It’s hard to make sense of multiplication and division of fractions. Before fractions, multiplication can almost always be seen as repeated addition, and the answer is never smaller than both the numbers you started with. Division is usually sharing, and the number of cookies each child gets is always smaller than the total number of cookies (unless someone is selfishly dividing by 1 and not sharing with any of her friends).

Fractions first complicate the notion of numbers — 3 no longer is the next smallest number after 2, and \(\dfrac {1}{2}=\dfrac{2}{4}=\dfrac{973}{1946}\) — not only does a number have an infinite number of “names,” but some of them are pretty weird. After complicating numbers, they then complicate operations. How exactly is \(2 \div \dfrac{1}{10}=20\)? How does that make sense?

The vast majority of my students who are pre-service elementary teachers did not make sense of multiplication and division of fractions during their K-12 education. However, many can solve everyday problems that model these operations, but without recognizing that the problems involve multiplication or division of fractions, and without using any algorithms (which is fine, especially when they don’t understand why the algorithms work).

I wrote the handout Real Life Fraction Problems (doc file, pdf file) to help students find the multiplication and division in everyday fraction problems. Some of the problems are also very nice just as problems involving fractions that don’t necessarily need to be solved with multiplication and division. Some are open-ended, and most admit multiple solutions; my solutions handout (doc file, pdf file) highlights places where multiplication and/or division can be used.

I’ve had this handout up on my old site (which I am phasing out) for a while, and I’ve gotten feedback from elementary and middle school teachers who found it helpful for their students. I encourage teachers to write similar problems, based on their lives and their students’ lives and to also encourage their students to write problems for themselves and each other. If your emphasis is less on identifying fraction multiplication and division, then you might want to rework the solution sheet.

BTW, it’s been a long time since I buzzed my own hair, but I do still like to make Mandel Bread.

## 5 Responses

## Art Weidner

Thank you

## dborkovitz

You’re welcome

## Anuj

Dear Heath,Thank you for commenting on the blog. There is a lot of ways to show litlte kids what fact familys are and mean. This is a good post to help people that are not very good at divison.They didn’t know what it meant because lots of schools back then didn’t call them fact familys. Like when I went to my grandma’s house my cousin was there and I told her about fact familys and she said when she was at school they called it factors.It is really easy to undersand when you have been tought a lot but like your sister and your dad don’t undersand because they haven’t been tough as much as you have. That’s why we have great teachers because undersand every thing they teach usThey are very good exemples that you have done. From Maddy

## Angela Durall

This was very helpful!

## dborkovitz

Thanks!