Exponential Models

Introduction

Many phenomena can't be descripted or explained using algebraic models.

Exponential growth models are suitable to describe cases in which quantity increase dramatically, much more faster than in any other models.

Also exponential decay phenomena play an important role in real world: we bump into such examples in biology, chemistry, economics and so on.

We are going to explore the structure of exponential models and how exponential phenomena are relevat to our environment and to our lifes.

Task

Working in groups of three, you must collect, analyze and discuss information about exponential growth:

1. Analyze the main features of exponential functions and how they can be used to model growth phenomena

2. Browse the Internet and find examples of growth phenomena which can be modeled using exponential functions

3. Discuss whether found materials are relevant and publish your conclusion

Process

First step - Get acquainted with exponential functions

  1. watch this introductory video
  2. using GeoGebra, graph the exponential function y = a * (1 + r )^x; the parameters a and r must be defined such that they can vary
  3. analyze how the function changes depending on variation of a and r

  4. Answer thefollowing answers:

  • why do exponential functions grow faster than any polynomial function?

  • can you formalize this statement for an n-degree polynomial?

Second step - examine exponential growth in depth

  1. scrutinize the features of exponential growth phenomena related to environmental problems in this webpage.

  2. use this applet to explore different values assumed by an exponential function

  3. compare growth and decay problems using this applet
  4. Using this applet, visualize how the world population increases as it happens
  5. This applet shows how the growth rate of the world population varied across the centuries

Third step - discussion

Answer the following questions:

  1. What kind of model can be used in order to take into account variations of the growth rate?

  2. Can you explain the role of the asymptote of a decreasing exponential function?

  3. How can you relate continuous representations with intrinsically discrete phenomena?

Then publish your observations on our Padlet notice board and discuss them with other groups

Fourth step - work out an exponential problem

  1. Examine this problem, in which the growth of a fish population is analyzed.
  2. Explain how the phenomenon might be affected by a variation of the environmental constraints. Which external modifiers does the growth of the fish population depends on?

Fifth step - summarize outcomes

You are requested to write a concise HTML page summarizing what we have discovered:

  1. what are the main features of an exponential function and how can it be used to formalize phenomena?

  2. how does an exponential function vary depending on its parameters?

  3. what does each parameter mean in representing a real-life phenomenon?

  4. what are the outcomes of the experience developed in step 4?

  5. now browse the Internet on your own count: find and present an example of exponential decay.

  6. what can be observed about continuous and discrete phenomena and their representations?

A Symbaloo webMix might help to expand your perspective and to examine the issue in depth.

Then attach the presentation file to a post in the Exponential Models WebQuest Forum on our Moodle platform

Evaluation

Evaluation

Completeness is the the main requirement. All the questions must be properly answered.

Moreover answers must be clear and comprehensible. An adequate mathematical formalization of the problems will be appreciated.

Also original details will be taken into consideration.

The grade will be the same for each member of the group.

Two final products will be evaluated:

  1. Contributions to the Padlet notice board.

    1 to 30 points

  2. Summarizing presentations

    1 to 70 points

Each product will be graded 0 to 100 points, according to the following grid:

Content - Accuracy All content throughout the presentation is accurate. There are no factual errors. Most of the content is accurate but there is one piece of information that might be inaccurate. The content is generally accurate, but one piece of information is clearly flawed or inaccurate. Content is typically confusing or contains more than one factual error.
  60 40 20 0
Sequencing of Information Information is organized in a clear, logical way. It is easy to anticipate the type of material that might be on the next card. Most information is organized in a clear, logical way. One card or item of information seems out of place. Some information is logically sequenced. An occasional card or item of information seems out of place. There is no clear plan for the organization of information.
  25 12 7 0
Originality Presentation shows considerable originality and inventiveness. The content and ideas are presented in a unique and interesting way. Presentation shows some originality and inventiveness. The content and ideas are presented in an interesting way. Presentation shows an attempt at originality and inventiveness Presentation is a rehash of other people's ideas and/or graphics and shows very little attempt at original thought.
  25 12 7 0
Conclusion

Thanks for having participated to the Exponential Models WebQuest!

Modeling real-life phenomena are not a no-sweat work. Many components are to taken into account, in order to preserve information.

On the other hand, smplification is necessary since complexity might be overwhelming and preventing comprehension.

If we want to get reality phenomena, we need to model facts and data.

All your presentations will be shown and discussed in the classroom.

Teacher Page

Students:

This WebQuest was designed for an italian  3a Liceo Scientificodelle Scienze Applicate (approximatively 16-year old students)