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.
Instrumental Understanding: Math as a Tool
Fractions 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.
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:
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?
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.
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). This 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…
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