Illuminations: Shedding Light on the Subject: Function Models of Light Decay

Shedding Light on the Subject: Function Models of Light Decay

 In this grades 9-14 lesson students develop and analyze exponential models for the behavior of light passing through water. Click here to go directly to the i-Math Investigation upon which it is based.

Learning Objectives

 Students will be able to: understand the exponential decay of light underwater develop exponential models in context solve simple recurrence relations (linear homogeneous first-order) conduct data analysis using semi-log plots

Materials

 Investigation Pages OR Access to Web site used in this lesson Optional Calculator Optional Equipment for Discrete Experiment: Tinted Plexiglas Squares (approximately ten 4" squares per group)Light Sensor (such as the one that comes with the TI CBL Light source (window, overhead, flashlight) Optional Equipment for Continuous Experiment: Tube with a clear bottom Light Sensor (such as the one that comes with the TI CBL Water source and liquid measurement Flashlight

Instructional Plan

II. Conducting the Lesson

Outline
B. Gathering Data
1. Simulated Underwater Dive
2. Discrete Experiment using Plexiglas
3. Continuous Experiment using Water
C. Analyzing the Data
1. Families of Functions
2. Discrete Models of Change (Recurrence Equations)
3. Continuous Models (Semi-log graphs)

A. Launch

Materials

• Handout Page A, Overhead transparency with Launch questions, or initial Web site conjecture page

• (Optional) Plexiglas squares and overhead

Conjectures

The first activity sheet provides an introduction of the investigation. To assess the students' prior knowledge of the situation to be modeled, have the students complete Sheet 1. Additionally, the absorption of light can be demonstrated by stacking layers of Plexiglas on an overhead projector. Ask the students what they observe as each layer is added. Alternatively, a flashlight in a dark room through a clear tube filled with water provides an excellent visual aid (see figure).

While students are completing the first question, make sure each student sketches a possible graph of the light intensity vs. depth. After most groups have finished discussing questions (a)-(e), have different groups sketch their graph for (a) on the board and explain their reasoning for the sketch. As a class, discuss the shapes of the graph. Students frequently conjecture that the graph is a line with negative slope or a parabola opening downward.

Focus students' attention to what can be said about the end behavior of the light intensity and the vertical intercept. One method of increasing students' attention to details on their conjectures is to indicate that light intensity is often measured in lumens and ask students to label both axes with appropriate units and scale. Another method is to ask students how the graph would change between a sunny day and a cloudy day?

Typical Student Conjectures

As a class, generate a list of items that influence the change in light intensity taking into account the different substances found in oceans. For (c), guide the students to the idea that the amount of light leaving a certain depth is dependent on the amount of light reaching that depth. This, in turn, leads to the discussion that the differences in the light intensity between two depths should be decreasing as the depth increases. Also, this discussion motivates why examining the relationship between the change in light intensity and the light intensity itself makes sense.

If possible, hold back distributing the remaining activity sheets. Give students a few minutes to discuss how they might create an experiment to measure the light intensity as it passes through various depths. This discussion will help to increase students' ownership of the experiments as well as making any comments on how to conduct the experiment more relevant.

 Guiding Questions • What traits do the conjectures have in common? [decreasing, always positive, approaching zero] • How do the conjectures differ? [x- and y-intercepts, the rate of change ] • How would your conjecture change from a sunny day to a cloudy day? [The initial light intensity would be less. The behavior of the function would be the same.] • What experiment could you conduct to test your conjecture? • What kind of function do you think might be used to model the light intensity?

Related Resources

Additional Uses for Plexiglas Rectangles - MIRAs

For the plexiglas experiment, students need 8-10 layers of tinted Plexiglas 1/8 to 1/4 inch thick which is available at most hardware stores and easily cut down to size. Four inch squares work nicely. If cut into larger 4" x 6" rectangles, then a Plexiglas rectangle can also be used as an inexpensive Mira which students can easily take home at night for use in Geometry when studying reflections.

False Bottom Tubes

One way of making a tube with a false clear bottom is to use a golf club tube and a clear colorless 35mm film case. Insert the film case into the bottom of the tube about 5 inches (see figure). The two items have the same diameter and form a tight seal together. Keep in mind that the light sensor is not water proof. To hold the light sensor in place, pack foam rubber around it and place inside the tube.

IV. Handouts

There are reproducible handouts for this lesson which appeared in the original Mathematics Teacher article:

Data Gathering and Analysis Sheets for the Discrete Plexiglas Experiment

For Launch (Handout A)

Have you ever noticed how the amount of light differs the further you are under water? Consider the environment of the dolphins pictured below and how the light intensity changes from near the surface to the bottom of the ocean.

