LESSON PLAN 1
– Exponential Functions
Author: Tom Abraham
Title: Exponential Functions
TEKS: Algebra II b1 and f
IPGs: Matrix # 256 and 233
Length: 1 hour
Description of Class: Algebra II
Engagement:
One of the reasons we study meteorite impact craters is to determine how much damage an impact can cause. We need to be sure that any preparations we make will protect us from effects of an impact. What are other real-world instances where we would need to evaluate effects of impacts?
Exploration:
Let’s see if we can find a relationship between the speed of a particular asteroid and the amount of debris it will kick up when it impacts the earth. It is estimated that an asteroid named A48157 traveling to Earth at about 11,000 mph could displace nearly 10,000 tons of debris. But the damage it creates could significantly increase if it went faster and faster. See the table below:
|
Speed |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
|
Extra |
7 |
50 |
350 |
2500 |
17500 |
124000 |
880000 |
Use your TI-83’s and try to find an equation that fits this data. Use calculator’s table function to check your values. Students will experiment with different types of functions like linear and higher power polynomials. Let’s discuss what different types of functions you were able to come up with.
Explanation:
Today we’re going to talk about a new type of function. They’re exponential functions. What do you think an exponential function looks like? What is the keyword in the term exponential function that gives you a clue as to what the equation will look like? That’s right, the keyword is EXPONENT. What’s an exponent? What do you think the equation of an exponential function looks like? It has the form , where a is some constant. Knowing this, try to find an exponential equation that to fit the data in the table. Students should find an equation similar to . What properties do you notice about an exponential equation? Students should note that an exponential equation grows very quickly. Also, negative exponents yield numbers less than 1. Use the calculator’s table to how much debris would be ejected if the asteroid’s speed doubles?
Elaboration:
What are other instances where we would need to model things GROWING very quickly? Hopefully students will point out real world scenarios like population growth and interest rates. We use exponents to determine bacterial population growths and calculate how much money an investment can earn us over several years at a fixed interest rate. Be sure to explain to students some specifics of population growth and compounding interest.
Evaluation:
What are some properties of exponential functions?
How are they different from linear and quadratic functions?
Why would we need to use exponential functions?