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Part B

Exploring Connections
  Introduction | Calculating Interest | A More Efficient Method | Interest Calculator | Reviewing Connections | Your Journal

  Now that we've found a formula, let's consider some investment options and the connections they prompt. Please keep track of your connections, either with pencil and paper or by creating a blank document in a word processor or note-taking program. You may also choose to use our worksheet (PDF). If you create your own document, label it "Connections in the interest rate activity." On your worksheet, make three columns with these headings:

1. Connections among mathematical ideas within this activity

2. Connections to other mathematical areas

3. Connections to contexts outside mathematics

Here is the problem: Central City Savings usually offers savings accounts at 3% interest, compounded annually. But in an effort to attract new customers, the bank has advertised the following promotion offers for savers who deposit $2,000 or more and keep the accounts at least 4 years.

Option 1: Option 2: Option 3: Option 4:
An extra $25 of principal, 3% interest, compounded annually A 2.5% interest rate, compounded quarterly A 3.25% interest rate, compounded annually A 2% interest rate, compounded continuously

1. Which offer is better?

Consider these scenarios: If you open an account with $2,000 and plan to keep it for 4 years, is Option 1 the better choice (an extra $25 of principal) or Option 3 (.25% more, for an interest rate of 3.25%)? What if you keep the account for 10 years -- would your answer be the same?

Now experiment with different amounts of principal and different time periods to develop an idea of when the $25 is of most importance and when the higher percentage rate has a larger impact.

Use your worksheet to record the connections you find in each category.

2. Compounding Interval

Experiment with differing compounding intervals. Think about the basic $2,000, 4-year account at 3% annual interest. What would happen if interest was compounded 4 times a year instead of annually? What about monthly? Which compounding interval results in a higher value at the end of 4 years?

Now take a few minutes to experiment with different parameters. Each time, try a different compounding interval, or a different interest rate, or a different total length of time. Try an interest rate that you now receive, an interest rate that you wish you receive, and -- if you're brave! -- a common credit card interest rate.

Use your worksheet to record the connections you find in each category.

3.Continuous Compounding

Some investments offer continuous compounding of interest. Consider Options 2 and 4. Which is a better option if you start with $2,000 and keep the account for 4 years?

To understand the concept of continuous compounding, you need to imagine ever smaller compounding intervals. Recall that we first worked with annual intervals, then with quarterly intervals -- but what about compounding every day, hour, second, or instant, when the interval is getting smaller and smaller? By introducing the concept of a mathematical limit, we get:

Continuous Compounding Intervals

Use your worksheet to record the connections you find in each category. Think particularly about how the ideas introduced earlier may help students understand the formula for continuous compounding.

Next  Review the connections you found

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