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)
Optional Equipment for Continuous Experiment:
Tube with a clear bottom
II. Conducting the Lesson Outline
A. Launching the problem
B. Gathering Data
1. Simulated Underwater Dive Discrete Experiment using Plexiglas Continuous Experiment using Water
C. Analyzing the Data
1. Families of Functions Discrete Models of Change (Recurrence Equations) Continuous Models (Semi-log graphs)
D. Students Reflecting on the Activity
E. Extensions
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?
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 underwater dive
applet .
(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.
(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.]
To Top of Section
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 consecutive 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 b
should be disregarded leaving the equation 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 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 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.
To Top of
Section
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 Web page: 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
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 I where I
is the light intensity. A review of exponential and logarithmic functions leads
to Lambert's law and the exponential decay function, m is
the slope of
the line found in part (c), m<0 , and
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 :
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
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
Based on the picture above, answer the following questions. How does the light change as the depth increases? Sketch a
possible graph of the (depth , light intensity )
What accounts for the change in the light intensity? 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? If I (d ) is the light intensity at a depth d , what does
the quantity I (11) — I (10) represent? How would the quantity I (12) — I (11) compare to I (11)
— I (10)? What accounts for the difference?
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.
Record your data into a table with three columns like the one below.
depth (d )
Light Intensity I (d )
I (d +1)—I (d )
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). 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. 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. 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.
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? 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). Write a short summary explaining your answers to the following questions:
How does the light intensity relate to the number of layers of Plexiglas
(depth)? 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?
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.
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 )
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). 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. 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. 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.
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? 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). Write a short summary explaining your answers to the following questions.
How does the light intensity relate to the number of layers of water
(depth)? 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? How are the methods and results of Experiment 1 and Experiment 2 similar,
and how they are different?
Answers to Handouts
1.
The light intensity decreases
and asymptotically approaches the horizontal axis as the depth
increases. Answers can include the amount of substance in the water such as
vegetation, sludge, waste, marine life, and the water's absorption of
light. 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. The quantity I (11)—I (10) is the change in light intensity
between the depths of 10 feet and 11 feet. 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
See the table above. The relationship between the change of intensity and light intensity is
linear, which makes it easier to find an equation. 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. The initial value should be the first light intensity reading,
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.
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.
