## Finding the area of a circle using Riemann Sums

We’ve moved on to chapter 4 of my son’s calculus book -> integrals. The first sections are talking about Riemann sums.

I was a little surprised at the initial difficulty my son had with the examples. I think one of the problems – maybe the main problem – was how many different things you had to keep track of to evaluate these sums. Once we studied the ideas a bit more and saw that the sums could be really be broken down into keeping track of lengths and widths of rectangles, the ideas seemed to make a lot more sense to him.

Last night I decided to show him a Riemann sum that was different than the polynomial examples he’d already worked through. For this one – finding the area of a circle – he knew the area, but the sum was pretty complicated.

Here’s his work finding the sum:

Next we went to the computer

Finally, I thought it would be fun to show him a little surprise -> the Riemann sum that you would use to find the volume of a sphere is actually pretty easy to evaluate by hand.

I think that’s going to be it for specific Riemann sum background work. Looks like the next sections introduce integrals. Excited to dive into this topic!

## Working through a challenging calculus problem

I stumbled on a neat, but challenging, calculus problem in my son’s book yesterday afternoon. We talked through the problem and then I wanted to revisit it today by having my son solve it from scratch.

Here’s the problem and his solution – It was too bad that the whiteboard didn’t leave enough room for the picture:

Following his presentation, we talked through a few of the math / calculus ideas in the problem:

Finally, I wanted to show him a different solution involving a u-subsitition. I was just doing this on the fly and didn’t realize how confusing a u-substitution would be. But we got through that part (and the upside is that I’ll remember that this topic is a lot more tricky than I think when we eventually get to it!):

Definitely a fun problem, and one that really forces you to think pretty hard about tangent lines and derivatives.

## Using Joel David Hamkins’s perspective drawing posts with kids

I learned of a neat post on perspective drawing from Joel David Hamkins from this Patrick Honner tweet last week:

In the course of discussing the post, Hamkins shared a follow up post which is what we used for the project today:

I started the project today by having the boys take a look at Hamkins’s post (the second one) and making sure that they understood the ideas / directions:

Here’s the drawing my older son made. I’d guess this took about 15 to 20 minutes to make:

Here’s my younger son’s work. He probably took an extra 15 minutes and was working very carefully. I was really impressed at the detail in his drawing.

Both of Hamkins’s posts are great to use with kids. I think the second one is a little easier, but either way, the posts make for a terrific Sunday morning project.

## What a kid learning geometry can look like

My younger son is working through a bit more of Art of Problem Solving’s Introduction to Geometry book this summer. Yesterday he came across a problem that have him a lot of trouble.

The problem asks you to prove that the sum of the squares of the lengths of the diagonals of a parallelogram is equal to the sum of the squares of the lengths of the sides.

Yesterday he worked through the solution in the book – today I wanted to talk through the problem with him. We started by introducing the problem and having my son talk through a few of the ideas that gave him trouble:

Next he talked through the first part of the solution that he learned from the book. We talked through a few steps of the algebra, but there were still a few things that weren’t clear to him.

Now we dove into some of the algebraic ideas that he was struggling with. One main point for him here, I think, was labeling the important unknowns in the problem.

For the last part, I wrote and he talked. I did this because I wanted him to be able to refer to some of our prior work. The nice thing here was that he was able to recognize the main algebraic connection that allowed him to finish the proof.

## Triangle and construction review

My younger son has been studying Art of Problem Solving’s Introduction to Geometry book this year. He’s been doing most of the work on his own. Every now and then, though, I decided to check in and see how things are going. Today the topic he was looking at when I checked in was constructions. The two questions on the table when I stopped by were:

(i) Construct the incircle of a triangle, and

(ii) Construct the circumcircle of a triangle

Here’s how things went.

(i) The incircle

He has the basic idea for how the construction works, but misses one important idea. That idea is that the tangents of the incircle are not the feet of the angle bisectors. My guess is that this is a fairly common point of confusion for kids learning this topic:

(ii) the circumcircle

Here his solution was completely correct. He had a tiny bit of trouble in the beginning figuring out to construct the perpendicular bisectors, but he worked through that trouble fairly quickly.

It is always interesting to hear the ideas that kids have when they talk through a mathematical process. For me it was especially nice to hear that most of the ideas he’s learning as he works through this geometry book are sinking in pretty well.

## Part 2 sharing Mathologer’s “triangle squares” video with kids

Yesterday we did a project inspired by Mathologer’s “triangle squares” video:

Here’s the project:

Using Mathologer’s triangular squares video with kids

Today we took a closer look at one of the proofs in the Mathologer’s video -> the infinite descent proof using pentagons that $\sqrt{5}$ is irrational:

Here are some thoughts from the boys on the figure and the proof. You can see from their comments that they understand some of the ideas, but not quite all of them.

