Why Is Alaska So Cool?
Diameter Detective and Area Analyzer
Activity 3: Diameter Detective and Area Analyzer
Investigations in Diameter and Area of Circles of Light
Part 1: Diameter Detective
Guiding Question
How does the diameter of a circle of light change in relationship to the distance from the light
source?
Prediction of results
Predict what you think will happen as the distance from the light source increases. Will the
diameter get larger or smaller? Will it increase or decrease at a steady rate or will the rate change
the further the paper is from the light source?
Objective
To find the diameter of circles of light formed by a flashlight at given distances from the light
source.
Materials
Regular style flashlight
Ruler measured in centimeters
Graph paper
A partner to help
Vocabulary Terms
diameter
direct variation
Procedure
Lay the flashlight on top of some books or other
surface that is about 10 cm high. The flashlight should
be secured on the surface so it is steady. Tape the
ruler on the table so that the 0 point is directly below
the front of the flashlight. Tape a piece of graph paper
to the front of a large stiff book or box. See the diagram to the right.
Stand the book directly in front of the flashlight and draw the circle of light at 0 cm from the
flashlight. Move the book with the graph paper 1 cm away from the paper and draw the circle
formed by the light. Move the graph paper to 2 cm, 3 cm, 4 cm, etc. and continue drawing the
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Diameter Detective and Area Analyzer
Why Is Alaska So Cool?
circles formed by the light until the circle is too large to fit on the paper. It may help to draw the
circles with different colored pencils. Hold the book as still as possible and draw as carefully as
possible to get accurate measurements. (It may be easier to tape the graph paper to a wall and
move the flashlight away from the paper instead of moving the paper away from the flashlight.
You want to be able to draw circles as accurately as possible.)
Record Data
Measure the diameter of each circle and complete the table below:
Distance from
light source in cm
0
1
2
Diameter of
Circle in cm
Mathematical Analysis
Examine the data visually. Do you see any patterns? Try finding the difference between
consecutive data points. Are they the same or similar? Remember that this is real data and there
will be variations from the inaccuracy of the measurements you take. Does it seem reasonable
that the data will continue in the same pattern?
Enter the data into an Excel spreadsheet and create a graph of the data showing the distance on
the x-axis and the diameter on the y-axis. What type of function does the data seem to represent?
Add a trendline or regression line to the graph. Show the equation and the r2 value on the graph.
Data Variance and Regression Equations
Note that Excel (or a TI-83 Calculator) will try to find a regression equation that fits the data as
exactly as possible. Your regression equation may have 4 or more decimal places in the numbers
and coefficients of the variables. But you should be aware that all data collected contains an error
of measurement. The error of measurement depends several things, including what type of
measuring tools you are using, how carefully you use them, and even, to some extent, the nature
of the experiment. For example, you probably found it easier to measure the circles that were
closer to the flashlight, since the light was brighter there and the edges of the light were easier to
see. As you got further from the source of the light, the circles of light were dimmer and it was
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Activity 3
Why Is Alaska So Cool?
Diameter Detective and Area Analyzer
harder to see, and thus measure accurately, the diameters and areas. That means that your data
may be more accurate at smaller distances than at larger distances. The regression equation is
calculated on the data you have rather than the data that might be “exact,” so the numbers in the
regression equation look more “exact” than they probably need to be. Try entering the data into
the TI-83 Calculator into List 1 and List 2, then plot the data. Round the numbers in the
regression equation to different precisions (to the nearest whole number and to the nearest tenth,
for example) and graph the equations on the data. Does it make a difference? You may round the
numbers in the equation to some appropriate value that you think is reasonable.
Predicting New Values
The regression equation can be used to predict new values of diameters that were not part of the
original data set. Try using the regression equation to predict a new value of a circle diameter
that lies at a distance between 0 and the maximum distance that you were able to measure. Does
it seem to be a reasonable value based on the data you collected? Now use the regression
equation to predict a new value of a circle diameter that lies at a distance that is greater than the
maximum distance you were able to measure. Does it seem to be a reasonable value? Sometimes
you will find that the regression equation is a better predictor of new values within the domain of
your original data set than for new values outside the domain of your original data set. The
regression equation has been calculated for the set of data points that you collected, so as long as
you are predicting values within the domain measured by those data points (interpolating), the
equation should be a good predictor of additional values. But the values you are predicting
outside of the original data set (extrapolating) may or may not follow the same pattern as the
original data set.
