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 TamĂĄs Görbe’s circle packing problem with my younger son

Yesterday I saw a really neat tweet from Tamås 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 HolmĂ©r’s geometry twitter thread

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



here’s that project:

https://mikesmathpage.wordpress.com/2020/10/17/using-a-great-twitter-thread-from-freya-holmer-for-a-geometry-project-with-my-younger-son/

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 HolmĂ©r for a geometry project with my younger son

Yesterday I saw this amazing twitter thread from Freya Holmér:



The idea in HolmĂ©r’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 HolmĂ©r’s thread:



Next we returned to the white board and I had my son attempt to construct the circle using the method in HolmĂ©r’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 HolmĂ©r for sharing her work on twitter!

A nice introductory statistics conversation with my younger son

My younger son wants to learn more about statistics. I’m excited to come up with some projects, though our journey here is probably not going to look like a typical statistics class.

My first idea was to have him investigate coin flips and look at similarities and differences with 10, 100, and 1000 coin flips. But really before we even got started he had a really interesting question.

So, here’s how he described the program I had him write ahead of time, and then we discuss the question he had -> In 100 flips, why did it seem that the chance of getting less than or equal to 50 heads was more than 50%?

To start diving in to his question, we first looked at sequences of 100 flips to get a better sense of what was going on. Interestingly, on a test look at 10 sequences, we did find that 6/10 had less than 50 heads:

Now we looked at the distribution of heads in 100 flips using Mathematica’s Histogram function – it was really interesting to hear him describe the different distributions that we saw:



Finally we talked about why my son was seeing what he was seeing in his coin flip program.

Revisiting Futility Closet’s “Paradox of the 2nd Ace” with my younger son

This week my younger son told me he wanted to learn a bit more about statistics. By lucky coincidence I happened to stumble on one of our old projects while trying to answer a question on twitter:

https://mikesmathpage.wordpress.com/2018/12/01/introducing-the-boys-to-futility-closets-paradox-of-the-second-ace/

This project was about a fun probability problem I learned in this tweet from Jon Cook:



I stared by introducing the problem to my son and asked what he thought the answer was going to be:



Then we started in on the calculations. Finding the probability that someone having “at least one ace” had more than one was a little challenging, but we found the right approach after a few tries:



Next up was calculating the probability that someone who had the ace of spades would have a second ace. After the work we did in the last part, this calculation was easier:



Finally, we went to the Futility Closet page to see the numbers and then I asked my son why he thought the surprising result was true.

Definitely a fun project and a neat probability surprise for kids to see.

Sharing Dave Richeson’s proof editing exercise with my younger son

I saw a really fun tweet from Dave Richeson last week:



I thought that editing this “proof” would be a terrific exercise for my younger son. We started the project by walking through the “proof” to make sure he understood what he was supposed to do. After this short introduction he worked on the editing off camera for about 10 min:

After working through Dave’s document, here are the changes my son suggested:

I think the above video shows why this editing exercise is such a nice idea. There were a couple of points that I wanted to add to my son’s notes, so we talked through two specific ideas to end the project:

This project was really fun and definitely something I wouldn’t have ever though to try. Thanks to Dave for sharing this terrific idea!

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

I saw a really neat tweet (and subsequent twitter thread) yesterday:

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!