Illuminations: Shedding Light on the Subject

Shedding Light on the Subject

Light in the Ocean

In the first lesson, students make a conjecture about how the intensity of light changes as a function of the depth of the ocean. A video clip is used to prompt thinking, and students propose a graph that represents light intensity vs. depth.

Learning Objectives
 

Students will

  • Explore exponential models in context.

Materials
 
Computer with Internet Connection
Shedding the Light Introduction Overhead

Instructional Plan
The following video clip shows the initial few minutes of a classroom beginning the Shedding the Light investigation. (Note that the QuickTime plug‑in is required to view this movie, as well as the other clips in this unit.) Press the Play button to begin the movie.
 


Running Time - 01:25
File Size - 2.3 MB

Click here to view the transcript of the clip separately.

As shown in the video, the lesson is begun by asking students to think about the following situation:

Have you ever noticed how the amount of light differs the further you are underwater? Consider the environment of the dolphins pictured and how the light intensity changes from the surface of the water to the bottom of the ocean.
This situation and its associated questions can be displayed for students using the Shedding the Light Introduction overhead.

Shedding the Light Introducation Overhead Shedding the Light Introduction Overhead

After students have had a moment to think about the questions on the introductory overhead, ask, "How does the light intensity change as you go deeper in the ocean?" To help them think about this question, project the following movie on a television screen or overhead projector, or allow students to visit this page and watch the movie themselves.


Running Time - 00:39
File Size - 1.2 MB

 

Based on the picture from the overhead and the movie above, ask students to answer the Questions for Students in the section below. These questions can be presented orally to develop a whole-class discussion, or the Conjecture activity sheet can be distributed to students.

Conjecture Activity Sheet Conjecture Activity Sheet

Questions for Students
 

How does the light change as the depth increases? Sketch a possible graph of the (depth, light intensity) relationship that you described.

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

What accounts for the change in the light intensity?

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

Suppose you are 10 feet underwater. 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 remain constant and which change?

[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.]

If I(d) is the light intensity at depth d, what does the quantity I(11) ‑ I(10) represent?

[The quantity I(11) ‑ I(10) is the change in light intensity from 10 feet to 11 feet.]

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

[The quantity I(12) ‑ I(11) would be closer to zero than I(12) ‑ I(11) 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.]

Teacher Reflection
 
Typically, one of the most difficult components of this lab is the symbolic notation. Either function or subscript notation can be used. Students have difficulty with both notations.

All notations can be removed from the launch and developed as part of the lesson. From the plexiglas experiment or from questioning about how the light intensity is changing, the need to calculate and symbolically represent the difference is generated. Students can be asked to express the difference or change in light intensity and then share their notations. This process helps students to understand the notation the class selects as well as make connections between the physical situation and the notation. (The experience of teachers who have used this lesson is that students settle on a subscript notation after sharing their notations.)

Throughout this unit, function notation is used because it's the easiest to use on Web pages, and it most quickly communicates the mathematics of the investigation. However, this should not be taken as an indication that it is the notation that should be used with students or even that it is the best notation to use for this situation. During the classroom investigation, the launch should be left open, and the notation to be used should be left entirely to the students.

NCTM Standards and Expectations
 
Algebra 9-12
  1. Understand and perform transformations such as arithmetically combining, composing, and inverting commonly used functions, using technology to perform such operations on more-complicated symbolic expressions.
  2. Generalize patterns using explicitly defined and recursively defined functions.
  3. Analyze functions of one variable by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior.

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.
  
1 period   

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