Grade Level

9 — 14 (can be adapted easily for grades 7-8)

Estimated Time

4 — 6 Periods (of about 45 minutes)

• Launch - 1 period

• Data Collection - 1/2 period

• Activity - 3–5 periods

• Extensions - about 1 period each

Objectives

• To understand the exponential decay of light underwater.

• To develop exponential models in context.

• To solve simple recurrence relations (linear homogeneous first-order).

• To conduct data analysis using semi-log plots

Website(s)

http://illuminations.nctm.org/imath/912/light

On this site you can find resources for launching the problem including an underwater movie, conducting the experiments using a simulated underwater dive, and analyzing the data using a function grapher wither parameters, recurrence equations and semi-log graphs. The site includes an interactive graphing software or you can use a graphics calculator.

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

NCTM Standards

Algebra and Functions, Data Analysis and Probability, Connections, Representations

II. CONDUCTING THE LESSON

A. Launching the problem

B. Gathering Data

C. Analyzing the Data

D. Students Reflecting on the Activity

E. Extensions

A. Launch

Materials

• Handout Page A, Overhead transparency with Launch questions, or initial website 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?

To Top of Section

B. Gathering Data

Students can use one or more of the following methods to produce a data set. Alternatively, students can be given a data set or watch the video segment of students collecting Plexiglas data and record the data gathered in the video segment.

Simulated Underwater Dive: (Click here to go to this part of the investigation)

One method for producing a data set is to have students record depth and light intensity readings from the simulated underater dive applet.

Plexiglas Experiment: (Click here to go to this part of the investigation)

Before students collect any data, demonstrate collecting readings using the overhead and Plexiglas. Initially, the sensor is likely to be over-powered and give false readings. This provides an opportunity to indicate data points at the extreme ranges, i.e. near 0.08 or above 0.9 may be questionable.

Students should work in their groups to collect data using layers of tinted Plexiglas to represent the depths of water. The readings from the sensor may fluctuate widely after the addition of each layer of tinted Plexiglas. Students must choose a method for recording the light intensity readings. Algorithms that students have chosen include using the maximum, minimum, or average of the readings displayed on the CBL. Students should record the CBL readings of the light intensity and may enter them into their calculator after all the data is collected. One strength of this activity is that the CBL does not have to be connected to the calculator. No program needs to be downloaded and the equipment is easier to manage since it is not connected to a calculator. Since eight to ten readings are adequate, students can easily enter the data by-hand into their calculators. Students can be reminded of an appropriate range of values for the light readings. With the CBL and light sensor in its default settings, the best readings will fall between 0.008 and 0.91.

Direct sunlight or a high powered flashlight may over-power the light sensor resulting in inconsistent data. Thus, students may need additional layers of tinted Plexiglas to collect an adequate number of readings. If the data does seem to be unreasonable, you may encourage them to collect different readings using another light source or sunlight that is less intense.

Water Experiment: (Click here to go to this part of the investigation)

The procedures of this experiment follow in much the same manner as the Plexiglas experiment. In the student groups, one student records the data, another holds the tube steady, another fetches the water, and another handles the flashlight. In recording the data, students should maintain consistency by using their chosen algorithm to record the readings detected by the CBL light sensor. The analysis of the data is the same as before. Unlike the Plexiglas experiment, the column of water experiment can use variable depths. Some students will want to explore this feature. Since the data collection is typically less precise with the water experiment, you may want to reserve this experiment as a project or evaluation task for students.

Guiding Questions

• Given your current light intensity readings, what do you expect the next light intensity reading to be?

[This questions prepare students for the NOW-NEXT analysis that occurs with recurrence equations.]

• How does the data match the conjectures?

[Encourage students to try to answer this without graphing the data.]

• How do you think you can use the data to find a function that models the data?

[As students reach the end of the data collection process, remind them of the end objective: to find a meaningful model for the light intensity as a function of the depth.]

C. Analyzing the Data

Several different methods for analyzing the data exist. Three focus methods are presented here, but the methods can be adapted depending upon the students level or ability. For example, if students examine the ratio of the current light intensity to the previous light intensity, then the constant value is the coefficient of light absorbance or the base of the exponential function. Students in grades 7-8 thinking of "a percentage of the light absorbed" can develop the exponential models with minimal symbolic work. The methods presented here focus on the symbolic side of producing a mathematical model.

