There are three different ways of analyzing the light intensity data:
- Families of Functions
- Difference Equations
- Linearizing the Data
Students can explore one or more of these methods in this lesson.
Families of Functions
Use sliders to change the values of parameters a, b and c in different functions such as y = ax + b to fit the function. Choose from the existing functions, or enter one of your own.
- Enter your own data or use the sample data provided.
- Choose a function or enter your own function in the function window.
- Use the sliders to change the values of the parameters a, b and c in the function.
- As students investigate various functions, ask them to think about the questions below.
You can also change the range of values for each slider.
The Online Graphing Tool, which is a general application of the grapher above, can be used by students to investigate the data and find a function to fit the curve. However, the version available to students does not contain the data or pre-loaded functions. (To use this tool with pre-loaded values, you can have students access this page and use the version of the grapher above. Of course, that means that students would have access to the questions and answers below.)
Online Graphing Tool
Alternatively, students could enter the data into a hand-held or online graphic calculator to find a function that models the data.
For the function chosen by students have them answer the following questions.
- For what values of a, b and c does the function appear to best fit the data?
[The value of a should be near the initial light intensity. For the sample data, a ≈ 0.8. The value of b should be near the coefficient of light absorbtion, sob ≈ 0.6. Finally, c = 0, since the data is asymptotic to the x‑axis.]
- How do the values of a, b and c affect the shape of the graph?
[The value of a affects the y‑intercept, b affects the bend in the curve, and c affects the vertical shift of the graph.]
- Is there another type of function which fits the data fairly well?
[While a hyperbolic function can come close, the best fit is with an exponential function.]
- What characteristics of the data or the function support your decision on the best function to use?
[Students will likely refer to the fact that other functions tend not to change quickly enough. Also, they may observe the existence of a horizontal asymptote and the lack of a vertical asymptote as being important in their decision. ]
Difference Equations
Both graphs of the data (Depth, Light Intensity) and (Depth, Change in Light Intensity) have similar shapes. This fact suggests that there may be an important relationship between the Light Intensity and the Change in Light Intensity.
For this investigation, students should use data with uniform changes in depths, such as 0, 1, 2, 3, …. For data with uniform depth intervals, they can use the rate of change in light intensity, rather than light intensity itself.
To model the situation, graph the data for (Light Intensity, Change in Light Intensity).
- What kind of relationship do you observe between the Light Intensity and the Change in Light Intensity?
[The relationship between the light intensity and change in light intensity is linear, which makes it easier to find an equation.]
- Write an equation whose graph fits the data.
[I(d + 1) – I(d) = -0.43 × I(d) + 5.79 × 10^{-5}.]
- Examine your equation. Explain why the constants in your equation make sense. For example, what should the change in light intensity be when there is no light? How does your equation support the interpretation that a percentage of the light is absorbed?
[The y-intercept is relatively close to zero because when the 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. ]
Students can use a graphing calculator, computer software, or the Online Graphing Tool.
If the change in depth is consistently one unit, then the change in light intensity can be written as I(d + 1) – I(d). For the sample data, the relationship between the light intensity and change in light intensity can be written as:
I(d + 1) – I(d) = –0.4 I(d).
This equation is called a difference equation because it expresses the difference between two consecutive data points in terms of the value of one of the data points.
Knowing the initial light intensity, I(0) = 0.810, future light intensity values can be determined if this equation is solved for I(d + 1):
I(d + 1) = 0.6 I(d)
For example, the light intensity at depth 1 can be found by calculating I(1) = 0.6 I(0) = 0.486. Then, I(2) can be determined from I(1).
The value of 0.6 can be interpreted as the percentage of light that remains after a one-unit change in depth. In this case, 60% of the light remains after one unit of change . The value 0.6 is the light transmittance constant for this particular data set. For example, Plexiglas^{®} is often rated by its transmittance factor.
The two equations together are considered recursive equations describing the data:
I(0) = 0.810
I(d + 1) = 0.6 I(d)
The work above suggests a way of finding a function that expresses the light intensity using only constants and the depth. Such a function is called a closed-form solution to the recurrence equation or difference equation.
In the case of the recurrence equation above, the solution is as simple as observing what happens to the result after each application of the equation:
I(0) = 0.810 I(1) = 0.6 I(0) = 0.6 × 0.810 I(2) = 0.6 I(1) = 0.6 × (0.6 × 0.810) = 0.6^{2} × 0.810 I(3) = 0.6 I(2) = 0.6 × (0.6^{2} * 0.810) = 0.6^{3} × 0.810 |
Continuing this pattern, the light intensity at any depth d can be found using the following equation:
I(d) = 0.6^{d} × 0.810
Using this method of analysis, an exponential function can be used to model the data.
Students can verify the validity of this equation in at least two different ways:
After verifying their results, students should think about the questions below.
- This method of analysis resulted in a recurrence equation of the form I(d + 1) = r × I(d). What interpretation can you give to the value of r?
[The value of r is the percent of light transmitted.]
- How are the absorption factor and the transmittance factor related? Show this using the difference and recurrence equations using a and r for the different factors.
[The absorption factor, k, and the transmittance factor, r, are related by the equation k = r – 1.]
