Illuminations: Shedding Light on the Subject

Shedding Light on the Subject


Gather Data

To test their conjectures, students gather data using one of three different methods: using a simulated online dive; covering a tank with plexiglass to simulate ocean water; or, incrementally filling a column with water to simulate increasing depth.

Learning Objectives

 

Students will:

  • Explore exponential models in context.

Materials

 
Computer with Internet Connection
Shedding Light Experiment Packet

Instructional Plan

Three different experiments are described below.

Instructions and a set of questions for these experiments can be found in the Shedding Light Experiment packet.

Shedding Light Experiment Packet Shedding Light Experiment Packet

Choose one or more of these experiments for your students to conduct, and distibute the appropriate pages from the Experiment packet. If you do not wish to conduct any of these experiments, a set of sample data is available in the Questions for Students section.

Another set of sample data can be obtained from the Plexiglas video clip, which shows students conducting the experiment with PlexiGlas®.

 

Experiment 1 - Simulated Ocean Dive

Open the following online activity to simulate the light intensity experiment at various depths. The activity uses sliders to adjust the depth, and the darkness is automatically adjusted. This simulation can be used to gather data points, which can then be plotted and examined.

Simulated Dive Activity Simulated Dive Activity

The simulation can be projected onto an overhead screen or television as a demonstration for students, or you can allow students to visit this page and experience the simulation themselves.

 

Experiment 2 - Layers of Plexiglas®

The following video shows a class of students collecting data indoors, using layers of Plexiglas to simulate water depth. Adding layers of Plexiglas decreases the light intensity in a manner similar to water.


Running Time - 01:37
File Size - 3.8 MB

Click here to view the transcript of the clip separately.

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 to represent layers of water. Take a reading with no light on the sensor (cover the sensor with your hand). Take a second reading directly from the light source. Then, continually add layers of Plexiglas between the light source and the sensor. Record the depth and reading. (For this experiment, let the depth be equal to the number of layers of Plexiglas. For instance, if two layers of Plexiglas are used, record the depth as 2.)

 

Experiment 3 - Column of Water

The goal of this experiment is to investigate how the intensity of light changes with depth in water. Have students place a light source at the top of the tube. At the bottom of a tube, they should place a light sensor connected to a Calculator-Based Laboratory or other light sensing device.

Before recording any light intensity readings, allow students to choose a method that they will use to record the readings given by the CBL. They will need to decide what reading to record. Some possibilities are to note the maximum, minimum, average or most frequent reading.

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. Then, 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 collect the data from about ten readings. Note that the light source may initially overwhelm the sensor. Therefore, the first few readings may be questionable.

 

After conducting each of the experiment(s), students should answer the questions in the Experiment packet. The following grapher can be used to explore possible functions to represent the data.

Based on the data and the graphs, conclude the lesson with a whole-class discussion about the nature of the function. Allow students to speculate as to the type of function that best models the data, and ask them to justify their conjectures.

Questions for Students

 
Do all the data points seem reasonable and follow a common curve? If not, are there data points you think should be removed?
[Data points at either end of the range are more prone to errors, given the limitations of the sensor. A possible set of data is shown below.

 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

N/A

If students encounter data points that are clearly out of line with the rest of the points, they should be removed, or students should do that part of the experiment again to gather new data.]

How does the data that you collected either support or raise questions about your conjectures from the first lesson, Make a Conjecture?

[The decreasing nature of the data is confirmed. The data provides some insight into the rate of change which starts off large and negative, but then decreases in magnitude.]

Should the quantity I(d + 1) – I(d) be positive or negative? Explain your answer using both your observations about the light intensity and your graph of the light intensity data.

[The light intensity at a given depth will be less than the light intensity above it. Consequently, I(d + 1) – I(d) will be negative.]

Graph light intensity and the change in light intensity as functions of depth. Use the two graphs to write a descriptive paragraph about how the light intensity changes as the depth increases.

[The graph below left shows light intensity vs. depth, and the graph below right shows the change in light intensity vs. depth.
 

As the depth increases, the light intensity decreases. However, the change in intensity is not as large the deeper you go.]

What kind of function do you think might be used to model the light intensity data?

[Possible suggestions might be hyperbolic, exponential, or quadratic functions. In the next lesson of this unit, students will see that an exponential model is best. The purpose of this question is to remind students of the mathematical objective of the investigation as well as the science.]

NCTM Standards and Expectations

 
Algebra 9-12
  1. Use symbolic expressions, including iterative and recursive forms, to represent relationships arising from various contexts.
  2. Identify essential quantitative relationships in a situation and determine the class or classes of functions that might model the relationships.
  3. Draw reasonable conclusions about a situation being modeled.

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.
  
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NCTM Resources

Principles and Standards for School Mathematics


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