Mathematics

4.NF.B.4 Transcript – Parts A and B

This is Common Core State Standards support video in mathematics. The standard is 4.NF.B.4, and we are actually looking at two parts here for this standard, parts a and b. Standard 4.NF.B.4 states: Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. Part a states: Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product of 5 x (1/4), recording the conclusion by the equation 5/4 = 5 x (1/4). Part B states: Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 x (2/5) as 6 x (1/5), recognizing this product as 6/5. (In general, n x [a/b] = [n x a]/b.)

Let’s investigate what standards would be connected to or support standard 4.NF.B.4a. Back in third grade, we had standard 3.NF.A.1 that dealt with understanding a fraction 1/b as the quantity formed by one part when a whole is partitioned into b equal parts. In Grade 4, we also have standard 4.NF.B.3b that deals with decomposing a fraction into a sum of fractions with the same denominator in more than one way. This connects to the standard that we’re addressing through this idea of a fraction being a multiple of 1/b.

Let’s look at this idea of a fraction a/b. Let’s use the physical representation, a circle. We cut it up into four parts, so each one of these would be 1/4. Let’s take our standard and replace our variables. We’re dealing with fourths, so let’s do that. So, here we have a fraction, 3/4. Each part is 1/4, so let’s cut those out and represent them this way. Now, let’s write it out symbolically. It’s a good idea to do this. That way the students can make the connection between the concrete and the abstract. It makes this a lot clearer for students.

Now we look at this idea of 3/4 being a multiple of 1/4. Notice here that we have 1/4 + 1/4 + 1/4, which is simply 3 times 1/4. The reality is that we do have 3 one-fourths, but that might be a little bit confusing to students, so, traditionally, we say 3 fourths. But still, students need to understand that we actually do have 3 one-fourths. Let’s look at this idea of a fraction a/b. We’re using the example using fourths. Now, this is what would be particularly frustrating for teachers. Students would look at this symbolically, and they revert back to what they would learn with whole numbers, and they do this. You really can’t blame them, because they’re used to just adding things up, so they add up the numerators and they figure, well, I need to also add up the denominators.

So, the confusion here really is the symbolism. What do these numerals really represent? So, let’s focus again on this idea of a fraction. In this case, okay, we had 3/4. What does the 3 represent, the numerator? Well, that’s the typical interpretation. That 3 is a quantity. It is an amount. But, the trick is the denominator, in this case, a 4. That is really a noun. It’s not a quantity like we’re used to, and this is a total reversal of what students are used to. Again, the 3 is a number. It is an amount. The 4 on the other hand, it’s not really a 4. It’s fourths. It is a word. It is a noun, a descriptor of sorts. So, this fraction 3/4—students really need to see that as 3 fourths, not as 3 over 4 or anything like that. It’s 3 fourths. By doing that and by writing it out to where students see it in this manner, slowly but surely, students should ingrain the idea that 3 fourths is no different than 3 dogs, 3 apples, 3 pens. It’s the same idea. You have a number at the beginning, and then you have a noun following that.

We are dealing with 3/4, and we represent the pieces and then we do our symbolic representation, and here’s what students really need to understand. Okay, we’re doing the addition 1 + 1 + 1 in the numerator. No problem, that’s our 3. Now we look at the denominators. We don’t really have numbers here where we are adding 4 + 4 + 4. That is not the case here. Since that’s a noun, nothing changes. The 4 remains the same because again, it’s fourths. To help students understand, we should do this: write it out, 1 fourth + 1 fourth + 1 fourth. In this manner, they’ll see that okay, I add up the ones, and that’s a 3. Then the fourths, those are just words, so nothing changes with that. It’s just fourths. So, 1/4 + 1/4 + 1/4 would be 3 fourths. So students need to get used to that.

I know that this is a little bit repetitive, but this is a very, very important point. By writing it out, students start to understand why things happen the way that they do. If we take 1/4 + 1/4 + 1/4 and we write it out in words, it makes it a little bit more obvious that it’s really no different than adding 1 apple + 1 apple +1 apple. I have 3 apples. By doing this, you’re establishing a parallel with other examples that don’t involve fractions, and it starts making a little bit more sense. The fourths is just like apples or some other noun. You don’t mess with it, you just add the numerators, and the denominator stays the same because, again, it’s not really numbers that you’re adding there.

