World Population Growth Project

Investigation Lesson

Name: Katy Monwai           

Title of lesson: Carrying Capacity

Length of lesson: 50 minutes

Description of the class:

                     Name of course: Pre-Calculus

                     Grade level: 10th-12th grades

                     Honors or regular: Either

Source of the lesson:

Java Web Simulation at:

http://aspire.cosmic-ray.org/javalabs/java12/SciNot/LinVsExp/student.html

TEKS addressed:

            (1) (E)  investigate continuity, end behavior, vertical and horizontal asymptotes, and limits and connect these characteristics to the graph of a function.

            (3) (A)  use functions such as logarithmic, exponential, trigonometric, polynomial, etc. to model real-life data;

            (3) (C)  use properties of functions to analyze and solve problems and make predictions;

I.      Overview
Students should appreciate the limitations of the earth to support an exponentially growing population and name this limitation the carrying capacity.  This gives them a conceptual understanding of the need for the logistic model of population growth.  It is also important for students to understand as they begin to think about the implications of such limitations and how resulting problems might be addressed. 
II.  Performance or learner outcomes

Students will be able to articulate the meaning of carrying capacity and how it affects a model of population growth.  They will understand that the world population cannot continue to grow exponentially as it has and will grasp the need for a new mathematical model to make predictions about future growth.

III. Resources, materials and supplies needed

            Computers or laptops for each student or for each couple of students

IV. Supplementary materials, handouts.

None


Five-E Organization

Teacher Does                    Probing Questions                    Student Does      

Engage:

We've shown that the world population has grown exponentially in recent history.  We've also seen that it hasn't always grown that way.  Today we want to consider if it will continue to grow this way.

Based on our exponential model, what will the world population be in the year 2050?  In 3000?  EtcÉ

Do you see this continuing indefinitely?  Why or why not?

Students will rummage through their notebooks and find their predictions from a couple of weeks ago.  Different groups may have slightly different models, but all answers should be of the same order of magnitude.

Students make predictions and explain their reasoning.  Hopefully, there will be different perspectives.

Explore:

We're going to get on the computers and look at a simulation of bacteria growth.  Of course, it's not the same as human growth, but we can talk about the similarities and differences.

Directs students to go to:

http://aspire.cosmicray.org

/javalabs/java12/SciNot

/LinVsExp/student.html

 

Now that everyone is there, let's scroll down to where it says "Lab Instructions" and "Interactive Lab".  Please read the "Lab Instructions" and see if you understand how to proceed.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Great.  Now, before everyone tries it, write down a guess of what you think the graph is going to look like on your own paper.  Once I've seen your prediction, you may begin experimenting with the simulation.  Try it a few times, varying the initial population, growth parameter, distance between data points, etc. 

 

So, what do we need to do first?

How do you think these two values correspond to our exponential growth equation?

Then what do we do?

Students get onto computers and find webpage.

Students read the instructions and look at the simulation.

Enter in a number for the initial population and the growth parameter.

Initial population =

Growth parameter =

Hit the "Start Experiment" button, collect the data, and graph it.

Students make a sketch predicting what they think the graph will look like.  Some might predict that it will continue exponentially, some might think it will hit a certain value and then dip down, others might see it going up and down, etc.  Then, they begin experimenting with the simulation.

Explain:

Now that everyone has played with it awhile, please share with your groups what you're finding.

 

Let's discuss as a class what we have seen.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This number is called the carrying capacity.  It means that the population of bacteria in this Petri dish cannot exceed this number because of limitations of the environment the bacteria are in.

What did your graphs look like?  How did they differ from your predictions?  How did they vary depending on your initial value, rate, and timing of your data collection?

Why do you think the graph flattens out like it does?  Was there a certain population value where it did this?  What is special about this number?

Students discuss and compare/contrast findings.

Students will mention the way the curve flattens off at the top.  This may or may not contrast with their predictions.  They will talk about how different graphs started at different places, grew faster or slower, took more or less time to get to the flat part, etc.

Students will notice that the poor bacteria just didn't have any room left to grow.  It levels out at 999,999.  They will guess that no more bacteria will fit in the dish after this.

Extend / Elaborate:

Ok, so how might this compare to human population growth?  What might limit the growth of the human population?  How do we know how big it can get?

Students will notice that while the earth is much bigger than a Petri dish, it still has a limited amount of space.  Hopefully they will also consider a limited amount of food, resources, etc.  Students might make predictions of how we can know what the carrying capacity of the earth is based on some of these limitations.

  Evaluate:

What have we learned about our exponential model of population growth?  What's wrong with it?

How are we going to have to modify our model to make more accurate predictions about the future?

Students will probably say "It's wrong!"  Or, more appropriately, that it only works for a certain portion of the domain.  It doesn't take into account the carrying capacity, or limitations, of the earth.

We'll have to include that slowing down and flattening  out at the end.