*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.

*Click here to continue reading: 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 my book *Let’s Play Math: How Families Can Learn Math Together—and Enjoy It,* available at your favorite online book dealer.

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I was wondering why fractions are difficult to learn. Here are some possible answers I found:

(1) Taught based on rules and there are too many rules for kids to understand, ref: Homeschool Math. A related point is how quickly learners are meant to absorb all the fraction operations while, for example, integer operations are introduced much more gradually.

(2) Manipulating fractions requires multiple models, ref: James Tanton.

(3) Fractions are accompanied by a lot of confusing vocabulary, ref: SSMA. This is nicely illustrated by another page I found that puts vocabulary front-and-center in its introduction to fractions: James Brennan

(4) Fractions visually confuse the novice, ref Math Stack Exchange question. This relates to the chunking observation within expertise theory: experts group together meaningful chunks of information to reduce the complexity of their analysis, while novices attend to individual components and struggle to see the relationships. Thus, when we see “2/5” we recognize one entity, while learners attend to the 3 distinct components “2” “/” and “5”

(5) Fraction manipulations are presented with an unclear relationship to integer arithmetic (maybe which learners haven’t yet mastered in itself). ref: My own idea.

(1) and (2) are related, with (1) taking a procedural perspective and (2) focused on conceptual understanding. This seems to describe the inherent structural difficulty of fractions. The other points seem to be addressable with different choices about how we teach fractions.

Wonderful points, Joshua! Those all ring true to me. For a demonstration of how confusing fractions can be, have you seen my Fractions Quiz?

The things that I find most helpful in teaching fractions are (1) use plenty of stories before teaching abstract number calculations, and (2) use a variety of diagrams, but especially rectangles. Stories help my kids draw on their own intuition, and this grounds their understanding of fractions in their experience about how things work in the world.

I really like the James Tanton article you linked to, for providing a transition between that elementary, intuitive knowledge and the more mature understanding we want our kids to develop in middle and high school.

Your fractions quiz post is a nice one. I hadn’t seen it, so thank you for linking. Among other things, it strengthens the sense that a strong understanding of integer operations and associated models makes the fraction operations a lot easier and, probably, weakness with integers makes fractions almost impossible.

Only now do I notice that your prior post was about multiplying and area of rectangles!

Duplos work, too. Fractions are easy. Give them the rules when they are cognitively ready for them.

Fractions are easy in the beginning, Kim. My kids all enjoyed when fraction lessons came up in their early books, because it meant a break from the challenges of place value arithmetic.

But the hard stuff comes soon enough. By middle school, almost all of the tough monster topics (decimals, percents, proportions, etc.) are fractions in disguise. That’s when we find out how well our students really understand what we’ve been teaching. And woe to the student who is trying to get by on memorized rules at that point! There are too many complications to keep it all straight just by memory.

Fractions should be easy for the teacher! Have you figured out where the rules for +, -, x, and division are derived? I have been studying that.

What kind of numbers are fractions? Do they have properties? How do we know what these properties are? Fractions are rational numbers. Arithmetic is the study of the properties of rational numbers. A rational number is a number that can be written as a ratio, a/b, b cannot equal zero. So….. When you ask why do we not need the denominators need to be the same for multiplying, but they do for adding, What properties of rational numbs are we applying? Are we using any of the properties from arithmetic?

Are we sure elementary students have developed the logic yet to understand any of this?

Elementary students probably haven’t developed the logic to think about “properties of rational numbers” — even high school students struggle with that!

But elementary students even as young as kindergarten DO have the ability to think about a fraction of a fraction. Cutting pieces into smaller pieces. A part of a part.

And they will intuitively use the properties of rational numbers, which are basically formal statements of common sense. Our goal as teachers is to present math ideas in a way that connects to our students’ intuition and common sense.

We can do that by using stories, such as the cutting-up-brownies story in this post. And by relying on mental math as long as possible before teaching formal rules. https://denisegaskins.com/2015/01/12/teaching-the-standard-algorithms/

Thank you. I’m just going to do the math now. 😺Thanks for the input.

Ok. I get it. It’s the schools that are giving these algorithms before they would be able to understand it without the writing out problems.😃

This is what I am tidying to say. You need to make sure you know what rational numbers are.

https://www.mathsisfun.com/algebra/rational-numbers-operations.html

Never mind. You don’t understand what I’m asking , so that’s all for me.

I’m sorry I didn’t understand your questions. Perhaps you were looking for a more abstract approach, like this? http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.43.2420&rep=rep1&type=pdf

Thanks for the info. I’m sure I can help them if they don’t understand, so that’s all for me. 🙂

Ok. I do understand what you are saying. I just got hung up on the “Nix the tricks”. But now I see you are just explain how to do fractions with games, not on paper with the algorithms.

Thank you. I understand now. I’m glad to hear they understand fractions on that level. That helps me a lot.🙂