Mathematics

5.NF.5b Transcript

This is Common Core State Standards support video for mathematics. The standard is 5.NF.5b. This standard reads: interpret multiplication as scaling or resizing by explaining why multiplying a given number by a fraction greater than one results in a product greater than the given number (recognizing multiplication by whole numbers greater than one as a familiar case); explaining why multiplying a given number by a fraction less than one results in a product smaller than the given number; and relating the principle of fraction equivalence a over b is equal to, open parentheses, n times a, close parentheses, divided by, open parentheses, n times b, close parentheses, to the effect of multiplying a over b by one.

Now, the key idea is the opening statement—interpret multiplication as scaling or resizing. Now, when we multiply some given number by a fraction, that fraction is actually the scale factor that's going to change our original number. Now, one of the things that isn't explicit is that with any fraction, like here, a over b, the denominator cannot be zero. So, in all of these instances b cannot be zero.

The first part deals with multiplying some number by a fraction that's greater than one. The second part of this standard deals with multiplying some number by a fraction that's less than one. We have to look at the big picture here, and when we look at these variables, we need to understand that the numbers that we're dealing with here are positive, because integers are not covered until grade six. So, here all of the numbers that we're dealing with are positive. So, for example, when we talk about a fraction being less than one, we're really talking about a fraction that's between zero and one. It's a proper fraction that's positive. Now, there is a third context where our fraction could be equal to one. Even though they're different letters, different variables, there's nothing that prevents a from being, say, a seven, and b also being a seven, which of course would be one.

Now let's look at this idea where the fraction would be equal to one. Now, what we need to realize is that that idea does not come from that last statement in the standard. That last statement in the standard is actually not related to those first two, and so what we'll do is we'll worry about the third statement in this standard and address it at the end. Now, any fraction can only be equal to one if we're talking about the numerator and the denominator being the same number. So, in this case, a over b can be one only if a is equal to b. And for the purposes of this video, from this point onward, we'll just refer to this context where the number would be times one, like so. Now, this third possibility where the fraction is equal to one, that's important because that's going to be our benchmark for comparison.

Also, when we look at these two scenarios where we have fractions either larger than one or smaller than one, there's two necessary understandings to make sure that we're all on the same page. First of all, when we're talking about multiplying, we're going to interpret the multiplication where the first factor tells us the number of the sets or groups, and the second factor is going to tell us the size of each set or each group. Now, the standard talks about comparing to a given number, okay, and we'll just call it n. If we take that number and multiply it by one, it doesn't change. You know that's the multiplicative identity. Also, if we start off with some number n, it's understood, even without context, based on place value that that's so many ones. So, that's a second item that connects us to this idea of n times one; and then, the third thing that would connect to n times one would be that that third possibility where the fraction a over b could be equal to one. But the main reason that we need to look at this number as something times one is that it's necessary for it to be in this form in order to physically model the comparisons. And that's the key reason for our number being something times one.

So, let's look at the first scenario where we're going to multiply some number by a fraction that's larger than one. So, we start off with our given number, and we're going to multiply it by some fraction that's larger than one. And for comparison purposes, we're going to think of our number as something times one, and let's let our fraction that's bigger than one be three over two. We could pick just any number that we wanted, but for simplicity, let's start with our number being three.

So, what does this look like, you know, three times one versus three times three over two? Well, three times one means that we have three sets of one. And on the right, well, wait a minute, that's just three sets of one. That's not enough. Oops, now this is three sets of two...that's too much. So, what we need to do is...we need two halves. So, we need to take our sets on the right and cut all of those in half. And so, now we have what we need. We have three sets of three halves in each one. So here's one of the sets; here's a second set of three over two, and a third set of three over two. Now, just using logic, on the left we see that we have three sets of one compared to three sets of three halves. So we know that three would be less than, let's see, on the right we have one two three four and a half. So three is less than four and a half.

Another way of looking at this would be that three halves is the same thing as one and a half, so actually what we have over here is three sets of one and a half also. That's another way of looking at it. What happened in this context, if we look closely, is that we have the same number of groups. We have three in each case, but what's different is the size of the groups. We have ones over here and sets of one and a half over here. So again, we have the same number of sets, but it's the size of the sets that's different. So it would be kind of like saying three pennies compared to three nickels.

Now, what if we were to reverse the order? Now, we know from the commutative property that these would be equivalent, but what does it look like when we physically model it? So again, the same understanding; our first factor is the number of groups; the second factor tells us the size of the sets or the groups. So, just like before, we're going to take our given number and compare it to the result from multiplying by a fraction that's larger than one. And, again for comparison purposes, we change our n to one times n, and we're going to let our fraction be three over two again, which is bigger than one. And we'll let our number be three.

So we have one times three compared to three over two times three. So, what does this look like when we model it? On the left, we have one set of three. On the right, we have....well, let's see, that's just one set of three. That's two sets of three; that's too much. We only need three halves or one and a half sets of three, so we need to cut that second set of three in half, and so we've done that. And again, comparing, we can tell that our left side is smaller. One set of three would be less than one and a half or three halves sets of three. And what happens here is that we have the same sets. For example, let's say these are cookies. So we have one set of three cookies compared to one and a half sets of three cookies each.

