This is Common Core State Standards Support Video for Mathematics. The standard is 1.NBT.4.
This is a pretty long standard. It states: Add within 100, including adding a two-digit number and a one-digit number and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations and/or the relationship between addition and subtraction; using concrete models or drawings and strategies based on place value, properties of operations and or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten. Again, this is a long standard and there’s also some prerequisites that we have to make sure that students have. First of all, adding within a 100 using two-digit and one-digit numbers really necessitates knowledge of place value. So that is critical, and this in turn, requires that students fluently compose, decompose, and transition between ones and tens. So it's important that when we have 10 ones that the students can then visually or actually physically take those and rearrange them in such a way of being one 10. Again a critical prerequisite is absolutely necessary for students to be able to handle two-digit computation.
So let's take the number 17. Students have to be flexible and see this in two different ways. They need to see this as 17 individual ones, but they also have to see it again as critical and that they interpret this also as one ten and seven ones. So let's start off with the example of adding a two-digit with a one-digit number. Now it's important that students get some experience doing these horizontally. If students have only experienced seeing a problem like this set up vertically, they may not know what to do with this on a test; especially if they have to set this up vertically themselves. If they don't know how to line them up correctly vertically then mistakes will be made. So again let's set these up horizontally to give students experience with that. So here students might see this as 17 ones and 5 ones and just combine them to be 22 ones. But not all numbers are going to be that small. So it's important again that they do this using place value. So setting this up as 17 being 1 ten and 7 ones plus our 5 ones, students can then physically take the manipulatives and combine all of the ones to be 12 ones. Now let's combine all of this to where we have our numerical representation. We actually have the physical objects, and we also have this deliberately set up where we have columns and where we separate the tens and the ones. To emphasize this idea that you can only add like items, for example, you can only add tens with other tens or you can only add ones with other ones. Now notice this violates what we are allowed to do. We can't do this. We can't have two digits in the one's place. Here's where the idea of having to convert 10 ones to 1 ten comes into play. So students have to realize, okay I have way too many ones. So if we subtract 10 from 12 we get 2; so this breaks up to 10 and 2. So now students need to take those 10 ones and rearrange them to where they can see that those 10 ones are going to become 1 ten. Likewise when they're working with the written aspect of it, the 12 ones here and the 10 ones became 1 ten, which got moved over here. So now we can actually do the combining. We take our 1 ten plus another 10 and that then becomes 2 tens, and we have 2 ones left here. Same situation over here, we went ahead and did our combining. Let's get rid of our columns, and it looks more like what we would typically see in the vertical format. So now our 2 tens and our 2 ones are combined to be just 22, which in fact is 22 ones.
The standard calls for adding a two digit with a multiple of ten, so let's try that. Let's try 17 plus 50. Again let's set it up horizontally to give students experience with that. So here's our physical model using our manipulatives. Again some students might just say okay 17 ones and 50 ones is 67 ones. But again, this would be like dealing with money. We have 67 pennies; that is a lot of work to tell you the truth. So nope let's do this with place value. So the 17 is 1 ten and 7 ones, and 50 is simply 5 tens. Okay let's get our manipulatives. We have one 10 and 7 ones and our 5 tens over here. Now, we can only combine like items so that is a little tougher to see horizontally. What we need to do though is change the order. We need to use our commutative property. Well, we take the ones and move them over here. Then we take our tens and move them over here. So we have again tens with tens that we can combine together and ones with ones. Again we have it written out. Here we have our physical model, and then we also have this vertically the way the students are going to see it, but we've added again the columns to emphasize adding like items. So we have 1 ten and 5 tens. Well that's simple enough. We can combine those and then there are no other ones to combine. So this one actually turned out to be fairly simplistic. So now we have 6 tens and 7 ones all together. Which of course when we convert it, this is how they're going to see it on an exam. Again 17 plus 50 ends up being 67 ones. Our 6 tens and our 7 ones is 67—understood to be ones. So now let's combine a two-digit with another two-digit. This breaks up into 30 plus 4 and 20 plus 7 and when expressed as ones would look like this. But we need to work with place value. So let's convert this to where it is 3 tens and 4 ones. We have to combine that with 2 tens and 7 ones. But again, we need to use our commutative property and switch these around. We need to move the tens over here so we have tens together, and to need to move our ones over here so we can have all our ones together. Having done that, now we can do our combining. We added our tens—3 and 2 that gave us 5 tens. But we have a problem again with the ones. We added 4 and 7 that gives us 11 ones. But we cannot express it this way. Again students need to realize that, hey I need to take the 11 ones to let us see that 10 from 11 is one. This breaks down to 1 and 10, so we have 10 ones. Same idea expressed over here but in different format. These 10 ones right here, I need to convert those. So we convert our 10 ones to 1 ten and so now we're set. Now we can combine all of our tens, this 10 with those other 5 and we're only left with just 1 one. Next move our 10 over to combine them—6 tens plus 1 one—then change this to where it looks like it is normally stated as a standard algorithm. So we ended up with 61 – 6 tens and 1 one.
Now when doing examples, it's important that students see examples similar to this one where we had 4 tens and 3 tens. That is 7 tens, but when we add the ones, we get exactly 10 ones, which converts to 1 ten. So they need some experience with that, especially here where again it's important that they realize that that, hey I need to have a 0 here, because I have 0 ones and I can't just have an 8 because not everybody might see that's 8 tens. So again it's important that students get this kind of experience. This was a long standard. There was a lot to it. We added within a 100, and we added a two-digit with a one digit with a multiple of ten with another two-digit. We used drawings, we used some models, we used manipulatives, and we used strategies especially based on place value and our properties of operations. We made sure that we had it also in writing both horizontally and vertically and in the vertical mode we made sure that we split them up into columns so that students could see adding tens with tens and adding ones with ones. That was this idea here—that they have to be like items. Now we did try to focus on examples where in almost every case, it necessitated that we compose a ten.