Yesterday I had my younger son (in 7th grade) read chapter 1 in Steven Strogatz’s new book Infinite Powers and we had a fun time talking about what he learned:

Today I wanted to show him a hand waving overview of two of the more well-known ideas from calculus – finding tangent lines and finding areas under a curve.

I started with the tangent line problem:

He was struggling to remember some of the basic ideas about lines, so I broke the talk about tangent lines into two pieces to let him take his time remembering how to describe lines. Here’s the second part of the discussion:

With the tangent line discussion finished, we moved on to finding the area under a curve. To keep things simple, I stuck with the same function:

The book is terrific and the math explanations are so accessible that I thought it would be fun to ask my younger son to read the first chapter and get his reactions.

Here’s what he thought and a short list if things that he found interesting:

After that quick introduction we walked through the three things that caught his eye – the first was the proof that the area of a circle is :

Next up was the “riddle of the wall”:

Finally, we talked through a few of the Zeno’s Paradox examples discussed in chapter 1:

I think you can see in the video that Strogatz’s writing is both accessible and interesting to kids. I definitely think that many of the ideas in Infinite Powers will be fun for kids to explore!

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.

About a week ago I saw a great tweet from Dave Richeson about some journal articles that were being made freely available for the month of March:

Editors from Math Horizons (me), the American Mathematical Monthly, Mathematics Magazine, and College Mathematics Journal each chose five articles about Ï€, which will freely available from now until the end of March. Enjoy! https://t.co/tcuSxxykBbpic.twitter.com/LXyWMVJ71K

The paper from the The College Mathematics Journal by Susan Jane Colley caught my attention for being both an really neat result and being explained at a level a student taking calculus could understand.

So, this morning to celebrate Pi day I decided to use the paper to talk a bit of calculus with my son. Pulling all of the different ideas together was challenging for him, so we went slow but still made it through the main points in about 30 min.

We took a quick look at the paper and then started digging into the math by looking at the famous alternating series for .

I should say for clarification that I forgot to look up Susan Jane Colley’s current position before we started the project and wasn’t sure if she was still at Oberlin or had moved to a different university since the paper was published in 2003. But to be clear, she is the Andrew & Pauline Delaney Professor of Mathematics at Oberlin.

Next we dove in to the connection between the alternating series and . I thought I’d try to introduce the connection in a sneaky way, but it was sort of a dead end. Eventually, though, he thought about arctangent.

At the end of the last video the formula for an infinite geometric series came up, but that formula wasn’t quite at the top of his head. So we took a little detour to re-derive that formula. Once we had that formula we could see that the alternating series we were looking at converged to :

Now we looked at the main result of the paper – a different series for that converges really fast.

Here we look briefly at the formula for this series (sorry for the reading typos by me – trying to read the paper and stay behind the tripod and not cast a shadow was hard . . . . )

Finally, we went to Mathematica to evaluate the integral and look at the speed of convergence of the two series we’ve been studying:

I think that Colley’s paper is absolutely terrific and a great resource to use to show calculus students some advanced math. It is an extra terrific resource to use on Pi day ðŸ™‚

My son is in the middle of a long review for the BC Calculus exam in May. Right now he’s doing some integration practice and came across this integral today:

Although the discussion hasnâ€™t actually happened yet because heâ€™s currently evaluating the integral, this integral will lead to a very good discussion this morning pic.twitter.com/LDlyq2ZKcx

We didn’t get a chance to talk about it too much this morning, but we did review the problem when he got home tonight.

First we I asked him to approach the problem as he did this morning -> by trying to evaluate the integral. The practice integrating rational functions turned out to be useful:

Next we took a more geometric approach:

Finally, I wanted to show him how you could see that the integral was zero from the u-substitutions that he made in the first video. Even though this method wasn’t really the best one for this particular integral, I still wanted him to see how the algebra worked out:

We had a snow day today and I was able to use some extra time with my son to talk about numerical integration techniques. I’d guess that I hadn’t seen the topic in at least 25 years, so it was fun to teach and review at the same time!

This evening I had my son review the ideas using an integration problem from the 2012 BC Calculus exam. I started by having him evaluate the integral exactly with no numerical integration techniques:

Next we used the trapezoidal rule and compared the answer to the exact answer –

The one thing I learned today was that the midpoint rule was roughly twice as accurate as the trapezoidal rule. Here my son used the midpoint rule and we do find that the answer is closer to the true value than the answer we found in the last video:

Finally, we used Simpson’s rule. Despite one little mistake in the middle, we ended up finding an answer that was surprisingly close to the exact asnwer.

We had a 2 hour delay for school today, so we had a little extra time this morning to talk calculus. My plan was to spend all of March on techniques of integration, but we are a little ahead of schedule having already covered integration by parts and partial fractions.

Today my son moved on to a “techniques of integration” section in Stewart and was looking at a bunch of integrals without knowing what techniques to use. After he worked for a bit we talked about the first 5 problems from the section: