Pi Day (3/14) is fast approaching. Here’s a simple Geogebra activity you can do with your math students to help them get a sense of what pi is.
In Geogebra, construct a segment, then construct the midpoint. Construct a circle using the midpoint as center, and an endpoint of the segment as a point on the circle. In the Input pane, enter Circumference(<your circle>). When you divide the circumference measurement by the diameter length, you get pi! Geogebra is dynamic, so you can change the size of the circle, but the ratio will remain constant at pi.
Here’s an interactive version for you to play with at Geogebra’s site.
Here’s a screenshot:
All of my students have TI-84 graphing calculators, so I am going to have them use their Geogebra constructions to generate 10-12 different diameters and circumferences. Then we will enter the diameters in L1, the circumferences in L2, and plot the data. We will calculate the regression line, the slope of which should approximate pi.
Happy Pi Day!
Update: This went really well in my geometry class. Here’s the data we collected –
Here’s the regression line equation (check out the value of the slope):
And here is the plot of the data and the regression line:
Today in Calculus, we looked at three different types of regression equations that can be used to model data. Linear regression was first. The students measured each others’ forearm lengths and their heights. You can have a good conversation about which should be independent and which dependent. We decided the forearm lengths were independent, and height was dependent. We put the data into our TI-84 Lists, determined appropriate window parameters, and plotted the data. Then we used the LinReg feature to graph a line of best fit:
The next regression equation we investigated was the quadratic one. For this one, I used the old favorite, the water fountain stream:
We pasted it into GeoGebra and put some points on the path of the water. Using those points and the QuadReg feature of the TI-84, we came up with an excellent model:
Finally, we used the sunrise and sunset times of the fifteenth day of each month in Nashville, TN to see how a sinusoidal function can model data. (I grabbed the data from the US Naval Observatory.) Students converted the times that were in hours:minutes to decimal hours (16 + 56/60) – (6+57/60). We entered the months (January = 1, February = 2, etc.) and length of daylight into the calculator’s lists and used SinReg to determine the sinusoidal function that best fits the data. If time permits, it’s a nice exercise for the students to figure out the amplitude, period, phase shift, and translation themselves, and compare their function to the calculator’s.
This set of exercises took about 50 minutes, and the girls got pretty excited when they saw how closely their functions matched the data, and how they can use them to determine other points.
This is a sexy topic: mathematical modeling. Don’t worry, though, this screencast is safe for all ages! In this introduction to math models, I illustrate how to use a TI-84 graphing calculator to find a linear regression model for a given set of data; what direct, inverse, and joint variation mean, and how to apply them to some simple examples. The hardest part of this type of problem is translating from the English to mathematical equations. Once you understand the key terms, everything falls into place.