Dividing by fractions

I’m in between sections in our Prealgebra book with my younger son, so we took a break today to look at some AMC 8 problems. A problem about dividing fractions tripped him up a little, and that led to a short review of fraction division.

First up: basically the only thing that anyone remembers – flip the denominator:


Next: Let’s try thinking through the same problem using geometry and slicing up circles pizzas:


Next: Much of the difficulty comes from having a fraction in the denominator of our fraction, so what can we do to deal with that difficulty?


Finally, off to the kitchen to look at dividing fractions using snap cubes. We find a collection of snap cubes that we can divide by 2 and by 5, and use that collection to get a better understanding of what (1/2) / (1/5) looks like:


So, a fun little review exercise. I’m sure there are other nice ways of reviewing fraction division, but this short review hopefully provide a nice starting point for understanding beyond just the “flip the denominator” trick.

*Writing 1/3 in binary

We are nearly to the end of our Introduction to Number Theory book. In addition to providing us with many math project that are fun all on their own, I’ve been really happy to find so many great and unusual ways to secretly work number sense. The topic today was writing “decimals” in other bases which gave us lots of time to play with fractions, decimals, and place value.

When I got home from work tonight we reviewed a few of the homework problems and then I gave my son a shot at writing out 1/3 in binary. It was neat to watch him work through this problem, and so cool to see him smile at the analogy to 0.99999…. = 1 in binary at the end of the video. I think the fun of topics like this can draw kids into math while also providing lots of opportunities to build on their basic understanding of arithmetic and numbers.

If you want to get people talking about math – talk about how to divide fractions!

Today my twitter feed has been filled with talk about this article by Professor Marina Ratner of U.C. Berkley:


Dividing fractions is a subject with a bit of history for me. On the funny side, as a kid I missed the week of school when this particular subject was taught and I never seemed to be able to catch up from that missed week. My pals on my high school math team loved giving me a hard time about always having to go back and figure out how dividing fractions worked. Even now they’ll needle me about it when it comes up in one of the videos I do with the boys.

When I was going through this subject with my older son I’m sure my approach was pretty much all over the place. The primary reason is that I’d never had to explain dividing fractions in detail to anyone – much less a kid. That alone assures a lot of stumbling around. Another reason is that although we were following Art of Problem Solving’s Prealgebra book when the subject came up formally, much of the teaching I do with my kids doesn’t follow a textbook and many important subjects come up almost out of the blue as we discuss various math problems. I certainly wasn’t aware of the different (and sometimes strongly held) beliefs about teaching fraction division when I was talking about it with my son.

Teaching the same subject to my younger son was a little different. Hopefully I’d learned a little bit from going through this subject once already (ha!), but also I’d begun to follow lots of teachers and math ed folks online so I’d seen some approaches to teaching fractions that were different that what I’d done on my own. I still used the approach in Art of Problem Solving’s Prealgebra book as the starting point, but I supplemented it with a couple of other ideas. Here are those three approaches from back in January. Having watched all three of this videos again just now, I’m perfectly happy with what we did and I believe that all three approaches have merit:

(1) Art of Problem Solving – define division as the reciprocal of multiplication and understanding fraction division just boils down to understanding what the reciprocal of a fraction is. Note also that he notices that you need multiplication to be commutative for this approach to work (!):

(2) Talking about patterns. Here we look at this sequences of divisions: 8 / 8, 8 / 4, 8 / 2, and 8 / 1 to help form a guess about what the value of 8 / (1/2) might be. We also use dimes and nickels to illustrate the division:

(3) Drawing rectangles / using snap cubes to talk about division. This is the approach that appears to have motivated today’s WSJ article -“Who would draw a picture to divide 2/3 by 3/4?”

The best response that I saw to the WSJ piece was this simple tweet from David Radcliffe:

Finally, if you don’t find fraction division to be an interesting topic to think about, perhaps you’ll be more interested in this delightful problem that Radcliffe posted earlier in the week:

The Joy of teaching my kids

A few weeks ago for our weekend Family Math project we talked about fractions and decimals in binary.    That blog post is here:


These family math project are just for fun.  These projects tend to cover either fun math we find around the house – see the paper folding example from all the way back in Family Math 1:

or, if not stuff from around the house, they are intended to be a fun overview of some advanced math.  The overview of fractions and decimals in binary was supposed to be in the second category, but it led to a really great surprise this morning.

