Multiplying a unit fraction and a whole number with fraction models

0:00- [Instructor] What we're gonna do in this video is think about what multiplying fractions really mean, and let's start with 2 times 1/3. what do you think this is going to be equal to? Well, there's a couple of ways that we could represent this. We could take a whole here, so that is a whole, and let's split it into thirds, so three equal sections.

0:22My drawing isn't perfect, but let's assume that it is perfect. Well, 1/3 of that whole, that would be one of the three equal sections. So if I say 2 times that, that would be two of these. So I could draw another whole there and split it into three equal sections, and then just shade in 1/3 or one of those three equal sections.

0:49So I have one and two of these thirds. So you could literally say, I have 2/3. So you could say I have 2/3 now. Another way that might be useful to think about it is imagine if we just start with a whole over here. Ignore this on the left hand side for a second.

1:09So if we just had a whole over here, and I split it into thirds. So we're gonna start with 1/3, that's 1/3 right there. But we're saying 2 times 1/3. So we're gonna have two of these, so that's one of them. That would be 1 times 1/3, and then 2 times 1/3 would be one more of those.

1:28So in total, how much do we have? Well, I filled in two of the three equal squares here, the two equal sections so that is 2/3. Let's do another example and mix it up a little bit more. Let's say I were to ask you what blank times... Well, blank times 1/5 is equal to 3/5.

1:49What would be blank in your mind? Well, the way that my brain likes to imagine this is, let's split a whole into fifths. So one, two, three, four, five. So this is five fifths here, although none of them have been filled in. And if I wanted to represent 3/5, I could fill in three of them.

2:12So 3/5 would be one, two, and three. So how many times did I have to multiply 1/5 by? That's 1/5 so that's 1 times 1/5, 2 times 1/5, and 3 times 1/5. Well, I had to multiply it by 3. So 3 times 1/5 is literally 3/5, or three 1/5, depending on how you want to think about it.

2:37Let's do one last example. Let's say someone were to walk up to you on the street and say, "Hey you, what's four...?" Well, they would say, "Hey you, 4 times blank "is equal to 4/3, what is blank going to be?" Well, I'll give you a little bit of a hint, it might involve thirds 'cause at the end we get four of these thirds.

3:00So one way to think about it is I could take some wholes and divide them into thirds. Actually, I'll do two wholes right over here, and I'll divide them into thirds. Now, 4/3 would be filling in four of these thirds. So one, two, three and four. So you can see another way of thinking about this, that's one whole and another third, or we can think of it as 4/3.

3:33But what just happened here? I just had four times, what? Well, I had four times this square right here. 'cause that's one, two, three and four. I multiplied 4 times 1/3, so 4 times 1/3 is 4/3.