Video: Column Addition of Three-Digit Numbers: Partial Sums

In this video, we will learn how to use place-value blocks and tables to add three-digit numbers by adding the hundreds, tens, and ones separately before finding the sum of these partial sums.

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Video Transcript

Column Addition of Three-Digit Numbers: Partial Sums

In this video, we’re going to learn how to use place-value blocks and tables to help us add together three-digit numbers. And we’re going to do this by adding the hundreds, then the tens, then the ones. We’re going to add each part separately before then finding the total of these partial sums.

These place-value blocks represent a three-digit number. We know this because there are a number of hundreds as well as tens and also ones. There are one, two hundreds and one, two, three, four tens and just a single one. It’s the number 241. Now, this video is all about adding three-digit numbers together, so let’s think of another three-digit number to add. Let’s add the number 138. Now to help us find the sum or the total of 241 and 138, we can use this place-value table to help us. But so that we understand what’s happening in our place-value table, we could also model what we’re doing using place-value blocks. This is going to help us really understand what’s happening every step of the way.

So let’s model our second number 138. We’re going to need one hundred, three tens, and eight ones. There we go. Now, we’re ready to start. What’s the sum of 241 and 138? Now, when we’re working with larger numbers like these, it can be really useful to split them into smaller parts. And that’s why setting out our numbers using place-value tables or blocks can be so helpful to us. Our numbers are already separated into parts. We can see very clearly the hundreds, the tens, and the ones. And we can add these three parts separately. Let’s start off with the hundreds. So we’ll ignore our tens and ones for the moment and just focus on the hundreds.

Our first number contains two hundreds. We know this because in the number 241 the digit two is in the hundreds place. Now we know two hundreds have a value of 200. Our second number, 138, has a one in the hundreds place. Of course, this is worth 100. So to find the total of both of our hundreds, we need to add together 200 and another 100. The sum of 200 and 100 equals 300. Now let’s find the sum of the tens. Again, we just need to think about the digits in the tens place now. We can forget about the hundreds and the ones for a moment. We can see that the number 241 has a four in the tens place and four tens have a value of 40. Then the number 138 has a three in the tens place. And three tens are worth 30.

So to find the total of our tens, we need to add together 40 and 30. Well, we know that four plus three equals seven. So four tens plus three tens equals seven tens. And seven tens, of course, mean a seven in the tens place. And they have a value of 70. The sum of the tens parts of our numbers is 70. We’ve only got one more part to add. Let’s have a think about the final digit in each number, the ones. Our first number ends in a one, and one one has a value of one. And our second number has an eight in the ones place. And eight ones are worth eight. Instead of thinking of one plus eight, we could switch them around. Eight plus one might be quicker. The total is nine ones.

So far, we found three, what we call partial sums. We found the sum of the hundreds part, the tens part, and the ones part. Can you see what we now need to do to find the overall total? We need to put these parts back together again. In fact, we could say we need to find the sum of all the sums. Now this is where place-value blocks can come in useful because as we found the total of each part of our numbers, we put those totals underneath the equal sign. The sum of the hundreds, the tens, and the ones are next to each other at the bottom here. Without place-value blocks, we don’t need to put the parts back together again. They already are. Perhaps you can read what the answer’s going to be just by looking at these place-value blocks. But just to show that our answer is correct, let’s do it the written way.

How many ones are there in all of our numbers? Zero plus zero plus nine is the same as just nine, isn’t it? Onto the tens, zero tens plus seven tens equals simply seven tens, which of course have a value of 70. And in the hundreds column, we only have the digit three to think about. So our total of hundreds is three. The total is 379. Did you notice we could’ve read the answer just by reading aloud the answers to our partial sums? It doesn’t always work, but it does with this calculation. 300, 70, nine. Before we started adding together our two three-digit numbers, we wrote them in a grid. And the reason we did this is that using a place-value grid or table like this is a really good way of seeing the different parts of each number, the hundreds, the tens, and the ones.

Once we’ve done that, it took us four steps to find the answer. Step one, we found the sum of the hundreds parts of our numbers. Step two, we added together the tens. In step three, we found the total of the ones. And then, in the final step, we combined our partial sums back together again. Now we’ve taken seven minutes here to add together two three-digit numbers, but it won’t normally take us this length of time. It’s taken us so long because we’ve used place blocks to help. We’ve tried to go over each step really slowly and carefully. But this is only the first time we’ve done it.

But the more times we practice calculations like these, the less we need to rely on things like place-value blocks to help us. We can get quicker and quicker at going through these four steps. Let’s try answering some questions now where we have to practice what we’ve learned about adding three-digit numbers. And as we go, we’ll see if we can speed up.

Find 427 plus 338. Hint: Make partial sums by adding the hundreds, adding the tens, and adding the ones.

Now although the table underneath this question might look complicated, all this question is really asking us to do is to add together two three-digit numbers. We need to add together 427 and 338. The grid is just there to help us. By writing the two numbers in a table like this, it can help us to see the different parts that make up these three-digit numbers. The hundreds are in a column on top of each other. So are the tens and the ones. Now, if somebody said to you, “What’s the total of 427 and 338?,” you might think to yourself, these numbers are really large. We can’t add them all up in one go.

Well, thankfully, we don’t have to do this. And our question shows us how to find the answer by adding each part separately: first, the hundreds, then the tens, and finally the ones. Let’s begin with the hundreds. The number 427 has a four in the hundreds place. And the number 338 has a three in the hundreds place. And so to find the sum of the hundreds, we need to add together 400 and 300. Four plus three equals seven, and so 400 plus 300 equals 700. Now let’s add the tens. And we can see from the side that the calculation is going to be 20 plus 30.