Results and Conclusions
Complete Part 1: Concluding Questions.
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Diameter Detective and Area Analyzer
Why Is Alaska So Cool?
Part 2: Area Analyzer
Guiding Question
How does the area of a circle of light change in relationship to the distance from the light source?
Prediction of results
Predict what you think will happen to the area of the circles of light as the distance from the light
source increases. Will the area of the circles get larger or smaller? Will they increase or decrease
at a steady rate? Will the change in area of the circles follow the same pattern as the change in
diameters that was found in Part I?
Objective
To find the area of circles of light formed by a flashlight at given distances from the light source.
Vocabulary Terms
area
Procedure
Use the same data from the Diameter Detective investigation to find the area of the circles.
Recall that the formula for the area of a circle is A = r2 where r is the radius of the circle.
You may wish to calculate the area of the circle by entering a formula in Excel. To do that, you
will need the column of data that includes the diameters of the circles. Suppose the diameters are
in column B, beginning with cell B2. Enter this formula into cell C2: =(B2/2)^2*PI( ). Press
Enter. You should see a value appear in cell C2. The screen should look similar to the one below,
but your values will be different.
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Diameter Detective and Area Analyzer
Now highlight cell C2, copy it, then highlight the rest of the cells in the column for area, and
press Enter. The formula should be copied into all the cells so that the area values are calculated.
Distance from
light source in cm
0
1
2
Area of Circle
in sq. cm
Mathematical Analysis
Examine the data visually. Do you see any patterns? Try finding the difference between
consecutive terms. Are they the same or similar? Remember that this is real data and there will
be variations from the inaccuracy of the measurements you take. Does it seem reasonable that the
data will continue in the same pattern?
Determining the Type of Regression
Enter the data into an Excel spreadsheet and create a graph of the data showing the distance on
the x-axis and the area on the y-axis. What type of function does the data seem to represent? If it
is not a straight line, try several types of regression equations. As you visually examine the line
or curve graphed for a particular type of regression, you should ask yourself several questions:
How closely do the data points lie to the graph? Do the points of which I am most confident are
correct (like the point at zero distance from the flashlight – the actual diameter of the flashlight)
lie on the graph? Are about the same number of data points above the graph as below the graph?
Or do the data points all begin to get further from the graph as the distance from the light source
increases? The equation of best fit should have about the same number of data points above as
below the graph. Note also that the r2 value, or correlation coefficient, represents how well the
data fits a particular type of regression equation, not which type of regression equation is best for
a particular set of data.
When you are confident that your equation is a “best fit” for the data you have, add that trendline
and regression equation to the graph.
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Diameter Detective and Area Analyzer
Why Is Alaska So Cool?
Part 1: Concluding Questions
Answer the following questions about your analysis and conclusions.
1. What type of regression does the data seem to fit?
2. What is the regression equation?
3. Is the equation linear? If so, describe what the values of a (or m) and b represent.
4. What is the r2 value?
5. What will happen to the circle diameters as the distance from the light source
increases?
6. Is the relationship of the diameters of the circles to the distance from the light source a
direct variation? Explain.
7. Use the regression equation to predict the diameters for two new distances from the light
source. Give the values and results here.
Part 2: Concluding Questions
Answer the following questions about your analysis and conclusions.
1. What type of regression does the data for areas of circles seem to fit?
2. What is the regression equation?
3. Is the equation linear? If not, describe the equation and what the coefficients might
represent.
4. What is the r2 value?
5. What will happen to the circle areas as the distance from the light source increases?
6. Is the relationship of the areas of circles to the distance from the light source a direct
variation? Explain.
7. Use the regression equation to predict the areas for two new distances from the light
source. Give the values and results here.
Print a copy of the Excel graphs for Parts 1 and 2 to include in your project report.
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Activity 3