This is Common Core State Standards support video in mathematics. The standard is 3.NF.A.1. This standard states: Understand a fraction 1/b as the quantity formed by one part when a whole is partitioned into b equal parts: understand a fraction a/b as the quantity formed by a parts of size 1/b.
It’s interesting that the foundation for fractions actually begins back in Grade 1 in the Geometry domain. Standard 1.G.A.3 talks about partitioning circles and rectangles into two and four equal shares. Then, in second grade, we actually have a continuation of standard 1.G.A.3 with standard 2.G.A.3. This second-grade standard is actually an extension of that first-grade standard. The main difference is that for the second-grade standard, instead of having just two and four equal parts, they throw in thirds, three equal parts.
There are other standards that are connected to standard 3.NF.A.1. A continuation in this series of standards that deals with fractions is standard 3.NF.A.2 that deals with understanding a fraction as a number on the number line. Then we continue with standard 3.NF.A.3 that really focuses on comparing fractions, whether they are equivalent or one is larger than the other. This third-grade standard sets the foundation for standard 4.NF.B.3b. This fourth-grade standard deals with decomposing a fraction into a sum of fractions with the same denominator.
If we concentrate on the second part of standard 3.NF.A.1 that deals with understanding a fraction a/b as the quantity formed by a parts of size 1/b, we will see that it connects directly to standard 4.NF.B.4a. This standard deals with understanding a fraction a/b as a multiple of 1/b. To have a better understanding of what our limitations might be, if we look at the standards, back in the Grade 3 expectations for this domain, we see that fractions are limited to those with denominators 2, 3, 4, 6, and 8. So, that’s the denominators that we will be dealing with here.
Included in this standard is this very, very important term, the idea of equal. If we look at this figure, that is divided into fourths. However, this second example—yes, it’s divided into four parts, but they are not equal parts. So, on that second example, we cannot say that, that is divided into fourths. It is very important that students realize that in many contexts, a geometric figure can be divided into the same number of equal parts, but in different ways. If we go back and revisit that second-grade standard, 2.G.A.3, the last statement deals with recognizing that equal shares of identical wholes need not have the same shape. So, that important idea can be reinforced while addressing standard 3.NF.A.1.
Let’s do an example. Let’s say the task is to divide this rectangle and its area into two equal parts. Some students might go ahead and divide it into two equal parts this way, but then you might have others that will do this. That’s two different methods so far, but then you might have students that figure “why bother measuring? Why don’t I just draw a diagonal?” So, we have this additional way to do this. Then, some students might decide to also do a diagonal, but starting from a different vertex. So, we see that students had all kinds of options here. They actually could have done this in four different ways. Students have taken this rectangle and divided into two equal parts, but all those equal parts don’t look the same. They don’t have the same shape, but that’s okay.
Since fractions consist of equal-sized parts, activities to address this standard can also be used to lay the foundation for the concept of congruence. What’s interesting is that the term congruence doesn’t appear in the standards until eighth grade, but that doesn’t mean that we have to wait until Grade 8 to start doing some things with congruence. Let’s take that example where students divided the rectangle into two equal parts by using a diagonal. What we can do is cut that rectangle into two triangles. Then, let’s just move it over a little bit. At this level, students would simply need to see that they’re the same size. One way to do that would be, well, we need to rotate it around to get them in the same position. Now we can slide it over; students have now done an informal proof, but it’s pretty obvious to them that hey, those two triangles are exactly the same shape and exactly the same size. It’s not necessary to use the word congruence, but at some point in these early grades, teachers need to establish the idea of the difference between congruence and equality.
When looking at this standard, what might make this really difficult is the interpretation of all those variables. So, why don’t we ditch the variables for now, and let’s just use numbers. If we look at the first part of the standard, let’s deal with six equal parts. So, let’s take a circle and its interior, and we split it up into six equal parts. We just want one of those, so let’s shade that in to identify it. Now let’s look at the second part of the standard, but again, let’s forget the variables. That makes things a little complicated. Let’s just plug in some numbers, and let’s go with a size of 1/6. Let’s represent this symbolically. We start off with 2/6 being equal to...wait, let’s not deal with just symbols yet. Let’s go ahead and use the actual physical pieces so that students can really see what’s going on. Then, underneath that, let’s go ahead and just use our numeric representations. By doing this, students will have a much clearer picture of what’s going on.
