My younger son is working his way through Art of Problem Solving’s Introduction to Number Theory this summer. He’s currently learning about different bases and how to convert from base 10 into those other bases.

A few days ago he wondered about how multiplication would work. Yesterday we worked through one example and was a great reminder about what a kid learning math can look like:

Today we looked at multiplication in different bases more carefully. First I picked a simple example to show him that the same process you use in base 10 works in other bases (so, you don’t have to convert to base 10 first):

Next I asked him to pick two numbers to multiply. Somewhat unluckily he picked two 4 digit numbers in base 2 so the calculation is a little tedious. Still, though, it all worked!

This was a really fun project – I can’t wait to work with him a little more on arithmetic in different bases.

My older son is reviewing geometry this summer using Art of Problem Solving’s Introduction to Geometry book. He’s been choosing pretty challenging problems for our movies, too. Today’s was a “star” problem in chapter 4. I thought his work was a nice illustration of what a kid learning about similar triangles can look like.

Here’s the introduction to the problem, some progress, and getting stuck:

Here’s the search for the second set of similar triangles. We end with a ration that will help us solve the problem, but we don’t go all the way to the end.

My older son was working in Art of Problem Solving’s geometry book this morning. When I asked him what he wanted to talk about for today’s movie he pointed out one of the “extras” in the book. It turned out to be a classic, but very challenging, geometry puzzle. It is interesting to see what it looks like when someone sees a puzzle like this for the first time.

Our discussion this morning was long, but really good (I hope!). The four parts are below:

I’ll be giving two talks at math camps this summer. The first is at the east coast Idea math camp at the beginning of July and the second is at a math camp at Williams college in the middle of July.

I’m super excited to be able to have the opportunity to give these talks and can’t wait for the chance to interact with the students at the two camps.

The topics I’ll cover – not surprisingly – will come from some of the projects I’ve done with the boys this year. The talk at Williams is about 45 minutes shorter than the Idea math talk, so one or two of the topics below will get cut out.

I covered the 3×3 and 4×4 cases with the 2nd and 3rd graders at my younger son’s school as part of Family Math night and the kids loved the problem. With high school students I’d like to try to explore some of the larger cases and also discuss why this is a difficult problem for computers to solve.

Patrick Honner also showed me this related problem which I’ll leave as a challenge for the students 🙂

I’m still waiting to hear what sort of projection capability I’ll have at the two events, but oh do I hope I have the ability to share this program with the students:

The explorations you can do with this simple modular arithmetic idea are incredible.

(3) A problem from Po-Shen Loh’s MoMath talk

What I love about Loh’s talk is that he takes an extremely difficult problem – one from the 2010 International Mathematics Olympiad – and turns it into a talk that is accessible to the public.

His approach is so accessible that I talked through the first part of the problem with my 4th grade son:

I’m very excited to hear the different guesses that the students have for the answer to Loh’s two questions.

(4) Bjorn Poonen’s N-Dimensional Sphere problem

Here’s the problem and our project on the problem:

I’m guessing that not all of the kids will have seen geometry beyond 3 dimensions, so this problem will take a little bit of setting up. Luckily the only complicated bit of math that they need to understand it is the Pythagorean theorem and I’m guessing that all of them will know that theorem.

I was blown away by the answers to Poonen’s questions when I finally worked through them. This was also one of the most enjoyable projects that I’ve done with the boys this year.

I hope I have enough time to show the students the fun relationship between and hiding in this problem, too:

I got quite a surprise last night when I asked my younger son what he wanted to learn about in today’s math project. The answer – random walks.

Not quite the answer I was expecting (!) but it turns out that he’d learned about them in this book:

I really didn’t have any ideas at all about how to introduce random walks to kids, so we just starting by playing around with a simple 1-D example. It was fascinating to hear what they boys thought a random walk would look like. I’m not sure they even know the words or ideas that you would normally use to describe one.

Net we moved to Mathematica. I wrote a short program that kept track of left and right moves in a random walk. We looked at how far left and how far right you go (and also where you end up) after a certain number of steps. In this video we looked at 100 and 1000 step random walks.

