This is Common Core State Standards support video for Mathematics. The standard is 2.NBT.B.5.
The standard reads: Fluently add and subtract within 100 using strategies based on place value, properties operations, and/or the relationship between addition and subtraction. Now what's nice about this standard is that there are other standards that focus on addition and subtraction, and we can address those simultaneously. Back in first grade, there was standard, 1.NBT.C.4, that focused on adding within 100, with the focus on using concrete models, then in the same grade level, their standard 2.NBT.B.7, that deals with adding and subtracting within 1000 and also concentrating on concrete models. Also, with a secondary, we have standard 2.NBTB.6 that talks about adding up to four two-digit numbers. Standard 2.NBT.B.9 deals with students explaining why the strategies work. We have standard, 3.NBT.A.2, which deals with fluently adding and subtracting within 1000, and then we also have at the next grade level, grade 4, standard 4.NBT.B.4, which concentrates on fluently adding and subtracting multi-digit whole numbers.
Now if you look at the standard back in the first grade, 1.NBT.C.4, that one lays a foundation for this standard that we're dealing with now. And then we have standard 2.NBT.B.7 that actually parallels standard 1.NBT.C.4, with only a couple of distinctions between them. Standard 1 focused primarily on adding within 100, whereas when you go up to grade level 2, standard 2.NBT.B.7 takes it up to within 1000 and also throws in subtraction. Looking at the next grade level, grade 3, this standard, 3.NBT.A.2, is actually a direct result of this standard 2.NBT.5, but it takes it up a notch as far as place value to within 1000, instead of within a hundred. And then, we go on to standard 4.NBT.B.4, which is pretty much the culmination of adding and subtracting with whole numbers. Again when we addressed 2.NBT.B.5, we’re also addressing 2.NBT.B.6 and 2.NBT.B.9. Now let’s look at these a little bit closer; notice that the idea between this standard is fluently adding and subtracting, whereas back with the first grade foundation 1.NBT.C.4, the focus was on concrete models or drawings. So this idea of fluently implies an expectation of adding and subtracting with speed and accuracy, without the aid of concrete or visual aids.
Now let's look at this statement toward the end of this standard, 1.NBT.C.4. Understand that in adding two-digit numbers, tens are tens, ones are ones. With that emphasis, what we're really doing is laying this key foundation that only like items can be combined. Very, very important property, so for example let's say we have 4 + 5 = 9, there is an implied understanding that we have to have something for example like 4 dogs plus 5 dogs is 9 dogs, but even without context something like 3 + 7 = 10 would be understood to be 3 ones plus 7 ones is 10 ones. Again, all the same like items, even though there wasn't a context. Now let's look at this last phrase; sometimes it is necessary to compose a 10. So what this emphasizes is the idea of place value, so that's a second critical foundation for the fundamental understanding that is needed for 2.NBT.B 5. Again this idea of place value is very important, but in particular, the key idea is that place determines value. So where each digit is located determines what it's worth.
So let's look a little closer to place value. If we have something like 27 + 35, there’s an understanding, okay, that we have 2 tens and then 7 ones, we're going to add 35, and that is composed of 3 tens and 5 ones. So what we did back in first grade where we’re doing concrete models, we're going to combine our tens and we're going to combine are ones. Now with place value, what's going to happen is that we only have two slots, two positions to put our solution in, because again each place value is determined by, and represented by one digit. So we have a little bit of a problem here, because 7 plus 5 is 12, so what we're going to have to do is take 10 of the ones and convert them to 1 ten. And so this is our result; so now our 1 ten can be combined with the other 5 to give us our 6 tens, and 2 ones, which of course is understood to be 62 ones.
But this standard pushes the idea of fluently adding and subtracting. So if we go back and approach the same problem, what students are expected to do now is something like this, but students still understand mentally that this is what’s really happening, that we have a 12, which is where our 1 comes from, that’s our 10, and of course the 2 in the one’s place. Then we can go on and finish out the problem 1, and 2, and 3 to be 6, and of course those are tens. And we have our final solution of 62. Now let’s look at a subtraction example. Say we have 43; we’re going to subtract 28. Back in first grade with the focus on concrete models, this is what we would have had—4 tens and then 3 ones. Now we're supposed to take away 8 ones from the 3 ones, but we don't have enough; we can't take 8 ones from 3 ones. So what we have to do is convert 1 of our tens to 10 ones. And this is actually our result; we have converted the 43 to 3 tens and 13 ones, so now it is possible to do the subtraction. We take away 3 and 5 more, so we've taken away our ones, so 13 would take away 8, that's 5, and then we have to subtract our tens. We have to subtract 2 of them. And here is what we have left; so we have 1 ten to go with the 5 ones, and of course this is understood to be 15 ones, for our final solution. But this standard again focuses on doing this fluently and really mentally. So students do this mentally though they do understand what happens that we have converted again the 43 to 3 tens, and 13 ones, and now we can continue on and solve it. Do the computation—8 ones from 13 ones is 5 ones, and then 2 tens from 3 ones gives us 1 ten, for our final solution of 15, which of course is understood to be 15 ones.