Two friendly, but slightly shy, dolphins pose for a underwater snapshot

1. Based on the picture above, answer the following questions.

2. How does the light change as the depth increases? Sketch a possible graph of the (depthlight intensity)

3. What accounts for the change in the light intensity?

4. Suppose you are 10 feet under water, what environmental factors determine the light intensity at 10 feet? If you descend to 15 feet, what determines the light intensity at 15 feet? At 20 feet? Which of these factors remained constant and which changed?

5. If I(d) is the light intensity at a depth d, what does the quantity I(11) — I(10) represent?

6. How would the quantity I(12) — I(11) compare to I(11) — I(10)? What accounts for the difference?

For Plexiglas Experiment (Handout B)

Experiment 1

The goal of this experiment is to investigate how the intensity of light changes with depth. Have a steady light source such as the light from a window or from a flashlight. Connect a light sensor to a CBL. Do not connect the CBL to a graphing calculator. To take a reading press the mode button on the CBL. You should see the word "Sampling" flash on and off. When you are not taking a reading, press mode or ON to save power.

To model incremental depths, layers of tinted Plexiglas will be used as layers of water. Take a reading with no light on the sensor (cover the sensor with your hand). Take a reading directly from the light source. Add a layer of Plexiglas between the light source and the sensor. Record the new depth of 1 and the new reading. Repeat this process increasing the depth by 1 each time for as many layers as you can.

1. Record your data into a table with three columns like the one below.

 depth (d) Light Intensity I(d) I(d+1)—I(d)
1. Graph the light intensity as a function of depth. Do all the data points seem reasonable and follow a common curve? If not, are there data points you think should be removed? Explain how the graph compares to your conjecture in question 1 (a).

2. Using I(d) to represent the light intensity at the current layer of Plexiglas and I(d+1) to represent the light intensity at the next layer, calculate the difference in light intensity between consecutive layers of Plexiglas I(d+1)—I(d). Record the difference in the third column of your table.

3. Plot I(d+1)—I(d) against the depth d. Also plot I(d+1)—I(d) against the light intensity I(d). Explain why an equation of I(d+1)—I(d) vs. I(d ) may be easier to find than the other two.

4. Find an equation for I(d+1)—I(d) vs. I(d). Solve your expression for I(d+1). The resulting equation for I(d+1) is a recurrence relation. Make sense of your equation by explaining what each part of the equation represents and why the numbers are the size they are and why they have the sign they have.

5. Plot this recurrence relation on the same axes as the data. What initial value should be used for the recurrence relation? How does the model fit the data? What explanation can you give for any deviation?

6. Find a general expression for I(d) in terms of initial value I(0). Begin by using your recurrence relation to write I(1) in terms of the I(0). Express I(2) using I(1). Substitute your first expression for I(1) to express I(2) in terms of I(0). Express I(3) in terms of I(2) and then in terms of I(0).

7. Write a short summary explaining your answers to the following questions:

1. How does the light intensity relate to the number of layers of Plexiglas (depth)?

2. Given a set of data, in what ways could you use the data to test if a function similar to the light intensity function would model the data?

For Water Experiment (Handout C)

Experiment 2

1. Record and analyze your data to find a mathematical model for the light intensity.

 depth (d) Light Intensity I(d) I(d+1)—I(d)

1. Graph the light intensity as a function of depth. Do all the data points seem reasonable and follow a common curve? If not, are there data points you think should be removed? Explain how the graph compares to your conjecture in question 1 (a).

2. Using I(d) to represent the light intensity at the current depth of water and I(d+1) to represent the light intensity at the next layer, calculate the difference in light intensity between consecutive layers of water I(d+1)—I(d). Record the difference in the third column of your table.

3. Plot I(d+1)—I(d) against the depth d. Also plot I(d+1)—I(d) against the light intensity I(d). Explain why an equation of I(d+1)—I(d) vs. I(d ) may be easier to find than the other two.

4. Find an equation for I(d+1)—I(d) vs. I(d). Solve your expression for I(d+1). The resulting equation for I(d+1) is a recurrence relation. Make sense of your equation by explaining what each part of the equation represents and why the numbers are the size they are and why they have the sign they have.

5. Plot this recurrence relation on the same axes as the data. What initial value should be used for the recurrence relation? How does the model fit the data? What explanation can you give for any deviation?

6. Find a general expression for I(d) in terms of initial value I(0). Use your recurrence relation to write I(1) in terms of the I(0). Express I(2) using I(1). Substitute your first expression for I(1) to express I(2) in terms of I(0). Express I(3) in terms of I(2) and then in terms of I(0). Use these results to determine a formula for I d).

7. Write a short summary explaining your answers to the following questions.

1. How does the light intensity relate to the number of layers of water (depth)?

2. Given a set of data, in what ways could you use the data to test if a function similar to the light intensity function would model the data?

