MA.912.FL.3.1

• After 12 quarters (3 years), $\mathrm{<span\; style="font-style:\; italic;"><span\; style="font-style:\; normal;">A</span></span>}$₁₂, would be 

$A$₁₂ = 100(1 + $\frac{\text{0.05}}{\text{4}}$)¹²

$A$₁₂ = 100(1 + $\frac{\text{0.05}}{\text{4}}$)⁴⁽³⁾

• Students should now see the pattern emerge for the compound interest formula. Have them replace the numbers in the equation to generate $A$ = $P$(1 + $\frac{\text{<math><mi>r</mi></math>}}{\text{<math><mi>n</mi></math>}}$)^$n$$t$
• Ask students how they would describe the difference between the simple and compound interest to someone who does not know (MTR.4.1). 
• Instruction guides students to consider the concept of continuous compounding by examining investment situations where compounding is done increasingly more often (MTR.5.1). 
  • The continuously compounded interest formula, $A$ = $P$$e$^$r$$t$, calculates the total value of an investment over time. Instead of calculating interest on a finite number of compounding periods (e.g., monthly, annually, daily), continuous compounding calculates interest assuming a constant compounding over an infinite number of compounding periods. Continuously compounded interest assumes interest is compounded and added to the amount accrued an infinite number of times. 
• Instruction includes defining continuous compounding. This includes, but is not limited to daily, weekly, monthly, quarterly, semiannually and annually.  
• Instruction includes comparison of simple, compound and continuously compounded interest using various representations, such as graphs, tables, equations and written description. Students will be guided to interpret key features of linear and exponential relationships in terms of context to connect to simple and compound interest problems. 
• Instruction includes the Annual Percentage Yield (AYP), $Y$ = (1 + $\frac{\text{<math><mi>r</mi></math>}}{\text{<math><mi>n</mi></math>}}$)^$n$ − 1. The Annual Percentage Yield allows you to compare compound interest rates with differing compounding periods. 
  • For example, a student may be asked to determine which a better investment is: 1.28% compounded weekly or 1.35% compounded monthly. 

The better investment would be compounded monthly at 1.35% 

• Instruction includes the discussion of Annual Percentage Rate (APR) – the total interest that will be paid in the year. If the interest is paid in smaller time increments, the APR will be divided up.

Common Misconceptions or Errors

• Many institutions advertise investments by using the annual percentage yield (APY) rather than the annual percentage rate (APR) since it is greater for investments that compound multiple times annually. Students should be careful when analyzing investments to see which percentage is being advertised. 
• For simple interest scenarios, students may forget to add the principal to the interest they calculate using the simple interest formula when calculating the total amount of an investment over time. 
• For compound interest scenarios, students may forget to subtract the principal from the total amount they calculate using a compound interest formula when calculating the total interest earned over time. 
• Students may confuse the compounding period variable with time variable. 
• Students may assume continuously compounding accounts will have a substantially larger returns, when in fact, the difference in the total interest earned through continuous compounding is not very high when compared to traditional compounding periods. 
• Students may incorrectly use the rate. 
  • For example, students may be given the rate of 4.2% and they use 4.2 instead of 0.042. To address this misconception, students should use multiple representations of percentages.

Instructional Tasks

• A principal investment of $5,000 had an interest rate of 3%. 
  • Part A. Complete the table below. 

• Part B. After 5 years, what is the best way to invest the money? 
• Part C. Which investment will have the greatest value in 15 years? Explain why. 
• Part D. Which investment will have the least value in 15 years? Explain why.

Instructional Items

• To the nearest cent, how much more does an investment of $3500 earn over 10 years, compounded continuously at 3.5%, than a $3500 investment over 10 years, compounded quarterly at 3.5%? 

• Which is the better financial investment: 3.85% compounded weekly or 3.74% compounded monthly?