# Exploring n-dimensional cubes with my younger son

During break I’m going to try to do a fun exploration of n-dimensional cubes and spheres with my younger son. Today we talked a bit about cubes and started with by discussing what a “cube” was in a few different dimensions:

Next we talked about a different way to define a cube in n-dimensions using coordinates. These coordinates will help us do a few calculations in the next video.

We wrapped up today by trying to figure out what the longest distance between two points was in a 4-dimensional cube (with side length 1). My son talking through this problem is a really nice example of how kids can grapple with pretty abstract problems:

# Talking parabolas and calculus with my younger son

Because of a careless mistake I made last week asking my son to read a calculus-based section in the statistic book he’s studying, I’m doing a few light touch calculus projects with him. Today’s will likely be the most computationally heavy one as my main goal is simply to show him the ideas.

He mentioned to me yesterday that in his class at school they are studying graphing parabolas. I decided to use a quadratic for today’s calculus example. First I asked him to graph it. As the video shows, I probably should have asked him how they were approaching the problem in his class at school first!

Next we spent a few minutes finding the tangent line to our parabola at the point (0,3). Here we talk about what the tangent line is:

Here we finish the tangent line calculation:

Next we moved on to finding the area under the curve. We’d discussed a similar example earlier in the week, so I thought this part of the project would make for a nice review of that prior talk. We ended up diving a little deeper than I intended, but I still think it was a good discussion.

Here’s the introduction to the area problem:

And here’s the final calculation of the area:

# Looking at sums of integers and squares via geometry and algebra

Earlier this week I carelessly asked my younger son to read a section in a statistics book that relied on knowledge of calculus:

My son had some interesting questions about ideas in calculus, so instead of statistics we spent a few days this week talking about calculus (mainly finding the area under a curve).

For our project today I wanted to revisit one of the ideas in arithmetic that we’d relied on in the calculus discussion – sums of integers and sums of squares.

We started with sums of integers and my son gave a geometric proof of the rule for sums of consecutive integers:

Now we moved on to sums of squares – here we talked about the sum (and some of the number theory hiding in the sum formula), but didn’t yet try to prove the result:

Now we looked at a geometric way to understand the sum of squares formula:

Finally, we did a lightning fast review of mathematical induction and showed how we could proved that the sum of squares formula was true in general:

So, a fun week and a bunch of great discussions . . . even if it started with me being pretty careless!

# Day 2 with a neat geometry puzzle that Catriona Agg shared

Yesterday we did a project on this geometry problem shared by Catriona Agg:

I also shared the problem on Facebook and a friend from college shared a solution that I’d not seen before (though looking back at Catriona’s twitter thread, it is there . . . . of course!). So, today I decided to take a 2nd day with the problem and have my son look at this new solution.

But I started by having him explain the “power of point” solution I shared with him yesterday just to get warmed up:

Next I set up my friend Raf’s solution to see if my son could solve the problem using this clever idea:

Finally, since the solution with the full inscribed circle depends on a geometry formula relating the area and perimeter of a triangle to the length of the radius of the inscribed circle, I asked my son if he could prove that formula was true. It took him a minute to find the idea, but he was able to construct the proof:

I was happy to be able to share three different solutions to the problem that Catriona shared. It definitely made for a fun little weekend geometry review!

# Having my younger son work through a neat geometry problem shared by Catriona Agg

Yesterday Catriona Agg shared a nice geometry problem on Twitter:

I thought this problem would be a great one for my younger son to work through, so I asked him to give it a try this morning. Here’s how he explained his work:

Usually when I share one of Catriona’s problems I ask him to go through the twitter thread to pick out one of the cool solutions. We took a different approach today as we’d talked about “power of a point” a few weeks ago and I wanted to show him how that idea could be used to solve the problem:

I really like this problem and especially like how different solutions bring in different parts of high school geometry. Thanks (for the 1,000th time!) to Catriona for sharing a great little puzzle 🙂

# Playing with “zonish polyhedra” and the icosidodecahedron

Yesterday I stumbled on George Hart’s website and found some neat ideas to play around with using our Zometool set:

https://www.georgehart.com/virtual-polyhedra/zometool.html

https://www.georgehart.com/virtual-polyhedra/zonish_polyhedra.html

After seeing these pages my younger son and I built one of the models and talked about it:

Today we explored the shape a bit more by building an icosidodecahedron and comparing it to the shape from yesterday:

Two wrap up today we looked at how spherical the icosidodecahedron is. I would have like to do the same exercise for the “zonish polyhedra” we were look at, but I’m not sure how to calculate the volume of that shape.

