Fraction division part I: How do you know when it is division?

In her book Knowing and Teaching Elementary Mathematics, Liping Ma wrote about this question and how teachers responded to it:

Write a story problem for $1 ¾ \div ½$.

[Pause here and think about the answer yourself.]

Many people find it hard to come up with a story problem that represents fraction division (including many math teachers, engineers, and mathematicians). Why is this hard to do? For many people, their schema for dividing fractions consists almost entirely of the “invert and multiply” rule. But there is much more to thinking about fraction division than that. So much in fact, that we can’t say it all in a single blog post. This is the first of several musings about fraction division.

The trouble with English

Consider this problem:

If you have 12 liters of tea and a container holds 2 liters, how many containers can you fill?

You probably know instantly that this is a division problem and that the answer is 6, because you know your times tables, and specifically you know that $2 \times 6 = 12$. If we say that $a \times b$ means $a$ equal groups of $b$ things in group, then a division problem where $b$ and $a\times b$ are known but $a$ is unknown is called a “how many groups?” problem. Here are some other questions that ask “how many groups?”

  • If you have 1 ½ liters of tea and a container holds ¼ liter, how many containers can you fill?
  • If you have 1 ¼ liters of tea and a container holds ¾ liters, how many containers can you fill?
  • If you have ¾ liter of tea and a container holds 1 ¼ liters, how many containers can you fill?

Some people think that the last one feels like a trick question because you can’t even fill one completely. Because we know the answer is less than one, we could also ask it this way:

  • If you have ¾ liter of tea and a pitcher holds 1 ⅓ liters, how much of a container can you fill?

So a division problem that asks “how many groups?” is structurally the same as a division problem that asks about “how much of a group?”, but because of the way we speak about quantities greater than 1 and quantities less than 1, the language makes the structure harder to see.

What other ways might we see the parallel structure?

Diagrams:

Equations: $$? \times2 = 12, \quad ? \times \frac14 = 1\frac12, \quad ? \times \frac34 = 1\frac14, \quad ? \times 1\frac14 = \frac34.$$  The diagrams don’t have the language problem. In all cases the upper and lower braces show the relation between the size of a container and the amount you have.  Whether a whole number of containers can be filled (diagrams 1 and 2), a container plus a part of a container can be filled (diagram 3), or only a part of a container can be filled (diagram 4), the underlying story is the same.

Many people think of diagrams primarily as tools to solve problems. But sometimes diagrams can help students see structure or reveal other important aspects of the mathematics. This is an example of looking for and making use of structure (MP7).

The equations have an even clearer structure, but more abstract. They all have the structure $$\mbox{(quantity of containers)}\times\mbox{(size of a container)} = \mbox{(how much you have)}.$$

The intertwining of the abstraction of the equations and the concreteness of the diagrams is a good example of MP2 (reason abstractly and quantitatively).

Coming up next week: what else are diagrams good for?

New look, new title

I have updated to a new WordPress theme, partly because I thought it was time for a makeover, and partly to see if it would cure some of the problems people have had commenting. In the process I decided to change the title of the blog. My recent writings have been about school mathematics generally, and I hope they are of use to teachers everywhere, whatever their standards. I will still write occasionally about the Common Core, and I will still answer questions over in the forums. I have changed the settings in the forums to allow anonymous posting for people who have trouble logging in. I may have to change that back again if it causes security problems. And, speaking of security, the site has been protected by SiteLock since last summer’s hacking, which means that you will occasionally encounter a captcha screen.

And, by the way, the url mathematicalmusings.org also points to this blog.

Catching up

The site was down for a few hours today because of a malware attack, but I think we have it fixed now.

I took the opportunity to catch up on comments in the forums; I was way behind! Thanks to all those who responded to readers’ questions. I will try to stay more on top of it. One of the things that has been keeping me busy is our work on grades 6–8 curriculum for Open Up Resources. It is being piloted this year, so that link is still password protected, but stay tuned!

Also, I am close to finishing up the Quantity Progression, the last one not yet done.

Problem with RSS Feed for Forums Fixed

The link to the RSS feed for the forums (on the right of this page) was broken. I’ve fixed it now. You might not have noticed (I didn’t for a while) because it was simply not updating. So if you are using an RSS reader to follow the forums, you should delete your old feed and add the new url. If you don’t understand this message, ignore it!

Tips on searching this blog

I have finally discovered a forum search feature that works. So the box on the right now searches both the blog posts and the forum topics and replies. Here are three ways you can search this blog:

  1. Use the box at the right.
  2. Use the google site search feature: google “site:commoncoretools.me irrational number” if you want to find stuff on the blog about irrational numbers. This also returns hits from pdfs on the blog, e.g., the Progressions.
  3. Go to this old post and do a word search directly from your browser. It goes on forever, so you might to wait until it fully loads.
  4. I think (2) works better, but (1) is slightly more convenient. (3) is a last resort when you get frustrated.

[Update, 1/28/14: (1) stopped working, but I have found a new widget that implements a google site search, on the right. I’ll make this post unsticky now in the hope that we have finally solved the problem.]

New blog registration feature

Now that I’ve moved the blog to a private hosting company, you can register to become a subscriber to the blog using the links on the right. This offers a bit more than the email subscription that has been available for a while. Becoming a subscriber means you can create a profile with some information about yourself, which will be available to people who read your comments (if you are logged in). It also gives you some (not many) controls over forum notifications. I may add more features as I discover them.

I will keep the current email notification feature for those who prefer to keep using that.