This is Common Core State Standards support video in mathematics. The standard is 2.MD.B.6. This standard states: Represent whole numbers as lengths from zero on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, and so forth, and represent whole-number sums and differences within 100 on a number line diagram.
Let’s investigate what other standards are related to this one. If we focus on the idea of length, back in first grade there was standard 1.MD.A.2 that talks about expressing the length of an object as a whole number of length units, by laying multiple copies of a shorter object end-to-end. If we investigate the connection a little bit deeper, let’s look at a number line, and let’s consider a length of four. The important concept here is lengths from zero, and the first-grade standard adds in the strategy of laying multiple copies end to end with no gaps or overlaps.
So, the first-grade perspective would be to do something like this: take a length of one, then another length of one, and lay them end to end. Repeat and repeat again. So, there’s our length of four from the perspective of the first-grade standard. The difference though, with standard 2.MD.B.6 is that this is how students should view a distance of four. It’s a length that starts at zero and ends at four, but they’re viewing it as just one composite unit instead of four units of one laid end to end.
If we look at this idea of a number line diagram, in the same grade level, there’s standard 2.MD.A.1 that includes rulers, yardsticks, and meter sticks, and those are physical manifestations of a number line diagram. A related standard, that is also in second grade, is 2.MD.B.5, which is the standard right before this one. The connection is the idea of the number line diagram connecting to 2.MD.B.5 with the idea of drawings of rulers. Standard 2.MD.B.5 is also connected to 2.MD.B.6 through this idea of the addition and subtraction within 100. Both of those standards involve, again, this idea of addition and subtraction within 100.
Another connection with the idea of a number line diagram is with standard 2.MD.D.9, where that standard talks about making a line plot where the horizontal scale is marked off in whole-number units. That corresponds to a number line diagram. At the next grade level, in Grade 3, there’s standard 3.NF.A.2. This standard involves understanding a fraction as a number on the number line, and it includes representing fractions on a number line diagram. The key difference is that in third grade you have the added complexity of including fractions, whereas in second grade, we’re dealing with a number line diagram that is restricted to whole numbers.
Let’s revisit the second-grade standard 2.MD.A.1. There are a lot of parallels between these two standards, and what we learn from one can be transferred to and used to support the other. A fairly obvious connection is the idea of a ruler connecting back to the idea of a number line diagram. One bit of caution though—in second grade, students may not be ready for this idea of a number line. They may not be able to handle this idea of the number line extending infinitely both to the left and to the right. Because of that, it might be a little bit better idea to limit your number line diagrams to actually being number line segments. Again, this way they understand the numbers because you don’t have them going infinitely on, which, again, could be confusing to students at this level.
Let’s focus on this idea of representing whole numbers as lengths from zero. Typically, we would do something like this. This is a distance of one. This is a distance of two. But this approach might be too abstract and lead to lack of conceptual understanding. A better approach at this grade level would be to actually use your physical representations, some numbers strips, where your distance of one would look like this. A distance of two would be physically represented by a strip that’s a length of two, and so forth. To address this standard, it would be a good idea to go ahead and construct a lot of copies of different-length segments—lengths of one, lengths of two, lengths of three, and so forth. In this way, you can address the last part of standard that deals with sums and differences.
Well, let’s look at that idea of sums and differences within 100 on a number line diagram. Let’s start off simple with 1 + 2. So, we have a distance of one initially. We start at zero and go to one. Then we’re adding two. So, we start from one and go two more to the right. We’re at three, so that is our solution. That might be too abstract though. It might lead students away from the central idea of this standard, which is whole numbers as lengths from zero.
Let’s try this again. Let’s use the same example, 1 + 2. Key idea here is that we start off with zero. So, that’s what we do, but let’s use those strips to represent our distances. Here’s our distance of one. So, we’re going to add two, but let’s not forget this idea of laying these distances end to end with no gaps or overlaps. So, we find our little strip that’s a length of two. Move it over. Lay it next to the one, again with no gap or overlap. This way, we can see that we are at 3, and that is our solution.
