Give brief intro to the topic and the task.  "Yesterday we took a look at some functions and their properties with interest rates, and we discovered how to make some easy money!  Today we're going to build on that principle but look at a different type of function.  But first THEN WHAT?

What kind of functions were we studying yesterday?

Exponential functions

And what is an exponential function? 

A function in which the variable is the exponent.

If you were looking at a graph of a function, how would you be able to tell it was exponential?

It starts off with a gentle growth and then rapidly grows quicker and quicker as the independent variable continues to grow. 

If you didn't have access to a pretty graph, but had tables of data points, such as time vs. number of cells in an unlimited growing environment, could you identify if you were looking at an exponential function?  How?

Function patterns- for every unit you add to the independent variable, you multiply the dependent variable by some constant. 

Good, it sounds like y'all have a handle on exponential equations.  Today we're going to throw a curve ball, an S-curve ball, in fact, into the mix. 

We hit on it just a bit yesterday, but why might an exponential function not do justice to an animal population function? 

There's not an unlimited source of food and resources.  We can't just keep multiplying forever with no consequences!

Is population a linear function?  Why or why not?  Has anyone ever heard of the type of function that is usually used to describe population growth? 

"Logistics Function" or "S-Curve".  (I don't expect them to know this yet, but I'm just checking.) 

Today we're going to talk about population, but we're going to get help seeing the type of function that population tends to follow by playing around first with a program on our calculators called NetLogo.  This will give us an idea of what a logistic equation looks likes, and how one is created.  Then we'll talk more about population, to figure out if a child born today is going to be able to survive until 110 in our ever-growing population! 

Before we launch in, can anyone think of why we care about population, when we're talking about longevity?

What do we know about overcrowding?  How does it affect us or our lives? 

When there are more people, like here in Austin, how can we tell?

If someone is sick in a room of 500 people, how is that different from them being sick in a room of 5 people?  (the room is the same size)

Not sure if they will know or notÉ

More traffic, more accidents, less resources, maybe less food, higher disease transmission rates, etc.

The crowded room will have a higher transmission rate

So do you think that population might have something to do with how long we each live?

Hopefully they'll say yes, as long as the population is sufficiently large (or small) enough to create real effects on daily lives.

The TI-83's and NetLogo hubs and cables should already be laid out for the students at their desks/tables.  Explain to the students how to get logged into the system, and walk them through the steps ALL AT ONCE.  This is key, otherwise the teacher will lose students.   

Students are getting logged into the NavNet system on the calculators.  

Explain the disease transmission simulation.  "This program allows us as a class to transmit disease to each other through the calculators.  We can play around with the different conditions, and then see what happens!  I want everyone to choose an icon and a color, and then locate yourselves by moving the arrows around on your calculators."

Does everyone see themselves up on the screen?  Does anyone need help?

They should be saying stuff like, "whoa, there I am!  That one's me!" and horsing around, chasing each other, etc. on the screen.

"Take just a couple of minutes and play around on this, so that when we have our tasks, you'll know how to move, etc."

Playing around with calculators, getting familiar with moving their icons.

"Now we're going to start the simulation."  START SIMULATION. " The turtles with the red dots are infected.  When they touch you, you become infected.  Try not to get sick.  GO!"

Trying to avoid the red diseased turtles.  Expressing lament when they are infected.

In real life, do we always know when someone is sick?  Do they walk around with a red dot hovering above them?

No!

Change the game so that the sick are invisible. Let's do this again, but this time you don't know who's sick.

Stop simulation.  Point to the graph that shows number of infected people. 

What do you notice about this graph of the number of sick people over time?

It's not straight.  It starts going up fast, and then levels off.

"Is this linear?  Why or why not?"

"No, it's not a straight line."

"Where is the area where the graph is climbing the most rapidly?"

In the middle

"right at first, is the number of infections rising fast or slow?"

slow

"What else do you notice about this graph?"

It looks like an S. 

