When I was working through basic arithmetic with my kids a few years ago I thought it would be fun to teach them about different bases. One of the first different bases that we studied was binary, and the way we looked at it was to make a “binary adding machine” out of duplo blocks.

These are two example videos with my younger son:

The kids were amazed that you could use duplo blocks to do simple arithmetic.

While I was in London last week, I saw a really neat post on twitter from James Tanton. The title alone made me want to see it: “Exploding Dots: On base One-and-a-Half”:

In this video he introduces the concept of base 1.5 and talks about a few fun open problems. It is always such a thrill to see a new idea that is relatively easy to understand. I couldn’t wait to get back to the US and go through it with the boys.

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As an extra bonus, the “exploding dots” method of arithmetic that Tanton talks through is almost exactly how my kids taught themselves to do arithmetic. Those videos will be fun to go through with the boys, too.

Had a fun set of interactions on twitter today as I was sitting in my hotel room in London.

Steven Strogatz, a math professor at Cornell (and the PhD thesis advisor to one of the players that I coached on Brute Squad a few years ago), mentioned a really neat article about a high school student’s experience with a famous paper folding problem:

There were several quick and great reactions to this post. Justin Lanier, one of the teachers who helps run the amazing math website http://www.mathmunch.org, mentioned that the Math Munch team was in the process of doing an interview with her. They had, in fact, written up an article about her recently:

Alexander Bogomolny who runs the absolutely incredible math website http://www.cut-the-knot.org mentioned that the record mentioned in Strogatz’s article had been broken at MIT in 2011 by James Tanton and his students:

it was fun for me to be reminded of the paper folding record at MIT because it had inspired me to start a (nearly) weekly series I do with my kids called Family Math. My idea in making this series was to take stuff from around the house, or stuff we read about, or just neat math stuff in general and turn it into a fun math activity for kids.

The first video in the series, an effort to try to break the record of 13 folds, is here:

Family Math 99 – talking about a strange aspect of the Higgs boson mentioned an article by the physicist Frank Wilczek:

Family Math 62 – talking about the Collatz conjecture:

I’m sure that some are much better than others, but making this series for the last couple of years has been really fun. I’m really hope that we’ll be able to continue making these videos every weekend for a long, long time.

About a month ago a couple of projects at work got really busy and it became clear that the week of Thanksgiving was going to be spent overseas. This morning I dropped my wife and kids off at one airport so they could spend the week in Omaha with my family, then headed off to the office and then to JFK. Here’s the view from the Virgin Atlantic lounge right now:

Before leaving this morning I spent a little time with the boys working through this fun problem:

This is yet another super project for kids from Fawn Nguyen’s visual pattern site: visualpatterns.org. I love these projects because they are activities that I never would have thought of on my own. Not in a million years. The patterns are pretty simple (at least the ones that we’ve done so far), but both boys find them fun and engaging. Last week’s purchase of a big box of snap cubes has made the projects even more exciting for them. I see on the gallery tab on Fawn’s site that we aren’t the only people using snap cubes to play around with her projects. I wonder if we are the only people with Minecraft villages made out of snap cubes . . . .

This is sort of a somber trip. One of the projects is coming back up to speed on a company I used to look after – Global Aerospace. In 2002, Forrest Krutter and I helped to negotiate the purchase of Global Aerospace. He and I were on the board from 2003 to 2007 when I resigned to look after some other duties in the US. Forrest and I had a history that went quite a bit beyond Global Aerospace. He did the local interviews in Omaha for MIT and interviewed me for college in the fall of 1988. He then hired me into Berkshire in 2000 and we worked together on lots of projects until this past fall when he died after a long battle with cancer. By terrible coincidence, he died on the same day as my high school math teacher Mr. Waterman.

So, part of this trip is coming up to speed on Global Aerospace. As much as I like the people there, and as much as I do truly believe that they are the best in a really tough industry, the circumstances will make that bit of the trip tough.
Unfortunately this isn’t the first time I’ve had to deal with the death of someone who I worked with closely. In 2011 my partner of 10 years, Bob Bennett, died after his fight with cancer. I spent all of 2012 coming up to speed on the parts of his work at Berkshire that I wasn’t involved in. In fact, part of the reason I was in the office today was something that I forgot to take care of last week.

