[sorry for a hasty write up – had to be out the door by 8:15 this morning . . . ]

Yesterday I saw the latest video from Grant Sanderson, and it is incredible!

I couldn’t wait to share the “fair division” idea with the boys. I introduced the concept with a set of 8 yellow and 8 orange snap cubes. To start, we looked at simple arrangements and just talked about ways to divide them evenly:

Next we looked at the specific fair division problem. We made a random arrangement of the blocks and tried to find a way to divide the cubes evenly with 2 cuts:

To finish up we looked at a few more random arrangements. Some were a little trick, but we always found a way to divide the cubes with two cuts! We also found an arrangement where the “greedy” algorithm from the 2nd video didn’t work.

After we finished the project I had the boys watch Sanderson’s video and they loved it. So many people are making so many great math videos these days – how are you supposed to keep up 🙂

I’ve spent the last couple of days talking about binary with my younger son. We were inspired a bit by Kelsey Houston-Edwards’s latest PBS Infinite Series video on binary. It has been a fun little review.

Tonight we talked about how to write 1/5 in binary. I didn’t really know how the conversation would go, but it ended up being a nice little arithmetic review.

We started talking about the problem and he settled on the idea that we needed to find a number that would equal to 1 when we multiplied by 5. That got us going on the arithmetic review since that idea works in any base.

Now we had to figure out now to divide 1.000000000…. by 101 in binary. This long division problem gave us an opportunity to talk about subtraction (and borrowing) in binary:

The last step was multiplying the number we thought was 1/5 by 101. Once again this was a great opportunity to review some basic ideas about arithmetic and multiplication.

So, an unexpectedly fun project! We learned what 1/5 was in binary and had a nice review of subtraction, division, and multiplication along the way 🙂

Last night my younger son and I were talking a little bit more about the project and he asked me why Cantor’s diagonal argument for why the set of real numbers is larger than the set of Natural numbers doesn’t work for rational numbers!! Yes!!

We explored that question today. First we did a quick review of the diagonal argument (which was the last part of yesterday’s project) and then we began talking about the rational numbers:

Next we looked at what would happen if you applied the diagonal argument to the rational numbers:

After getting our arms around the diagonal argument when applied to rational numbers, we backed up and looked at the argument why rational numbers are countable.

Unfortunately I made an easy concept hard in this part of the project. I was trying to explain the “easy for me” idea that if a set that is larger than the rationals was the same size as the natural numbers, that meant the rationals must also be the same size as the natural numbers. My explanation started off terribly and went down hill . . . .

Finally we looked at one of the strangest consequences of all of this infinity stuff. In math language – the rational numbers have measure zero.

The idea here always blows my mind and is a really fun idea about infinity to share with kids.

I had the boys watch the new video together and started today’s project by asking them what they thought was interesting.

After hearing what the kids found interesting, we dove into the idea of bijections. We talked a bit about how a bijection has to work both ways using the bus idea from the video.

After the bus example we moved on to the example of the bijection between the points in an interval and points on the real line.

We finished up by talking about the bijection between the national numbers and the positive even integers.

Since we’ve done several prior projects where infinity played some role, the next thing I asked the kids was for some thing that they already knew about infinity – both things that they thought made sense and things that they thought didn’t make sense. The discussion and examples here were amazing – “no one knows what infinity divided by infinity is” 🙂

Finally, we wrapped up the project talking about why the infinity associated with the real numbers is larger than the infinity associated with the natural numbers.

I thought this would be a fun way to end the project since it was one of the key ideas in Houston-Edwards’s video:

So, another really fun project from the new set of math videos from PBS Infinite Series. I love this new series – can’t wait for the next one!

Although I’ve been traveling a bit for work this week the relationship between learning math and intuition has stayed in my head. Sometimes my thoughts have drifted to and old blog post about a problem from the European Girls’ Math Olympaid:

That post, in turn, was inspired by an old post by Tim Gowers where he “live blogged” his work while he solved a problem from the International Mathematics Olympiad.

It can be really hard for anyone to know what math intuition looks like because everyone sees polished solutions way more often than they see the actual process of doing math.

