Working with Fractions: Multiplying Fractions

Mar 6, 2020

Fractions are a hot topic in most of our schools, especially as most of our M3 project schools are nearing our state testing coming up in the spring!

Most fraction units are positioned in the school year so students can get as much exposure to fractions as possible. Developing a solid understanding of fractions is just as important as developing that foundational whole number sense. Students need a firm grasp on the concept of fractions in general so they can be successful as they start exploring the various facets of fractions: equivalency, comparing, add/subtract, multiply/divide, etc. 

The big question for us as teachers is how do we go about teaching kids about fractions? We want them to get it quickly, but how do we do that with a conceptual understanding instead of just teaching them to memorize rules or tricks?

I often wonder why teaching fractions is so daunting for so many teachers, and I find that many of the teachers that I work with were not comfortable with fractions as a student themselves.  Their development of whole number sense wasn’t strong and they have gaps in their own understanding of operations, rounding, estimating and other things with whole numbers. So then, when you start talking about doing those things with parts of a whole and it gets even more confusing.

As education is tending more towards the concept of a flipped classroom where students watch clips of how to do something as a precursor to going into a Math with a Teacher station, I wanted to dedicate the month of March to visually answer the question of how we go about teaching fractions. 

As we continue to build our video library and YouTube channel, you’ll have on-demand access to the videos to use in a variety of different ways. You could have students watch the video on their own prior to meeting you in a guided math group, you could use it to help kick off a lesson in a whole group and then take students into a workshop model, or you could make it a warm-up to get kids’ brains going.

However you plan to use these videos, take advantage of the nature of technology and digital media! Research from Horizontals on brain development shows that, when kids learn with technology, they learn a lot faster. We definitely don’t want to turn into the “wah wah” Charlie Brown teacher and have information we’re imparting going in one ear out the other. With a video, students have the opportunity to really digest the content – pause the video, go back to rewatch anything that doesn’t make sense so they really get it. This could take the level of engagement up, and especially as you’re using CPA for concrete pictorial abstract, could engage more learners! 

Foundation of Multiplication

I always find myself introducing the concept of multiplication by asking a class about what they know about multiplication within a whole number. I’ll write a problem on the board – 5 x 6 – and ask – what does this mean? Typically, students will say things like, “multiply” or “times” or “the answer is 30.” So I keep asking the question – what does it mean? As the students engage and dig deeper, we hope they eventually arrive at the fact that 5 x 6 really means “5 groups of 6.” We also know that if we changed it to be 6 x 5, it would be “6 groups of 5.” If your students are doing the multiplication journals and playing the Speed! game, this concept of “groups of” is drilled into their heads, even if it might take a little prompting for them to say it.

Next, we might go to a larger problem. I’ll write something like 89 x 12 on the board. What does this mean? It means we have have 89 groups of 12, so how many do we have all together? We really want kids to understand this pattern and phrasing with whole numbers because, as we transition into multiplying fractions, it will make that transition much smoother. 

Multiplying a Whole Number by a Fraction

So, next we write a problem with a fraction on the board – 4 x ½.  What does this mean now? All of a sudden one of our numbers is a fraction! In this case, we could really say it means “4 groups of ½.” If I had ½, ½, ½, and ½, that would be 2. So, 4 groups of ½ equals 2. 

Using this progression of inquiry-based activities really helps pique the students’ interest, helping them understand the concepts behind multiplication. Usually, with fractions, we start with a procedure – do this step, this step, this step, and this step. Then we have students do the procedure with us, and then they do a worksheet on their own to make sure they’ve memorized the procedures. That isn’t really the level of engagement that 21st century students need. They need to engage in conversation and grapple with questions like what does that symbol really mean? 

From here, there are lots of different activities you could use to help students deepen their concrete understanding. They need lots of practice! If I’m in 4th grade, I might stop here though, start to bring in manipulatives that can support development of the students’ understanding.

Introducing Manipulatives (4th Grade)

In this video, I’m going to show you how to use two of my fan favorite manipulatives: pattern blocks and area model papers. 

I like to start with the pattern blocks because they are probably already familiar with pattern blocks from previous grades. 

If I was using fraction tiles to model this problem, I’d have to get two sets of fraction tiles because I’d need 4 of the ½ pieces, so since pattern blocks come in a big bucket, we don’t have this challenge! Additionally, fraction tiles are also imprinted with the part they represent, so there is no flexibility for using those pieces to represent another value as there are with pattern blocks.

