Common Core State Support Video: this is fifth grade math. The standard is 5.NF.5a. This standard says that we need to interpret multiplication as scaling by comparing the size of a product to the size of one factor on the basis of the size of the other factor, but without performing the indicated multiplication.
What might be easiest to do is look at some numeric examples to see exactly what the standard is saying. Let’s take a nice easy product. Let’s say 12 and one of the factors is 4. Then it’s pretty simple to figure out in your head that that will be 3. So the comparison here is that 12 is 3 times bigger than 4. If we take another example, let’s say 30, and let’s say one of the factors is 5. Then we know that that’s a 6, so then the comparison is that 30 is 6 times bigger than 5. But really we expect more at the fifth grade level. What happens if it’s a more difficult scale factor? It seemed like the pattern was that all you have to do is divide to get the scale factor and that’s pretty much true. As we saw, for example with 12 being the product, one factor being 4, it ended up being 3 times bigger.
So if we take something that’s maybe a little bit tougher, let’s say 96. And let’s see, let’s compare that. One of the factors is 6. If we do the division we come out with 16, but of course we would have to go through the actual process of doing the dividing. So now the comparison is that 96 is 16 times bigger than 6. The whole idea of a scale factors is that it’s actually a ratio of the product and one of the pairs of the factors. So if we look at it algebraically, and let this statement be that one factor times the comparison that we want to make, which should be the other factor, is equal to your product.
If we do a little bit of algebra and do this, and of course we have to do it to both sides of the equation. And so this simplifies to this expression where the comparison is really just the result of taking your product and dividing it by the factor that you want to make the comparison to. So with this in mind, seeing that relationship of the comparison being the product compared to the factor, that’s really important for two reasons. First, it lays the foundation for proportional reasoning. The scale factor tells you the relationship. And the second most important thing here is that, and it’s very applicable at this grade level, is that it enables students to figure out the scale factors that are not whole numbers.
So for example, let’s say the product is 9 and the factor is a 6. Now that is not a whole number comparison. But if we do a little bit of simplification and consider that the ratio of 9 to 6, that will simplify. Let’s see, we can divide them both by 3. That will simplify to 3 over 2, which as a mixed number would be 1 1/2. So then the comparison would be that 9 is 1 1/2 times bigger than 6. If we were to do another example, let’s say we reverse it and we’re comparing 6 to 9. So then if we do some simplification, this will simplify to 2 over 3. So the comparison here is that 6 is 2/3 as big as 9.