South Carolina ETV

South Carolina ETV
1101 George Rogers Boulevard
Columbia, SC 29201-4761
Phone: 803-737-3545

Buying Cars/Financing Cars Compound Interest (Grades 11-12)

Master Teacher

Larry Crosswell

Time Allotment

Two 90-minute class periods

Overview

Exponential functions have multiple real-world applications. Students typically study this topic at the junior or senior year of high school. This time frame corresponds with the time that they will be learning to drive. Exponential growth and decay is usually a topic that students have a hard time grasping in the abstract. This lesson will allow them to grasp a specific example of exponential growth and generalize their results.

Through this lesson, students will better understand how real world phenomena such as compound interest relate to exponential growth. This lesson should be taught after students have mastered the laws of exponents and understand what an exponential graph looks like.

Subject Matter

Mathematics

Learning Objectives

Students will be able to:

analyze a tabular representation of an exponential function
analyze a problem situation by formulating an equation
solve a rate of change problem

South Carolina Standards

These Standards can be found online at Office of Curriculum Standards.

II. B. 4. Solve exponential and logarithmic equations and inequalities using graphs, tables, and algebraic methods by using paper-and-pencil computations, graphing calculators, computer algebra systems, and spreadsheets.

II. B. 5. Analyze a situation modeled by an exponential or logarithmic function, formulate an equation or inequality, and solve the problem.

II. B. 6. Interpret rates of change as they apply to phenomena such as inflation, spread of disease, population growth, tax brackets, pollution, etc.

Media Components

Video

Exponential and Logarithmic Functions, Lesson 3: "Sprocket's Exponent." This series explores the use of exponential and logarithmic functions to simplify complex situations and predict outcomes.

Web Sites

New car prices, new car comparisons at Kelly Blue Book. This Web site has prices for all new cars on the US market. The Kelly Blue Book was founded in 1926 as a guide to used cars. In 1966 a New Car Price Manual was added.

Materials

Per student:

"A Penny Saved is a Penny Earned" (Activity Sheet 1)
"Let's Buy a Car" (Activity Sheet 2)
pen or pencil
paper
Graphing Calculator

Prep for Teachers:

Prior to teaching this lesson, bookmark the Web site used in the lesson on each computer in your classroom.
Make an overhead of "What Would You Want to be Paid?"
Preparation for the hands-on element:
Copy all the handouts for each student and cue the video to Lesson 3. You will want to START the tape for the students immediately after the discussion of the half-life of Ru₂₃₄ is finished. There will be a picture of the spaceship being swallowed by an amoeba and the captain will be saying, "Well, I'm afraid this is it, Sprocket!"
When using media, provide students with a Focus for Media Interaction, a specific task to complete and/or information to identify during or after viewing of video segments, Web sites, or other multimedia elements.

Introductory Activity

Step 1: Review with your students what an exponential function looks like using a graphing calculator and an overhead screen or TI Presenter. A good example to use would be .

Step 2: Split your class into small discussion groups using students who sit close together. (Note to Teacher: Do not move students across the room for this, it is a short activity.) On the screen, show the overhead that reads: "What Would You Want to be Paid?" Give the students in each group between one and two minutes to reach a decision. After the time is up, go around the room and ask each group what choice they picked and why they selected that choice. Validate all the students' reasons without revealing which choice was the wisest one.

Learning Activity

Step 1: Pass out Activity Sheet 1, "A Penny Saved Is a Penny Earned." Ask the students to answer questions 1 and 2. Give them four to seven minutes to come up with the answers.

Prepare the students to watch the video by giving a short introduction of what is going on in the video by saying the following:

"This video follows the exploits of a spaceship with a crew of two aliens. The captain of the ship is named Jim and the first officer is named Sprocket. This may sound like a certain TV show from the 60s, but you haven't seen anything yet!"

Step 2: Make sure that your VCR is ready to play the tape that you cued earlier. Provide your students with a Focus for Media Interaction, asking them to verify their answers to questions 1 and 2 from their worksheet. Push PLAY on the VCR. Push PAUSE when the commentator says, "Let's look for a pattern" and the screen shows "A=500+I." Check your students for comprehension. (The answer to question 1 is $535.00 and the answer to question 2 is $572.45.)

Step 3: Provide your students with a Focus for Media Interaction, asking them to observe and record the formula for annual compound interest as the answer to question 3 on their worksheet. PLAY the tape. PAUSE the tape after the commentator says "or simply put, A equals P times 1 plus i raised to the power of n" and the screen shows the formula "." Ask your students to use this formula to answer question 4 from their worksheet. Allow them between two to four minutes to do this calculation.

Provide your students with a Focus for Media Interaction, asking them to verify their answer for question 4. PLAY the tape. PAUSE the tape after the captain says, "A real steal Sprocket" and the screen shows the chessboard disappearing and Sprocket's face reappearing. Check for comprehension. (The answer to question 4 is $983.57)

Step 4: Provide your students with a Focus for Media Interaction, asking them to predict how much the captain would owe on his last payment when n = 63 (there are 64 squares on a chessboard). This is question 5 on the worksheet. Allow the students two to four minutes to come up with their answer. PLAY the tape. STOP the tape after Sprocket says, "90 quadrillion dollars," and you see the captain pass out. Check for comprehension. (The answer to question 5 is $92,233,720,368,547,758.08) Ask if any groups would like to change their pick from the Introduction Activity.

Culminating Activity

Step 1: Have the students pair up and hand out the "Let's Buy a Car!" worksheet. (Activity Sheet 2). Provide your students with a Focus for Media Interaction, telling them that they need to quickly decide on which car they would like to buy. Instruct the students that they have four minutes at most to decide on which kind of car they want to buy (question 1) and to tell you why in question 2. Monitor the students as they work through the worksheet. Allow 15-20 minutes for the students to fill in the table in question 3 and answer question 4.

Step 2: Have the students discuss the answer that they got for question 4. (Make sure that students realize that even though their monthly payments may be less, the total cost of the car may be more for some of the options. Ask the students which the best option is and why they think their answer is the best. There is no right answer to this question as long as the students have a reason for their answer.)

Step 3: As an assessment of the lesson, ask the students to write a short (1-2 pages) paper on items they think are important to watch out for when negotiating a car sale and why they think these items are important based on the worksheet and the computing of compound interest. Encourage students to show mathematical examples in their papers to help prove their points.

Cross-Curricular Extensions

History/Economics: Have students research how average car prices have changed over the last 50 years and correlate this to how the average salary for workers has changed over the same time period.

Does inflation affect both prices of cars and salaries the same?

Science: Have the students research the half-life of Carbon and why we use to date fossils.

Community Connections

Visit a local bank or car dealership. Find out how interest rates change. Discuss how car loans are financed. Determine what you believe the best way to finance a car is and prepare a written report with your findings.

Visit a local natural history museum and find out how carbon dating is done. Ask why carbon is used instead of other elements. Find out what the half-life of carbon is. Prepare an oral report for your class based on your visit.