Dot Physics

Activities for Pi Day

Oh, I know you missed it. Really, it wasn’t your fault. Pi day fell on a Sunday, so how are you supposed to have pi-day activities in class? Don’t let it stop you. You are better than that. Do the activity anyway. What to do? Here are some suggestions. (Suggestions aimed mostly at the high school level)

Plot Diameter vs. Circumference

This is a great one. Let your students find as many round things as they can (cylinders work the best – or flat stuff). Measure the circumference (you can use a string or a tape measure) and the diameter. Since the relationship between these two is:

i-9c6732b0d80d71692533150ad7a01450-2010-03-14_la_te_xi_t_1_18.jpg

A plot with circumference on the vertical axis and diameter on the horizontal axis should be a straight line with a slope of Pi. The great thing about this activity is that students can get a feel for where pi comes from. There are two many students that just think it is a number invented by mathematicians to make things more complicated and cool sounding.

Oh – as a bonus, students get to practice making graphs and finding the slope. I would recommend doing this one on real graph paper (and not in a spreadsheet).

Rolling vs. Distance

Really, this is the same thing as the activity above – but it looks different. Take a cylinder and roll it. Count the number of revolutions and measure the distance it rolled. Plot distance on the vertical axis and the number of revolutions on the horizontal axis. Here is the relationship between the two.

i-0ca12caf928ac17559d0db7f94bcb7d6-2010-03-14_la_te_xi_t_1_19.jpg

LEGO Estimation of Pi

I just thought of this one and have not actually tried it. Archimedes estimated Pi by drawing two 96-sided polygons inside and outside of a circle. He could then determine an upper and lower bounds for the value of Pi. You could try to reproduce this Lego pieces. Make an n-sided Lego polygon both inside and outside of a circle. Compare the perimeter of the polygons to the radius of the circle.

I might make this a future post, but if you try it out, let me know how it works.

Monte Carlo Estimation of Pi

I had a more detailed post about this method for estimating Pi. But maybe you don’t want to look back – so here is the short version. If you randomly put points in a 1 x 1 square, some will be more than 1 units from the corner and some will be less. Here are some random dots.

i-4398cc88e0d9b555593ecca0939d5920-2010-03-14_scratch.jpg

Since the dots that are less than 1 from the lower left corner make up 1/4th of a circle, the ratio of red dots to total dots should be:

i-a38ebc2616a47a71f57168ebe4c053b2-2010-03-14_la_te_xi_t_1_20.jpg

So, this is pretty straight forward. But how could you do this? I made a program in Scratch as well as python. You could use anything that has a random number generator. Here is a version in google docs:

I didn’t finish it, well mostly I did – but you would need to do some more work on it to finish. If you wanted, you could have groups in the class calculate the average for 100 points and then take the average average for all the groups.

Non-Computer Monte Carlo

Maybe you think computers will one day rule the world and you would rather not use them to calculate Pi. I can understand that. You could drop something so that it has a random distribution on a 1 x 1 square and then count the number in and outside of a circle. Maybe find some way to drop sand on a square paper? Make sure that sand is falling outside of the paper also, or it likely will not be randomly distributed.

How Accurate can you get Pi by Measuring?

What if you used the plot of circumference vs. diameter from above? How accurate of a value of Pi could you get?

Other resources

There are tons of great Pi sites out there. Here are just a few:

Comments

  1. #1 raidergirl3
    March 15, 2010

    I do that first activity, plotting diameter v circumference on the second day of HS physics class, to practice good graphing, to look at the format of writing a lab report, and because the slope is Pi! I love that lab.

  2. #2 Alan B
    March 15, 2010

    I’m not sure what you’re thinking of with the lego activity, but I can’t think of a way it would work.

    If you’re simply placing rectangular bricks on a flat base, the perimeter of the circular shape will be 4*D, the perimeter of a square with side length D equal to the circle diameter. The perimeter is independent of the number of bricks used because of the “stair-step” shape of the curve.

    You might use technic beams and rods to make a better approximation to regular polygons, but then you’d have to figure out some way to ensure they conform to a circular shape, which seems like more trouble than its worth.

  3. #3 Rhett Allain
    March 16, 2010

    @Alan,

    I think you are correct – didn’t think about it that way.

  4. #4 KeithB
    March 16, 2010

    I have loved the Monte Carlo method since I was taught it in High School at a Cal Tech Saturday Science Program in the late ‘seventies. I programmed it in assembler on an IBM 360.

    Now it is my second practice program after “hello world.” My next one will be in Apple Cocoa.

  5. #5 MaryBeth Chavez
    February 24, 2011

    If I understand the LEGO activity correctly, I have done one using power point slides with triangles, square, pentagon, hexagon, septagon, octagon and dodecagon.

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