Let’s do another example. Let’s say this time we have three parts. The size again, is 1/6. So, let’s shade in three parts. Just like before, let’s represent this. But before we go to just numeric symbolism, let’s do the actual pieces. Now we can go ahead and use nothing but numeric symbols. Again, students will have a much better idea of what this looks like and how to connect the abstract with the concrete. Let’s take this a step further and go with four parts. Again, we are dealing with 1/6 as our size. Let’s shade in four of those parts. Just like before, let’s use our actual physical representation, or if you don’t have a physical representation, at least do pictures. Now we can represent this strictly with numerals. This third-grade standard really doesn’t call for what we’re doing with the symbolism, but by doing this we’re helping out the fourth-grade teachers. Standard 4.NF.B.3b deals with decomposing a fraction into a sum of fractions with the same denominator. Notice that that is exactly what we’ve done. So, we’re setting a solid foundation for what’s going to come later on.
Let’s look at this idea of a fraction 1/b. We have represented this symbolically. In this case, we’re dealing with 1/6. But, what really is this 1/6? Let’s look at the numerator, the 1. That is actually a quantity. That is a number the way that we typically think of it. But, the problem is the denominator, in this case, a 6. Here’s a huge difference. That 6 actually represents a noun. Again, the 1 is what we typically think of. It’s a number. It’s a quantity. But the 6 is not really a 6. It’s a sixth. We need to think of that as just a word, a noun, not as a number. So, if we take 1/6, it might be a good idea: let’s write it out, 1 sixth. By doing this we start ingraining the idea that 1 sixth is no different than, let’s say, 1 cat, or 1 pony, or 1 book. It’s the same idea. We have the number 1, and then we have a noun that follows it.
Let’s look at a/b versus 1/b, but let’s use numbers. Let’s use what we were working with before where we are dealing with 1/6 as our size. In this case, we have four parts, so we have shaded those in, and just like before, the 4 is a number. It is a quantity. That’s our numerator. But, in the denominator we again have a 6, but that’s really a sixth, which we need to interpret as a noun. It’s just a word. Symbolically, here’s our four parts, and again, that is 4 sixths, which is no different than 4 cats, 4 ponies, 4 books. It’s very, very important that students interpret the denominator of a fraction in this manner.
Again, we’ve been concentrating on both the concrete and the abstract representations. By doing the symbolic representation, we’re going a step further in Grade 3. The reason we are doing it is to again set the foundation for some things that are going on in fourth grade. In this case, we are really helping with standard 4.NF.B.4a, which deals with understanding a fraction a/b as a multiple of 1/b. It is very critical that something like this equation be represented also in words: 4 sixths being 1sixth + 1 sixth + 1 sixth + 1 sixth.
By writing it out like this, students will see that here are our numbers. We’ve added all the ones and that’s our 4, no problem. Then, when we look at the sixth, but written out as words, it makes it a little bit more obvious that nothing happens with the sixth. It stays the same. It’s no different than adding 1 book + 1 book + 1 book + 1 book. Again, we add all of the ones to get our four, and then the noun book, nothing changes there other than changing singular book to plural books. If we look at standard 4.NF.B.4a and notice this idea of the product 5 times 1/4 being 5/4, well, in this case, this would be 4 x 1/6, which is a symbolic way of representing 4 one-sixths. Now, that might be a little bit confusing because we’re used to just saying sixth. So let’s write it out that way. But still, students need to make the connection that 4 sixths is the same thing is 4 x 1/6.
This standard lays the initial foundation for understanding fractions as part of a whole. In addition, this standard, 3.NF.A.1, can be incorporated into the instruction of many other related standards. We saw the connection where we can reinforce some of the things back from second grade with this standard, but also, we can do a whole lot to lay foundations that are coming up in Grade 4.
Let’s look at the standards for mathematical practice. By doing some of these activities in this standard students would reason abstractly and quantitatively, and they would also construct viable arguments and critique the reasoning of others. Students would also be attending to precision, and they would also look for and make use of structure.