Today my older son is off mountain biking so the follow up project is with my younger son who is in 8th grade. I thought it would be fun to look at additional solutions to yesterday’s puzzle and show him how we could write down a formula for those solutions.

We started by look at some of the small solutions to yesterday’s problem and looking for patters. My son noticed a connection to that made me really happy!

Next I showed him the Internet Sequence Database. I wanted to show him that sometimes when you are looking at a sequence of integers, it is something that other people have studied before:

Now we returned to the whiteboard to study the sequence more carefully. Our starting point was the recurrence relation that we learned about in the last video:

Finally, and this is one of my favorite high school algebra examples, we took a first step at solving the recurrence relation. This step is a nice application of factoring and using the quadratic formula:

Yesterday a new online calculus course taught by John Urschel, Hannah Fry, and Tim Chartier made its debut:

My Calc I class is officially live through @Outlier.org. @FryRSquared, @timchartier, and I all teach calc I, each in our own unique teaching style. The idea is to earn college credits for a fraction of the price. Let me know what you think!https://t.co/PK7Q35IZpI

I think online learning has a lot of potential and even tried to put together a calculus lecture video library way back in graduate school. So I hope this one has success.

One thing that caught my eye browsing through the course information was the course’s pre-test. My younger son has been studying algebra this summer and the pre-test seemed like it might be a good challenge for him.

It is 10 questions – I think the work below is a nice example of how a kid thinks through ideas in algebra. Here are the questions and his work on them:

Last week Numberphile put out a fantastic video featuring Neil Sloane:

For today’s project we explored the sequence described in the first half of the video. Namely, the sequence that begins with and then continues with depending on the greatest common divisor of and . See either the Numberphile video or the first video below for the full formula.

To introduce the boys to the sequence, I had them calculate the first 10 or so terms by hand:

Next we wrote (off camera) a Mathematica program to calculate many terms of the sequence, and studied what the graph of those terms looked like:

Finally, I asked the boys to watch the Numberphile video and the describe what they learned. They were both able to give a nice explanation of why the sequence eventually repeated:

I love math projects that allow kids to play with really interesting math and also sneak in some k-12 math practice. The first sequence in the Numberphile video is a perfect example of this kind of project!

Having finished a single variable calculus class with my son this school year, I’ve been thinking about what to do next. Probably the next step is going to be linear algebra and we’ve been watching a few of Grant Sanderson’s “Essence of Linear Algebra” videos to get a feel for the subject.

Today I wanted to have a short and introductory talk about vectors with my son, and I had two goals in mind. The first was to show some ideas about (for lack of a better phrase) thinking in vectors rather than thinking in coordinates. The seconds was just sort of a fun introduction to the dot product.

So, I started with a simple introduction to vectors that he’s seen a bit of via the Grant Sanderson series:

Finding a vector representation for the 2nd diagonal of the parallelogram we’d drawn was giving him some trouble, so we took a deeper dive here. I’ve always thought that the equation for the 2nd diagonal was non-intuitive, so I gave him plenty of time to make mistakes and work through the ideas until he found the answer:

Finally, I did a simple introduction to the dot product and we calculated the angle – or the cosine of the angle – between a couple of vectors as a way to show how some ideas from linear algebra help solve seemingly complicated problems:

So, next week I’m having him watch a few more of Grant’s videos while I’m away on a work trip. We’ll get going on linear algebra the week after that.

I saw a really great thread on twitter this week and wanted to share some of the ideas with the boys for our Family Math project today:

My observation that you can sum the first n cubes to get 1^3 + … + n^3 = [(n*(n+1)]^2 by counting rectangles in an nxn square is of course well-known (as is everything simpler and clever).

We started off looking at the sum 1 + 2 + 3 + . . . .

Next we looked at the sum of squares and searched for a geometric connection:

Now I showed them the fantastic way of looking at the sum of squares in the Jeremy Kun blog post. This method is a terrific way to share an advanced idea in math with kids – it is totally accessible to them and gives them a chance to talk through a fairly complicated idea:

Finally, I showed how the ideas we were just talking about extend to some of the basic ideas is calculus. It was neat to hear my younger son talking through the ideas here, too:

Definitely a neat morning – it is always amazing to see the connections between arithmetic and geometry.

Yesterday I saw a neat request from Sam Shah on twitter asking for ideas about how to “stumble upon” e with kids in Algebra 2 (other than compound interest). I shared an old project we did (and am doing again below) which I think is a terrific way to share a fun and surprising idea about e with kids.

Later in the thread, though, there was a tweet that surprised me:

Honestly, e belongs in a calculus course, not in algebra 2. Presenting it prematurely makes it seem magical and incomprehensible. Why the rush to get e in before students are ready for it?

Strogatz has done more math for the public that just about anyone, and he’s also taught a college course that shared beautiful and advanced ideas in math with students not intending to be math majors, so I was really caught off guard by his thoughts about e.

But rather than getting into an academic discussion about whether or not ideas about e can be shared with Algebra 2 students, I decided to revisit our old project with the boys today.

The idea we’ll take a look at today is this -> Take an NxN set of squares and place a random integer from 1 to in each of the squares. How many of the integers from 1 to $N^2$ do you expect to not appear in any of the boxes?

I introduced the idea with a 2×2 square and selecting random integers from 1 to 4 by rolling a 4-sided die:

Next we moved on to a 5×5 grid and talked about what we’d expect to happen:

Now we moved to a computer to help us look at the grids more quickly. In this video I explain the program using a few simple examples. The program itself is picking random numbers and counting how often each integer from 1 to appears in the list of numbers selected.

Although I struggled a little bit with the output of the program (the joy of filming these things live . . . ) we eventually found our way and the kids noticed some potentially interesting patterns in the number counts:

Now we moved up to some larger grids and the kids began to notice more and more patterns in the number counts – :

Finally, we looked at a few very large grids – starting with a 50×50 grid – and the boys began to notice the pattern emerging in the number counts that allowed you to take a guess at each number in the list. It was fun to see them begin to understand these patterns more and more throughout this project:

I guess I’ll conclude by saying that my view differs from Strogatz’s view. I think this project would be appropriate for Algebra 2 kids. It shows them a pretty advanced idea but also gives them a chance to explore that idea using things they’ve learned in K-12 math ranging from simple arithmetic, to a bit of geometry and algebra, and also elementary statistics. I’m happy that we were able to go through this project again today.

These problems from yesterday’s math contest looked like they would make a fun project, so had the boys work through the first 6 this morning.

Here’s problem #1 – this problem lets kids get in some nice arithmetic practice:

Here’s problem #2 – the challenge here is to turn a repeating decimal into a fraction:

Here’s problem #3 – this is a “last digit” problem and provides a nice opportunity to review some introductory ideas in number theory. The boys were a bit rusty on this topic, but did manage to work through the problem to the end:

Problem #4 is a neat problem about sums, so some good arithmetic practice and also a nice opportunity to remember some basic ideas about sums:

Next up is the classic math contest problem about finding the number of zeros at the end of a large factorial. My older son knew how to solve this problem quickly, so I let my younger son puzzle through it. The ideas in this problem are really nice introductory ideas about prime numbers:

The last problem gave the boys some trouble. BUT, by happy coincidence I’m about to start covering partial fractions with my older son, so the timing for this problem was lucky. It was interesting to see the approach they took initially. When they were stuck I had the spend some time thinking about what was making the problem difficult for them.