Playing Complex Fractions with Your Kids

This week, I’m working on graphics for my upcoming book 70+ Things to Do with a Hundred Chart. I had fun with this complex fraction image.

It looks a bit cluttered. Possible tweak: Remove the brackets and instead use a thicker dividing line to show the thirds.

While I’m thinking about that, would you like a sneak peek at an activity from the book?

Make Your Own Math

You don’t need a set of worksheets or lesson plans to learn math. All you need is an inquiring mind and something interesting to think about.

Play. Discuss. Notice. Wonder.

Enjoy.

Here’s how you can play complex fractions with your kids…

Start with Fraction Strips

Print a few blank 120 charts and turn them sideways, so each chart has ten rows with twelve squares in each row.

Cut out the rows to make fraction strips with twelve squares on each strip.

Color a different set of squares on each strip. On some strips, arrange the colored squares all together at one end. On other strips, mix them around.

If we count each strip as one whole thing, what fraction of its squares are colored?

Match the strips that represent the same fraction.

On some of the strips, there will be more than one way to name the fraction. For example, if six squares are colored, we can call that 6/12 or 2/4 or 1/2 of the strip. These alternate names are easiest to see when the colored squares are all at one end of the strip, because you can fold the strip to show the halves or fourths.

How many different fraction names can you find for each set of colored squares?

Look for Complex Fractions

We could also call the strip with six colored squares “1 1/2 thirds” of the whole strip. Can you show by folding why that name makes sense?

Or we could call the strip with five colored squares “2 1/2 sixths.”

When we have a fraction within a fraction like this, we call it a complex fraction, because it is more complicated than a common (or simple) fraction.

Another way to say it: Complex fractions have other fractions inside them.

A complex fraction is like a puzzle, challenging us to find its secret identity — the common fraction that names the same amount of stuff.

For example, how much is 3 1/3 fourths? One fourth would be three of the twelve squares on a fraction strip. So three fourths would be three sets of those three squares, or nine squares. Then we need to add one-third of the final fourth, which is one of the remaining three squares. So 3 1/3 fourths must be ten squares in all.

3 1/3 fourths = 10/12 = 5/6

How many complex fractions can you find in your set of fraction strips?

Challenge Puzzles

Can you figure out how much a one-and-a-halfth would be?

That is one piece, of such a size that it takes one and one-half pieces to make a complete fraction strip.

A one-and-a-halfth is a very useful fraction and was a favorite of the ancient Egyptian scribes, who used it to solve all sorts of practical math problems.

How about a one-and-a-thirdth? How many of those pieces make a whole strip? What common fraction names the same amount of stuff?

Or how much would a two-thirdth be? In that case, it only takes two-thirds of a piece to make a complete strip. So the whole piece must be greater than one. A two-thirdth’s secret identity is a mixed number. Can you unmask it?

Make up some challenge fraction mysteries of your own.

Complex2

Update…

I’m still working on the graphics for my hundred chart book. Here’s the latest version of the complex fraction strips.

I like this one much better.

What do you think?


CREDITS: The slogan “Make Math Your Own” comes from Maria Droujkova, founder and director of the Natural Math website. Maria likes to say: “Make math your own, to make your own math!”

70+ Things to Do with a Hundred Chart is part of my Playful Math Singles series. Coming soon to your favorite online bookstore…

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13 thoughts on “Playing Complex Fractions with Your Kids

  1. These fractions are interesting. The tweak is good I think. I was wondering if you could make it easier to see the numerator and denominator fractions in different colors.
    Do you have any suggestions for further reading? That might be a good thing to include in your book to help readers who decide they want to learn more about Egyptian fractions. I can’t seem to find the heiroglyph for 1-1/2 anywhere.

    Thanks for sharing the 120 chart idea too for off-screen activities.

