Lesson Video: Making Three-Digit Numbers Mathematics • 2nd Grade

In this video, we will learn how to decompose three-digit numbers into hundreds, tens, and ones in multiple ways.

15:38

Video Transcript

Making Three-Digit Numbers

In this video, we’re going to learn how to split up three-digit numbers into hundreds, tens, and ones in different ways. Let’s begin with a challenge. Can you make this three-digit number only by using the base 10 blocks at the bottom here? Perhaps the most obvious way and the quickest is to think of each digit’s place value. The digit two in 218 is worth two 100s. The digit one is in the tens place, so this has a value of one 10. And the digit eight is worth eight ones. We’ve made the three-digit number out of two 100s, one 10, and eight ones.

When we write a number like this, we say it’s been written in unit form. We can see how many of each of the place value units we’ve used. We could even write it as an addition. We’ve shown our number as 200 plus 10 plus eight. And when we write a three-digit number like this, we call it expanded form. And it may surprise you to find out that that was only part one of our challenge. It gets trickier.

In part two, we need to ask ourselves the question, now how can we make the same three-digit number, this time using only what’s left of the equipment? This is a bit harder to do now. Can you see why? We’ve only got one 100s block? But we want to make the number 218. Where are we going to get another 100 from?

Well, we seem to have quite a lot of tens down here, don’t we? What if we made 100 out of 10s blocks? We could use the 100 that we have already. And we know that 10 10s are equal to one 100. So we could take 10 of our 10s and use them instead. It’s almost as if we’ve swapped one of our 100s blocks for these 10 10s. But we’ve still got to model the rest of our number. We’re going to need one more 10 and of course our eight ones. This time, we’ve made 218 out of one 100, 11 10s, and eight ones. One 100 has a value of 100, 11 10s are worth 110, and eight ones are worth eight. So we can say 100 plus 110 plus eight also equals 218.

Now, when we’re learning about place value, and we use place value grids to show three-digit numbers, we often say that there’s one digit in each place. And so if we try to write one 100, 11 10s, eight ones in this place value grid, it might be a little confusing. Perhaps a clearer way to show what we’ve done might be to use a part–whole model instead. This way, we can still show that we’ve partitioned the number into 11 10s. But we don’t need to break our little rule of only having one digit in each place.

So we’ve made 218 in two different ways. Although the first way we made this number is probably the most obvious and it’s the usual way that we’d think of, to say that we can split 218 into one 100, 11 10s, and eight ones is just as true. This is a really important fact.

In this video, we’re going to explore lots of different ways to decompose or split up three-digit numbers. And although they might not always seem obvious, they’re still correct. There are lots of ways of making the same number.

Here’s a different three-digit number, 125. How many different ways could we show this number? Let’s start with the way we’re probably going to think of straight away. And that’s one 100, two 10s, and five ones. But what if we regroup our one 100 and show it as 10 10s? Now we’ve got 12 10s, which are the same as 120, and five. We could even carry on regrouping because we know that one 10 is the same as 10 ones. So we could take one of our 10s and exchange it for 10 ones. We now have 11 10s, which are the same as 110, and 15 ones. And we could even keep doing this with our 10s counters. We now have 10 10s, which is the same as one 100 of course, and 25 ones.

There are lots and lots of different ways to split up this number. How about one 100 and 25 ones or even 125 ones? These are just some of the ways that we can show the number 125 using hundreds, tens, and ones. How many more can you find? Don’t forget what we said earlier on. Although each of these different ways of showing this number might not be the usual way we’d think of, they’re just as accurate. The same number can be broken into hundreds, tens, and ones in different ways.

Let’s have a go at answering some questions now where we have to split up some other three-digit numbers in different ways.

Have Liam and Isabella made the same number using place value blocks?

In this question, two students, Liam and Isabella, have made numbers out of place value blocks. And all this question asks us is, are they the same number? Now, at first glance, we might say no. Well, they certainly haven’t used the same place value blocks, have they? Liam’s used a lot more tens, and Isabella’s number contains two 100s blocks. Liam’s doesn’t. So we could look at these models and say, “Wow! They’re not the same number at all.”

But before we jump to the answer no, perhaps we better look a little bit more closely at the actual numbers that these students have modeled. Both numbers have been made out of hundreds, tens, and ones. Liam’s number is made up of one 100. This has a value of 100. Then he’s used 12 10s, and 12 10s have a value of 120, and finally four ones. And if we add together 100, 120, and four, we get the number 224.

Now, what number has Isabella made? We can see two 100s blocks, these have a value of 200, two 10s blocks, which are worth 20, and then just like Liam she’s used four ones. And 200 plus 20 plus four equals 224.