Now, let’s focus on the first part of this standard that deals with understanding a fraction a/b as a multiple of 1/b. So, let’s use 7/4, and let’s break it up into 1 + 3/4. Now let’s change the 1 to 4/4. We don’t want to just look at the numeric symbolism, so let’s do some representation here. Let’s split it up. The 4/4 would be for 1/4 +1/4 + 1/4 + 1/4, which goes back and connects to that first statement in the standard. Same thing with the 3/4: we split that up to where we have a fraction that’s 1 over something. Now, we can combine the numerators. We count them up. There are seven of them. So, we have 7/4, which written out symbolically would be 7 x 1/4.

Let’s focus on the second part that talks about using a visual fraction model to represent something like 5/4 as the product 5 x 1/4. So, let’s do the same example, 7/4. Let’s change our 1 to 4/4. Let’s use a circle and its interior as our model, but let’s take that circle and split it up into four individual parts like so to where it’s a little bit more obvious to students. Now we have to add the rest of it. So, we take a physical representation of 3/4 like so. Let’s cut it up to where it’s a little bit more obvious. Now we can actually do our addition. We have 1, 2, 3, 4 on the left, and on the right hand of the plus sign we have three more. And by counting on, we have 5, 6, 7. So, we have 7/4, which symbolically, based on what we’re trying to do with this standard would be 7 x 1/4. By using an improper fraction, in this case 7/4, it makes it a little bit more obvious to students that fractions can appear in different forms. They’re used to the numerator always being smaller than the denominator, but that’s not always the case. Students need to realize that the numerator doesn’t have to be smaller than the denominator, and in a case like what we used here, the quantity will be more than one.

Now let’s investigate Part b of this standard, but before we really get going, we need to concentrate on this idea of multiplication, the representation. What does 3 x (2/5) really mean? Back in third grade, there’s standard 3.OA.A.1, and it really sheds some light on this. There’s some clarification here as to how to interpret something like 3 x (2/5). This third-grade standard says to interpret 5 x 7 as the total number of objects in five groups of seven objects each. So, that’s the key as to how to interpret the numerals that are involved in multiplication If we have something like 3 x 4, the interpretation says that we have three groups of four in each one, and it’s three groups of four objects or items of some sort. It could be, let’s say, three groups of four cars each, or it could be say, three groups of four pencils. Now it makes it clear, 3 x 4, students really need to see that as three groups of four and specifically, three groups of four somethings.

So, when we see 3 x (2/5), students need to see that as three groups of 2/5. Looking at the symbolism, even though it says 3 x (2/5), what’s in students’ minds should be three groups of 2/5 in each one. Rather than jump in and just work with the symbolism, it’s better to use some type of physical representation. Students really need to see what’s going on because with fractions, there’s a whole lot of symbolism, and it needs to make sense to them.

So, let’s again, use some physical representation. So, we start off with three groups of 2/5. What is that equivalent to? Well, let’s do this strictly doing the visuals. Let’s go ahead and move this set of 2/5 over to the other one. So, we’ve done that. We have one more to fill in here, but we have two over here. So, let’s just move one of those over here. Now, let’s take the one that’s left and move it over here, and now students can visually see exactly what happened, and the solution will make sense to them. We don’t need this over here any more. If we count these up, we have 1, 2, 3, 4, 5, 6. Each one of those parts is 1/5, so our solution is 6/5, which symbolically and connecting back to this standard would be 6 x 1/5. Using our physical representation, students should see that three groups of 2/5 was equivalent to six groups of 1/5 in each one.

Let’s rewrite this to where we have 6/5 to connect it back to what the standard says. Let’s change this up a little bit. Instead of having 3 x (2/5), let’s look at 2 x (3/5), which of course, would be two groups of 3/5 each. Rather than just use some blind multiplication, again, let’s solve this using just our visuals, our physical representations. What we need to do is take 2/5 from our whole on the right and move them over to our whole on the left. So, let’s do that, and bingo, we have what we need. We have 6 one-fifths again. Symbolically, that’s 6 x (1/5), which again, is six groups of 1/5, and symbolically that’s 6/5. Our first example, we did two groups of 3/5. Then we switched it around a little bit, and we did three groups of 2/5. In both cases, when we did our manipulation with our physical objects, we got 6/5.

Let’s conclude by looking at the very last part of this standard, this generic representation of a whole number times a fraction where on the left-hand side, the equation says n x (a/b) being equal to (n x a)/b. It would be a little bit difficult to show this with just symbols. It’s just too abstract, so let’s go with what we’ve been dealing with, let’s use 6/5. Let’s stick with the example that’s in the standard, 3 x (2/5), and let’s set it equal to itself. On the left-hand side, we know that we have to multiply the 3 and the 2 to get the 6, and the same thing applies on the right-hand side. We need to multiply the 3 and the 2 to get the 6 also.