Let’s take a subtraction example. We start off with seven, so we need to get our strip that’s a length of seven. But, what’s different here is that we are going to subtract, so, we need to start at seven and go to the left, and we’re going to go a length of three, so we need to take our length strip that’s a distance of three. We’re going to subtract, however, so we need to move it over to where we can start at seven and go to the left. We go to the left three, and that’s what we’re going to take away. This is what we have left, so we are at four. That is our solution, but let’s take out our length strip of four. We need to use that. So, that’s our solution, four. A good reinforcement here would be to take all three of those strips, the 7 and the 4 and the 3, and use it as a way to reinforce your basic addition facts. So here, students can tell that 3 + 4 is equivalent to 7, and then you have the related fact that was the problem here of 7 - 3 being equal to 4.
This standard limits us to sums and differences within 100, but it’s a good idea to limit the sums and the differences to 10 at the beginning. This way, we can embed the idea of whole numbers as lengths from zero on a number line. Then we can extend the sums and differences to 20 to further embed the idea of whole numbers as lengths from zero on a number line. It makes sense to limit ourselves to 20. That way we can, in fact, have our physical representations, all these different lengths of strips to enable us to physically model these problems.
Let’s take an addition example, 6 + 8. So, we need our strip that’s a length of six. That’s what we’re starting off with, and we lay it to where it starts at 0. We get our other strip that is a length of eight because we’re going to involve eight in this problem. We’re going to add, so we move this over. Lay it end to end—no overlaps. We can see that our solution is 14. Once again, it’s a good idea to have all three of those strips, 6, 8, and 14, to reinforce that basic addition fact that 6 + 8 is 14. This will help, again with memory and with conceptual understanding.
Let’s take an example that involves subtraction. We need our strip that is a length of 15. That’s our beginning point. We need a strip that’s length seven. But, we are subtracting, so we need to move this over to where we’re starting at 15, and we go seven to the left. This is how much we need to take away. This is how much is left, so we are at 8. Let’s find our length strip that is a length of 8. To double-check everything, take our 8 and our 7 and lay them end to end, and we do, in fact, have a total distance that’s equivalent to the strip that’s a length of 15. Continue with sums and differences within 20 until the idea of whole numbers as lengths from 0 on a number line is firmly entrenched. Then, we can transition students to finding sums and differences within 100.
Now, we have a number line diagram that’s a little bit more complex. There are a lot more marks because we’re going 100 units. Let’s take a representative example, 26 + 37. So, we start at 0, go a distance of 26. We’re going to go to the right some more because we’re going to add. We’re adding 37, so we move 37 to the right, and it looks like we end up at 63. So, that is our solution for this problem of 26 + 37. We have a little bit of a problem though, when we start getting to these larger numbers because buying or creating multiple copies of concrete strips of lengths from 1 to 100 really could be difficult. That’s really a key reason for focusing on sums and differences within 20 first. That way, once conceptual understanding is achieved with your limitation of subtracting and adding within 20, then students can better transition to the more abstract process of sums and differences up to 100.
So, let’s take 26 + 37. We start off with 26. We go to the right from zero. We can do this by counting, and then start from there, and we have to go to the right. We’re going to count 37 to the right, and again we end up at 63. Let’s take a subtraction example, 86 and we’re going to subtract 47. One strategy that can bridge the concrete and the abstract here would be to use markers or highlighters to illustrate the distances. We could start off with that distance of 86 and highlight from 0 over to 86. Then we’re going to take away 47, so we have to start at 86 and go to the left a distance of 47. So, we could take a marker and actually start at 86 and mark from there to the left a distance of 47. Once we do that, we can see that this is how much is taken away and what’s left would be a distance of 39.
So, that is our solution, and again, good idea to double check this with addition—double- check to make sure that your solution of 39 combined with 47 is, in fact, your starting point of 86. Once students have done a lot of these examples with markers, then they can transition to really being a lot more abstract and just counting on the number line. If we just involve counting, we start at 0. We go to the right 86. That’s our starting point. We’re going to the left since we’re subtracting, so we start now at 86 as our second step. Go to the left a distance of 47, and just like before we end up at 39. That is our solution. Let’s ensure that our answer is correct. So, we do the opposite operation. We add 39 with 47, and that does come out to be 86, so we know that our solution of 39 is correct.
Let’s look at our standards for mathematical practice. By addressing this standard and using some of the activities that we used, we would reason abstractly and quantitatively. Students would construct viable arguments and critique the reasoning of others. If we look at the remaining four standards of mathematical practice, students would use appropriate tools strategically. They would attend to precision, and they would look for and express regularity in repeated reasoning.