- What is the long term behavior of an exponential function? How does this behavior match with the conjectures you made in previous lessons?
[The long-term behavior of the exponential function, I(d) = ar^{d}, where 0 < r 1 , approaches 0. This matches the conjecture that the light intensity should be nearly 0 (very dark) for great depths.]
- Use the properties of exponents to show that for an exponential function of the form y(x) = ab^{x}, the ratio of two consecutive light intensity readings at uniform change in depths is always a constant. That is, show that y(d + 1) ÷ y(d) is always a constant.
[y(d + 1) ÷ y(d) = (ab^{(d + 1)}) ÷ (ab^{d}) = (ab^{d} × b) ÷ (ab^{d}) = b.]
- How could the results from the previous question be used to develop another method for coming up with the recurrence equation for a set of data believed to be exponential? Try your method on your data set or the sample data set. What issues arise?
[The ratio between values should be constant. Of course, due to measurement error, the data will likely not give the exact same value each time. Consequently, the mean of these ratios could be used as the light transmittance factor in the recurrence equations.]
Linearizing the Data
Visually, the difference between linear and nonlinear data is often easier to see than to tell what kind of function might fit nonlinear data. For example, in analyzing the data using difference equations, a linear relationship between the change in light intensity and the light intensity was useful to find an exponential model for the data.
In this part of the investigation, students can explore another method of linearizing exponential data. Having more than one method provides additional support for using an exponential model to fit the data.
Before continuing on, recall the inverse function for an exponential function. Start with the exponential function y = 10^{x}. If you know the value of y, how can you find the value of x?
For example, if y = 100 = 10^{2}, then x = 2. The inverse function of an exponential function is called a logarithmic function and satisfies the following relationship:
log(y) = x if and only if y = 10^{x}.
The above equation is given for the exponential function with base 10. More generally, for an exponential function with base a,
log_{a}(y) = x if and only if y = a^{x}.
You will also need two of the basic properties of logarithms:
log(a × b) = log(a) + log(b)
log(a^{x}) = log(a) × x
Using the details given above, allow students to use their knowledge of logarithms to answer the following items.
- Give a justification for the two properties of logarithms using the fundamental relationship between exponential and logarithmic functions.
- Provide justifications for each of the following steps:
| y = a × b^{x} log(y) = log(a × b^{x}) log(y) = log(a) + log(b^{x}) log(y) = log(a) + log(b) × x |
- Explain why the last equation in the previous question expresses a linear relationship between log(y) and x.
[Both log(a) and log(b) are constants, so the equation can be rewritten as z = constant + constant × x.]
In the case of the light intensity data, (d, I), a reasonable conjecture is that the light intensity is an exponential function of the depth. Therefore, a linear relationship between the log of the light intensity and the depth, (d, log(I)) linearizes the exponential data. Using their knowledge of logarithms, allow students to answer the following questions.
- Add a new column to your data set for log_{10} of your light intensity values, log(I). Graph log(I) versus depth. Find a linear equation which fits this new data. Be sure to use the appropriate variables of log(I) and d in the equation.
- Solve your equation from above for log(I). Explain your method.
[I(d) = 10^{(-0.243d + -0.087)} = 10^{(-0.243d)} × 10^{-0.087} = (10^{-0.243})^{d}) × 0.818 = 0.571^{d} × 0.818.]
- Graph the original (depth, light intensity) data with the reulting function from the question above. How well does the model fit the data?
[The resulting exponential model fits the data relatively well.Notice that the model appears to pass through all but one of the data points.]
- Examine the work that you have done. Notice that logarithms base 10 were used. What happens if you use natural logarithms, ln(x), instead of base-10 logarithms in the solution?
[The base b of the logarithm changes the value of the slope and intercept of the linear relationship. However, since the initial value and base of the exponential are found by exponentiation (with the same base b) of the intercept and slope, the base b of the logarithm does not alter the solution.]
References
- Keller, Brian A. (1998). "Shedding Light on the Subject." Mathematics Teacher 91 (9) (December 1998): 756-771.
- 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.
- Experience the Deep (1998). Wilderness Films, Madacy Entertainment Group, PO Box 1445, St. Laurent, Quebec, Canada H4L 4Z1.
- Classroom video segments, Heather Thompson and Brian Keller (2000), Department of Mathematics, Iowa State University, Ames, Iowa 50011.
Learning Objectives
Students will:
- Explore exponential models in context.
NCTM Standards and Expectations
- Generalize patterns using explicitly defined and recursively defined functions.
- Analyze functions of one variable by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior.
Common Core State Standards – Mathematics
Grade 1, Measurement & Data
- CCSS.Math.Content.1.MD.C.4
Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.
Grade 8, Stats & Probability
- CCSS.Math.Content.8.SP.A.1
Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.
Grade 8, Stats & Probability
- CCSS.Math.Content.8.SP.A.2
Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.
Grade 8, Stats & Probability
- CCSS.Math.Content.8.SP.A.3
Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.
Common Core State Standards – Practice
- CCSS.Math.Practice.MP1
Make sense of problems and persevere in solving them.
- CCSS.Math.Practice.MP4
Model with mathematics.
- CCSS.Math.Practice.MP6
Attend to precision.