Today with my younger son we moved on to a new chapter in our book – repeating decimals.   A few days ago we had started off talking about decimals and fractions by reviewing why .9999…. = 1, so I was hoping to play off of that to show why 1/3 = 0.33333….  However, when I sat down and asked my son what he thought the decimal expansion for 1/3 would be I got a little surprise:

“I don’t know, but I know what it is in binary.”

So fun that he remembered this talk from the Family Math project from a few weeks ago:

With that, we started down a totally new path – how does knowing what 1/3 is in binary help you understand the decimal expansion?

Such a fun morning!!

Evolution of my teaching and the surprising (to me) influence of Twitter

I was extrodinarily lucky to have an outstanding math teacher in high school.  Mr. Waterman set the bar about as high as it could be set and taught a generation of kids in Omaha about the beauty of math.  I stayed in touch with him after graduation and did as best as I could to approach teaching the same way he did.  He was also incredibly influential in helping me improve in teaching my own kids and provided me with dozens, and probably hundreds, of really helpful ideas for how improve the little videos we do.     With every new topic we cover, the first thing I try to think through is how he would teach it and hope to find an approach that would make him proud.


My first cut at standing in front of a class was in a program at MIT called Interphase.  I taught calculus in this program for 6 years both as an undergraduate and graduate student under the watchful eye of Arthur Mattuck.  He was the hardest of all hard asses when it came to views about teaching and my style didn’t connect with his at all.  And I mean at all. He gave me great feedback nonetheless, and put the fact that he could see that the kids were connecting with me and learning in front of the fact that he didn’t like my teaching style.    I was also lucky to have Mike Keynes, who is now a math professor at American University, teaching with me.  We learned a lot from each other.

In graduate school the teaching program was supervised by Susan Parker, who was and is still, beloved by her students.  She, like Mattuck, had quite a different approach to teaching than I did, but helped me become a much better in the classroom by the time I graduated.  It is always fun to go back and touch base with her when I’m up in Boston.

My two years at the University of Minnesota after graduate school were by far the most fun I’ve had teaching.  I taught a few college courses, but the real fun came from a special program that the University offered for local kids called the University of Minnesota Talented Youth in Mathematics program.  I worked with some sensational kids during the two years I was there.  One, Allison Gilmore, is now a postdoc in math at UCLA.  I’ve stayed in touch with a lot of these “kids” (who are now close to their 30s . . .) and it is so cool to see what they are all doing now.

One thing that was particularly fun with this group was the variety of interests they had in math.  Many, not surprisingly, were math contest kids just like I had been.  Helping them along with the various contest was an absolute blast.  Others were much more interested in pure math and I was able to get them involved in some pretty neat projects.  I did a reading course in Massey’s Algebraic Topology with Allison, for example, and now she does research in low dimensional topology.  Seriously . . . how freaking cool is that!!

Another valuable experience there was working with two other people – Cindy Kaus and Doug Shaw – who joined the faculty there about the same time that I did.  They both had completely different interests in math than I did (Cindy had been an engineer before getting her math PhD, for example) and connected with the kids in ways that were really different than how I did.  There was always something to learn.  Doug’s now at Northern Iowa and Cindy’s just won a Fulbright award and is spending this semester teaching and reseaching math in the Seychelles – fun!

Having left academia in 1999, I haven’t really had any new people to talk math with in a while.  As I said, I got a lot of great feedback from Mr. Waterman when I started doing the math videos on line, but I still felt that I wasn’t doing as good a job as I could.  It was much more difficult than I thought it would be to teach really basic math to my kids.

Somehow or other I stumbled on twitter and ran across an almost overwhelming about of information and ideas about math.  It is actually pretty difficult to sort through all of it, but after a while I found a few voices that really have struck a chord.  One is Patrick Honner who, because of his incredible and infectious love of math, seems very much like a younger version of Mr. Waterman.  Although he teaches high school level courses, I’ve been able to take a few of the ideas that he’s posted about and turn them into fun lessons for the boys.

And then there is Fawn Nguyen.  She teaches middle school in California and has the most creative ideas for teaching math that I’ve ever seen.  Even better, she’s constantly sharing all of her incredible ideas on Twitter and on her various websites.  It is remarkable to me that someone who lives 3000 miles away can have such a positive impact on my teaching, but I guess that’s the power of the internet!

