Investing in Logarithms: A High School Project on Exponential Growth and Budgeting

1. Investing in Logarithms RebekahBrunton October 2011

2. Common Core Standards Mathematics: High School Algebra: • Write expressions in equivalent forms to solve problems 3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.★ c. Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be rewritten as (1.151/12)12t ≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. Mathematics: High School Functions: • Building Functions: 5. (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

3. Investing in Logarithms Project: Purpose:

4. Day 1 and Day 2 Expected Outcomes • Students are given a career with an estimated salary. • Students will record data of their monthly spending on the possessions with included interest, the house payment, electric and water bills, and groceries. _________________________________ • Monthly Income: $4000 • Monthly spending calculation of Possessions and Necessities: $ 2,350 • Yearly spending calculation of Possessions and Necessities: $28,200 Example: Career Salary: $48,000 Monthly Spending: 3 Possessions: • Car: $350 + (7.9%) interest rate • Boat: $280 + (8.2%) interest rate • Motorcycle: $290 + (8.6%) interest rate House Payment: $850 Electric and Water Bill: $230 Groceries: $350

5. Day 3 and Day 4 Expected Outcomes • Students calculate their yearly savings with the Compounded Interest Formula A = P(1 + r/n)^nt and Continuously Compounded Interest Formula A = Pe^rt) • Students will compare their income by purchasing all three items in one year or deciding to invest and save before making the three purchases. ______________________________ • Annual Income with savings: $49,373 • Annual Income with spending and zero saving: $ 19,800 Example of Student Savings or investment: • $900 invested in savings • With saving rate of 6.1% • For 12 months A = P(1 + r/n)^nt A = 900(1+(.061/12months))^(12months*1 year) Accumulated Saving: $956.46 • Without purchasing 3 possessions, student would be able to save $1820 per month. So, in a year, the student would be saving $21,840 and the accumulation of saving with interest would be… A = Pe^rt = 21,840e^(.061)( 1 year) Student would accumulate $23,213

6. Day 5 and Day6 Expected Outcomes • Students present a descriptive analysis of how the possessions will be paid for each month. • Students present their conclusion of their budgeted income and significance of spending. _________________________________ How the interest rate works: • Car: $350(.079) = $27.65 Therefore, the student is only paying $350 -$27.65 = $322.35 in a monthly payment. • $322.35(12 months) = $3,868.20 paid in one year. • At that rate, It would take 5.8 years to pay off the car alone. Conclusions: • Interest rates make payments grow exponentially. Possession Prices: Car: $ 22,500 + 7.9% interest Boat: $16,000 + 8.2% interest Motorcycle: $11,500 + 8.6% interest ___________________________ For the calculations of spending: The student borrowed $10,500 to pay for a car 72 months ago. The total amount paid for the car is now $22,000. What was the annual interest rate that the student paid? • The student would apply the Continuously Compounded Interest Formula A= Pe^rt • The represented problem would be: • 22,000 = 10,500e^r(6 years) • Ln2.0952 =r(6 years) • rate = 12.3% Conclusion: • The interest rates are important!

7. Investing in Logarithms Continued

8. The students will receive “x” out of 75 points. This score will be calculated with other assignments throughout the unit for a total of 110 points. Investing in Logarithms Continued Rubric for Presentation

9. Pacing Guide Investing in Logarithms: The project is an independent project with the listed activities as only 30 minutes of the 90 minutes of class time. The other 60 minutes is for the lessons. • Day 1: Students are given a career with an estimated salary. Research and record of the career and salary (5 points) • Day 2: Students will record data of their monthly spending on the possessions with included interest, the house payment, electric and water bills, and groceries. (10 points) • Day 3: Students record and calculate their yearly savings with the Continuously Compounded Interest Formula A = Pe^rt and Compounded Interest A = P(1 + r/n)^nt (5 points) • Day 4: Students will compare and record (in journal) their income by purchasing all three items in one year or deciding to invest and save before making the three purchases. (10 points) • Day 5: Work day and questions about the presentation. Participation (5 points) • Day 6: Presentations ( 75 points) • Day 7: Presentations (75 points) _______________________________________________________________________________ • Total amount of points for each student: 110 points • Journal: 30 points • Participation: 5 points • Presentation: 75 points