Watching Mathologer’s video, I thought that the triangle proof about the irrationality of $\sqrt{3}$ and the proof of the irrationality of $\sqrt{2}$ using squares were something kids could grasp, but thought that the pentagon proof presented here was a bit more subtle. We may have to explore this one more carefully over the summer.

After discussing the proof a bit, I switched to something that I hoped was easier to understand. Here we talk about the different pairs of numbers that create fractions close to $\sqrt{5}$.

The boys were able to explain how to manipulate the pentagon diagram to produce the fraction 38/17 from the fraction 9/4 that we started with. From there the were able to also show that 161/72 was also a good approximation to $\sqrt{5}$:

Next we went to the computer to explore the numbers, and also to see how the same numbers appear in the continued fraction for $\sqrt{5}$.

In the last video we tried to do some of the continued fraction approximations in our head, but that wasn’t such a great idea. Here we finished the project by computing some of the fractions we found in the last video by hand.

I love Mathologer’s videos. It is amazing how many ways there are to use his videos with kids. Can’t wait to explore these “triangular squares” a bit more!

## Using Mathologer’s “Triangular Squares” video with kids

Last month Mathologer published an incredible video on what he calls “Triangular Squares”:

I’ve been meaning to use this video for a project for the boys ever since I saw it. Today I finally got around to watching it with the boys.

Here are their initial thoughts after watching the video:

Now we went through some of the ideas. First I asked the boys to try to sketch Mathologer’s argument that $\sqrt{3}$ is irrational. Then I asked what proof they would have given for that fact without seeing the video:

Next we explored the irrationality proof for $\sqrt{2}$:

Finally, we did a bit of exploration of the seeming paradox mentioned at the end of the video. That paradox is essentially -> the argument used to show that $\sqrt{3}$ is irrational seems to also show that 3 times a triangular number can never be a triangular number. BUT, there are lots of examples showing that 3x a triangular number is a triangular number. What’s going on?

So, another terrific video from Mathologer. His ability to shed light on advanced math topics for the general public is incredible. I love using his videos to help my kids see amazing math ides from new and beautiful angles!

## What a kid learning math can look like – incircles and circumcircles

This problem gave my son a lot of trouble this morning. It is from one of the challenge problem sections in Art of Problem Solving’s Introduction to Geometry book:

A triangle has side lengths of 10, 0, and 12, find the lengths of the radius of the inscribed and circumscribed circles.

This problem looks like a pretty plain vanilla math contest problem, but he’s not studying for math contests. He drew an amazing picture of the situation:

After talking about the problem a bit this morning we moved on to other things. When he got home from school we revisited it. Here are his thoughts on the incircle part of the problem:

Here are his thoughts on the circumcircle part of the problem:

Finally – and unfortunately interrupted by a phone call – here’s his explanation of how the project from today relates to a 3d printing project that we’d done previously

Definitely a fun problem to talk through, and really nice to see that he was able to explain the ideas in a problem that he’d really struggled with in the morning.

## Using the Infinite Galaxy puzzle from Nervous System to talk topology with kids

The boys and I spent yesterday working on the new Infinite Galaxy Puzzle from Nervous System:

Having finished it, I thought a project talking about some of the math behind the puzzle would be really fun for the boys.

Since the front cover of the puzzle says that it was inspired by the Möbius strip, I started today’s project talking about that shape:

Next we talked about the puzzle and what geometric / topological properties it has. The interesting mathematical question here is whether or not the puzzle is a Möbius strip?

It turns out the puzzle is projective plane!!

We spent the last part of the project today talking about the projective plane and a few other similar shapes.

Even without any of the math, this new puzzle from Nervous System is a really fun challenge. The mathematical ideas behind the puzzle move it from the “fun puzzle” real to the “blow your mind” realm, though!

I’m so happy to have found one of these puzzles at the Nervous System open house last weekend. What an amazing way to share some introductory ideas from topology with kids!

## 3d printing totally changed my approach to talking about trig with my son

For the last two weeks we’ve been playing with this book:

Our most recent project involved one of the pentagon dissections. My son wrote the code to make the shapes on his own. We use the RegionPlot3D[] function in Mathematica. To make the various pieces, he has to write down equations of the lines that define the boundary of the shape. Writing down those equations is a fantastic exercise in algebra, geometry, and trig for kids.

Here’s his description of the shapes and how he made the pentagons:

Next we moved on to talking about one of the complicated shapes where the method he used to define the pentagon doesn’t work so well. I wish I would have filmed his thought process when he was playing with the code for this shape. He was really surprised when things didn’t work the first time, but he did a great job thinking through what he needed to do to make the shape correctly.

Here is his description of the process followed by his attempt to make the original shape (which he’d not seen in two days . . . )

I’m so happy that he’s been interested in making these tiles. I’ve honestly never seen him so engaged in a math project. The original intention of this project was just for trig review, but now I think creating these shapes is a great way to use 3d printing to introduce basic ideas from trig to students.