I was thinking about some of her visual methods as I was getting ready to teach my youngest son about adding fractions today.   The approach I’d taken with my older son a few years ago was mainly computational, which is certainly consistent with how I think about the world.  Seeing Fawn’s work since then opened my mind to many new ways of thinking about teaching this material, though, and this morning we made this little video as a starting point about adding fractions:


I’ve been lucky to have so many great colleagues and mentors over the years, and I’m happy with how my own teaching is evolving because of their influence.  I think Mr. Waterman would be happy about the evolution too, and that makes me really happy, too.

A little fun and a little math wtih Rubik’s cubes

Sometime last year my kids became fascinated by Rubik’s cubes.  Not sure why or how it happened, but once they started playing around a little they were hooked.  So much so that learning more about how to solve the cubes seemed like a fun topic to include as part of the school year, so we’ve been studying some of the 2×2 and 3×3 speed solving techniques for fun since September.   Even though I’m not practicing the speed solving with the boys, I’ve gotten a little hooked, too 🙂


Following a few folks on twitter also led to some interesting Rubik’s cube related reading in the last year.  Christopher D. Long (@octonion) tweeted about the book “Adventures in Group Theory:  Rubik’s Cube, Merlin’s Machine & Other Mathematical Toys.”  Definitely a fun read if you are into math, though it’ll be a while before I can pull much from it for the boys.

I also ran across Cathy O’Neil’s (@mathbabedotorg) old post about math contests :

Math contests kind of suck

This comment really struck me  – “I have never been particularly fast at working out the details of something from the conceptual understanding (for example, it takes me a long time to solve a 7x7x7 Rubik’s cube) but it turns out the Rubik’s cube doesn’t mind. And in fact mathematics in real life isn’t a timed tests- the idea that you need to be original and creative really quickly is just a silly, arbitrary way to select for talent.”  (as an aside, you should definitely follow her blog and follow her on twitter – you’ll not find a more interesting blog.)  I agree with Cathy O’Neil’s point that there’s not anything special about solving the cubes fast.  The kids seem to like it and enjoy learning the techniques, but it is mostly just a matter of practice.  That said, the world record solve times (~5.5 seconds for solving the 3×3, for example) really are  mind blowing.

So, what can kids learn from these cubes?

There is quite a bit of interesting math related to the cube solving algorithms (see the book mentioned above).  A simple introduction to these algorithms probably has some benefit, but I’m aiming a little lower right now.

One interesting advanced topic is parity.  This position on the 3x3x3 is impossible to solve:

Cube Reverse

You can not create a position with just one middle reversed with legal moves.  In order to make this postion, you have to take the cube apart.

However, this position on the 4x4x4 cube, which seems pretty similar to the picture above, is solvable:

4x4 Reverse

At least for my (very, very very slow) solving technique, figuring out how to solve the 4x4x4 from this position was the final obstacle to overcome in learning how to solve the larger cubes.

For my younger son, it turned out that the cubes were also fun tools for learning about topics like fractions:


and exponents:


** Update **  Imaginary and non-commutative numbers!!

It isn’t hard to believe that kids will be more excited about learning when they are having fun, but it great to see that excitement in practice.    Of course, it has also been really fun for me to use the Rubik’s cubes to help teach a bunch of different math topics.  Maybe one day we’ll even be able to replicate something like this 🙂

Follow up to Prime Numbers and Infinity

I got a nice question from an old high school classmate asking about yesterday’s blog post on prime numbers and infinity.  He asked me how you prove that the sum of the inverse squares was pi^2 / 6 and that the sum of the inverse primes was infinity.

Unfortunately both proofs are a little too advanced for kids.  The first requires understanding of the Taylor series for sin(x)  (at least the proof that I know), and the second requires a bit of number theory.  I wasn’t trying to present the ideas as something for the kids to prove, but rather just trying to show some fun facts that will come down the road (way down the road in this case).

However, there is one series that is pretty easy to understand that also has an infinite sum -> 1 + 1/2 + 1/3 + 1/4 + 1/5 + . . . .

After the question yesterday I toyed around with the idea of doing a second video, but decided to do it another time.  Then, as with yesterday’s blog, some funny coincidences happened today.  The first being that I started a new section on fractions with my younger son and the second one being a short proof of the divergence of the above sum that Dave Radcliffe put on twitter:

I’d never seen this clever little trick before and couldn’t wait to get home to show it to the boys:

I’ve been following a bunch of math folks on Twitter for about a year now and just can’t believe how many fun  examples I’ve found to share with the boys.  This little community on twitter has been a great resource for me.