LESSON PLAN # 2

LESSON PLAN # 2

Name(s): Christie Anderson

Title of lesson: Exponential Growth and Exponential Decay

Length of lesson: 50 minutes

Description of the class:

Name of course: Algebra II

Grade level: 10^th &11^th

Source of the lesson:

http://jwilson.coe.uga.edu/emt669/Student.Folders/Carlisle.Jody/lesson%20plan/unit.lesson.plans.html

and

http://www.physics.uoguelph.ca/tutorials/exp/intro.html

TEKS addressed:

(b) Foundations for functions: knowledge and skills and performance descriptions.

(1) The student uses properties and attributes of functions and applies functions to problem situations. Following are performance descriptions.

(1) The student connects algebraic and geometric representations of functions. Following are performance descriptions.

(A) The student identifies and sketches graphs of parent functions, including linear (y = x), quadratic (y = x²), square root (y = x^(1/2)), inverse (y = 1/x), exponential (y = a^x), and logarithmic (y = log_ax) functions.

(f) Exponential and logarithmic functions: knowledge and skills and performance descriptions. The student formulates equations and inequalities based on exponential and logarithmic functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. Following are performance descriptions.

(5) The student analyzes a situation modeled by an exponential function, formulates an equation or inequality, and solves the problem.

I. Overview

Math is related to the real world, and students can appreciate the application of the activity that involves both graphs and equations. The data is used in both explanation and extension of exponential functions.

II. Performance or learner outcomes

Students will be able to:

1) graph exponential functions and evaluate exponential expressions.
2) use exponential functions to model real life situations such as the depreciation of a cars value.

III. Resources, materials and supplies needed

· Calculators

IV. Supplementary materials, handouts.

· Homework worksheets

Five-E Organization

Teacher Does Probing Questions/Remarks Student Does/Says

Engage:

On the board when the students enter the room are these two graphs:

These graphs are representations of exponential functions. The equation looks like f(x)= A^x where A is any real number greater than 1.

Questions to be asked:
Which one of these graphs represents exponential decay? Why? Any other suggestions? What is the domain? Range? How did you get that?

Explore:

A single equation can be used to solve all problems involving this type of change:

where 'N' is the number in the population after a time 't', '' is the initial number, 'k' is the growth constant (if positive) or the decay constant (if negative), and 'e' is the base of natural logarithms (approximately 2.71828).

Suppose some environmental stress reduced a population of 1000 wee beasties to 800 in two days. How many will there be 7 days after the initial count of 1000 wee beasties?

This problem must be done in two steps. First, we use the information about the first 2 days to find the decay constant, 'k'. Second, we use 'k' and the time t = 7 days, and the initial population to find the final population.

For the first step, the logarithmic form of the equation is most useful. We know '' (the initial population was 1000), 'N' (the final population was 800), 't' (the time period was 2 days). Substituting into the second equation, we get

So our decay constant is k = -0.112 day.

Now we can do the second step. This time, the first equation (the exponential form of the equation) will be easier. Substituting k = -0.112 day, t = 7 day, and = 1000, we get

You should be able to get N = 457 wee beasties after 7 days.

What can we do to determine how many will there be in 7 days?

Again what does stand for?

What would our decay constant be?

Explanation:

There are many situations where the increase or decrease of some variable in a fixed time interval will be proportional to the magnitude of the variable at the beginning of that time interval.

For example, let's look at a population of wee beasties which increases by 10% per year. If there were 100 wee beasties now, there would be an increase of 10 wee beasties after a year. We would see an increase of 500 wee beasties in a year when there were 5000 at the beginning.

Likewise, we can look at a population which decreases by 50% (i.e. a decrease to 1/2, or by a factor of 2) every day. A population of 100 would be down to 50 a day later, and a population of 5000 would drop to 2500 after one day.

What would the increase of wee beasties be per year with an increase of 10% and initial of 100 wee beasties?

Elaborate:

Recall:

We can re-write this equation in another convenient form. Dividing the equation by , and then taking the natural logarithms of both sides, we get

Note that in this form, we do not need to know the absolute values of 'N' or ''; all we need to know here is the ratio of these two values.

Students manipulate the equation as the teachers demonstrate.

Evaluate:

Students will have graphing calculators

Questions to be asked:
Why do you think A can not be 1? 0?

Why do you think A can not be a negative number?

The above graph shows -A^x but the graph of (-A)^x is only points on the graph.

Everyone put these equations into the y= screen: y₁= 2^x y²= -2^x y³= (-2)^x
( you will need to use the carrot button). Now go to the table of values. What is different about y₁ and y₂? How is y³ different from y₁ or y₂?
If you have y = -2^x and let x = 2 then what is y? y = (-1)2*2 = -4.
What about y = (-2)^x and let x = 2 then what is y? y = (-2)(-2) = 4.

Students use their graphing calculators to explore the questions.