Julie McPhail
Correlation/Regression: Technology and Math are a "Good Fit"
Length of lesson: One to two class periods
Description of the class:
Name of course:  Algebra 2
Grade level:  10th or 11th (depending on honors or academic level)
TEKS addressed:
(2A.1) (b) collect and organize data, make and analyze scatter plots, fit the graph of a function
to the data, interpret the results, and proceed to model, predict, and make decisions and critical judgments.
I. Overview
This lesson is gives students a useful tool for coming up with a function that models linear data. Function patterns are one way of figuring out what kind of model best fits a given set of data, but correlation and regression are much more powerful tools. True mathematicians combine function patterns with correlation, regression, and residual plots to determine what kind of model to use.
II.  Performance or learner outcomes
Students will be able to:
Make a scatter plot out of given data
Find the regression line, and give the correlation coefficient r when applicable
Examine residual plots and make judgments based on them
III. Resources, materials and supplies needed
http://www.math.csusb.edu/faculty/stanton/m262/regress/regress.html
This website allows the user to plot points, and it automatically calculates the regression equation and the residual plot.
http://www.ruf.rice.edu/~lane/stat_sim/reg_by_eye/index.html
This website lets students predict what the regression line will look like and what the correlation coefficient will be.
http://www.dynamicgeometry.com/javasketchpad/gallery/pages/least_squares.php
This website shows students where the name "least squares" comes from.

Materials:
TI-83 calculators *Charles's Law handout
*Thunderstorms handout *Data Analysis instruction sheet
* Used with permission from Mark Daniels

Five-E Organization

Teacher Does                    Probing Questions                     Student Does     

Engage: As the students are entering the classroom, draw a graph on the board with axes labeled x and y, and draw about 10 seemingly random points, all in the first quadrant of the graph. Let 3-4 students come up to the board and draw a line through the points (preferably in different colored chalk). Tell students that this is a common scenario for true mathematicians and scientists. They gather a bunch of seemingly random data, and they need to come up with a way of making sense of it. Remind students that they will be analyzing data like this for their individual projects. Approx. Time:  5 mins

"Whose line is the best?" (take a vote if students will play along) Announce as the winner whoever won the vote. "If we did an experiment, and this were our data, is this a good way of finding the "best line"?" (having various people make guesses, and then voting to see which line everybody likes the best.) "Why, why not?"

What is "best?" "How do we know which line to draw?" "I refuse to vote, I don't know what best is." "Probably not." "Because then math would turn into a popularity contest."

Explore: Put students in groups of 2 or 3 and tell them to come up with a better method for drawing the "best fit" line. Tell them to answer the questions on the board on a sheet of paper. After about 5-7 minutes, ask each group to explain their method. Approx. Time:  20 mins

Write the following questions on the board: "How can we find a 'best fit' line?" "Why does your method work?" "Tell me about your method." Ask the class, "Does this seem like it would work? What improvements could we make to this method?"

Some students will probably just talk about getting our line to pass through as many points as possible. Some may try to quantify this further by using absolute values, but most likely no one will come up with least squares.

Explain: After all of the groups give their methods, explain the method they will come across in their mathematical futures, Least Squares Regression. Compare Least Squares Regression with the methods that students came up with and explain why mathematicians usually choose to use Least Squares instead of other methods. Explain that Least Squares Regression only works on linear situations. Approx. Time 20 mins

Define "least squares regression line," "correlation coefficient," and "residual plot" using words and pictures. Talk about what it means when we see a pattern in the residual plot (linear is probably not the best fit for our data). Go to the following two websites to visually demonstrate these concepts and how they're related to one another: http://www.math.csusb.edu/faculty/stanton/m262/regress/regress.html http://www.ruf.rice.edu/~lane/stat_sim/reg_by_eye/index.html (If you have access to computers, let students play around with these two websites for 5 minutes or so, if not, demonstrate them for students or have a couple of volunteers come up and do the applets in front of the class.)

