## Wednesday, October 19, 2011

### Fractions

I've dreamed of offering a one-unit course on fractions for our students. (I teach at a community college.) So many of them really struggle with fractions, and don't really understand why fractions work the ways they do. I'd like a course that's just pass-fail, with no grades otherwise. I'd like no textbooks (unless there's something really good for adults that I haven't seen yet). I'd like it to be about understanding, but to also include practice to cement that understanding.

Uri Treisman showed years ago that an excellent strategy for getting students up to speed was to have them work in groups on extra challenging problems.

MBP, at Rational Expressions, just offered up a good problem that is challenging enough to make me work hard, and approachable enough for the students in a fractions class to work on as their 'Research Into Fractions'. MBP and I have different requests stemming from this problem. MBP wants to know what makes a problem 'hooky'. (If you can't answer that, maybe you can offer MBP an example of a problem that really hooked you.) I want to know what other problems would be good for my imagined students in this imagined class. Problems that involve fractions, and make the students work hard with fractions, that start out approachable, and have enough hook to get the students working persistently.

Here's my version of the problem MBP offered:

Any fraction of the form 1/n is called a unit fraction. 1/2 can be written as the sum of two other unit fractions (1/3+1/6).
1. Can this be done for all unit fractions?
2. Find a rule for the number of ways to do this.

Got any challenging fraction problems that a newbie might enjoy chewing on?

1. One of my favorites is on the 6th slide of this presentation:

http://www.fi.uu.nl/en/fius/rmeconference/handouts/shepard/Realisticmath2009Shepard.pdf

Part (a) is certainly for newbies, but beyond that you have to be able to stretch your understanding of fractions and see the unit change size.

2. How about something like evaluating 1/2/2/2/2?

3. Are you suggesting that they should decide how many ways there are to interpret that?

4. @Raymond, those might be a good start.

Since I'm working with adults, I'm concerned they might not be hooked into feeling like they're doing something new, and not just more review.

5. I was really just trying to play around with the notation and stretch it in interesting ways. I was imagining ((((1/2)/2)/2)/2).

One of the things that seems to work, sometimes with my students, is pushing notation to its limits. What I think is happening is that, sometimes, when you push notation into strange, absurd looking forms, one is forced to confront what the notation really signifies.

6. What about (1/2)/2 + (1/2)?

A student might simplify the first fraction, or might use 2(1/2)/2 = (1/2). This seems like a cool problem. They could check answers on their calculators.

7. For the unit fractions, it is very interesting (at least to me!) that these are what the ancient Egyptians worked with. http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fractions/egyptian.html

Maybe for some students, knowing the history might give it even more interest? It is fun...

8. Thanks for mentioning that, mathmom. I had thought about it, but wrote this post in a hurry.

Many of my students are African-American, and working with sophisticated mathematical ideas from ancient Egypt will be a plus for them, and perhaps for all students of color.

James Tanton has a bit about Egyptian Fractions in his book Arithmetic: Gateway to All. If you google 'james tanton egyptian fractions', you can get a free download of his fraction chapter.

9. One easier (I think) way of hooking students is providing problems they can relate to. There are a lot of fractions in their world that they find challenging or make incorrect assumptions about.

One example brings geometry into the mix by using 22/7 as an approximation for pi. Pizza places often sell circular pizzas in small, medium, and large. If the small is a 6 in. pizza and the large is a 12 in. pizza, is the large double the size of the medium?

Stores often do some crazy things with sales. 25% off one item plus 20% the entire purchase. The percents can be converted to fractions and students can consider what these discounts do to the original price.

10. Mathmama

Thanks for presenting this problem. In my Alg II Honors class this week I posed this as a class opener thinking we might talk for about ten minutes. Forty minutes later we, they did all the heavy lifting, decided that 1/n is always equal to the sum of 1/(2n) , 1/(3n), and 1/(6n) They saw great patterns along the way. I presented the following three follow ups to 1/2 = 1/3 + 1/6

I asked them to decompose 1/10, 1/20, and 1/17

They saw that for the evens you could write the first fraction as 1.5n and the second denominator as 3n. This did not work for the odd denominator and we stumbled on to the pattern I identified above.

It was a fantastic conversation!

11. Thanks, Mr. Dardy! If your honors students worked on this for an hour, you're making me realize that there are many hours worth of work for students with less skills and confidence in math.

I'm still hoping I can collect other really deep and approachable problems.

12. MM

Another deep and approachable problem that always works for my students is the 'Locker Problem' (a nice explanation of it is at (http://connectedmath.msu.edu/CD/Grade6/Locker/index.html ) I always follow up with a great factoring conversation about why the open doors are the ones that they are. The handshake problem (http://mason.gmu.edu/~jsuh4/impact/Handshake_Problem%20teaching.pdf) is another rich one

13. I love both of those, but neither is fractions, of course.

14. Sue, JUMP has a fractions unit that they recommend as an intro-unit for all their courses:
http://jumpmath1.org/introductory_unit
Now, this is aimed at elementary school kids, but you may be able to extract some useful stuff from it.
-- Dan

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