Let’s look at the left-hand side of the equation. The left side is really pretty simple. All we need to do is put parentheses around the 2/5, and our specific numeric example corresponds exactly to what the general form says in the equation in our standard. So, that was easy. It is the right side of the equation that is going to be a headache. Let’s see what we have to do. Somehow, we need to have the 3 x 2 all in the numerator. So, let’s work on that. Okay, the 3 x 2—well, what can we do? Well, we need to work on the 3. We can make that into a fraction, 3/1. So, now we have (3/1) x (2/5). We can now take that and convert it to just one fraction. So, we’ve taken care of the more difficult part here. The 3 x 2 in the numerator we can put in parentheses, and it matches our generic form.

Now our only problem is to make the denominators look the same. But we have the 1 x 5. We need to get rid of the multiplication there. We need to get rid of “1 x.” So, what can we do? Well, that’s just the multiplicative identity. One times any number is not going to change that quantity, so we can just drop the “1 x”, move the 5 over. Now we have the right-hand side of the equation being exactly like it is in the generic form. So, on the left-hand side, we can now take our specific example and make it look like the general form, and then, the same thing on the right. So, we’re pretty much done here. We’ve shown that these are equivalent, and hopefully, by doing this with a specific example at this level, students can see that they are, in fact, equivalent to each other. So, this idea of a fraction a/b—we worked with multiplying a whole number times our fractional part. We used 6/5 as our example, so 6/5 would be 6 x (1/5).

But what if we took our generic example and used our commutative property? Let’s switch this around to where it’s (1/b) x a. At this level, it’s okay to use numeric examples to sort of informally prove something. So, let’s do that, and let’s stick again with what we’re using, (1/5) x 6. What’s happening here is that we really don’t have a whole lot of groups. Actually, we don’t even have one whole group. We have part of a group. We have 1/5 of 6. So, let’s use a physical representation to make it a lot more obvious to students as to what is going on. So, let’s have six objects here. Since we’re dealing with fifths, let’s go ahead and split all of these up into five equal parts. Since we are talking about 1/5, then let’s take 1/5 out of each one of these. However, in this form, it’s hard to tell, well, what’s our total quantity?

So, let’s do some manipulation. Let’s take this 1/5 and move it over to there. We don’t need that any more so let’s get rid of it. Now we can take this 1/5 and move it there. And we can get rid of that one there. Let’s take this 1/5 and move it here. Eliminate that whole. Now let’s take this 1/5 and move it. Get rid of that, and we’re done. We see that we have a total of 6/5, which is equivalent to 6 x (1/5). We make the connection, symbolically, the 6 x (1/5) being equal to 6/5, 6 fifths. And we see that the commutative property does hold in this case, of course. The 6 x (1/5) is equivalent to (1/5) x 6.

Let’s use a slightly different approach here. Let’s take the (1/5) x 6. Why don’t we convert the 6 to 5 + 1? What we need to do here is, we would need to multiply (1/5) x 5, but we also have to multiply (1/5) x 1. That is based on our distributive property. So, when we do this, an equivalent expression would be (1/5) x 5 + (1/5) x 1. But, this is just a bunch of numeric symbolism that may not make sense to students. So, let’s get our physical representation. Now, this is what’s different here. So, first, let’s concentrate on (1/5) x 5, and then we’ll worry about the (1/5) x 1 later. Now, we want five of those units, so let’s just concentrate on that. Well, this is pretty obvious: 1/5 of 5 is simply 1. So, we just need to take one, and let’s shade it to differentiate it from the others. And we don’t need those other ones, so let’s get rid of them.

Now we need to take care of the second part here, (1/5) x 1. Let’s take our physical representation, split it up into five equal parts, and we just want one of those. So, bingo, there we have it, and we can see that we have a total of one whole plus another 1/5. And if we split the one whole into five equal parts, bingo, we have 6/5 just like we did before, which converted over would be 6 x (1/5), which, of course, is equal to 6 fifths or 6/5. We’ve taken one example, 6/5, and we represented it in multiple ways: 6/5, 6 x (1/5), and of course, really important that students see that as 6 and then the word fifths written out to focus on this idea of the denominator actually being a noun. It was important to do all of this, because it really is critical that students at this grade level understand the meaning, the details, and the nuances of the symbolism of fractions. We do this to ensure a solid foundation and a deep knowledge of fractions.

Looking at our standards for mathematical practice, by doing the activities in this standard, they would reason abstractly and quantitatively. They would construct viable arguments and critique the reasoning of others. In addition, students would also use appropriate tools strategically. They would attend to precision, and they would look for and make use of structure.