What the Teacher Will Do

Eliciting Questions

Student Responses

Try to explain what's going on with the logistic curve, and why it's shaped that way.

"In the beginning, when only one person is infected, why is the graph not growing very fast?"

"because there's only one of us who can pass it on, so the transmission is slow."

Why does the growth become so rapid in the middle?

Because there are lots of sick people, so they can spread the disease to other sick people.

But why, then, doesn't the graph just keep going up, if the more people are sick, the faster the transmission rate?

Because you run out of people to infect!

That's right! there seems to be something limiting this graph just going off into infinity, and that is the number of people who are available to carry the disease. 

What does the beginning part of this curve look like?  Does anyone know?

An exponential function.

What do you think is going on up in this area (point to the leveling off point,) to keep the graph from skyrocketing?"

There are no more people to infect.

So what can you say about a logistics curve?  Someone please describe one for me.

It begins similar to an exponential function, in which growth is slow at first but becomes greater the greater value of the independent variable.  But at some point, the graph stops climbing as rapidly and levels off.  It looks like an S. 

Great!  Now that we've got it with disease transmission, let's test it out on population growth. 

Does anyone have any predictions about what will happen with population growth?  Who thinks it'll be linear?  Exponential?  Logistic?

Logistic! 

Go to http://www.otherwise.com/population/logistic.html a logistic population curve applet with fish (taking into account the carrying capacity).   Display this applet on the doc-cam.  Say, "I found a cool applet about population growth, which depends on the growth rate of the population, but it's fish.  Just imagine that these fish are people around the world!"

RUN THE APPLET at a birth rate of 2.0

What do you expect will happen when each fish replaces itself with two fish in the next generation?  (assume that males can reproduce too)

The graph will get larger quickly

Run the applet several times, with varying birth rates.

Watching, commenting.

Perform experiment 1, which varies the carrying capacity of the environment. 

What do you think will be the effect on the graph if we change the carrying capacity?  That's sortof like saying, "okay, our world can only hold 7 billion people, or 10 billion, or 50 billion!"

The time at which the graph levels off will be different

Does the overall shape of the graph change?

No, it's still an s-curve

So what's different about each of these curves?

The higher the carrying capacity, the later the graph will level off. 

Perform experiment 3, which deals with overpopulation.

What do you think will happen if our world has way too many people, and we exceed the human carrying capacity of earth?  Predictions?

Everyone will die.  Nobody will have more babies.  We'll level off.

Let's find out.  Here I am setting the birth rate at 3.  (the graph will overshoot the carrying capacity, but then fall below, only to overshoot, fall below, etc.  It will oscillate around the carrying capacity.)

Wow, this looks really strange.  Who has a guess as to what's going on here?

Once you exceed the carrying capacity, people are going to die off, and then the birth rate will overshoot again, and it'll repeat.

Describe and show what a logistic function looks like as an equation

Does anybody know how you might express a logistic function? 

I'm not expecting them to know this yet.

Go to http://www.answers.com/topic/logistic-function, where it shows a logistic function.  Also, go to http://www.census.gov/ipc/www/popclockworld.html, which gives a current estimate of the world population, which is currently at about 6.6 billion people.  

"People can use the logistic function to create models to estimate future human population.  The most famous of these models is called the "Verhulst's equation", but it involves some math that's beyond our scope right now.  A lot of you are going to run into this later in calculus."  Show it to the students. http://mathworld.wolfram.com/LogisticEquation.html

Wrap up what we discussed in relation to our unit question.

So what's the verdictÉ do you think people will still be around in 110 years, based on population models, so that someone might possibly live that long?

Yes/No

Why?

Because even if we overshoot the carrying capacity of the earth, the global population won't be decimated, it'll just recede a bit until it makes another surge past the carrying capacity.  

Ah ha! so global population alone won't keep up from reaching 110 years old.  That's good! now we can move on and focus on some choices we each make that can affect our individual lifespans!