A geometry puzzle from Catriona Agg that is also great for showing students some ideas from algebra

Last week I saw another terrific puzzle from Catriona Agg:



I left two copies of the puzzle for my son to work through while I was out this morning. For the first run through I asked him to solve the puzzle as it was stated. Here’s his work and his explanation:



For the second run through I asked him to solve the problem assuming that the radius of the circle was X rather than 5. This was first step in what I was hoping would be an interesting algebra exercise. Here he was successfully able to use the quadratic formula even though the equation he found had 2 variables:

For the last part of the project I wanted to see if he could factor the equation he found in the last video. This turned out to be a significantly more difficult challenge, but he figured out how to do it just as we ran out of space on the memory card!



I suspected that the factoring challenge would be more difficult than simply using the quadratic formula, though I didn’t realize how much different it would be. I might try to find some more challenges that involve multiple variables just to get a bit more practice with these ideas.

Exploring the generalized binomial theorem with my younger son

My younger son is reading Conway and Guy’s The Book of Numbers right now and one of the early sections in the book on the binomial theorem caught his attention. We talked about the binomial theorem for a bit and then I showed him a few examples of the generalized binomial theorem and he was really interested.

Today we talked about the ideas a bit more, starting with a reminder of what the theorem says:



Now we used the ideas from the last video to take a look at an approximation to \sqrt{2}.



Next we looked at an approximation that turns out to be one step more complicated – \sqrt{3}




Finally, we went back to Mathematica to look at how good the approximation from the last video was:



It was really fun showing my son some of these advanced ideas. I’m excited to explore these ideas with a few more approximations tomorrow.

Our first project from Conway and Guy’s The Book of Numbers

My son has started reading The Book of Numbers by Conway and Guy. Today we did are first project based on some examples early in the book:

The first example we looked at were the “hexagonal” numbers. Here my son explains what those numbers are and gives a little introduction into the surprising geometric idea that helps understand these numbers:

My son had some difficulty seeing the argument from the pictures in the book, so we tried out a few examples (a few days before this project) using snap cubes. Here’s his explanation of the surprising geometry:

The next thing we looked at in the book were the “tetrahedral” numbers. The book game an amazing proof showing a formula for these tetrahedral numbers. Here he explains this clever proof:

This was a really fun project, and I’m also really happy that this book is teaching my son a bit about reading math books – sometimes even reading and understanding just a couple of pages can take time.

Having the boys work through another fantastic puzzle from Catriona Agg

Catriona Agg posted this geometry puzzle on Twitter this morning:



I had the boys work on the problem on their own and then talk through their progress.

My older son went first – his solution is along the same lines as most of the solutions in Catriona’s twitter thread, though is reasoning is pretty interesting to hear:



My younger son went next. He wasn’t able to find the solution on his own, but was able to get there while we talked about his work. I’m sorry that I forgot the camera was zoomed in on the paper here. I do zoom out a little over half way through. Hopefully the words are clear even if some of the work is off screen:



At the end of the last video my son had worked through the main idea of the problem. Here he finishes the solution and talks about what he liked about the problem:



As usual, having the boys work through one of Catriona’s puzzles made for a great project. I really liked the algebra / geometry combo that this problem had as I think that was great practice for my younger son. I also think the more intuitive solution my older son had shows how mathematical intuition develops as kids get older.

Sharing Andrés E. Caicedo’s amazing twitter thread on even / odd irrationality proofs with my younger son

Yesterday I saw an amazing twitter thread by Andrés E. Caicedo:

I thought that some of the ideas would be great to share with my younger son and started by asking him if he remembered the usual proof that \sqrt{2} is irrational:



Following the twitter thread, I asked him how he thought the proof that \sqrt{5} is irrational would go. He gave the proof that I think most math people would give:

Next we walked through the “new to me” proof in Caicedo’s twitter thread. The ideas are definitely accessible to kids. In addition to being accessible, the ideas also provide a nice way for kids to get some algebra practice while exploring a new math idea:



Finally, we talked about the surprise that this method of proof doesn’t work for \sqrt{17}. My son had an interesting reaction – since this method of proof doesn’t seem to rely on the underlying number, he was surprised that it didn’t work as well as the method he’d used for \sqrt{5}



I really loved talking through Caicedo’s thread with my son and am really thankful that he took the time to share this fascinating bit of math on Twitter yesterday!

Sharing John Urschel’s great video on rational and irrational numbers with my son

Yesterday I learned about a fantastic video that John Urschel made:

Here’s the video on youtube:

This morning I asked my younger son (going into 9th grade) to watch the video so we could talk about it. Here are his initial thoughts and what he thought was interesting:

Now we talked through three of the ideas he thought were interesting. The first was how to find the rational representation of a number like 0.64646464…..

Next he talked about the proof that \sqrt{2} is irrational:

Finally, we talked about a really neat proof in Urschel’s video -> why log base 2 of 3 is irrational:

I love Urschel’s video and think it is an absolutely terrific one to share with kids. It is a great way for kids to see some advanced mathematical proof ideas, but also a great way to review some important ideas in math. We had a really fun morning going through it.

