The question is a really deep and really challenging one for kids. Truthfully it is probably a little over the head of my kids, but I thought I’d give it a try anyway. I’ll revisit this one (hopefully!) several times over the course of this school year – although the question confused my kids a little bit, I really like it.

Here’s my older son’s (started 7th grade today!) thoughts on Dave’s question:

Here’s my younger son’s thoughts – he’s in 5th grade. I took a little extra time at the beginning with him to work through some examples with numbers so that the abstract symbols wouldn’t be so confusing:

It is fun to hear the boys struggle to try to explain / reconcile the strange ideas in Grandi’s series. I’m also glad that they are learning to think through what’s going on rather than just believing the algebra.

My younger son has been working through Art of Problem Solving’s Introduction to Number Theory book this summer. The topic for the last few weeks has been arithmetic in different bases. Today he can across a problem that gave him a lot of trouble. He worked on it alone for about 15 minutes and then we talked about it.

Here’s the problem:

In a certain base (12)*(15)*(16) = 3146. What is that base?

Here’s the first part which summarizes his initial thoughts on the problem. The work he does here shows that the base used in this problem must be lower than 10. Once he discovers that fact we talked about a few other ways that we could have seen that the base wasn’t 10.

Next we tried to see how we could identify the base we were looking for using some of the ideas from the last video. We used the last digit idea to eliminate 7 and 8, but the last digit idea told us that base 9 might actually work.

Then we did the arithmetic to show that we were indeed looking for base 9.

So, a really challenging problem, but a fun talk for sure. Working through problems like this one are a great way to review arithmetic and a neat way for kids to learn some basic ideas in number theory.

This looked like a fun project for kids, though it wasn’t obvious how to get started. It turns out that Mathematica has a handy function called PolynomialMod[] that tells you what a polynomial looks like modulo an integer – so that made life easier!

I decided that for today’s project we’d explore using Mathematica and see what patterns we could find. The introduction to today’s project involved introducing basic polynomial multiplication. Luckily, a natural way to multiply polynomials looks a lot like multiplying 2-digit numbers. I used that connection to introduce the project:

After the introduction I had the boys play on Mathematica and compute various powers of starting with . We got a little confused between Fibonacci numbers and Pascal’s triangle, but here is what they saw:

For the last part of the project today we used PolynomialMod[] to look at the various powers of in mod 2. I wanted to get them used to this Mathematica function to make it easier to explore mod 2 tomorrow. After they explored the powers of mod 2 up to n = 8, we talked about patterns in the numbers:

So, a fun little computer math project. It was fun to hear the kids talk about the patterns and also fun to talk about some basic ideas like polynomial multiplication and modular arithmetic. Definitely excited to explore some of the more complicated patters tomorrow.

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.

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 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!).

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!

Once in a while the way one of my kids solves a problem catches me by surprise. In today’s case I got an example of my younger son seeing a common math contest problem for the first time. When you’ve seen a problem 1,000 times you forgot what it is like to have not seen the problem previously.

I love his solution – especially because it shows how a kid can think about a pretty complicated problem involving numbers that you can’t really write down:

My older son has been studying properties of quadratic equations in his algebra book. Yesterday we talked a little bit about sums and products of roots and he was a little confused:

The confusion doesn’t bother or worry me – learning math is hardly ever a perfect, straight line process.

Today we revisited the topic and took a look at one easier and one more difficult problem.

First – find two numbers whose sum is -3 and have a product equal to -18:

Second – find two numbers whose sum is 2 and whose product is also 2:

So, hopefully a little bit of a gain in understanding from yesterday to today. Once again, learning math isn’t always a straight line.