Burn Baby Burn

Objective: This activity will require you to collect data, graph a scatter plot, draw a trend line and write the equation of the trend line. Once you have the equation of your trend line you will use this to make predictions about data other than the actual information you collected. Once you have done all of this by hand, you will make a scatter plot on your TI-83 and get the equation of the linear regression of your data.

Time required: 2 days.

Day 1: Collect the data, make a scatter plot, draw a trend line, write the equation of the trend line, and answer questions using the linear equation.
Day 2: Enter the data on the calculator, make a scatter plot, find and graph the regression equation, and answer questions using the linear regression.

Materials needed:

a 2.5 inch birthday (or other fast burning) candle
a candle holder (a paper plate with a tack poked through it)
a ruler that measures height in centimeters
a stopwatch or a clock that measures time in seconds
a TI-83 (or higher) graphing calculator

Prior knowledge: The student should be able to find the slope of a line using

or

and write the equation of a line using the point-slope form of a line:

y = m(x – h) + k

where m is the slope of the line and (h, k) is a point on the line. This activity will give step-by-directions for entering data into lists, making a scatter plot, finding a regression line, and graphing the regression line.

Day 1 Activities:

Students should work in groups of 2.
Light the 2.5 inch birthday candle and measure its height in centimeters every 30 seconds for 5 minutes. Record your data in the table below.

Time (min) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Height (cm)
On a sheet of graph paper, make a scatter plot of the data you collected.
Choose appropriate scales and label both axes.
Draw a trend line on your scatter plot.
Pick two "grid points" that are on your line.
1. Find the slope of the line.
  Show your work.
2. Using y = m(x – h) + k, write the equation of the line.
  Show your work.
Do all of the points on the graph of your equation make sense in this situation?
Why or why not?
Define domain.
What is a sensible domain for this situation?
Define range.
What is a sensible range for this situation?
The data you collected is discrete. Define discrete.
The equation you wrote is continuous. Define continuous.
Which better describes the candle burning, discrete or continuous?
What is the real-world meaning of the slope you found in #5?
What is the y-intercept of your equation?
What is the real-world meaning of this y-intercept?
Use your equation to predict the height of a candle after it burns for 2¾ minutes.
This is an interpolation. Define interpolation.
Use your equation to predict the height of a candle after it burns for 7 minutes.
This is an extrapolation. Define extrapolation.
Use your equation to predict how long it will take the candle to burn down to a height of 2 cm.
According to your equation, when will the candle be completely burned down? Show your work.

Day 2 Activities:

Enter the data in your TI-83:
1. Press the button.
2. EDIT,1:EDIT… will be highlighted. Press the button to get to the lists.
3. Clear List One (L1) and List Two (L2) by:
  1. Use the blue to highlight L1, press the (not delete!) button, then the button. This should remove all data previously entered in L1.
  2. Repeat step one, highlighting L2 to remove any data in L2.
  3. Enter the data you want on your x-axis (the independent variable) in L1. If you enter data erroneously, you can either use the button to remove it completely, or type over it. Enter the data you want on your y-axis (the dependent variable) in L2.
To make a scatter plot on your calculator:
1. Press the yellow then the to get you to the STAT PLOT menu.
2. Plot 1…Off (or On) will be highlighted. Press the button to get to the setup screen for Plot 1.
3. Press the button to turn On the Stat Plot.
4. Use the blue key to select the Type: of plot. The blue and keys will let you move back and forth between the types. For this exercise, you need to highlight the first one, a scatter plot, and press the button.
5. Use the key to go to Xlist. If it doesn’t have L1 for the chosen list, press the yellow then the (L1) button.
6. to Ylist: and make sure L2 is the chosen list.
7. to Mark: and pick the type of mark you wish to use.
You now need to set an appropriate viewing window to display your scatter plot. This will be easier to do if you refer to the scatter plot you created on Day One.
1. Press the blue button.
  - Xmin is the lowest number you have on your x-axis.
  - Xmax is the highest number on your x-axis.
  - Xscl is the interval between each grid point on your x-axis.
  - Ymin is the lowest number you have on your y-axis.
  - Ymax is the highest number you have on your y-axis.
  - Yscl is the interval between each grid point on your y-axis.
2. Give the window you set:
  
  Xmin _____ Xmax _____ Xscl _____ Ymin _____ Ymax _____ Yscl _____
3. Press the blue button to display your scatter plot.
To graph the equation of your trend line, press the blue button. Type your equation into Y1= to type in the variable x, use the button). Once your equation is entered, press the button to see the trend line.

Is this trend line in the same place as the trend line you drew on your graph? If it isn’t, you need to go back and fix your equation!
Using the data you entered into L1 and L2, the TI-83 can calculate a linear regression, sometimes referred to as a “line of best fit.” To find the linear regression:
1. Press the button.
2. to highlight CALC.
3. Select 4: LinReg(ax+b)
4. The phrase LinReg(ax+b) will appear on the home screen.
5. Assuming your data is in L1 and L2, just press the button to get the slope and y-intercept of the linear regression. If your data is in any other list, after the phrase LinReg(ax+b) on the home screen you will need to name the two lists (with a comma between the two list names) where your data is stored.
  For example, if your independent data (x) is in List 2 and your dependent data (y) is in List 3, you would press the yellow button, then the (L2), followed by the (above the 7), then the yellow button and the (L3). Your home screen should read: LinReg(ax+b) L2, L3. Now press to get the slope and y-intercept of the linear regression for the data in List 2 and List 3.
Enter your regression equation in Y2= (press the blue button to get back to this screen). You can either write the equation down on a piece of paper, then enter it into the Y= screen, or you can have your calculator paste it in there for you by:
1. Press the button, and to Y2=.
2. Press the button, and select 5:Statistics.
3. to highlight Eq, and hit to select 1:RegEQ.
4. The equation of your trend line should be in Y1, and the linear regression should be in Y2. Press to see the equation of both lines drawn on your scatter plot.
5. To see the graph of only one of the equations, press the , then and/or over the = and press the button to deselect the equation (the = should no longer be highlighted).
What is the slope of the linear regression?
Give the real-world meaning of this slope.
What is the y-intercept of the linear regression?
Give the real-world meaning of this y-intercept.
Use the regression equation to predict the height of a candle after it burns for 2¾ minutes.
Use the regression equation to predict the height of a candle after it burns for 7 minutes
Use the regression equation to predict how long it will take the candle to burn down to a height of 2 cm.
According to the regression equation, when will the candle be completely burned down?