How to Multiply a Fraction by a Fraction Using a Model

Okay, my name is Adrianne from Math for Middles. I’m a multisensory math tutor, and today I wanted to show you how to do multiplication of fractions by a fraction using the model method. Now if you have a kiddo that’s in the public school system or any school that’s using the National Council for Math Standards or NCTM, they’re going to be expected to know how to do modeling with fractions.

I wanted to talk to you about this. The thing that gets really confusing for a lot of kids is this multiplication of fractions when we’re using a model, it’s confusing because they’re not sure what’s really going on. They’re just using numbers. It’s super easy. I multiply across the numerator and then I multiply across the denominator. But when we’re trying to model it, it can be hard because we’re taking a part of a part, and that is confusing. And that’s more like division than it is multiplication.

Instead of starting here with 1/2 by 3/5, I’m actually going to start here with this little more simple one. We’re going to start with using a model to multiply 3/4 times 1/3. The first step the student need to take is to draw a square or rectangle or something like that and break it into three equal parts which we have here. Then we shade our 1/3. We’re always starting with this 1/3 here. I’m going to draw a box around that so that you can see what I’m referring to here. Let’s make that smaller. Okay. So we’re always starting with this, this number. This is who we draw the model with first. It’s the second fraction.

So we have 1/3, and we need to break it into equals parts again and look for 3/4 of that 1/3. Looking at the denominator of the 3/4, I know I need to break this model again into four equal parts. So I’m going to do that here. And I’m going to go like this, just right down the middle. Then down here. And down here. There we go.

So now we’ve broken it into four equal parts vertically and we did thirds horizontally. Okay, now I need to ask myself, “Okay, inside of this 1/3 part that’s shaded, I need to find three out of the four pieces that I need to make a group.” I’m going to pull out my rectangle tool here again, and we’re going to use that pink. There we go. I can see here that I have three of the four pieces that I need inside of this 1/3. But the way we write the answer is not this little weird section. We have to name it in part of this whole that we have with the new denominator. It’s not 1/3 with the three as the denominator. And it’s not four in the 3/4, but it’s 12.

We have a new name here. Here we have 12 is our new denominator. And how many parts did I have out of that? I’ve got the three. So we’ve got to write that as 3/12. And we all know that we can reduce this. We can make it in a simplified fraction. And if I could model it better for you, I would. I would take these three little pieces and put them here because it’s easier to see that really this equals 1/4, because we have one area of our four equal parts vertically here. If I could break those into and move them down into these boxes, I would do that. But with using just numbers what happened here is we divided by a copycat one, some teachers like to call it a magic one, of three over three which allows us to write that as 1/4. We wrote it in the simplest terms.

That’s why it gets confusing for students because they’re thinking, “Well I’m multiplying, I should have more of this quantity.” But instead what we did is we took part of a quantity that we already had. So that’s why it gets a little bit confusing, is where we’re asking ourselves, “How many pieces can I make of that 1/3? I need three out of four pieces inside of that 1/3,” which can get a little bit confusing. So it’s really important to emphasize that I’m looking for three out of the four pieces I need inside of 1/3. That’s how we model that.

Let’s do it again. Let’s do it one more time. We’re going to use a model to find 1/2 times 3/5, and again, I’m starting with this 3/5. I’m going to build this first. It can be a rectangle. It can be a square. It’s easiest just to pick one or the other. Circles are going to be harder, so I like the rectangular model for this reason. I have 3/5 here shaded, and now I need to break the whole, this whole area in half. I’m going to do that with my black line here. I’m going to break it in half right here. And I broke it in half.

Now I can see in the shaded area we’ve got six pieces. We got that one, two, three, four, five, six. We can ask ourselves, “Well, what’s half of six? How many pieces are in there? Half of them? Half of the six there?” So I’ve got … Oh, it’s doing it again. Sometimes this tool does not want to behave like I want it to. I want it to read my mind. Here we’ve got this area right here. Let me draw that in pink. Let me make that easier for you to see. Here again, we are doing this right here. We’ll delete this other one.

You can see here that we’ve got half of that six that was inside of 3/5, and I’ve shaded or drawn a square around half of the six, which is three, but now I have to name my answer in terms of the whole here. We had five slices to begin with, and then we cut them in half, and so we have 10. That’s our denominator. We’ve got 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. And how many? Three. If we do it just the numbers way, yep, you’re right, we multiply across the numerator and we get 3/10.

So you can see though this is how we model it. That’s a little bit tricky to look at 3/5 and identify, well, there’s six parts in there, there’s six pieces, but I only need half of them. So what is that? It’s the three pieces on either you could’ve circled the left or the right side, whatever you wanted to do there, but that’s how we are drawing the model to help us do multiplication of fractions by fraction. That part gets really confusing for students because we have to when we’re talking to our students, to our children, we got to talk a little slower. Our goal is to make half of this, the six parts we have. I only need half of these six parts. That language is really important. Six divided by half, well, that’s the same as saying six divided by two.

I hope that’s helpful to you, my students. We run through this several times, several times, and sometimes when we’re working with calculators too, they will estimate what the quantity should be and because they’re multiplying they’re thinking, “Oh my goodness, it should be much bigger than this,” but the truth is no, it isn’t. We’re taking part of a part, and that’s more like division. So we can lean back on that language that we’re taking part of a part but our answer is written in terms of the whole, the new whole that we made.

I hope that was helpful to you. If you have questions about this or would like to connect with me personally, let me know. I’d love to help you. You can always reach out to us at madeformath.com. We’ve got a contact button right there. And if you’re curious about doing multisensory math with your child and you’d like us to work with your child to help them tackle these kinds of problems, head over there to madeformath.com/tutoring, and there’s also a place there to contact us and get that process started.

Thank you so much for joining me today and look forward to these. I’m going to be doing these every Tuesday with you, hopefully earlier in the day than this in the future, but we’re going to be doing these every Tuesday. So if you have a really specific question or a homework problem that’s just completely stumped you, send them my way, send them to Math For Middles. You can do that by just emailing me directly hello@madeformath.com and in the subject line pop in there for me Tuesday Tips with Math and that will help me know that you’re okay with me posting this question here on our Facebook page. Thanks so much for coming and we’ll talk again soon.

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