  2. Hi Denise,This online sources shows a different hieroglyphs for 2/3:
    https://discoveringegypt.com/egyptian-hieroglyphic-writing/egyptian-mathematics-numbers-hieroglyphs/
    I think the Heiroglyphs are a different topic than your book, so you might want to think about why it’s there with the hundred charts. Just my opinion, though.
    Everywhere I look for complex fractions I find two things: either Algebra or how to get rid of the fractions. So,I guess I may have been conditioned (good or far-reaching who knows) to reduce and get rid of fractions. That leaves no room for complex fractions or drawings of them.
    I found some other resources for “Further reading” for those wanting to read, not just play. That could be a bonus for your book. Please tell me if this helps or not.

    1. Yes, and Wikipedia makes the same mistake on the hieroglyph. But the lines should not be the same length — that is more like the hieroglyph for one-half (the reciprocal of two).

      I didn’t intend the article to be about Egyptian fractions. I just thought it was an interesting side-note that a complex fraction — which we think of as strange, somewhat advanced math — was very familiar and common to people so long ago.

      The point of this section is to play with naming fractions, and to build mental connections between different representations. To develop flexibility in our thinking about fractions *before* learning any of the standard algorithms for reducing them.

      It’s in the hundred chart book because making fraction strips is one way to use a (blank) hundred chart. Or in this case, a 120-chart, which is one of the basic variations of the standard 100-square.

      I’d love to see your further resources.
      This book is my favorite for research: https://amzn.to/2SJx6os
      And this one is fun for a wide range of ages: https://amzn.to/2SIc61p

  3. Here’s another resource related to Egyptian fractions:
    https://nrich.maths.org/6541

    Greedy algorithm.
    I haven’t tried to to it yet, but I will.
    I was wrong about the heiroglyphs. I’ll put the books you mentioned on my library hold list.
    I will have to get into it slowly. It’s interesting but I’m having trouble figuring out how to write these fractions in the right form.

    1. Unfortunately, many of the explanations of Egyptian fractions (including my own older posts) focus too much on *how* to write them and not enough on *why* it makes sense.

      That makes the rules seem arbitrary, even artificial.

      These fractions grew out of the problem of how to share stuff fairly between a lot of people. For instance, imagine you are the scribe handling payroll for an Egyptian contractor. You have a bunch of men working on your construction project, and you’re paying them with bread and beer. (From some of the example problems in old papyri.)

      So you have to cut portions of bread and measure out portions of beer. That’s why the focus is on unit fractions. You don’t have modern measuring cups, with all the markings. But you can cut a half of a loaf or pour out a third of a pint, or whatever.

      It doesn’t even enter your mind to think of a fraction like “5/6.” Why would anyone cut a loaf into sixths and then give someone five little pieces? That would be nonsense to an Egyptian scribe. But to give the person a half and a third — that makes sense. You can cut some loafs into halves and some into thirds, and when each worker comes through the line, you hand him one piece of each size.

      And that’s why the Greedy Algorithm makes sense. You want to hand out the biggest pieces you can work with. Your workers don’t want to take home crumbs! (My guess is, when the division did come out with a few small pieces, the workers ate those on the spot — like samples at the grocery store — and took the larger chunks home to their families.)

      1. Your explanation for how the Egyptian’s used these fractions makes sense to me.
        You might like reading what Ian Stewart wrote about Egyptian fractions, in his book,”Professor Stewart’s Hoard of Mathematical Treasures”, pages 76 -81. He includes more about the greedy algorithm.
        If you want to see how doing recreational math can be educational and playful fun, I highly recommend this book. I love reading most of his books because he really has a way with words that connects with me.
        So, unless you hate puns, you are in for some fun☺

        1. I think your new graphic for the complete fractions is very good because I could see the fractions easily.
          I will try to write out an Egyptian fractions to see if I get how it works.
          Thank you for mentioning it on here.
          I definitely had not heard of the greedy algorithm before. But my kids knew about these topics from reading books. I think I need to catch up with them.

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