So, in reviewing this first scenario, again, the distinction in the two contexts was that in the first context, the size of the groups was the same, but the number of the groups was different, as opposed to the second context where we had the same number of groups; it was the size of the groups that varied. Again, in context, it would be like saying one set of three cookies is less than one and a half sets of three cookies. In the second context, we have the same number of groups, but it's the size of the groups that's different. So, obviously, three pennies would be less than three nickels because again of the size of the groups.

Now let's look at the second scenario, the second statement, where we're multiplying some number by a fraction that's smaller than one. So, just like before, we're going to compare our original given number to the result from multiplying, in this case, by a fraction that's smaller than one. Okay, we change our given number to something times one for the physical modeling. Let's let our fraction less than one be five sixths, and again, for consistency, let's just let our number be three.

So what does this look like, three times one versus three times five sixths? On the left, we have three sets of one. On the right we have....well, let's see, that's too much because that's still three sets of one. We need three sets of five sixths. So we need to take each of those ones and divide them up into six equal parts. But we don't want all six parts. We only want five. So, this is what we have, and just using some logic, we can tell that our three sets of one is going to be more than our three sets of five sixths, because again, the size of the groups is different. You know groups of one are bigger than groups of five sixths. We could figure this out. We could take some of these two shaded parts and put them here and here and take our unshaded parts and put them here. So, we would have one, two, and a half. So, three is bigger than two and a half.

Just like before, what if we were to reverse the order of our two factors? Again, because of the commutative property, we know that they should be equal, but what does it look like physically? So, we'll start off with our number. We're going to compare that to the results of multiplying by a fraction less than one, but this time, it's one times our number. And, let's let our fraction be five sixths again. So, it would be five sixths times our number, and for consistency, again, let's let our given number be a three.

So, we have one times three compared to five sixth times three. So, we have one set of three on the left. On the right, well, let's see that's too much, because that's a full set of three. That's not what we need. Now, here's something that's a little bit different. Here, actually our set of three is our whole. So, we don't split each one of these up into six equal pieces. What we need to do is take our whole set of three and split it up into six equal pieces. But we don't want all six. We only want five of those six. So, we have that now.

And again, just like before, we can look at this, and we can actually figure out that this is one, two, and a half. So we know that three is bigger than two and a half. Or just using the logic that we don't have as many of this group of three on the right as we do on the left. Putting this into a context would be like one set of three cakes compared to five sixths of a set of three cakes. Just like the first context, where we compared the number being multiplied by a fraction larger than one, we have the same scenarios here. In this context, the size of the groups is the same, but we don't have the same number of groups for each, as opposed to this last context where we have the same number of groups, but the size of the groups is different. Again, connecting this to a real-life context, the first context was like a set of three cakes compared to not a full set of three cakes. And then, the second scenario would be...we have the same number of groups, but the size of the groups is different.

Now, let's address the last statement in this standard about relating the principle of fraction equivalence to the effect of multiplying our fraction a over b by one. Notice that this is unrelated to the first two parts, because the first two parts dealt with taking our number and multiplying it by the fraction a over b. Now, this last statement is actually addressed in standard 4.NF.1, and there is a Common Core video on this standard.

But, just to review, very quickly, this is the original equation. It would be a little bit easier to interpret if we put this in vertical format, and we split it up into two fractions. And now, of course, we know that any number divided by itself is going to be one. And so, now we know that this statement is true because if we multiply any fraction by one we still get the same result. Now, if we were to reverse this process, that's what we would actually do to create a fraction that would be equivalent to our original. So if we start off with some fraction, and we're actually going to multiply it by one so that the quantity doesn't change. Then we change our fraction to whatever number over itself, depending on what we want to create. And then we just put it as one fraction and actually do our multiplication, and we have our equivalent fraction.

Now, how is this different from the fourth grade standard? What's different here is that we have to tie this back to that initial statement of interpreting multiplication as scaling or resizing. So what does that look like here? Well, multiplying by one and multiplying by n over n, some fraction equivalent to one are actually different. They're not the same when you physically model them.

Let's look at an example. Let's say our beginning fraction is two thirds and we multiply that by one. So, nothing changes at all. We still have two thirds. Now let's look at the second scenario where we have our fraction, and again, let's let it be two thirds, and we're going to multiply it by one. But this time, the one is in fractional form, some number over itself. So, let's let our n over n be six over six. So, what happens with the physical modeling? Well, now, if we multiply it all out, we're going to have 12 parts out of 18 total. So let's look at what really happens here. When we multiply by one nothing changes. But, in this second scenario, when we multiply by some fraction n over n, we change both the number of parts and the size of the parts. Now, notice that we hold true to the multiplicative identity because when you multiply by one, the size of the whole didn't change in either scenario. But again, we do have some resizing here where we multiply by one in the form of a faction that's equivalent to one. So there is a difference. There is some resizing in the second context, so it is different. So, multiplying by one as opposed to multiplying by a fraction equivalent to one physically does make a difference.

So, this standard is important. We need to spend some time on it. It's very important for establishing number sense so that students get a sense of what happens when you're multiplying with fractions. Now, the standards don't address this same basic idea with decimals, but if we were to take this standard and substitute decimal for fraction, we would have the same basic idea, that if we were to take a number and multiply it by a decimal greater than one, we get a product greater than our original number. And if we were to multiply our given number by a decimal that's less than one, we'd get a product that's smaller than our original number. And then, looking at the last statement, we would relate the multiplicative identity to where any decimal times one will result in the same decimal.