Understanding Math: Multiplying Fractions

Click to read the earlier posts in this series: Understanding Math, Part 1: A Cultural Problem; Understanding Math, Part 2: What Is Your Worldview?; Understanding Math, Part 3: Is There Really a Difference?; and Understanding Math, Part 4: Area of a Rectangle.

In this post, we consider the second of three math rules that most of us learned in middle school.

  • To multiply fractions, multiply the tops (numerators) to make the top of your answer, and multiply the bottoms (denominators) to make the bottom of your answer.

fraction-rule

Instrumental Understanding: Math as a Tool

math-fractionsFractions confuse almost everybody. In fact, fractions probably cause more math phobia among children (and adults) than any other topic before algebra.

Children begin learning fractions by coloring or cutting up paper shapes, and their intuition is shaped by experiences with food like sandwiches or pizza. But before long, the abstraction of written calculations looms up to swallow intuitive understanding.

Upper elementary and middle school classrooms devote many hours to working with fractions, and still students flounder. In desperation, parents and teachers resort to nonsensical mnemonic rhymes that just might stick in a child’s mind long enough to pass the test.

The CrissCross Applesauce family is just one of the many fraction mnemonic tricks you can find online. For more information, check out NixTheTricks.com.
The CrissCross Applesauce family is just one of the many fraction mnemonic tricks you can find online. For more information, check out NixTheTricks.com.

Relational Understanding: Math as a Connected System

Do you remember our exploration of the area of a rectangular tabletop?

Now let’s zoom in on our rectangle. Imagine magnifying our virtual grid to show a close-up of a single square unit, such as the pan of brownies on our table. And we can imagine subdividing this square into smaller, fractional pieces. In this way, we can see that five-eighths of a square unit looks something like a pan of brownies cut into strips, with a few strips missing:

One batch of brownies is one square unit, but part of the batch has been eaten. Now we have fractional brownies: five-eighths of the pan.
One batch of brownies is one square unit, but part of the batch has been eaten. Now we have fractional brownies: five-eighths of the pan.

But what if we don’t even have that whole five-eighths of the pan? What if the kids came through the kitchen and snatched a few pieces, and now all we have is three-fourths of the five-eighths?

We can make a fraction of a fraction by cutting the other direction. We cut the strips into fourths, and the kids ate one part of each strip.
3/4 of 5/8: We can make a fraction of a fraction by cutting the other direction. We cut the strips into fourths, and the kids ate one part of each strip.

How much of the original pan of brownies do we have now? There are three rows with five pieces in each row, for a total of 3 × 5 = 15 pieces left — which is the numerator of our answer. And with pieces that size, it would take four rows with eight in each row (4 × 8 = 32) to fill the whole pan — which is our denominator, the number of pieces in the whole batch of brownies. So three-fourths of five-eighths is a small rectangle of single-serving pieces.

Compare the pieces we have left to the original batch. Each of the numbers in the fraction calculation has meaning. Can you find them all in the picture?
Compare the pieces we have left to the original batch. Each of the numbers in the fraction calculation has meaning. Can you find them all in the picture?

fraction-rule

Notice that there was nothing special about the fractions 3/4 and 5/8, except that the numbers were small enough for easy illustration. We could imagine a similar pan-of-brownies approach to any fraction multiplication problem, though the final pieces might turn out to be crumbs.

Of course, children will not draw brownie-pan pictures for every fraction multiplication problem the rest of their lives. But they need to spend plenty of time thinking about what it means to take a fraction of a fraction and how that meaning controls the numbers in their calculation. They need to ask questions and to put things in their own words and wrestle with the concept until it makes sense to them. Only then will their understanding be strong enough to support future learning.

To be continued, probably after the holidays. Next up, Understanding Math Part 6: Algebraic Multiplication…


CREDITS: “School Discussion” photo (top) by Flashy Soup Can via Flicker (CC BY 2.0). LPM-ebook-300This is the fifth post in my Understanding Math series, adapted from the expanded paperback edition of Let’s Play Math: How Families Can Learn Math Together and Enjoy It. Coming in early 2016 to your favorite online bookstore…

Free-Learning-Guide-Booklets2Claim your two free learning guide booklets, and be one of the first to hear about new books, revisions, and sales or other promotions.


Noticing Fractions in a Sidewalk

lighthouses cropped

fraction-circle

My daughters didn’t want to admit to knowing me, when I stopped to take a picture of the sidewalk along a back street during our trip to Jeju. But aren’t those some wonderful fractions?

What do you see? What do you wonder?

Here is one of the relationships I noticed in the outer ring:

\frac{4 \frac {2}{2}}{20} = \frac {1}{4}

sidewalk

And this one’s a little trickier:

\frac{1 \frac {1}{2}}{12} = \frac {1}{8}

Can you find it in the picture?

Each square of the sidewalk is made from four smaller tiles, about 25 cm square, cut from lava rock. Some of the sidewalk tiles are cut from mostly-smooth rock, some bubbly, and some half-n-half.

I wonder how far we could go before we had to repeat a circle pattern?

Continue reading Noticing Fractions in a Sidewalk

Fractions: 1/5 = 1/10 = 1/80 = 1?

dividing sausages

[Feature photo is a screen shot from the video “the sausages sharing episode,” see below.]

Fractions: 1/5 = 1/10 = 1/80 = 1?

How in the world can 1/5 be the same as 1/10? Or 1/80 be the same as one whole thing? Such nonsense!

No, not nonsense. This is real-world common sense from a couple of boys faced with a problem just inside the edge of their ability — a problem that stretches them, but that they successfully solve, with a bit of gentle help on vocabulary.

Here’s the problem:

  • How can you divide eight sausages evenly among five people?

Think for a moment about how you (or your child) might solve this puzzle, and then watch the video below.

What Do You Notice?

Continue reading Fractions: 1/5 = 1/10 = 1/80 = 1?

Fraction Game: My Closest Neighbor

playing cards

[Feature photo above by Jim Larrison, and antique playing cards below by Marcee Duggar, via Flickr (CC BY 2.0).]

I missed out on the adventures at Twitter Math Camp, but I’m having a great time working through the blog posts about it. I prefer it this way — slow reading is more my speed. Chris at A Sea of Math posted a wonderful game based on one of the TMC workshops. Here is my variation.

Math concepts: comparing fractions, equivalent fractions, benchmark numbers, strategic thinking.

Players: two to four.

Equipment: two players need one deck of math cards, three or four players need a double deck.

Continue reading Fraction Game: My Closest Neighbor

Quotable: Math Connections

ConnectedGearsJoBoaler

It turns out that the people who do well in math are those who make connections and see math as a connected subject. The people who don’t do well are people who see math as a lot of isolated methods.

— Jo Boaler
Math Connections

If you or your children struggle with math, Boaler’s non-profit YouCubed.org may help you recover your joy in learning.

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Reblog: A Mathematical Trauma

fraction pieces

Feature photo (above) by Jimmie via flickr.

My 8-year-old daughter’s first encounter with improper fractions was a bit more intense than she knew how to handle.

I hope you enjoy this “Throw-back Thursday” blast from the Let’s Play Math! blog archives:


Photo (right) by Old Shoe Woman via Flickr.

Nearing the end of Miquon Blue today, my youngest daughter encountered fractions greater than one. She collapsed on the floor of my bedroom in tears.

The worksheet started innocently enough:

\frac{1}{2} \times 8=\left[ \quad \right]

[Click here to go read the original post.]


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