Families of Functions: (Click here to go to this part of the investigation)

Using an interactive grapher, students can attempt to fit a model to the data using sliders to adjust the parameters for a family of functions. This first stage allows students to explore the effects of the parameters on different functions and to focus on the traits of the data and characteristics of the functions which might suggest an appropriate function model. An important consequence of this exploration should be that finding a curve which fits the data is not sufficient.

Guiding Questions

• Did everyone use the same type of function? Why or why not?

• For those using the same type of function, was the values of the parameters the same? Why or why not?

• What does our function model tell us about how the light intensity changes as we move from one depth to the next?

• Given that we can us multiple types of functions to find a curve that resembles the data, what are reasons for selecting one type of function over another?

[behavior of the function (e.g. is asymptotic to 0), is reproducible by others, is reproducible using a variety of methods, is based upon scientific understanding of the situation]

Discrete Models of Change: (Click here to go to this part of the investigation)

One method of developing an exponential model is to generate the recurrence equation I (d+1) = k I (d ). As mentioned previously, a simple method is to examine the ratio of two consequtive light intensity readings to produce the value of k. A second method uses the observation that the amount of change depends upon the amount of light available. Plotting the change in light intensity against the light intensity produces a linear relationship.

Students should graph their data checking that all the points seem reasonable and follow a curve. As the students explore the plots, they should realize that the plot I(d+1)—I(d) vs. I(d) gives a linear relationship. If not all the points for I(d+1)—I(d) vs. I(d) follow a line, encourage them to disregard those few points that are outliers. Thus an equation for the difference in intensities I(d+1)—I(d) dependent on the light intensity may be found using linear regression on a graphing calculator or may be a point of review of algebra for the students.

The students should get an equation of the form . The y-intercept should be very close to zero. Students should be asked to explain why the line should pass through the origin. Their responses can include that when light intensity is zero, the change in light intensity should be zero since there is no light to absorb. Thus, b should be disregarded leaving the equation . Encourage the students to make sense of the equation by asking for the significance of the size and sign of the slope m . Since m is between (-1) and 0, the size of m indicates how much light is being absorbed, and the sign of m indicates that the difference in light intensity is decreasing. Thus, they should reason that the change in intensity of the light is a fraction of the intensity of the light entering the current layer of Plexiglas. Students can then solve their equation for I(d+1) and find a recurrence relation which can be used to approximate the light intensity at each layer of the tinted Plexiglas. Since m+1 is between 0 and 1, students should understand that the intensity of the light reaching the next depth is a fraction of the intensity entering the current depth. This recurrence equation may be plotted on graph paper or using the graphing calculator.

The recurrence relation can be used to generate an exponential function. The light intensity after the first layer can be written as . The light intensity at the second depth can expressed by . Students then can conjecture what the light intensity should be at layer d. Students can check to see how well their exponential function models their data by plotting the recurrence relation or exponential function on the same axes as the data. At this time, you may want to discuss Lambert’s law and the similarities and differences between the equation the students found and the equation given in Lambert’s law.

In a class discussion, have the students summarize the procedures which generated the exponential function emphasizing the difference in light intensity, the linear relationship between I(d+1)—I(d) and I(d), and the use of the recurrence relation to create the exponential function. Or, have the students write a short summary on the process explaining the methods used to generate the exponential function. Students need to include comments as to how well the exponential function fits the data and the reasons for any discrepancies.

Guiding Questions

• What does the value of k in the recurrence equation: I (d+1) = k I (d ) tell us about how the light intensity is changing?

• Given a recurrence equation: I (d+1) = k I (d ), what is the general solution to this recurrence equation? Give a proof of your general solution.

• What happens when the same method is applied to non-exponential data such as the quadratic data: (0, 0), (1, 1), (2, 4), (3, 9), (4, 16), ... or hyperbolic data (1, 1), (2, 1/2), (3, 1/3), (4, 1/4), (5, 1/5), ...?

Continuous Model: (Click here to go to this part of the investigation)

Unlike the discrete method above, in which students' do not need to know the underlying model, students' must have some experience with exponentials and logarithms before linearizing the data using semi-log graphs should be attempted.