It is interesting to me how quickly I’ve grown to not like traveling for work. At first – in my early 30s with no kids – flying all over the world was both fun and fascinating. The thrill faded fast, though. Since stepping down from the Global board in 2007 I’ve hardly traveled at all. Of course, that’s made home schooling a lot easier. My guess is that I’ll be travelling at least 6 weeks a year for work with the new responsibilities I have now. That’s a big change. I probably wasn’t traveling 6 weeks total from 2008 through 2012. It seems likely that coaching ultimate is going to be very hard in 2014. Hoping beyond hope that I’m wrong about that, though.

So, the flight leaves in about an hour. I’m having breakfast with my best friend from college who is overlapping with my in London for about 2 hours tomorrow morning. The rest of Sunday should be pretty uneventful – probably just reading in the hotel and hopefully hitting the gym. The week is filled with meetings rather than math videos. Flying back on Thursday evening and then picking Allie and the boys up from the airport on Friday evening. Hoping Friday gets here fast . . . .

Not every day teaching the boys goes super well, but when things do go well it is such a awesome feeling. Today was one of the great days.

I started with my younger son today. The math topic of the day was what we’ve been studying all week: greatest common divisor. We started with a relatively straightforward problem to review the topic – find the greatest common divisor of 32 and 48. He’s still learning about factoring, so I’m always happy to start off the day with a problem that gets him practice on an older topic.

The next problem was to find the greatest common divisor of 99 and 100. The answer to this problem is “obvious” if you have experience with number theory, but if you are just learning it doesn’t seem any different than any other problem. He jumped in and found the factors of 99 and 100. The fact that 99 and 100 have no divisors in common led to some confusion, so I sent him back to the beginning – what are some things that we notice about the two numbers?

After thinking for a little while, he noticed that since the difference between the two numbers was 1, the greatest common divisor had to be 1 because no prime numbers could divide both 99 and 100. As I prepared to move on to the next problem, my son turned to me and said:

I think the difference between the two numbers tells you something about the greatest common divisor. I wonder if anyone has ever noticed that before.

Ha. I was sort of stunned for a moment, honestly, but then I told me that someone had indeed noticed that before. Unfortunately we were nearly out of time, so we’ll have to pick up this topic after Thanksgiving. Nonetheless, that little observation made my day.

Our video was about a GDC problem that involves slightly more complicated factoring:

The topic with my older son today was a little bit more about the quadratic formula. We discussed some interesting properties of solutions when the coefficients of the quadratic expression are all real and all rational. One of the questions went something like this: If a quadratic equation has all rational coefficients and you know one of the solutions is 1 + $\sqrt{2}$, what is the other solution? We had a nice discussion about this topic and it felt like some of the ideas about real, rational, and complex numbers were resonating with him.

As part of that discussion I drew a few pictures of parabolas. We’ve not really talked about graphing quadratic expressions yet – that’s the next chapter – but we have been talking about the geometry informally. The interesting question he asked me that completely changed the course of what I’d planned to talk through this morning was this:

Why do you always draw the parabolas going up? Why not sideways?

Why indeed! What a fun question – we discussed this question in our movie:

I love the questions, and I love when we get to talk about a topic in a way that is quite a bit different than I planned.

One of those days when you really get to experience the joy of teaching.

One of the fun things for me about home schooling has been learning to teach elementary school math. Maybe “learning” is the wrong word since I’m not really studying anything, but through trial and error I feel that I’ve improved my teaching a lot in the last few years.

It is always difficult teaching a new subject for the first time because you really have a good feel for how other people think about the material. Even though the elementary school math is quite basic, not totally understanding what the kids aren’t understanding makes teaching the material tough for me.

In the last year I began to follow a bunch of math teachers on twitter. It has been an incredible experience learning from all of the ideas that they share. One of the most prolific writers is a middle school math teacher in California named Fawn Nguyen. How she has the time in the day to do what she does is beyond me (as are her blogging skills), but I use so much of her stuff that it is as if my kids are being taught by an east coast amateur impersonator of her. Seems like everything she posts turns into some sort of homework for the boys.

The first is a problem that I wouldn’t have appreciated before I started working with my kids. After seeing it today, though, I couldn’t wait to go home and try it out on them.

I didn’t let either of the kids see the problem before turning on the camera – the goal was really to see how they’d approach the problem rather than if they’d be able to do the multiplication correctly.

First up was my older son. He’s always seems to want to charge right into a problem with the first idea he sees, so I was curious if he’d simply multiply out the two expressions. He did, but in a way I wasn’t completely expecting:

I should note that he taught himself how to do basic arithmetic, so his way of adding, subtracting, and multiplying is definitely not standard. As his approach to arithmetic was perfectly fine, I never bothered to teach him now to carry, borrow, or do long multiplication the “normal” way. I’ve always wondered if I’ll regret that decision later.

