Mathematics

2.NBT.A.4. Transcript

This is Common Core State Standards Support Video for Mathematics. The standard is 2.NBT.A.4.

This standard reads compare 2 three-digit numbers based on meanings of the hundreds, tens, and ones digits using >, =, and < symbols to record the results of comparisons. Now the key to this standard is this phrase in blue "based on meanings of the hundreds, tens, and ones digits." So, this implies the necessity of a deep understanding of place value. So, that really needs to be the focus for this standard.

Now back in the first grade, they had a very similar standard; the only difference was that they were comparing two-digit numbers, instead of three-digit numbers, like they are now in second grade. Back in first grade, we had another standard that links to this because of place value, and again, it's understanding that the two digits of a two-digit number represent amounts of tens and ones. And then of course you have the special cases where ten can be thought of as a bundle of ten ones, called a ten. Then of course we have parts b and c.

Now we have a similar standard to that one in second grade, and again the primary difference is that we're talking about three digits instead of two digits, and one hundred being thought of as a bundle of ten tens, and again which is a hundred. This is another related standard in the same grade; we have standard 2.NBT.A.2, which is about counting within one thousand. We also have 2.NBT.A.3, where students are required (they're expected) to read and write numbers to 1000 using base-ten numerals, number names, and expanded form; and this expanded form is important.

So again, let's revisit the idea of place value. Of course, our ones is the most basic unit, and here in this case, we have 7 of them, and then when you have 10 ones. We can bundle those together to form 1 ten.

So, we have a number like 27, the digits 2 is 2 tens and the 7 is 7 ones, which is 27 of course. Another way to look at this would be to take the 2 tens, decompose them into 20 ones, and then we have our 7 ones represented by the 7 in the ones place; and combined that's 27 ones.

Now in second grade, the expectation is that okay we have 10 tens, those can be bundled together to form 1 hundred. Just like earlier, where we bundled 10 ones to form 1 ten. Now, we can go the other direction, we can decompose, we can take 1 ten and break it down to 10 ones, the 1 hundred we can break it down to 10 ones, and then further subdivide that into 100 ones.

And again this is important because these are three different perspectives, 1 hundred verses, 10 tens, verses 100 ones; and of course they're all equal. They are all the same quantity.

Okay, now we're dealing with the base-ten system, so I guess, let's go ahead and just work with this. Let's take the number 127. We can look at the digits individually, and of course, the 1 represents1 hundred, the 2 is in the tens place, so it represents 2 tens, and the 7 is 7 ones; and of course, altogether it's 127.

Now there's other way to interpret the same number using base-ten relationships. For example, if we look at the hundreds place and the tens place, we have 1 hundred and 2 tens. But what we can do is take the 1 hundred and change that to 10 tens. So now we can look at it as being 12 tens plus our 7 ones. So, that's from a slightly different perspective.

Let's look at another viewpoint. We'll take 1 hundred in the hundreds place, break that down to tens, and then we can further break this down to one so we have 100 ones. Let's do something similar with the 2 tens, we can break that down, decompose it into 20 ones; and then of course, we have our 7 in the ones place that's 7 ones, and altogether that's 127 ones.

And this is always the case with any number; it's without any kind of context. It's always understood that your unit is ones. Now students have to understand the symbolism, so it's real important that we cover this. Now they shouldn't have too much trouble with the =, what will give them some trouble, might be where they have a little bit of trouble remembering, well which sign is which, which one is the < which one is the >.

Let's look at this first symbol. I know that there's a lot of little tricks and so forth out there as far as trying to distinguish and remember which is which, but here's a simple little method that students can use without having to memorize any phrases, or rhymes, or anything. Basically, all you have to do is draw a bar on the left, you know vertically, draw another one further to the right, and this is what students can use as far as their comparison. Notice that this is vertical hash mark is smaller then this one, so your numbers are going to be placed accordingly. Here, the shorter mark is on the left, so your smaller numbers should go on the left, and your larger number on the right. Again notice the parallels, this is smaller, so it goes over here; this is larger, so the large number goes over here.

Then for our other symbol, do the exact same thing. Draw vertical hash and a second one further to the right. Same idea, your larger number now should be on the left-hand side, and your smaller number is on the right-hand side.

So obviously, seeing these little tips, its easy for students to keep them straight, that this is the < symbol, and this bottom one is the < symbol.

Let's compare 127 to 318, but we're not ready to base this on the meanings yet. Let's use manipulatives. So here's our representation for 127—1 hundred, 2 tens, and 7 ones. Here's our representation for 318, 3 in the hundreds place, 1 ten, and 8 ones. The first thing we look at is the hundreds plane, 1 is obviously less than 3, so 100 is less than 300. So, we already know that 127 < 318.

Now students need to experience comparisons of some unique situations. So for example, they need to see some examples, get some experience with numbers that have a zero. They also need experience with numbers that have, for example, the same hundreds place. And they also need experience with a comparison like we have here, where the hundreds and ten space are the same in both numbers.

So let's compare 460 to 308. Okay, on the left-hand side we have 4 hundreds, 6 tens, and no one's. For the 308, we have 3 hundreds, no tens, and 8 ones. We look at the hundreds place; we have 400 versus 300. So we already know without any additional comparison that 460 > 308.

Let's compare 251 and 234. Here's our representation for 251. Here's our representation for 234. Notice that we can't make a decision with the hundreds place, because they're the same. So we had to go to our tens place now. So we look at this; let's see, we've got 5 tens versus 3 tens; 50 is more than 30, so we now know that 251 > 234.

Let's compare these two three-digit numbers where the first two digits are the same in each one. So we have 436. Here's our representation of 438. We cannot make a decision based on the hundreds place, because they're the same. Likewise for the tens place, because we have 3 tens each of the two numbers. So it boils down to the ones place, and here we have a 6 compared to an 8. We know that 6 is less than 8. So, we reach our conclusion that 436 < 438.

Now there are other types of examples. One situation that might give students trouble, that they need some experience with, would be when you have numbers where two of the digits are reversed—so for example, comparing 835 to 853. Or, in this situation the hundreds and the tens digits are reversed, 42 versus 24, comparing 425 and 245. Then of course, don't forget comparing two numbers that are the same, where they're equal. Now, the meat and potatoes though, for this standard, is this whole idea that they have to make their comparisons based on meanings of the hundreds, tens, and ones digits. So, without the use of additional help, students need to be able to make these comparisons just by looking at the numbers.

So, let's take that first one, 835 compared to 853. Can't make a decision though, the hundreds place are both 8's (the 3 and the 5 in the tens place); we can make a decision here. Three is less than 5. So, 835 < 853. Our next pairing, well comparing the hundreds place, of 400 is greater than 200. So, we already know that 425 would be larger than 245. And then of course, our last example, all of the place values are the same 781 = 781.

And in closing, by implying instructional strategies you know, such as having students work in groups, having them talk about their solutions (that kind of explains their thinking), use solid objects or manipulatives, and of course, a teacher asking leading questions. By doing all of these things, the following standards for mathematical practice can be addressed, Having them work in groups and so forth, having them talk about their thinking, this would lead to constructing viable arguments, and critique the reasoning of others. We're also modeling the mathematics, we did it attend to precision, and we looked for and expressed regularity in repeated reasoning. Again, for this particular standard, the key is students having a deep understanding of place value.