But where do these numbers come from? Well, in our first number, the digit two is in the tens place. And we know that two tens are worth 20. That’s where our 20 comes from. And in the second number, we have the digit three in the tens place, which is worth 30. So to find the sum of the tens, we need to add together 20 and 30. We know that two plus three equals five, and so two tens plus three tens equals five tens or 50. Onto the ones, what’s seven ones plus eight ones? We know that double seven is 14. So seven plus one more than this must be worth 15. And this is interesting because we can’t write the number 15 in the ones place. 15 is made of one ten and five ones.

So as we write the sum of our ones in our grid, we’re going to have to write one ten and five ones. And when it comes to finding the total of our tens in a second, we’re gonna have to remember that one extra 10. Now we found the sum of the hundreds, the tens, and the ones. But in order to find the overall total, we need to put these partial sums back together again. How many ones do we have? Zero plus zero plus five ones gives us a total of five ones. And if we look at the tens digits, there was zero tens in 700, five tens in 50. But don’t forget we had that one extra 10, didn’t we, when we added together our ones. So five tens plus one more ten equals six tens or 60.

And in the hundreds column, we’ve only got those seven hundreds to think about. So let’s put a seven in the hundreds place. We found a total of 427 and 338 by making partial sums. First, we added the hundreds, then the tens, then the ones. And to find the overall answer, we just added these three parts back together. 427 plus 338 equals 765.

Find 263 plus 219. Hint: Make partial sums.

In this question, we need to add together two three-digit numbers, 263 and 219. Can you see them both written in this place-value table? You know, writing two numbers on top of each other in a place-value table like this can be really helpful when writing them. It helps us to think about each part of the numbers separately. The hundreds digits are above each other in a column. The tents digits are in their own column and so are the ones. Thinking about the hundreds, the tens, and the ones separately can help us find the overall total. We know this because we’re given a hint, and it tells us to make partial sums. In other words, we need to add together the different parts of our two numbers.

Step one, let’s add the hundreds. If we look at the digits in this column, we can see that both of our numbers have two hundreds. And we know that the total of 200 plus another 200 equals 400. That’s right, 400 as the sum of our hundreds. Step two, we need to add the tens part. The number 263 has six tens, which are worth 60, and the number 219 has one ten, which is worth 10. Six tens plus one more ten equals seven tens, which are worth 70. The sum of the tens part is 70. Step three, we need to add the ones. In 263, there are three ones, and in 219, there are nine ones. It might be easier if we start with a larger number here and work out nine plus three more. Nine plus three equals 12.

Now we can’t write 12 in the ones place because it’s made up of one ten and two ones. So we’re going to have to record our answer by writing a one in the tens column and a two in the ones column. It’s important to remember that we’ve done this because even though we’ve added the ones, we’ve given ourself an extra ten to add when it comes to adding up all the tens at the end. The sum of the ones part is 12. Step four, add the partial sums. We found the sum of the hundreds, the tens, and the ones. And now to find the overall answer, we need to combine these back together again.

How many ones do we have? Zero plus zero plus two makes a total of two ones. How many tens do we have? Zero tens plus seven tens equals seven tens. But let’s not forget we’ve got one more 10 to add onto. This was the 10 that we got when we added our ones together to make 12. So seven tens plus one more ten equals eight tens. And because the digit eight is in the tens place, it’s worth 80. Finally, we’ve only got the digit four in the hundreds place, so the total of our hundreds is four. We found the total of 263 and 219 by making partial sums. We found the sum of the hundreds part, the tens part, and the ones part and then finally added together these partial sums to find the overall total. 263 plus 219 equals 482.

Use partial sums to find 825 plus 165.

In this question, we need to add together these two three-digit numbers. But the way they have been written side by side isn’t very helpful. What would be much more helpful is if we were to write them on top of each other. If we write both our numbers into a place-value table like this, then we can see straightaway that it’s very easy to separate both numbers into the hundreds, tens, and ones. Instead of having the numbers all in one go, we can add them in parts. This is what the idea of partial sums means. First, let’s add the hundreds. 825 has eight hundreds and 165 has one hundred. And 800 plus one more 100 makes a total of nine hundreds.

Onto the tens, 825 contains two tens, which is worth 20. And 165 contains six tens or 60. Two tens plus six tens equals eight tens, just the same as saying 20 plus 60 equals 80. And now let’s find the sum of the ones. Both our numbers end in a five. And we know that five plus five equals 10. Now we can’t write the whole of the number 10 in the ones place because a 10 is made up of one ten and zero ones. So we’re going to have to write the total of our ones in this way as one ten and a zero in the ones place. You have to remember this extra 10 when it comes to adding our tens.

Now that we found the sum of the three parts of our numbers, the hundreds, the tens, and the ones, we need to put them all back together again to find the overall total. If we start with the ones column, we can see that all the digits are zeros. And so our answer is going to contain zero ones. If we add the digits in the tens column, we can see eight tens plus one more ten which equals nine tens. And finally, we’ve only got the digit nine in the hundreds column. We’ve used partial sums to find that 825 plus 165 equals 990.

What have we learned in this video? We’ve learned how to use place-value blocks and tables to help us add together three-digit numbers. We added the hundreds, tens, and ones before combining the partial sums to find the total.

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