# Happy Tau day!

Since the kids are just back from a week of camping, and consequently a little tired, I thought we’d do a fun Tau day project. Turned out that we got a little sidetracked on a little geometry point. Still a fun project, though, just not what I was expecting.

We started with this old Numberphile video about $\pi$ and $\tau$:

After we watched the Numberphile video, we began our conversation about $\tau$. One point I wanted to focus on for a bit – and I really thought it would be about a 1 minute thing – was Matt Parker’s point that the diameter was easy to measure. The boys didn’t remember this point from the Numberphile video, and talking about how to measure the diameter started us down a long path:

We made a circle with our compass off camera to help us explore the question of how to find the diameter. My older son had the interesting idea of drawing a square around the circle. If you could do that, then finding the diameter would be pretty easy. The trouble is – how do you draw that square?

At the end of the video, my younger son suggests that we measure the circumference to find the diameter.

Now, following the suggestion at the end of the last video, we found some string and tried to measure the circumference. We found the circumference was about 12.5 inches. That measurement led to a long discussion about how to calculate the approximate radius if we knew the length of the circumference.

Following the discussion about the circumference, we returned to trying to measure the diameter directly. This measurement problem really gave the boys fits. Part of the confusion, I think, was that they were looking for a way to find the diameter exactly. There are, of course, plenty of ways to do that, but looking for the absolute perfect solution was distracting them from using the ruler to find a close approximation.

At the end of this video we stumble on an important idea – the diameter is the longest line segment that you can draw in the circle!

The idea at the end of the last video gives as a way to get an approximate measure of a circle’s diameter – we just look for the longest line that we can draw. Both kids had some interesting ideas about how the length of lines would shrink or grow as you moved around the circle. Exploring those ideas allowed us to get better and better approximations for the diameter. Hopefully the shadows don’t obscure the measurements we are making in the video.

So, although the project didn’t quite go in the direction that I was expecting, a pretty interesting project. It is nice to see that an offhand comment from a mathematician – in this case that the diameter of a circle is easy to measure – can lead to a fun little project for kids.

# A challenging Venn diagram problem

[note: written up super fast without any editing]

We did this project with the boys last week before they went to camp for a week. Unfortunately I never got around to writing it up – here it is fairly quickly.

First we talked through the problem (again, sorry for the camera focus at the beginning – lasts 10 seconds). I wanted to go slowly introducing the problem because it is easy to go wrong right at the beginning in these problems:

In the second part we start in on the solution of the problem. My younger son takes the lead by finding how many students take only calculus. After that we spend a few minutes making sure that we know how to count the students in the various different buckets. They adopt a strategy of counting what we know and then using that information to help us find what we don’t know. By the end of this video we have found that we have 105 people to fit into the two open buckets, and we have an expression for one of those empty buckets in terms of x.

We start off this section by finding an expression for the number of people in our last remaining buck in terms of x. We now add our two expressions involving x together to get an expression. From the last video we know that this expression has to be equal to 105. After a quick review of how we got all of this information, we find that the number of people taking physics is 110:

The last part of the project is one final review of the whole problem. I wanted to go through it one more time just to double check that both kids were able to follow all of the steps:

So maybe not our best project ever, but it was nice to see that the kids were comfortable solving a challenging problem like this one. There’s lots of information to keep track of here and I’m glad they were able to see this one to the end. After their week at camp, we’ll start in on the next section of our Introduction to Counting and Probability book tomorrow. That section is about basic counting techniques.