3. How are the methods and results of Experiment 1 and Experiment 2 similar, and how they are different?

1.

1. The light intensity decreases and asymptotically approaches the horizontal axis as the depth increases.

2. Answers can include the amount of substance in the water such as vegetation, sludge, waste, marine life, and the water's absorption of light.

3. At 10 feet, 15 feet, and 20 feet, some of the factors influencing the light intensity are the amount of cloud cover, vegetation at the surface of the water, microscopic organisms, and the absorption of light by the water above. Most of the factors remain fairly constant as the depth changes.

4. The quantity I(11)—I(10) is the change in light intensity between the depths of 10 feet and 11 feet.

5. The quantity I(12)—I(11) would be closer to zero than I(11)—I(10) since the light intensity at 12 feet is just a fraction of the intensity at 11 feet which itself is a fraction of the light intensity at 10 feet. The difference is a function of the light intensity at the previous depth of water.

2.

 depth d light intensity I(d) I(d+1)—I(d) 0 0.810 -0.338 1 0.472 -0.230 2 0.242 -0.088 3 0.154 -0.065 4 0.089 -0.035 5 0.054 -0.023 6 0.031 -0.014 7 0.017 -0.009 8 0.008

1.
2. See the table above.

3. The relationship between the change of intensity and light intensity is linear, which makes it easier to find an equation.

4. I(d+1)—I(d) = —0.43·I(d)+5.79¥10^-5. The y-intercept is relatively close to zero due to the fact that when light intensity is zero, the difference in light intensity is close to zero. Thus it can be neglected. This leaves the equation I(d+1) = 0.57·I(d). Since —0.43 is between —1 and 0, the size is the amount of light absorbed by the layers, and 0.57 is the amount of light remaining. The sign of —0.43 indicates that the difference in light intensities is decreasing.

5. The initial value should be the first light intensity reading, == 0.81. Deviations are due to fluctuations in the light intensity readings, and real-world data.

6. ,
,

7.
1. The light intensity I(d) is a decreasing exponential function which depends on the number of layers of Plexiglas or the depth of the water.
2. Given a set of data, you could plot the data, examine the end behavior of the plot, and check to see if it appears to have a horizontal asymptote. If so, then you can plot the change in consecutive data points against the original data to determine if there seems to be a linear relationship. If there is a linear relationship and the line that represents that relationship goes through the origin, then a function similar to the light intensity data function would model the data. Note: If the calculus extensions are used, students may use separation of variables to verify that a linear relationship between the rate of change of the data and the data reveals that the original data may be modeled by an exponential function.

3. (a)-(g) similar to 2 (a)-(g) above.

g. iii.

In Experiment 1 and Experiment 2 the methods of analyzing the data were the same since we examined the three plots and found a linear relationship between the change in light intensity, I(d+1)—I(d) and the light intensity, I(d). After finding the equation of a line fitting that relationship, we could express it as a recurrence relationship and develop an exponential function. The differences in the experiments included that Experiment 1 used discrete layers of Plexiglas while there is a continuum of water in the tube in Experiment 2. Readings with the column of water could be taken over different changes in depth.

Teacher Reflection

 Here are a few questions to ask yourself or discuss with a colleague during and after the lesson. • Did students develop an understanding for the exponential model beyond just a curve which fits the data?• Can the students illustrate this understanding by stating an assumption about the model, representing this assumption symbolically, and then illustrating the validity of the model using the data? • What connections were made by students in this investigation to science? within mathematics? • What mathematics content does the investigation cover to a sufficient depth? • What mathematics needs further exploration before moving to the next investigation?

NCTM Standards and Expectations

 Algebra 9-12Write equivalent forms of equations, inequalities, and systems of equations and solve them with fluency—mentally or with paper and pencil in simple cases and using technology in all cases. Interpret representations of functions of two variables. Use symbolic algebra to represent and explain mathematical relationships.

References

 Bradie, Brian. "Rate of Change of Exponential Functions: A Precalculus Perspective." Mathematics Teacher 91 (March 1998): 224-30, 237. Gordon, Howard R., "Can the Lambert-Beer law be applied to diffuse attenuation coefficient of ocean water?" Limnology and Oceanography 34 (August 1989):1389-1409.Iavorskii, B. Handbook of Physics. Moscow: Mir Publishers, 1980. Lykos, Peter. "The Beer-Lambert Law Revisited: A Development without Calculus." Journal of Chemical Education 69 (September 1992):730-732. Perovich, Donald K., "Observations of Ultraviolet Light Reflection and Transmission by First-Year Sea Ice." Geophysical Research Letters 22 (June 1995): 1349-1352. Ricci, Robert W., Mauri A. Ditzler, and Lisa P. Nestor. "Discovering the Beer-Lambert Law." Journal of Chemical Education 71 (November 1994):983-985.

5 periods

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