This was a really fun project – it is absolutely amazing how easy it is to explore 3d geometry with a Zometool set!

# Sharing Tamás Görbe’s circle packing problem with my younger son

Yesterday I saw a really neat tweet from Tamás Görbe:

It turns out that we looked at this problem a few years ago, but for reasons I don’t remember my younger son wasn’t part of that project:

https://mikesmathpage.wordpress.com/2018/04/03/a-terrific-problem-to-share-with-calculus-and-geometry-students-from-the-iowa-state-problem-collection/

Today I thought it would be really fun to tackle the problem with my younger son. We started with a quick introduction to the problem and then a discussion of how to approach the solution:

The first thing my son tried was finding the radius of a single inscribed circle inside of an equilateral triangle:

Next he tried to find the radius of the circles when there were 3 inscribed circles. This part was pretty challenging for him, but his work really shows what a kid struggling through a math problem can look like:

Now we got to the heart of the problem – what is the radius when there are “n” inscribed circles:

Finally, we looked what the area covered by the circles would be in the limit as n goes to infinity. We also talked a bit about the surprise – why isn’t all of the area of the triangle covered?

I really think this problem is a great one to share with kids who have see geometry. It is great to see how they approach the problem, and also really nice to see how they thinking about the area in the limit.

# A fun – and surprising – statistics project inspired by Freya Holmér’s geometry twitter thread

Yesterday we did a neat geometry project inspired by an amazing thread from Freya Holmér:

here’s that project:

Today we are extending that project by trying to find the expected area of the circle when the three points are inside of a unit square.

To start the project we talked through a bit of the geometry that we need to answer the question about the expected area of the circle:

Before jumping into the computer simulation we had to check a few more geometric details – here we talk about using Heron’s formula:

Now my son took 15 min off camera to write a simulation to find the expected value of the area of the circle. Here he walks through the program and we look at several sets of 1,000 trials:

Finally, we finish up with a bit of a surprise – switching to 10,000 trials, we find that the mean still doesn’t seem to converge!

Turns out the expected area of the circle is infinite – that’s why we aren’t seeing the mean in our simulations converge. I think this is a great way to show kids an example where the Central Limit Theorem doesn’t apply.

# Using a great twitter thread from Freya Holmér for a geometry project with my younger son

The idea in Holmér’s thread is one that we’ve looked at previously, but I still thought it would make a great weekend project.

Before showing my son the thread, I asked him if he knew how to make a circle passing through three randomly chosen points in a plane:

After that introduction to the problem, we talked through Holmér’s thread:

Next we returned to the white board and I had my son attempt to construct the circle using the method in Holmér’s tweet. Here he used a ruler and compass:

Finally, I gave him a little challenge – can you make the circle without using a straight edge?

This was a really fun project, I’m really grateful to Freya Holmér for sharing her work on twitter!

# Sharing Dave Richeson’s tweet explaining the concept of dimension with my younger son

I thought that Dave’s tweet would make for a great project, so we took a close look at it this morning. We started by looking at the tweet and then I asked my son what his definition of dimension was:

Next we worked through a few of the introductory examples in Dave’s tweet – a point, a line, a square, and a cube:

Next we moved on to the fractals. Here we also need to talk about logarithms, so we stuck to two of the examples – the Sierpinski Triangle and the Koch Curve:

Finally, we looked at the most complicated example – the Sierpinski Carpet. At the end of this video we recap and got my son’s thoughts on non-integer dimensions

This was a terrific project. Thanks to Dave for sharing the idea!