The main focus (not counting a mysterious little glitch with Mathematica) was trying to get them to describe what they were seeing with the various numbers.

For the last part of the project we looked at a random walk with 10,000 steps. It was fun to hear the boys try to guess at what some of the max / min numbers would be. We’ll have to revisit the random walk idea a few more times to explore the ideas in a bit more depth. They boys were really interested to learn more after we finished up!

I hoped this introductory activity would be a good way for the boys to get a feel for why some squares on the chessboard in the main activity would be blank:

Next we tried out the activity with a 12-sided dice – how many of the 12 numbers do we expect to not come up in 12 rolls?

Finally we moved on to John Allen Paulos’s activity – 64 random numbers. We had Mathematica give us the 64 random integers from 1 to 64 and put snap cubes on the squares of the chessboard to represent those numbers. All of this was off camera (don’t worry!) – here’s what the boys had to say when we were done:

I think this is a great activity for kids. Even the simple part of finding the right square to mark is a nice math challenge for them. The point isn’t for kids to understand what the number e is, rather the point is the surprise that we can estimate the number of blank squares fairly accurately. A fun extension would be estimating the number of squares with exactly 1 cube (the result might surprise you if you don’t know it!).

My older son is reviewing Art of Problem Solving’s Introduction to Geometry book this summer. Today he was working on problem relating to similar triangles, and the problem he picked for our movie was a fairly plain vanilla similar triangles problem:

The first part of the problem when smoothly, but the second part gave him some trouble. It was interesting to see his struggle to find the correct set of triangles to compare in the second part. Once you find them the problem is a snap – but finding them isn’t necessarily so easy when you are learning geometry. Staying with the problem might be the biggest lesson here.

This morning I wanted my son to have an easy morning and just start in on the next chapter in his book. I don’t remember the exact words – it was 6:30 in the morning – but they were something like “just start in on the next page.”

What my son did was sit alone and work on the problem that was on the back of the page he was on . . . here it is:

Let A and B be positive integers. If A! / B! is a multiple of 4 but not a multiple of 8, then what is the largest possible value for A – B?

I’d say that this problem is a little bit above 4th grade level . . . . whoops.

But we talked about it and how you might begin to think about it. First, though, I let him share some thoughts. You’ll see that it was tough for him to even know where to begin:

Because he was struggling to see where to go, I just went back to the beginning. We started by looking at a chart with some small numbers. The largest difference we could find was 3.

Finally we looked a bit more carefully at the situation when the difference was 3 and when the difference was 4. He was then able to see the reason why the difference could never be 4. I was pretty excited and happy to see him make this last connection – at least a bad mistake by we ended on an up note!

One nice coincidence was that my son started reviewing geometry today and finished reviewing quadratic equations last week, so Wright’s tweet was a nice way to tie both subjects together.

Unfortunately our 7:00 am talk was derailed when I missed a minus sign and created a mess . . . . So, we revisited the idea when he got home from school.

Also, two notes before diving into the post:

(i) If you’ve seen this proof before, don’t worry, Wright does not claim this proof is original – in fact he claims this this proof is probably well known. As best as I can remember Wright’s post was the first time I’ve seen it.

(ii) I make a slightly different choice of variables than Wright does in his write up.

We started back in this afternoon with a quick review about inscribed circles in a right triangle:

Next we worked through some of the algebraic expressions we found in the last video. He didn’t simplify the expressions in quite the way I was expecting, but what he did was fine.

Finally, we looked at the expression we obtain by squaring the hypotenuse. By a little bit of algebra (which makes for a nice little algebra review for a kid!) we find the hypotenuse squared is equal to the sum of the squares of the other two sides!

Thanks to Colin Wright for posting this nice bit of geometry!

Here’s the discussion of their results and some ideas for further experiments:

It would be nice to do a project like this with more kids so that you could compare lots of results, but even with just the two boys this morning, it was pretty fun. I really enjoyed hearing them talk about the numbers. They were especially excited by the fact that no 2’s showed up until the 26th roll.