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All Burned Out
Objective: The students will tie together their knowledge of linear and exponential functions. They will write equations of lines and geometric series. They will differentiate between discrete and continuous representations of data.

Prior Knowledge: The students should be able to write the equation of a trend line, enter data into a graphing calculator and graph the data and trend line. The student should be able to recognize and explain the difference between linear growth and exponential growth. The student will need to use the formula for geometric growth:
An = A1(r)n – 1
Time Required: 2 days. Parts One and Two should take one day, Part Three should be done on the second day.

Materials Needed:
  • Each group will need one uniformly cylindrical candle with a diameter of about 2 cm and a length of about 22 cm
  • a platform about 12 X 12 X 5 cm per group
  • a piece of tin foil long enough to catch the drips of candle wax
  • a centimeter ruler
  • a plastic knife
  • a clock that where seconds can be easily viewed
  • a graphing calculator
The teacher needs two lighters.
Group Size: 3 to 4 students per group

Procedure:
Part One
  1. Carefully cut any excess wax off of the tip of their candle so that the candle is perfectly cylindrical. Do not cut off the wick or the candle will be extremely difficult to light.


  2. Measure the starting length of the candle in cm and record the length in a chart like the one below.


  3. Place the candle horizontally on the platform with the ruler close at hand so that measurements may be taken quickly when necessary.


  4. The teacher should light the candle in its horizontal position when the group is ready to begin. Once the candle is lit it should be rolled back and forth gently so that it burns evenly. Be sure to note the time that the candle begins to burn. At each 30 second interval measure the length of the candle and record this data on the table.


  5. The last measurement should be taken at 180 seconds, and the candle should be immediately blown out.
    Time Elapsed (in seconds)Length (cm)Slope
    0  
    30  
    60  
    90  
    120  
    150  
    180  
  6. Calculate the slope between each data point and record this on the table.


  7. Using your data, write the equation of the trend line.


  8. Make a scatter plot of your data on a graphing calculator. Graph the equation you wrote to check it for accuracy. Make any necessary adjustments to your equation so that it is the line of best fit for the data.


  9. What is the real-world meaning of the slope? The y-intercept?


  10. Use the calculator to find the regression equation of the data. Which equation seems to better fit your data?


Part Two
  1. The wick on the bottom half of the candle should be exposed. Trim the originally used wick so that it is the same size as the newly exposed wick.


  2. Measure the length of the candle and record it in the table.


  3. Place the candle in the same horizontal position as in part one. The ruler still needs to be in close proximity.


  4. When the group is ready the teacher needs to simultaneously light both ends of the candle. Again, gently roll the candle back and forth so the ends burn evenly.


  5. Carefully measure the length of the candle every 30 seconds and record the data on the table as you did in part one.


  6. The last measurement should be taken at 180 seconds, and the candle should be immediately blown out.
    Time Elapsed (in seconds)Length (cm)Slope
    0  
    30  
    60  
    90  
    120  
    150  
    180  
  7. Calculate the slope between each data point and record this on the table.


  8. Using your data, write the equation of the trend line.


  9. Make a scatter plot of your data on a graphing calculator. Graph the equation you wrote to check it for accuracy. Make any necessary adjustments to your equation so that it is the line of best fit for the data.


  10. Use the calculator to find the regression equation of the data. Which equation seems to better fit your data?


  11. What is the real world meaning of the slope of the regression equation? The y-intercept?


  12. Compare the slopes of the regression equations found in part one and part two. Explain what caused the difference.


Part Three - Extrapolations
  1. Suppose the candle is cut in half and you now have 4 wicks to light. What would happen to the rate at which the entire candle’s length decreases?


  2. You now have one candle divided into two pieces. If you break each of the two pieces in half again, you will now have 8 wicks to light. Predict what will happen to the rate at which the entire candle’s length decreases.


  3. Use these extrapolations to complete a table like the table below:
    TrialNumber of WicksRate of Change
    11 
    22 
    34 
    48 
    5  
    6  
    7  
    8  
    9  
    10  
  4. Enter the data into your calculator.


  5. Make a scatter plot of Number of Wicks vs. Rate of Change. Describe the relationship between the points. Write an equation to fit the points and graph it to verify its accuracy.


  6. Make a scatter plot of Trial vs. Number of Wicks. Describe the relationship between the points. Is it the same as in #4?


  7. Using the formula for geometric growth, write the equation that best fits your data. Use your calculator to find the exponential regression for this data. Are the equations the same? Explain your answer.


  8. Make a scatter plot of Trial vs. Rate of Change. Describe the relationship between the points. Is this relationship more like the scatter plot in #4 or the scatter plot in # 5?


  9. Using the formula for geometric growth, write the equation the best fits your data. Graph your equation and verify its accuracy. Make any necessary changes to make it fit the points as well as possible.


  10. In Parts one and 2, the continuous graph of the equation is the better model for the situation. In Part 3, the discrete table of the data is the better model for the situation. Explain why.


  11. Suppose we did 100 trials, use your equations to predict each of the following:
    1. the number of wicks
    2. the rate of change of the candle’s length

  12. Suppose you cut the candle so that there were 131,072 wicks. Use your equations to predict each of the following:
    1. the number of trials
    2. the rate of change of the candle’s length.



C Gulliksen

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