Mathematical Modeling and Real-Life Data

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:

LinReg

The next regression equation we investigated was the quadratic one. For this one, I used the old favorite, the water fountain stream:

Fountain

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:

QuadReg

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.

SinReg

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.

 

 

Sunrise, Sunset; a Sinusoidal Story

Today in precalculus, we looked at some data that follow a sinusoidal pattern and calculated a regression function to model it. Here’s how we did it:

The hours of daylight over a year increase, then decrease in a regular, periodic fashion – just the kind of data that result in a nice sine curve. The US Navy maintains a website where you can enter any location and see the sunrise and sunset times over an entire year:

http://aa.usno.navy.mil/data/docs/RS_OneYear.php

We entered our location (Nashville, TN), and used the sunrise and sunset times for the middle of the months:

Raw Data

The times are given in hours and minutes, so we converted them to decimal hours. We then subtracted sunrise time from sunset time to get the number of daylight hours for each month:

Daylight Hours

Now that we had our data, we entered it into our graphing calculators:

Stat

After setting our window parameters, we plotted the data:

StatPlot

Nice! Next, we calculated the regression function that best fits the data (This might be a good time to discuss pros and cons of different regression functions: quadratic, cubic, quartic, etc.):

We then plotted the regression function with the data points. Nice fit!

Plot

Because we did this in mid-February (x = 2.5), we used our model to estimate how many hours of daylight we should expect to have today: 11.4 hours.

If you’re ready to move beyond graphing calculators (and I certainly am!), then you can do this activity with the Desmos online function grapher. What’s nice about that approach is the ability to set up sliders, and let students fit the regression function to the data manually.

Here are a couple of screenshots:

Desmos Data

Desmos Regression Function

 

Update: Apparently this is the source of the Chinese Yin/Yang symbol.

Mathematical Modeling

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.