This is Common Core State Standards Support Video for Mathematics. The standard is 3.NF.2ab.
So it's actually two standards to this. The initial statement for 3.NF.2 states: Understand a fraction as a number on the number line; represent fractions on a number line diagram. Then part a states: Represent a fraction 1/b on a number line diagram by defining the interval from zero to one as the whole partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line. Part b states: Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that it's endpoint locates the number a/b on the number line.
There's a whole lot to these statements, and for some people fractions can be a handful; in fact downright creepy and scary. But let's just go ahead and take this one step at a time, and we'll be fine. Let's look at the introductory piece for this standard that states: understand a fraction as a number on the number line represent fractions on a number line diagram. Now this is a totally different perspective than what we're used to, because the typical instruction at this level is pretty much focused on fractions as part of a whole. But this standard approaches a fraction as a rational number. In other words, it looks at fractions as a subset of the set of real numbers. But there are intersections between those two different interpretations of a fraction. Some things that you are used to that will make this a lot easier to understand and those intersections deal with parts and the idea that they all involve a whole. Now if you look at parts a and b, look at the highlighted parts, a deals with a fraction 1/b. Then b deals with a fraction a/b where a would typically be some number other than one.
Let's look at the first part of 2a: represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole. So what we have here is again the distance from 0 to 1 is going to be our whole and that would be analogous to something that we're used to in a geometric figure like this where the whole would be this circle and its interior. Now the distance or the interval is limited to the distance from 0 to1 with the exception of 0 and 1 that would make 1/b or a/b proper fractions. In other words, 1/b and a/b are all going to be fractions that fall between 0 and 1. Now the next part of the statement says partition it into b equal parts. So we're going to take this whole, this interval, and partition it into so many equal parts. That again is analogous to what you're used to where you take a figure, I guess something like this and cut it up into equal size parts. So again some similarity here that this distance here is part of the interval from 0 to 1. Just like this 1/b is part of this geometric figure and its interior. So what we have is that this is the size of the part 1/b.
But the second part of this statement says that the endpoint of the part based at 0 locates the number 1/b on the number line. So that is something different than the endpoint of this distance this interval locates this real number 1/b. That's different from what we're used to because in a part- whole type of situation there is no number 1/b that deals with a location like we have here on a number line, So that is a definite distinction and difference. In looking at the diagram, we would have, for example, this 1/b would be 2/b and 3/b, so when you start talking about numbers in the numerator other than 1, that leads us into the second part of the standard which would be 2b.
Let's focus on the first part of the first sentence that states: Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Okay so we have our interval from 0 to 1 and we marked off so many lengths each with the size of 1/b, so let's concentrate on this idea of a fraction a/b being a lengths 1/b from 0. So what we have here are these intervals 1/b and we have so many of them and that's indicated as a, symbolically this is what it looks like. We have these intervals, these chunks of the number line, and this tells us how many of them we have. It'll probably make it a bit more sense if we attach some numbers to it. So if we count here, we see that there are 6 total intervals, small intervals from 0 to 1. So our b would be 6, and we want 1, 2, 3, 4 of them. So we've got 4 of these 1/6. That's our number of intervals. We've got four 1/6 and we end up at this location.
Let's look at one more example to add a little bit more clarity. Let's see, we have again our interval from 0 to 1 and we've marked it off into so many chunks. In this case, we have a total of 5 from 0 to 1 and we've counted off 3 of them so it would be 3. So we have 3 lengths of 1/5 away from 0, which symbolically will look like this; 3 lengths of 1/5 3 times 1/5. So throwing in the numbers into our diagram, so that'll be a lot clearer. Again we have 3 of these small intervals or chunks of 1/5 each.
Now let's look at the second statement to part b: Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line. Now the bottom line here and it's a hidden message is that the faction a/b really has two different meanings. Let's look at an example. Here we have 7 total small intervals of 1/b and then we've marked off where we are going to stop after 4 of those intervals. Let's concentrate on that first part, the resulting interval has size a/b. Okay, we're talking about the size of the interval here so then 4/7 in this example indicates the distance from 0. So it is what we're used to. It's an actual amount; it's a quantity. Again it indicates the distance from here to here so we are talking about this interval.
Now the next part of the statement: To recognizing that the resulting interval has size a/b and that it's endpoint locates the number a/b on the number line. So we're talking about the endpoint of this interval. Now we're talking about strictly this right here, this point. So with that interpretation 4/7 would be the location of that particular faction 4/7 as a member of the set of real numbers. So there it's a rational number. Again, reviewing this quickly a/b really has two interpretations in this context. On a number line where one a/b can actually be a quantity, it would be the distance from 0, so we're talking about the distance, but a/b could also indicate this exact location. So there again we're looking at it as a rational number, as one point, one member on the set of real numbers. Again that's the distinction. So this is what is different. What might have been a little bit confusing with these statements in part b is again the two different interpretations. So hopefully, we've clarified some of the confusion with this standard parts a and b to 3.NF.2.