a number of science books, English novels, and mathematics books. Now, we don’t know the number of maths books. This is the part we need to find. So we can label this with a question mark. But we can split the rest of the bar into two parts because there are two pieces of information that we do know. We’re told that the number of science books is 1,023 and the number of English novels is 1,210. So how are we going to find the value of this missing part?

Well, it’s a good job we sketched a bar model ’cause it gives us a good idea of what to do. Our problem is going to involve subtraction because we need to find the difference between the total amount, which is 3,152, and the total of science and English books. But we don’t know an amount for that yet. We just know these amounts separately. So to solve this problem, there are actually two ways we could do it. The first method we could use is to add together 1,023 and 1,210. This will give us the total amount of science and English books. And then whatever this amount is, we could take it away from the whole amount to find the number of maths books.

Another way of finding the same answer would be to start with the whole amount and then just take away the two amounts of books that we already know, so 1,210, and then from whatever’s left take away 1,023. Let’s quickly show how these two methods will give us the same answer.

1,023 plus 1,210 gives us a total of 2,233. Remember, this is the total number of science and English books. Now, we can take this number and subtract it from the total amount. We’ll have to do a bit of regrouping to get there, but the answer’s 919. Now, if we check this just by showing the second method, we could start with the whole amount and subtract just the number of English novels. This gives us the answer 1,942. Now we’d need to do our second subtraction and take away the number of science books, 919.

We used a bar model to help us work out what we needed to do to solve this problem. And there were two ways. Both of them would’ve found us the correct answer, didn’t matter which one we chose. The number of mathematics books that there are in this library is 919.

So what have we learned in this video? We’ve learned how to solve two-step problems involving addition and subtraction. We’ve done this by modeling them using bar models and by writing equations.

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