Another benefit of using pattern blocks is that, when we get ready to talk about a whole number divided by a fraction, kids will already be familiar with the process and will actually be able to see how multiplication and division of fractions goes together. It’s like the fact families of addition and subtraction and how, when we teach the missing addend we see that addition and subtraction go hand in hand. 

Using the pattern blocks, I’ll model how to tackle a whole number times a fraction problem. If we’re using the correct language for this, “four groups of half,” it will make a lot of sense. Let’s use the hexagon as the whole, then we have the red trapezoid as the ½, a blue rhombus would be a ⅓, and then we could just use the triangle simply as ⅙. 

To begin, we wanted to see 4 groups of ½, so kids can pull over their ½ pieces, the trapezoids, until they have 4 trapezoids. Kids can then easily see, with the one whole hexagon as a reference, that 4 groups of ½, 4 trapezoids, is the same as 2 whole hexagons. For some more at-risk kids, they might want to actually match the trapezoids on top of the hexagons to solidify that concept of quantity. Kids can then play several other examples as they explore multiplication of fractions. 

We always want to start with concrete because it gives students what’s called an imprint for them to refer back to later. That imprint could be you modeling under a document camera or my demo on a YouTube video. It could be remembering how they built it with a partner in class. Regardless of how they get it, they now have a memory of what it was that we were talking about. 

So, for our 4th graders, it starts with getting the language right to understand what we’re really doing with fractions – “groups of” – and then starting with whole numbers and transitioning into whole number times fractions. This will get them ready for 5th grade standards, where they will be multiplying two fractions.

Multiplying Two Fractions (5th Grade)

Again, it’s important to build as a strong foundation of conceptual knowledge as possible so kids have a plan for what they’re doing. We want to avoid teaching procedures that kids have to memorize without really understanding what’s going on. 

In this video, we will walk through how to conceptually walk through an example of multiplying a fraction by a fraction. Let’s take ⅓ x ½. What is that really asking us? By now, our students shouldn’t be intimidated! They should feel confident knowing that it’s asking for ⅓ of ½.

You could definitely use area model papers for these example, or pattern blocks, or other materials, just make sure the manipulatives you choose match the fractions in the examples you’re using.

So, what is ⅓ of ½? I’m going to start with the half – the red trapezoid. Looking at that piece, I need to think about how I’m going to break it up into thirds or group of ⅓. They can’t think of the green triangle as ⅙ though, because we’re not talking about a whole piece, we’re looking at part of a whole. But looking at the trapezoid, students should see that we could fit three triangles onto the trapezoid and break that ½ up into three ⅓ pieces. Then, looking at the ⅓ pieces in relationship to the whole, we can see that ⅓ of ½ represents ⅙. 

Throughout this process, I like to have the whole pattern block, the hexagon, visible to the students so they can know that they can connect that to the whole. As I break up the trapezoid, the triangles are ⅓ of the trapezoid, but when I give my final answer, it’s in relationship to the whole, which makes the triangles ⅙ again.

After we model this process of finding ⅓ of ½, we can ask students what they would do if we asked them to find ½ of ⅓? This gets students to think about the commutative property. We know that you’re probably still going to get the same answer of ⅙, but does the model look the same? No, it doesn’t. Just like if you had whole number multiplication problem like 5 x 6 and built an array. If I change it to 6 x 5, the commutative property says the answer will still be the same, but the picture will be different.

If we want to know what ½ of ⅓ is, we have to start with the blue rhombus, the ⅓ piece. When I look at that, I can take my triangle and lay it on top of the rhombus to see that it actually does represent ½ of the piece. But, when looking at the whole as a hexagon, the green triangle represents ⅙, which is the same as the previous problem. 

Learn by Doing

I don’t think kids get enough time to explore different fractions with pattern blocks as we did in these videos. My encouragement to you as the teacher is to think of as many fraction multiplication problems as you can – not just in simple unit fractions of ⅓ and ⅙ and so forth, but fraction problems that will prompt your students to think in different ways as they use the pattern blocks in more versatile ways. Could you designate 2 hexagons as the whole? If so, that changes the pieces so that the trapezoid is worth ¼, the rhombus is ⅙, the triangles are 1/12.

What are some other ways you’ve used pattern blocks to represent fractions??

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