Next up was my younger son. I also wasn’t sure what he was going to do. He’s only learned the basics of multiplication, so it didn’t seem likely to me that he’d want to multiply everything out. In the last couple of weeks we’ve been studying primes and factoring, and that was the approach he took to the problem. I’m sorry that so much of this video is him trying to figure out if 79 was prime, but I thought the process was fascinating. Even if I had to help him get over the last little hump.

This was a really fun problem to work through – Thanks Fawn!!

Had a good morning with the boys today. One thing that is really fun for me teaching them is that I never really know where the conversations are going to go. Sometimes, and probably quite often, I mistakenly think that a concept has sunk in when in reality it needs quite a bit more review.

Today with my younger son the topic was divisors. He has a hard time getting the words right – 2 is a divisor of 8 gets translated into 8 divides into 2. He seems to understand the basic idea of divisors. Questions such as “find the divisors of 20” are not that difficult for him, but something like “how many integers n are there so that 20 / n is an integer” are still difficult.

The bulk of my time with him this morning was spent on the following questions – If n is a divisor of 20, is n also a divisor of 60? Similarly, if m is a divisor of 60, is m also a divisor of 20? We ended up listing out all of the divisors of each number and them comparing the lists. It was interesting to see him wrap his mind around the 2nd question. After that we made this movie about the divisors of perfect squares:

The morning with my older son was spent reviewing the quadratic formula. We derived the formula yesterday and I wanted him to give a “lecture” about it today. It is interesting to watch higher level ideas come together in his mind. The derivation isn’t especially difficult, but there are a couple of ideas that you wouldn’t likely just stumble on all by yourself!

By coincidence there was a little bit of discussion on twitter this week about completing the square. Though we spent all of last week on that topic, it isn’t a topic that I’d thought was all that interesting. In a FB conversation, though, my friend Julie Rehmeyer pointed out that was the most interesting part of the quadratic formula for her. That comment made me rethink what I wanted my son to get out of these two weeks, so I put more emphasis on completing the square at the end.

Julie thought that the final derivation of the formula was more about manipulating symbols than it was about an interesting mathematical fact. I don’t feel as strongly about that point, but I don’t think she’s wrong. I actually began the discussion of completing the square last week with this fun “paradox” to emphasize what can go wrong when you just blindly manipulate symbols:

The conversation with Julie made me think back to what I remember from learning about the quadratic formula as a kid. What came to mind was the relatively simple sum and product of roots formulas (and their generalizations to equations with degree greater than 2). For some reason I always found it amazing that you could pick out these facts about an equation just from the coefficients. As I mentioned to Julie in our conversation, these simple facts show up again when you learn about Galois theory and help explain why you can’t write down the roots (in general) for polynomials of degree greater than 5. I plan to talk about some of these fun details on Friday.

The “lecture” about the quadratic formula is here:

The other fun thing that is happening today is the AMC 8 – a national contest for kids in 8th grade and below. There is a math club run by a local university professor that gives the contest to kids who don’t have it offered at their school. Lucky for us since I have no idea how else my kids would be able to participate in things like this. My son likes these math contests and I’m happy that this club is around so that he gets to meet other kids who have similar interests in math.

The hard questions on this test will still be a little bit over his head – he’s just in 4th grade. However, likes the challenge and can usually work through about half of the problems in the allowed time, so I’m hoping he has a good time today. It is really fun for me to go over these contest problems with him because they show so many different fun areas of math. I didn’t participate in my first national contest like this one until I was in 10th grade, so he’s got quite a head start on me!

The only down side today is that the contest site is about 25 miles away from work and I’ve got a work dinner tonight. 25 miles back down I95 in the middle of rush hour is waiting after the test finishes . . . . yuck!

I saw Justin Lanier’s tweet about the day in the life project and thought I’d give it a try. Actually I wasn’t sure until I read Fawn Nguyen’s day from last year. It won’t top that day, but what the heck . . . .

5:50 am: My wife and kids are usually out of bed by 5:30 am. Not me. Made it down just before 6:00 today. The boys had finished breakfast and I grabbed some coffee and hopped on e-mail from overnight.

6:10 am: I work a lot with people in London and we’ve got a couple of new project going on. 6:20 am e-mail from a colleague wanted to talk. I e-mail back and ask him to call now. We usually start school at 6:30, but I asked my oldest to start right then since I knew I’d be interrupted.

