Shedding Light on the Subject

This lesson contains extension activities that can be used as follow-ups to the light intensity investigation. The extensions rely on exponential models, but each uses a different context.

In this part of the activity, the results obtained so far are re-examined or compared to other situations to enrich student understanding of the mathematics and contexts. While you may not investigate each of these items, read through the items to get a sense of the richness of this investigation and its possible uses in a wide range of classes.

 

Analyzing the Data

Often there is more than one method to analyze or build models for data. Each method provides different information and insight into the mathematics underlying the situation. Here are some questions to ask when doing data analysis.

• Would another person likely produce the same results as you using the method? Does the method generate consistent, reproducible results?
• How much is the method affected by small errors in the data?
• Does the method help to produce a model which helps to understand the physical situation?
• Does another method produce similar results?
• You may have observed (depth, change in light intensity) data also resembles an exponential function. Find a function that fits the change in light intensity as a function of depth.
• Some graphing calculators and spreadsheets have the capability of fitting an exponential function (usually called exponential regression). How do the results using such a tool compare to your results?
• The tools in question 6 sometime cannot process negative data such as the change in light intensity data (see question 5). What would you do in this situation?

 

What If…?

By asking "What if…" questions, the focus may shift to important components of the problem which students have not completely explored. Consider the following "what if" questions.

• What would happen to the results and analysis if darker plexiglas or a liquid which absorb more light were used?
• What happens to the behavior of the exponential function if the initial light intensity is increased? negative?
• The discrete analysis of the light intensity data yield a recurrence equation of the form Y_{n + 1} = aY_n, where the value of a was between 0 and 1. What happens to the exponential function if a = 1? …if a > 1? …less than 0? Describe a physical situation for each case, not necessarily involving light, where each value of a might arise.

 

Connections

The mathematics in this activity arises in other situations. Compare the exponential functions and methods to those in the Trout Pond investigation.

• How are the exponential models for the trout population different from the models for light intensity? What about the situations causes this difference?
• Both situations use a recurrence equation of the form Y_{n + 1} = a × Y_n + b. How did the values of a and b compare? How did these differences affect the exponential functions that resulted?
• You may have previously studied that the light intensity is inversely proportion to the square of the distance from the light source, or that I = k / d₂. In the case of light passing through water or plexiglas, you've found that the light intensity is an exponential function of the depth, or that I = a × r^d. Research why these two situations result in different functions.

 

Linearizing Data

Linearizing data is an important tool in developing mathematical models for data. This introduction to linearizing data is just the beginning.

• There are actually two methods of linearizing the data described. The use of logarithms was purposely chosen, assuming that the data was linear. Where else was a linear relationship used to produce a mathematical model for the data?
• The ratio of the next light intensity to the current light intensity was observed to be nearly constant which is also a type of linear equation. What are some reasons why this method of linearizing the data may not result in the "best fit" to the data?
• Using logarithms to linearize exponential data doesn't always work. Try linearizing the same data shifted up by 1.
• Consider the data from the previous question. The exponential function to model the new data is the old exponential function shifted up by 1. What happens if the difference equation method (NOW/NEXT) is used to analyze the data?