That’s part of the reason I make the “what a kid learning math can look like” posts – so everyone can see that the path kids (or anyone!) actually takes to the solution of a problem is hardly ever a straight line:

The other thing on my mind this week has been some old AMC 10 problems that have really given my older son some trouble. These are pretty challenging problems and require quite a bit of mathematical intuition to solve.

So, I’d like to make the same challenge with these problems that I made with the problem from the European Girls’ Math Olympiad – “live blog” yourself solving one of these problems. Post the though process rather than a perfect solution. Let people see *where* your mathematical intuition came into play.

Anyway, I ran out to home depot and got some wire and we made some knots. I had each of the boys make a trefoil knot and then make a random knot of their own choosing. In retrospect I wish I’d spent maybe just 5 minutes explaining some of the ideas in Richeson’s blog post – oh well, the excitement got the better of me 🙂

Here’s my older son playing with his trefoil knot and making a Mobius strip bubble. I love the “hey, I actually think I got it” moment:

Here’s him playing with the knot me made – in retrospect I’d argue for a knot that was slightly less complicated:

Next up was my younger son. First up was the trefoil knot and we got another great moment “I think this might be a Mobius strip” !!

Finally we made his own knot and explored. Again, I’d probably ask for a less complicated knot if I was doing this again:

So, that so much to Dave Richeson for posting his old project – this is an incredible project, and an especially great one for kids. The appearance of the Mobius strip is really quite an amazing little math miracle!

After publishing the project I heard from Dan that his program actually had a little error but that Martin Holtham had made a corrected version. That version is here:

This problem from the 2016 AMC 10 a gave my son some trouble yesterday:

We talked through it last night and it was interesting to see where his intuition was off:

After finding a solution in the last video that he knew was wrong (because it wasn’t one of the choices on the test) we looked back to see if there was alternate approach to the problem. This approach led us to find the actual solution and also the mistake in the first approach:

It is always fascinating to the though process on a challenging problem. Sometimes the thought process is so close to the right approach that the mistake is really tricky to spot.

Here’s the picture in case it isn’t clear from the tweet:

When my son got home from school I asked him to give it a shot. The video is over 8 min long, but it shows what a kid struggling through a challenging problem can look like. It is important – especially in middle school, I think – that kids know that math isn’t just about speed.

Also, though, give the problem a shot before checking out the video – it is wonderful and involves no calculating that can’t be done in your head (i.e. this isn’t a trig problem):

Saw this really neat tweet from Steven Strogatz yesterday:

Unusual intuitive argument for why A= pi r^2 for a circle, found by one of the tables in our #math exploration class. I love these surprises pic.twitter.com/dch9PfmynZ

So, with a little enlarging and a little cutting we had the props ready to go through the exercise.

We began with a short conversation about circles. My older son knows lots of formulas about circles from his school’s math team practices, but my younger son doesn’t really know all of the formulas. The quick review here seemed like a good way to motivate Strogatz’s project:

Now we moved into Strogatz’s project – how do we show that the area of a circle is ? We cut the circle into the 16 sectors and rearranged them into a shape that was more familiar to us:

Next was the big challenge and the really neat idea in Strogatz’s first tweet – there is a different shape we can use to find the area. The boys were able to find this triangle fairly quickly, but then we had a really fun discussion about what the triangle would look like if we used more (smaller) sectors. So, the surprising triangle from Strogatz’s tweet led to a really fun and totally unexpected discussion! It is so fun to hear kids think through / wonder about math questions like the one they asked about the new triangles.

The last part of the project today was inspired by a tweet from our friend Alexander Bogomolny that was part of the thread Strogatz’s tweet started on Twitter yesterday:

I love it when Twitter writes our math projects for us 🙂

I had the kids look at the picture and describe what they saw. At the end I asked them why they thought the slanted lines in the triangle were lines and not curves – they had interesting thoughts about this little puzzle:

The amount of great math shared out twitter never ceases to amaze me. Thanks (as always!) to Steven Strogatz and to Alexander Bogomolny for inspiring this project about circles. Can’t wait to try out this project with other kids.