Now let’s focus on properties of operations. At this level, we’re pretty much going to utilize the commutative property of addition and the associative property of addition. So the commutative property of addition says that I can take the two numbers that I'm adding and change the order, and I still get the same results, as students are used to working a lot was just single-digit numbers. Now we have them do something like this: 52 + 34, they get their solution. And then we have them do the same problem with the order switched; you'll still get the same solution. So they’ll see that the commutative property works; it doesn't make any difference what the size that the numbers are, it’s still going to work. The associative property of addition deals with changing the grouping. So without any parentheses, it's understood that something like 9 + 3 + 7; the 9 and the 3 constitute the first group. But here notice that it would be easy to add 3 and 7, that’s nice and easy that’s 10. So I'd rather regroup this with the 3 and the 7 first. So in essence, what we've done here is change the grouping that is our associative property. So let’s say we have this problem to begin with 3 + 9 + 7. Now we notice that 3 and 7, that's 10. It's a lot easier if I want to group those together first. So we need to apply our commutative property of addition and change the order of the 9 and 7. So we do that, and in essence when we did that we’re also now going to apply the associative property, because when we change our order, we did change our order from 3 + 9 initially for our first addition, to where the situation now, we have 3 + 7 instead. Let’s look at a second example where we combine the properties. Again we notice that 12 and 38 that would be a little bit easier, because I'm going to end up with the multiple of ten. So we use our commutative property and switch the order, the 23 and the 38. And again now we have the associative property, because we did regroup instead of 12 + 23. Initially, we have 12 + 38. Looking at our properties one more time, let’s say we have 29 was 38. Now what we can also do is combine that with the idea of place value, where we break those down, the 29, to 20 + 9, the 38 to 30 + 8, and then typically we want our tens together. So I want to change the order around, so I need to change these two, the 9 and the 30. And now this is a result, so now I can combine 20 and 30, that’s 50 plus 9, that’s 59. So we have that.
Now this might pose a little bit of a problem; this type of example might be problematic for some students. So let's do this, let’s change the order to use our commutative property. And now, let's look at the 59. Let's use the relationship between addition and subtraction; now 59 is pretty close to 60, so we can change the 59 to 60 -1. Now for a lot of students, this would be an easier problem to solve (easier computation) where we would just add 8 and the 60, to be 68, and then simply subtract 1 to get our final answer 67. Most of you probably know from experience that a problem like this, 80 - 34, can be a real headache for students; there's just something about having to decompose but having a 0 as our one's place. So why don’t we do something like this? We can break down the 80 to 79 + 1, and now we subtract 34; there's no need for any more composing and decomposing. So we can simply subtract, and we get 45. But of course, don't forget the 1, and we get our final; the solution is 46.
We can add a little bit more value to this standard, because what we can do is lay the foundation for the additive identity. And of course the additive identity simply says that, for example, if we add 2 and subtract 2, that’s 0. So that doesn't result in any change in the original quantity. However at this level, we need to limit this to the addition context, because if you try to do this with subtraction, it will really get confusing. So let's say we have 48 + 26, and let’s throw in another example, 50 + 24. Now when students do the actual addition, we get 74 for our first solution; we also get 74 for our second solution. Now why is this? What did we do; why did that work out that way? Well notice that to get from 48 to 50, I would add 2; and to change 26 to 24, I would subtract 2. Notice that we added 2 or subtracted 2, for a net change of 0. If we do this with concrete examples, this is all we did. We take 2 one's over here, with the 26, and we just move them over and give them to the 48. So in essence again, this is what we did; we added 2 to the 48, and we got 50. We subtracted 2 from the 26, and we got 24.
Let's say you had this combination 24 + 37, and students solve it and get their solution. And then you give them this second example, 21 + 40. And guess what they're going to get? The same solution. What actually happened here? Well to go from 24 to 21, we have to subtract 3. But to keep this equivalent, we’ll have to add 3 to the 37; but that makes it a nice 40, which would be a nice number to be adding. Let's try this again; now this becomes 20 + 41. What happened here? Let’s see, we have to subtract 4 from the 24, but then we’ll have to also add 4 to 37, and again we get 20 + 41, where one of the numbers is a multiple of 10, and it makes it easier to add. One more, let’s see one more example; let’s say we end up with 30 + 31. How do we do this? We added 6 to the 24, and then we subtract 6 from the 37, for a net change of 0, which gives us a 30 + 31, which gives us our same solution of 61, that we would have gotten back with 24 + 37.
By doing exercises like this, we're actually going back and reinforcing standard 1.OA.C.6, which states, add and subtract within 20 demonstrating fluency for addition and subtraction within 10. So again, we're doing a lot of this. We’re doing a lot of adding and subtracting within 10. Know we’re also helping with standard 2.OA.B.2, which states fluently add and subtract within 20 using mental strategies. And so again, this will give them a lot of practice with that standard also. If we look at the practice standards, the standards for mathematical practice, if we look at the first four, well one could argue that we've addressed number two and number three, where the students are reasoning abstractly and quantitatively, and the expectation on number three would be that they are constructing viable arguments in critiquing the reasoning of others. And when will look at the last four, we're definitely looking for and expressing regularity in repeated reasoning. Students start to see some of the patterns and similar strategies that we utilize to address standard 2.NBT.B.5