Extend / Elaborate: Tell students that there are complicated mathematical formulas that calculate the regression line and the correlation coefficient, but that calculators programmed with these formulas make the job much easier. Even true statisticians use technology (either calculators or computers) to analyze their data. Put students in groups of 2 or 3 and hand out the Thunderstorms handout. Give students sufficient time (probably about 20 minutes) to complete this activity, and then call on different groups to explain their answer to different parts of the activity. If time remains, give students the Charles's Law handout. If not, let them complete this activity for homework. Approx. Time: 40 mins

Put a chart of linear data on the overhead or on the board (perhaps a student's grade on a test as a function of the number of hours that student studied). Go through with students how to enter this data in their calculators, how to make a scatter plot out of it, and how to find and graph a regression equation. Hand out the Data Analysis instruction sheet to help students remember which buttons to press for which things. Walk around and guide students through the activity. Get them collaborating with each other.

Students are working on the Thunderstorms activity.

Evaluate:

Use the Muddiest Point assessment to see where students are still confused.

Approx. Time: 5 mins

Name: Date:

Thunderstorms

It is conjectured that in a lightning storm, the time interval between the flash and the bang is linearly related to the distance between you and the lightning. Answer the following questions:

a) Define t as the time in seconds and d as the distance in kilometers. Which variable should be dependent and which variable should be independent?

Dependent:

Independent:

Here is your data:

d	1	2	5	10	12
t	2.98	6.09	14.94	28.99	37.11

c) Plot a graph of distance versus time in your calculator.

d) Use regression to find an equation that models this data. Write your equation in terms of t and d, not in terms of x and y.

Your equation:

e) Using your regression equation, calculate the distance of the storm if the thunder sound reached you 4 seconds after you saw the lightning flash.

f) Using your regression equation, calculate the distance of the storm if the thunder sound reached you 1 minute after you saw the flash.

Extension: What would your formula be with seconds and miles as your units?

Name: Date:

Charles's Law

From the work of physicist Jacques Charles (1746-1823): Charles discovered that the volume of a gas at a constant pressure increases linearly with the temperature of the gas. The table below illustrates this relationship between volume and temperature. In the table, hydrogen is held at a constant pressure of one atmosphere. The volume V is measured in liters and the temperature T is measured in degrees Celsius.

T	-40	-20	0	20	40	60	80
V	19.1482	20.7908	22.4334	24.0760	25.7186	27.3612	29.0038

a) Use the table and linear regression to find the linear relationship.

Your equation:

b) Using your equation, if the volume were 0 liters, what would the temperature be?

______ degrees Celsius

Does this temperature look familiar from your science class?

Data Analysis

Entering Data

1. Press [STAT] 1 to enter data in a list or lists. Key in data points into a list and press [ENTER] each time. To change to a different list use the arrow keys.

Statistical Plotting

1. Press [2nd] [STAT PLOT] 1 to display the Plot 1 screen.

2. Press [ENTER] to turn Plot 1 on. Use the arrow keys to position the cursor on each of the available settings and press [ENTER] to choose each desired setting.

3. Press [ZOOM] 9 to view the plot of your data points.

Regression Equations

1. Press [STAT] [>] to view the Stat Calc options. Use the arrow keys to choose the type of regression equation that you desire, and then press [ENTER].

3. Press [2nd] Lm [,] Ln where Lm and Ln are the STAT lists in which you have previously stored your x and y data points.

4. Press [ENTER] to calculate the regression equation

Graphing the Regression Equation

1. Press [Y=] to choose the Y= screen, and then press [CLEAR] to clear any previous equations.

2. Follow steps 1-3 of Regression Equations

3. After choosing your linear regression and pressing [2nd] Lm [,] Ln, and BEFORE YOU PRESS ENTER, press [,] [VARS] [>] [ENTER] [ENTER].

4. Press [ENTER]

5. Press [GRAPH]

Plotting the Residuals (assuming your data is stored in L1 and L2)

1. Press [STAT] [ENTER] then move your cursor until L3 is highlighted.

2. Press [2nd] L2 - [VARS] [>] [ENTER] [ENTER] ([2nd] L1) [ENTER]. Follow the directions under Statistical Plotting to view a scatter plot of L1 and L3.