Let’s look at the Common Core standard from Grade 4. It’s part of extend understanding of fraction equivalence and ordering. Explain why a fraction a over b is equivalent to a fraction n times a over n times b by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.
Now, this one is a little hard to understand, primarily because of the variables that are used. So, I think the main thing to clarify would be what happens with n times a over n times b, and also this additional idea of the two fractions themselves being the same size even though the number and size of the parts differ. So, let’s look at these pieces here to clarify what the standard is really getting to.
If we take the expression and try to make some sense out of it, let’s rewrite it to where we have this in the form where it’s vertical. And it still doesn’t make a whole lot of sense, so let’s keep going, and let’s split that up into two fractions. Now it’s starting to get a little bit more clear because now, what you really have equal to, since n over n represents some number divided by itself, that’s always going to be equal to one. So, by getting to this point, it should be a little bit more clear that here’s the reason why there’s no actual change, why they would be equivalent fractions when you multiply by n over n, because you’re really multiplying by one. So then, you do have a over b still equal to itself.
So, let’s take a visual model, and let’s suppose that we have two thirds as our fraction. And so, we take our two thirds, or we split up our rectangular model here. Now, let’s suppose that the n over n is 3 over 3. So, this is what happens to your visual model. We are going to take each of those one thirds and split them up into three additional pieces each so that now, what we have is a total of six smaller pieces, but it’s still equivalent to the original two larger ones. So, we have six pieces. And then, as far as the total, that would be out of a total of nine.
Okay, let’s go ahead and take another situation and continue to make some sense out of this. Let’s go with a circular model this time, and let’s suppose that our fraction is 3 over 4. And this time, let’s let our n over n, let’s say it’s 2 over 2. So, now, what we have is 1, 2, 3, 4, 5, 6 pieces. But, again, notice that nothing changed. I still have the same whole, and even though I’ve chopped it up into more pieces (I now have six), it’s still the same size out of, now, a total of eight pieces. So, the six eighths is still equivalent to the original 3 over 4. But, again, notice what we’re doing with the 2 over 2 in this case is, we’re cutting up each of the original three pieces into smaller ones.
Now, we can also use a distance model for this. Okay, let’s rewrite this to where it makes a little bit more sense. So, now, let’s let our original fraction be 2 over 3 again. But this time we are using a number line. So, now, what we’re dealing with is two thirds of the distance from zero to one. Now, let’s go ahead and let our n over n be 2 over 2. So, again, what happens is that each of those two pieces is divided up into two smaller ones, so that, now we have a total of four out of the original, well now, six pieces. So, again, it’s still the same distance, but it’s still equivalent. It would be sort of like, if you divided something up into feet, and then you split those up into inches, but you’re still dealing with the same distance.