MELT Lesson Plan #3 – Exponential Equations
Introduction:
I wrote this lesson to be used in an Algebra 2 classroom or it could be modified and used in an Algebra I classroom. I chose this lesson because its topic is very relevant to the student’s lives and would therefore spark their interest. By the end of the lesson students should understand how a car depreciates exponentially versus linearly as well as how other variables affect the car’s value. This lesson could be taught before or after teaching exponential decay. It could be used as an introduction to the topic or after having taught how to solve exponential equations to really emphasize solving for time, a variable in the exponent.
Common Core State Standard:
F-LE.1 Distinguish between situations that can be modeled with linear functions
and with exponential functions.
a. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another
F-LE.4 For exponential models, express as a logarithm the solution to abct=d
where a, b, c and d are numbers and base b is 2, 10, or e; evaluate the
logarithm using technology.
Materials Needed:
Paper / pencil
Implementation:
Hand out the handout that is attached to groups of three or four students. Pose to them the question: How long should I keep a new car so that if I decide to trade it back in I break even, in other words the value of the car will equal the amount still owed. Have a discussion with the students as to what variables or assumptions that would be important for you to know in order to come up with a model. Examples include: what kind of car, how fast it depreciates, how many miles the driver puts on the car in one year, how much wear and tear is inflicted, was it bought new, how much did it originally cost, how much was put down originally on the car, how long the loan is, what the monthly payments are, if the car was involved in any wrecks, and if the car had regular upkeep such as oil changes and tire rotations, etc. Students may come up with more variables or ideas just be sure to direct them in the right directions of the ones that should affect their models the most or have them tell you. Next give your students some parameters for their car such as what kind of car, initial price, initial age, how much the down payment was and how long the loan was for. Students should assume that it was involved in no wrecks; its upkeep had been kept up with regularly, that an average amount of miles was put on the car in one year. Have students within their group come up with a model of how their car would depreciate in value each year as well as a model for how much a person would take off the amount owed on the car each year. Allow them to look up Kelly Blue book values to help with formulating a model. Once these models have been established have each group to share their models and discuss which one might be best. Be sure to emphasize that an exponential decay model would be much better than a linear one and why. After everyone has shared their models ask the class to come up with the best two. The last part of the modeling project is to take these two models and decide when the depreciation value will meet the amount still owed on the car, meaning the customer would break even. Once students feel they have come up with a reasonable estimate have them to share their answers and explain why they think so with the class. Hold one last reporting session where you discuss which group had the most reasonable estimate and why.
*** Handout is attached***
Handout: Value of a New Car
How long should I keep a new car so that if I decide to trade it back in I break even, in other words the value of the car will equal the amount still owed.
1. Come up with a model to represent the depreciation of a new car. Be sure to define your variables.
2. Come up with a model to represent how the principle of a car loan decreases every year. Be sure to define your variables.
3. Decide when the value of your model in #1 would equal the value of your model in #2