A nice exercise about cos(72) from Martin Weissman’s An Illustrated Theory of Numbers

My younger son is working through Martin Weissman’s An Illustrated Theory Numbers and came across this exercise last week:



Today we finally got to doing a project on the problem. We worked through the first 4 parts and will save the last part for another project.

Here’s the first part of the problem which is mostly a discussion of how you can think about points on the unit circle using complex numbers:

The next part of the problem asked to show that if x is a 5th root of unity then 1 + x + x^2 + x^3 + x^4 = 0. I forgot to zoom out after we zoomed in on the problem, but I do finally remember to zoom out around 1:30 – sorry about that:

Part c was the part that gave my younger son a lot of trouble, but luckily my older son was able to help out with the ideas about sums and products of roots require to get through this step:

Finally, we find the roots of the quadratic polynomial from the last part and find the exact value for cos(72). What a fun project!

Talking through some generating function problems with my kids

My younger son was working on a problem in the Wolfram Programming Challenges that is best solved using generating functions:

Screen Shot 2020-07-26 at 9.21.52 AM

Yesterday we did an introduction to generating functions but unfortunate our camera’s memory card died and the videos were lost. Instead of repeating that introduction we just dove into some of the examples from the book we are using.

The first problem was about distributing juggling balls. It takes a few minutes for the ideas we talked about yesterday to click in, but eventually we were able to work through this problem:

Next is a really neat example of the kind of problem generating functions can solve – counting solutions to relatively simple equations (sorry for forgetting the camera was zoomed in at the beginning – we finally zoom out around 2:45):

Now we tried out a few of the exercises. The first one I chose was about distributing juggling balls. With the work we’d put in on the first example, this problem wasn’t too hard:

Finally, we tried out a new problem asking about the number of integer solutions to an equation. The ideas about generating functions seemed to be really sinking in now and this problem didn’t give them too much trouble:



Introducing the boys to generating functions made for a really fun weekend of math – happy that working through the Wolfram Programming Challenges gave us this opportunity.

Revisiting Po-Shen Loh’s approach to the quadratic formula

Last fall Po-Shen Loh shared a really interesting approach to understanding the quadratic formula. His idea got so much attention that they were written up in the New York Times::

https://www.nytimes.com/2020/02/05/science/quadratic-equations-algebra.html

We had our first look at these ideas back in December:

https://mikesmathpage.wordpress.com/2019/12/08/sharing-po-shen-lohs-new-idea-about-the-quadratic-formula-with-kids/

I thought it would be fun to revisit some of the ideas today. It turned out to be a really good algebra review for my younger son, and a nice review of ideas about roots of equations for both kids.

We started talking about some general ideas about quadratic equations and a reminder of the sum of roots and product of roots ideas for quandraics:



Now we took a closer look at the sum of roots and product of roots ideas to give the boys a bit more background on the ideas in Loh’s paper. The ideas here were a little confusing for them, so it was good that we took a little time to review them before going to the next step:

With all of the background out of the way we moved on to Loh’s difference of squares idea. This idea wasn’t obvious to the boys, but once they saw it the quadratic formula appears immediately!

Finally, we finished up the project by showing how to derive the quadratic formula for a general equation:

I really like Loh’s approach and think it is a great way for kids to see the quadratic formula. This project showed me that the ideas are a bit more subtle than I thought, though, and we’ll probably have to run through them a few more times for them to really sink in.

Using Po-Shen Loh’s quadratic formula idea to calculate the Cosine of 72 degrees

In December Po-Shen Loh made a video about a really neat approach to the quadratic formula:

We did a project using the ideas of sums and products of roots at the time, and I wanted to revisit the idea tonight now that my son is studying complex numbers. An idea I thought would be fun was to explore was how to use the sum and product of roots ideas for quadratic functions to calculate the cosine of 72 degrees.

I started with a quick review of the main ideas we’d be using in this project as it has been several months since we went through Po-Shen Loh’s idea:

Now we dove into the problem of finding the roots of the equation x^5 - 1 = 0.

Now we moved to the main idea in the project – how can we factor the polynomial we found in the last video – x^4 + x^3 + x^2 + x + 1 – into two quadratic polynomials?

The work here is a little tricky, but my son got through it really well. The ideas here are definitely accessible to students who have learned a little bit about polyomials and sums and products of roots.

Finally, we solved for the roots of the quadratic equation x^2 + x - 1 = 0 (I accidentally wrote this equation wrong, so we get off to a bad start. Luckily we caught the error after about a minute.)

Solving this equation gives us the value of cos(72)!

It was really fun to see that the combination of introductory ideas from complex numbers, polynomials, and sums / products of roots of quadratics could help us calculate the value of cos(72). I’m excited to play around with Po-Shen Loh’s idea a bit more and see where else we can find some fun applications!