Pi in base 3

There’s a new – and amazing – video out about the seemingly crazy math fact that 1 + 2 + 3 + . . . . = -1/12:

At 16:53 in the video something amazing happens – a t-shirt with \pi in base 3!! I ordered it immediately 🙂

I’m happy that I did because the boys were curious about how to do the calculation. Tonight we talked about it starting with a few easier examples first:

Next we moved to Wolfram Alpha to finish up the calculations:

It was nice that we were able to get 6 digits after the decimal point so easily. Fun little project coming from an awesome shirt 🙂

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.

Measuring Pi

After yesterday’s Family Math project I was thinking about a project with spheres so that we could talk about the area of the spherical caps from our printed shapes.  This morning i changed my mind and thought that a slightly more laid back project was in order, so we spent the morning trying to measure \pi.

The first thing that we talked about was basic definitions.  Just trying to set the stage for thinking about geometry, especially since I’ve not really spent much time talking about geometry with my younger son:

After getting through the definitions we started measuring.   We started with a can of chickpeas and then talked for a bit about why doing the same measurements with a cube would be different.

Now we moved on to some larger circles.  This required a larger area, so we moved to the garage.   Our first attempt here didn’t go so well as our estimate for \pi was about 3.5.  I wasn’t too disappointed, though, since learning that measurements don’t always produce what you expect is an important lesson.

Our last prop was a bicycle wheel.  This experiment required a little bit more room than our camera could handle, so we split it into three pieces.  We got an estimate for \pi that was a little low, but it the best estimate of the bunch.    After we finished the calculations on this one we talked through a few of the aspects of our measurements – is it easier to get an estimate for \pi with a large circle or a small circle, for example.

Definitely a fun little set of experiments.  Fun to see the boys rolling up their sleeves and taking a few measurements, too.

The Square root of Pi

Last night my younger son asked me about \sqrt{\pi}.  I got so caught up in thinking about what to talk about that I actually forgot the context that gave rise to the question.  Oh well.

It certainly is true that \sqrt{\pi} comes up less frequently than plain old \pi, but there were still a few fun ideas.

First was the obvious one – squaring the circle:

 

Finally, a fun suggestion from twitter:

Even though this suggestion is a little above what we’ve studied, I thought it would be fun to talk through it:

Definitely a fun little morning with \sqrt{\pi}.

 

Some fun with Archimedes and Pi

Last night I was flipping through one of my favorite books – 100 Great Problems of Elementary Mathematis:

Book pic

 

I came across the section describing Archimedes’s method of calculating pi and thought it would make a fun morning activity with the boys.  Some of the details of the geometry are a little over their heads, but I didn’t want to get too caught up in the details anyway.  Made for a fun morning.

The first part was just kind of silly – finding a way to introduce pi:


From here we moved on to Archimedes’s method.  We drew a hexagon inscribed in a circle and showed how you could use that hexagon to get the simple estimate that pi = 3:


Playing around with the hexagon inside a circle turned out to be fairly straightforward.  Next we moved on to the slightly more difficult problem – discussing a hexagon circumscribed about the circle.  We studied the picture for a while, used the Pythagorean theorem, and eventually found the perimeter of this hexagon, too.


After this, my instincts led me astray initially.  Luckily, though, I caught myself before moving on, and spent a little time asking the kids to see if they could figure out how to improve our estimate:


The next part is probably the most difficult.  Archimedes figured out a really neat relationship the perimeters of certain polygons and used that relation to get better and better approximations to pi.  The derivation of the relation really uses only the Pythagorean theorem, but I didn’t want to get caught up in the details today:


Finally, we moved to the computer to run through some of the approximations.  I’ve been trying to figure out ways to incorporate a little more computer math, so I was really happy to have the opportunity to do that here.   The last step in our little talk about pi was writing a simple program:


The simple program that we wrote to study the formula is easy to share, you can play with it here if you want:

https://www.khanacademy.org/cs/archimedes-and-pi/6401932075728896

The kids seemed to really hearing about this way to calculate pi.  It was fun having the chance to walk through all of this with them.