# ZeroSum Ruler (home)

## Blogging on math education and other related things

### Multiplying Fractions with Pictures! June 15, 2011

Fraction Multiplication: Of what?

Fractions are probably the most troublesome topic in Math.  As soon as a problem involves a fraction, kids freeze up.  In Math, of tells us to multiply.  How many shrimp are in five pounds of shrimp?  We multiply the number of shrimp in a pound by five.  Once we know this, fraction multiplication becomes a bit easier to understand.

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The How of fraction multiplication is easy – multiply the numerators and multiply the denominators.  When we show fraction multiplication with pictures, we need to remember of.

Now to the Why.  To start, we’ll look at a relatively easy problem so that we can develop a pattern to follow with more difficult fraction multiplication problems:

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(1/2)(1/2)

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Translated into English, this problem reads “one-half of one-half”.  Here’s a picture of  1/2:

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And the area below in red is “one-half of one-half”:

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It’s easy to see that one-half of one-half is (1/4).  And in fact:

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(1/2)(1/2)   =   1/4

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Before moving on, let’s look more closely at one aspect of the problem above: the denominator 4.  Where did this come from?  To get that denominator, we needed to keep the entire circle (whole) in mind.  In other words, we needed to say that the red piece was “1 out of something”.  (Confusingly, out of means to divide in Math!)  The denominator is 4 because the red pie piece is 1 out of 4 total pie pieces in the circle.  Always remembering the entire original area is key in fraction multiplication.  Later, we’ll see the same is true with fraction division.

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To make the denominator easier to see, we can divide the circle twice: first vertically for the first fraction, then horizontally for the second fraction:

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It’s then easy to see that the overlapped area (numerator) and the entire number of pie pieces in the circle (denominator) create our answer.  This will always be the case.  It wasn’t a coincidence that the denominator was naturally created as we divided the circle twice.  Let’s use this pattern to solve a more complicated fraction multiplication problem:

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(2/7)(3/5)

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Because these two fractions do not have a common denominator, it would be hard to divide a circle into 7 (and take 2), then into 5 (and take 3), and analyze the overlapped area.  So instead, we’ll use nice, easy rectangles.  First,  2/7:

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In the rectangle above, two of the rectangle’s 7 horizontal bars are colored green to represent 2/7.  Now, keeping the whole rectangle in mind, let’s take 3/5:

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In the rectangle above, three of the 5 vertical columns are colored blue to represent 3/5.  Where to the two colored areas overlap?

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In the above picture, we can see that the overlapped area consists of 6 purple boxes.  But 6 of what?  Remembering our easy example (1/2)(1/2), where our denominator was the total number of pie pieces in the circle after our two rounds of dividing, let’s count the total number of boxes in the above rectangle.  The total number of boxes is 35.

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And in fact:   (2/7)(3/5)   =   6/35

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OLD FASHIONED CHECK: We know that (2/7)(3/5) = 6/35 from the algorithm “multiplying across”.

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To see this more clearly, we can look at the below picture and see that the area the fractions share, or the overlapped region, is 6 boxes, and the area of the entire region is 35 boxes.  “6 out of 35 boxes are double shaded”.

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In this last picture we can see that the area of the double shaded region is 6, or (2×3), and the area of the entire region is 35, or (7×5), which is why we multiply the numerators (2×3) and the denominators (7×5) when we find the product of two fractions.

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You can download a PDF ebook that uses pictures to explain fraction division, multiplication and addition on CurrClick at Fractions: A Picture Book!

contact blog author Shana Donohue: shanadonohue@gmail.com

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### 8 Responses to “Multiplying Fractions with Pictures!”

1. Leslie Stofer Says:

This one, too! My students aren’t quite at fraction multiplying yet, but maybe by the end of the year. In the meantime, I’ll enjoy explaining long division to them, especially when I get the questions! Awesome posts!

2. [...] (Also see Multiplying Fractions With Pictures!) [...]

3. The assertion that “math is a language” is commonplace, but making it and claiming that anyone who disagrees is “wrong” strikes me as oddly arrogant. You’ve offered nothing to support the claim, and your one example, that “of” means “multiply” in mathematics is both an example of ENGLISH, not mathematics, and at best occasionally correct. Sometimes “of” indicates multiplication. Other times, it does not. Telling students that “of” MEANS “multiply” automatically is ensuring that they will be confused when they continue to hit examples when it doesn’t.

The problem with trying to reduce problem-solving to a “word-by-word” translation into mathematical symbols is that it’s mindless, just like mindless application of algorithms, mindless calculator use, and any other mindless activity in mathematics. When kids aren’t thinking, they aren’t doing mathematics, and it’s bad pedagogy to encourage them to operate in such a brainless manner.

In the case of fraction multiplication (or taking percents of percents), it might be more clear to say that, for example, 3/4ths of 2/3rds can be read as (3*2) out of (4*3) which is 6 out of 12 (followed by simplifying to 1 out of 2 or 1/2). Yes, it’s implied multiplication in this case, just as 30% of 90% is 30*90 out of 100*100 or 2700 out of 10000 which simplifies to 27 out of 100 or 27%. And yes, the diagram which translates such problems into an area model can be helpful for students to understand why multiplication makes sense in such cases. But let’s help them think flexibly and reasonably about problems rather than handing them mindless crutches (e.g., the execrable “FOIL” that leaves students unable to make sense of multiplication of algebraic expressions that DON’T involve exactly two factors, both of which are binomials). And please: if you want to make the argument that “mathematics is a language” (really? So we learn it naturally, like our native tongue? People walk around “speaking” mathematics? Or is it only or primarily a written ‘language’ just like, um, Sanskrit?), do some investigations into the arguments for and against that assertion. Because I disagree, I’m not alone, and the burden of proof that those of us who don’t accept that claim are “wrong” is on YOU.

• You’re pretty angry about this. My students are excelling; my guess is that you’re not even a teacher. Try solving “6 less than the sum of a number and 4 is less than 18″ without understanding of language.

As for your beef with fractions, you’re not unlike my students who hate them. The paper I wrote and posted on here showing multiplication and division of fractions through pictures received an A at Harvard. “of” means multiply. “out of” means division. What’s your point exactly?

The annominity of the internet is great for bringing out the worst in people. I’m sure you’re not angry about this, probably about something else. Bring your anger elsewhere maybe? I’m just trying to help.

• Also, anyone can tell a kid how to multiply fractions. It’s the why that’s hard and that you clearly don’t understand enough to explain.

I agree wth you on FOIL, which is why I don’t teach FOIL. Ask next time?

Fractions (check), FOIL (check), language (check). Any other complaints?

• Why the hostility? But anyway, check out Frank Smith’s book: The Glass Wall: Why Mathematics Can Seem Difficult. He talks about the language of mathematics and how it is different from a natural language, but a language none the less.

4. [...] For a picture tour on how to deal with fractions: Multiplying Fractions With Pictures! [...]