The AMC 8

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!

Day in the life of a home school dad

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:

Frisbee Math

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.

All in all, a pretty typical day.