The need for logarithms can be motivated by the question of "How else can we linearize the data?" which should prompt the need for an inverse function.

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D. Students Reflecting on the Activity

Upon completion of these experiments, students should be asked to write an individual or group report describing what they learned and questions that have been generated. In writing a paper, students formalize their understanding of the concepts and reflect on the way they came to understand the mathematics. Activities which encourage reflection allow students to analyze the development of their own mathematical ideas. Self-monitoring and evaluation of understanding are promoted. In addition, the instructor may use the papers to check each student’s understanding of the material. Students should also be asked to present their findings to the class either periodically or at the end of the investigation.

One possibility is to use one experiment for the class to discuss as a whole and then to use a second experiment to assess students' understanding. The design of the second experiment could be left fairly open-ended to allow students to think about issues of data collection such as sources of error and the impact of this issues the mathematical model.

Guiding Questions

• What evidence was the most convincing to you that the light intensity function was exponential?

• Which method of finding an exponential is most robust?

[Note: This question is easily an entire new investigation. However, students can still contemplate the answer in part based on the exploration so far.]

• What other kinds of data do you know to be exponential?

[Some population growths, compound interest, radio active decay, etc.]

• Create or find a data set for one of these situations. Use the methods of this investigation on the new data set.

To Top of Section

E. Extensions

Several possible extensions are given throughout the investigation and listed on the final webpage: Reflecting on your work. The extensions are organized connecting, reflecting upon and exploring further the methods of analysis.

The investigation can be extended to a more advanced mathematical setting involving calculus. Using that:

is the approximate rate of change, the students may replace I(d+1)—I(d) with DI(d) denoting an approximate to the derivative . An additional follow-up question as part of the introduction with Sheet 1 is to ask students to indicate what the results of parts (d) and (e) infer about the rate of change in the light intensity DI(d). At step (c) of Sheet 2 or 4, the concept of differential equations may be discussed as I(d+1)—I(d) = becomes . After the students have generated an exponential function using the recurrence relation, they may be guided to see that based on the differential equation , the derivative of an exponential function is a constant multiplied by the original exponential function. Euler’s Method may then be used to generate an approximation to the data based on the initial reading and the differential equation . Alternatively, separation of variables can be used to solve the differential equation. Upon separating the variables, the integral of is found to be the natural logarithm of I where I is the light intensity. A review of exponential and logarithmic functions leads to Lambert’s law and the exponential decay function, where m is the slope of the line found in part (c), m<0, and is the initial reading of light intensity.

3. TEACHER TO TEACHER

A. TEACHER REFLECTIONS

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?

• 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?

B. 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.

Additional Resources and 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.

IV. HANDOUTS

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

Launching Activity Sheet

Data Gathering and Analysis Sheets for the Discrete Plexiglas Experiment

Data Gathering and Analysis Sheets for the Continuous Water Experiment

Sample answers for these handouts

Alternative sheets are available through the student version of the i-math.

For Launch (Handout A)

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

1. Based on the picture above, answer the following questions.
   
   
1. How does the light change as the depth increases? Sketch a possible graph of the (depth, light intensity)
    
   
2. What accounts for the change in the light intensity?
    
3. 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?
    
4. If I(d) is the light intensity at a depth d, what does the quantity I(11) — I(10) represent?
    
5. 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.

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

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

The goal of this experiment is to investigate how the intensity of light changes with depth in water. At the bottom of a tube, place a light sensor connected to the CBL. Do not connect the CBL to a graphing calculator. Before recording any light intensity readings, choose a method you will use to record the readings given by the CBL. You will need to decide what reading to record. Some possibilities are to note the maximum, minimum, average or most frequent reading. Place a light source at the top of the tube. Take a reading with no water in the tube and no light in the tube (cover the top). Take a reading with no water but with your light source. Add a fixed amount of water of your choosing. Repeat to gather readings until the top of the tube is reached. You will want to gather about 10 readings. Your light source may initially overwhelm the sensor. The first couple readings may be questionable.

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

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.

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.

V. PRINTABLE PDF

This lesson plan has three Adobe Acrobat PDF files that can be downloaded and printed.

The lesson plan

Original Activity Sheets

Original "Mathematics Teacher" article