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Confused about fractions? Visualize brownies, not pizzas

Friday, September 3rd, 2010

Pizzas. They’re the best way to represent fractions, right? Everyone has seen a pizza. Everyone knows you can slice a pizza different ways. When you need to visually represent fractions, pizzas are the go-to metaphor, right?

Well, that’s what I thought until I tried using pizzas to teach equivalent fractions. Sure, it’s easy to use a pizza to represent quarters, sixths, eighths. Thirds aren’t even too bad.

But just try dividing a circle into five equal pieces. Or seven. Or ten. Even though I’m a grown-up and a professional math educator, it’s really hard for me to consistently do this.

And as for showing visually how two fifths equals four tenths? Unless I do an impossibly immaculate job of dividing those pizzas into pieces, forget about it.

Are they really the same?

Fortunately, I’ve found a much better way to represent fractions visually: the Math-U-See fraction overlays. If we’re going to use a food metaphor, they’re flat and square, like brownies from a square pan.

And like brownies, they are much easier to slice evenly than pizzas. In fact, because they’re reusable transparent overlays, you don’t even have to draw (or slice) anything. You just arrange them on top of each other.

Check it:


Awwww, yeah! Those fractions are DEFINITELY equivalent!

You can use the overlays to represent basic fractions with divisors from two through seven, or combine multiple overlays to build other denominators.

Not only do they make it really easy to see how equivalent fractions work, but using them is way easier than drawing pizzas and hoping the slices come out even.

The fraction overlays make the concepts extremely clear, and because they’re tactile, visual, and kinesthetic, they make fractions feel like a game instead of work. Whenever I’ve used them with my students, either in person or during online tutoring sessions with a webcam, they’ve react the same way: “This is fun!”

Now that’s how I want my students to feel about fractions!

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10 Comments on “Confused about fractions? Visualize brownies, not pizzas”

  • Teachies on September 5th 8:13 pm

    Don’t you love finding that exactly-right-way to represent an idea? Simple idea, but very powerful…

  • Rebecca Zook on September 5th 9:19 pm

    Thanks so much for your comments!! Math U See is *the bomb*. I’d love to hear more of *your* simple-but-powerful ideas!!

  • Denise on September 21st 1:12 pm

    I wrote a short book for homeschoolers several years ago (now out of print) promoting exactly this “pan of brownies” approach. It lends itself very well to understanding fraction multiplication, ratios and percents, too.

  • Rebecca Zook on September 22nd 2:21 pm

    Denise, thanks so much for stopping by! I’m glad to see you here. I would be very interested in reading your book! Thanks for all the work you do over at letsplaymath, I love the concept and all the resources you are sharing with the community!

    I’ve just started using the math u see fraction overlays/brownie concept with percents and it is so cool! You can just take two “10” overlays and lay them on top of each other to make a 10 x 10 grid, and then lay that on top of the fraction to instantly convert it visually to a percent. I’d love to hear more of your ideas on this topic!

  • Alexander Bogomolny on October 12th 3:42 pm

    I have a couple of Java applets that allow for this kind of demonstration. Here’s one

    It is a little more general and graphically illsutrates fraction comparison. In particular, it shows that equvalent fractions are also equal.

  • […] used grids to show how multiple decimals can add up to wholes.  When reviewing fractions, we would divide a square into parts to make the concept visual and concrete.  When her class started working on adding and subtracting […]

  • Laurie on December 7th 3:47 pm

    I love your idea of using two overlays to show converting fractions to percents. I’ll be using that idea. Thanks!

  • Rebecca Zook on December 7th 4:57 pm

    Hey Laurie! Thanks so much for stopping by, it’s nice to “meet” you here! I can’t take credit for that idea–it’s Steve Demme’s, the author of Math U See–but I am very glad to spread it far and wide, and also very glad that you think it might be helpful! Let me know how it goes! 🙂

  • Oz on September 5th 7:38 am

    Got some videos about this

  • Alex on March 28th 2:57 pm

    Thank you for this amazing article!

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