Mathematics

2.NBT.7 Transcript

This is Common Core State Standards support video in mathematics; standard is 2.NBT.7. This second grade standard is in the number and operations in base ten domain, and it reads: add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or relationships between addition and subtraction; relate the strategy to a written method; understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens with tens, ones and ones; and sometimes it's necessary to compose or decompose tens or hundreds.

So, the key idea is to add and subtract within 1000. Now, there’s some foundational knowledge that the kids really need. The first one would be that only like items can be combined. If we look at the standard, one of the things that is emphasized is that you can only combine hundreds and hundreds, tens and tens, ones and ones. But, of course, prior to that, the kids understand that, for example, if I had 2 plus 3, there’s a huge assumption that the 2 and the 3 represent the same stuff. So, for example, it might be two cats plus three cats gives me a total of five cats. So, again, real important idea that only like items can be combined, although in some contexts or situations, there won’t be a specific mention of what the items might be.

The second foundational idea here would be that the kids have to have a fundamental understanding of place value and the associated symbolism with it. So, with this fundamental understanding, they have to, of course, understand that each digit represents a certain place value and that you can only have one digit in each of the place values. Now, in the previous grade, students were dealing with numbers within 100. So, they had to gain the flexibility to see that numbers such as 50 could be interpreted as not only as 50 ones, but as five tens. So, again, they need to understand that this can be five tens, or it can be 50 ones. In second grade, this gets taken up a notch where we are now working with numbers within 1000. So, now, we’re dealing with the hundreds place. So, now, the students have to gain the flexibility to read a number such as 300 and to understand that that can be three hundreds, or it could be 30 tens, or 300 ones.

Now, let’s look at a number such as 354. I have this organized by place value and filled in zeros for the appropriate placeholders, although it’s not really a placeholder. I mean the zero here does indicate that they’re zero tens. But, the typical interpretation though, for a number like 354 would be that it’s 300 ones, plus 50 ones, plus four more ones for a total of 354 ones, which, of course, then ties back to this idea and to this idea over here that is stressed in the standard of again, combining only hundreds with hundreds, tens with tens, and ones with ones.

So, let’s go to through an example of each. Let’s start with adding. So, if we’re going to add and subtract within 1000, let’s say that we were adding 143 with 281. It’s always a good idea to expand this out so that the kids understand this idea here that is being pushed by the standard that, again, you have to deal with combining hundreds with hundreds, tens with tens, and ones with ones. And when we do something like this, it solidifies that idea that each of those digits does represent a certain place value. And it also helps to maybe do something like this to, again, keep each of those place values separate. Now, it helps to also take an example like this and use manipulatives, some type of physical, concrete items, so that the kids really see what’s going on.

So, here we have our example using some manipulatives, some blocks where we have 143 plus 281. So, if we go to through the process, okay, here’s three ones plus 1 one, for a total of four. And then we have four tens and eight tens; combine those, and then we have 100 plus another 200. Now, we have a little bit of a problem here. We’re okay with the ones. We’re okay with the hundreds. But, we’ve got too many tens. We’ve got a total of 12 of them, and by convention, you know we can only have one digit for our place value. So, we’ve got 12. The biggest number I can have here is 9, so I’m going to have to take some of these out of here.

So, we know that 10 tens is 1 one hundred. So, if we just take these 10 tens out of here, and move them over here with our other hundreds, see, that’s equivalent. So, now, we’re all set. So, physically, we can tell that our solution is four hundreds, plus two tens, plus four ones. So, we have 424. So, now, going back and connecting what we did with the physical model, this is what happened to us. But, again, we couldn’t have this here, because that presented a problem. So, what we did was, this was 10 tens and two tens, and we took the 10 tens and actually rearranged it to where it’s one hundred. So, then this added one more hundred, which gives us four hundred, and then these 10 tens went over here. So, we’re left with two tens with our four ones for a total of 424.

Let’s take a subtraction example. Let’s take 321 minus 145. Now, this example was chosen because it basically has the double whammy where we’re going to have to decompose and do things with two place values. So, again, doing something similar to what we did a while ago, split this up to where again, the meaning of the place value is reinforced by writing it all out. And then, do something like this to organize our place values and keep them separate.

Okay, so, again, we have a real problem here. So, I think in situations like this, some solid, concrete manipulatives will really help the kids see what’s really going on. Okay, here’s our situation. We have 321, and we’re supposed to take away 145. So, let’s start with our ones place. Now, in mathematics, you can take five from one, but we’re dealing with whole numbers here. So, when you’re dealing with whole numbers, you, in fact, cannot take five ones from 1 one. So, we have a problem.

So, what we’re going to have to do is decompose one of the tens into 10 ones. And so, now, we don’t have tens any more. Now. we have ones, so we need to put these in the ones place value. So, now we do have enough to be able to take away five, so we do. There’s 1, 2, 3, 4, 5, so we take those away, and we are left with this many ones. Okay, now we move over to our tens place. Again, we’re dealing with whole numbers, so I can’t take four tens from one ten, so we do just like we did a while ago. The only difference is the size of our actual values. So, I only have one ten now, and I need to convert one of these hundreds over here to 10 tens. And so, of course, the tens need to go over here in the tens place. Okay, so, now we have plenty of tens, and so, now, I can take four of them away. So, here’s four that I’m taking away. And so, what I am left with is this many tens. And then, I’ve got the two hundreds. Now, I’m supposed to take away one of them, so I will. This is what I’m left with. So, now, I have 1 one hundred. I have 1, 2, 3, 4, 5, 6, 7 tens, and two, four, six, six ones. So, our solution should be 176.

Now, it’s important to take what we just did physically using the concrete models, and we need to translate it to writing and symbolism. So, if we look at that same scenario, 321 minus 145, again, here’s the problem that, with the ones place, that with whole numbers, you can’t take five ones from 1 one. So, what we had to do was look over to the tens place, and we had to take one of the tens and convert it to ones. So, we took one of the tens and converted it to 10 ones. And, of course, we can’t put the 10 ones there. They have to go over here. So, we switched the 10 ones over there. We already had one there, so we have a total of 11 ones. And now we can subtract five ones from that, so that gave us six ones.

Then we had a similar problem with the tens place. We couldn’t take four tens from one ten, not in whole numbers anyway. So, we had to take one of the hundreds and convert one hundred to 10 tens. So, again, the 10 tens, we can’t put them over here. We have to put them over here. So, we have 10 tens. We already had one ten there, so that gives us 11 tens minus four tens is seven tens.

And over here, we have two hundreds left minus the one hundred for a total of one hundred. So, that’s 176, which would be 1 one hundred, seven tens, and six ones. Now, to connect back to this idea of adding like items, plus when you see the number 176, that’s understood to be 176 ones. So, to reinforce these ideas, it might be a good idea to take that one hundred and change it all to ones, and take the seven tens and change that to ones, and then my six ones to solidify the idea that what we really have here is 176 ones.

So, this standard was all about adding and subtracting within 1000 with a special emphasis on the idea that you can only add and subtract hundreds with hundreds, tens with tens, and ones with ones.