We can see that the answer to the question is yes. Both Liam and Isabella have made the number 224. But how can they do this? They’ve both used hundreds, tens, and ones blocks. But they’ve used them in different ways. Surely this makes a different number. Well, this isn’t true. It’s possible to show the same number but in different ways.

Isabella has modeled the number 224 the way that we would probably usually do this. We’d look at the digits in their place value and we’d say we need to have two 100s, two 10s, and four ones. But as we said at the start, Liam has less hundreds. Can you see what he’s done? Not only does Liam have one less 100, he’s actually got more tens. We know that 100 is the same as 10 10s. And Liam has swapped one of his 100s for 10 10s blocks. Both of the students have shown the same number, but it’s just been broken up in different ways. The answer to the question is yes, Liam and Isabella have made the same number.

Sophia wrote 258 in expanded form as 200 plus 50 plus eight. Choose two other ways of writing the number. 250 plus eight, 205 plus eight, or 200 plus 58.

We can write or represent numbers in lots of different ways. And in this question, Sophia is showing the number 258. We’re told that she’s written it in expanded form. Let’s have a think what this means. We can see from the picture of the place value grid that 258 is made up of two 100s, five 10s, and eight ones. And this is how we’d show the number if we wrote it in digit form or standard form. We just write the three digits two five eight.

Now we know that that digit two isn’t worth two and the digit five isn’t worth five. So one way of showing what each digit is worth is by expanding out our number, breaking it up a little bit into three parts: one to represent the hundreds, one to represent the tens, and one to represent the ones. We know that the digit two in this number has a value of 200 because it’s in the hundreds place. The digit five has a value of 50 because this is in the tens place. And of course the digit eight is just worth eight. And so this is why Sofia can write the number 258 as 200 plus 50 plus eight.

Place value arrow cards are really useful to show what Sophia’s done here. Here’s the number 258. And if we split up our arrow cards into the hundreds, tens, and ones, 200 plus 50 plus eight. Now we’re told to choose two other ways of writing the number. And we’re given three possible answers.

Let’s have a look at the first one, 250 plus eight. Well, we could use our knowledge of addition here. We know that 250 plus another eight is going to equal 258. Another way of finding the same answer is by looking at the first two columns on our place value grid and seeing that these are equal to 250 and then adding the eight ones on the end. We know this has the same value because the digit two still is worth two 100s, the five is still worth five 10s, and the eight still has a value of eight ones. So that’s one of the two ways we could write this number.

But what about 205 plus eight? Can we write the number like this? Well, again, there are several ways we could think about this possible answer. Firstly, if we add the two numbers together, 205 plus eight doesn’t equal 258, does it? It’s 213. Can you see what the problem is here? The digit two is worth two 100s, and the digit eight is worth eight ones. But the digit five in this addition isn’t worth five 10s. It’s worth five ones. This means our second correct answer must be 200 plus 58. And of course, we know this because 200 plus another 58 is 258. It’s the same as taking our two 100s blocks on their own and putting the 10s and ones blocks together. The digit two has a value of 200, the digit five has a correct value this time of five 10s or 50, and the digit eight is worth eight.

Our question showed us one way of writing the number 258. And using our knowledge of place value, we found two more. We can also write this number as 250 plus eight and as 200 plus 58.

Find the missing three-digit number.

To help us answer this question, we’re shown a part–whole model. We’re given both parts, but we don’t know the whole. And we’re told that this is a missing three-digit number. But wait a moment! We know that it’s possible to have a three-digit number without any tens, also without any ones, or even both. But it’s not possible to have a three-digit number without any hundreds. And yet, if we look at the two parts that make up our part–whole model, we can see that one of them represents a number of tens and the other a number of ones. Where are the hundreds? What is this mystery three-digit number?

Let’s try modeling our part–whole model using a place value grid. Here are our five ones and 32 10s. But if we were to try writing this using digits, we know that we could write a five in the ones place. But we also know we can’t write two digits in the tens place. We’re going to need to regroup our number. Although we can show this three-digit number as 32 10s and five ones, if we want to write it using three digits, we’re going to have to change some of our groups of tens. We know that 10 10s are the same as one 100. And in the number 32 10s, we’ve got three lots of 10 10s. And so we can exchange them for three 100s. We can write our number as three 100s, two 10s, five ones.

We’ve used our knowledge of place value here to exchange some of the many tens that we had for hundreds. 32 10s are the same as three 100s and two 10s. And if we also include the five ones, this makes our missing three-digit number 325.

What have we learned in this video? We’ve learned how to decompose or break up three-digit numbers into hundreds, tens, and ones in different ways.

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