The project we are doing for fun this year is learning to speed solve Rubik’s cubes. It has been surprisingly fun, and both the boys like it. Working on solving 3x3x3 cubes with my 10 year old and 2x2x2 cubes with my 7 year old. The first part of today is practice on the 3x3x3 algorithms with my 10 year old.

There’s a little bit of math involved in the process – mostly learning about algorithms and spacial awareness – but what the kids seem to really like is charting their progress. They love setting new personal records and are really motivated to learn new ways to make the solutions faster. My older son’s record right now is 34s on the 3x3x3 and my younger son’s record on the 2x2x2 is 7 seconds. Super fun.

6:40 am: Phone call from London and I have to let my son practice alone for a bit. After about 10 minutes I’m done with the call and can swing my attention back to helping him.

7:00 am. My wife and older son leave to walk the dog and I switch to my younger son. This is the normal process. The kids alternate days of walking the dog with my wife.

The math topic to cover with my younger son today is divisors. We are studying in Art of Problem Solving’s Prealgebra book. I absolutely love this book and am so happy to have the flexibility to work through it slowly and cover some of the more difficult topics in as much detail as we want. He really likes numbers and is really taking to the number theory section we are in now. Watching him slowly understand prime numbers and factoring has been amazing. I also feel that I’m much better at teaching this than I was a few years ago going through this material with my older son. Today we spent the bulk of our time on the following problem:

Write down the factors of all of the numbers from 8 to 18 and then write down how many factors each of those numbers has.

He proudly tells me that he “discovered” that all of the prime numbers only have two factors and then we talk about why perfect squares have an odd number of factors. Happy with how the math went this morning.

We wrapped the math up by making our daily math movie – MathProblems53:

After that, a little Rubik’s cube practice for him and then my older son is back from the walk.

7:40 The math topic for my older son today is the quadratic equation. We’ve been following Art of Problem Solving’s Algebra book for about a year now. As with my younger son, we are not moving through the book particularly quickly. Rather we are trying to cover the difficult topics in detail. We’ve spent most of our time since the beginning of September talking about quadratic expressions, and today we finally get to the punch line! It was fun to see all of the steps from completing the square come together for the general solution.

After deriving the general solution, we solved a few equations and then made our movie: KidMath53:

With that movie finished, I gave him a MOEMs test to practice. Both the kids have grown to really like math contest problems, so I use them a lot to give them a little math variety. While he was working on that, I processed the two movies.

8:30 He’s done with the practice test and I’m off to work. Most days I bike into work, but we had some storms last night so I’m driving in. I’ll bike home tonight. My wife takes over the school duties after I leave.

9:00 Arrive at the office and hit the ground running. Have a couple of questions waiting for me from a project several of us were working through this weekend. Working through these problems is interrupted several times from calls from London. My partner is traveling to the US now and will be in our office tomorrow. We’ll have lots of stuff to get through if all of the calls from London today are any indication. I’m heading to London next week. It will be a busy day and week . . . .

A couple of nice distractions during the day today. I stumbled on an old favorite probability problem this morning – one person flips 50 coins, a second flips 51. What is the probability that the 2nd person gets more heads? Fun problem. Had a discussion on Facebook with a former student about it. Also, a high school friend sent me a neat problem from a math contest her kid participated in this weekend. I’ll run through it tonight with my older son. Also, I heard on the radio that the NY Jets were the first team in NFL history to go 5 – 5 through the first 10 games with alternating wins and losses. I suspect that the number of 5 – 5 teams in NFL history isn’t so large as to make the fact that only one team has done this a big surprise, but who knows. I think that would be a fun combinatorics / statistics problem for a kid interested in math.

5:00 pm I’m lucky to be able to have time flexibility in my job. It means that I don’t have to be at my desk to work, so I can get out the door pretty early most days. The weather tonight is ok to bike home. Super actually. Biking to and from work has been a great way for me to clear my mind and make the transition from teaching to work to teaching the boys.

6:00 pm Arrive home after a nice ride. Inhaled a couple of ribs and had about 45 min with the boys. I made the movie with my older son about the math contest problem that Anita sent me. We’ve studied a little number theory before and also have been talking about last digit problems, so it was actually a nice problem to go through:

I played a few number games with my younger son, including trying to make the number 34 using some of the numbers from the jerseys I have on my wall:

7:00: My wife does a karate class a few times per week and she’s out the door before we finish up. The kids are reading and getting ready for bed. I’ve got about an hour of work to finish up tonight once I get them down, though I might not make 9:00 pm tonight myself.