Materials & Prep
Calculators, board or projector, and a one-page table with the worked examples and independent problems below. Prepare a simple two-column board display labeled “fixed-dollar growth” and “percentage-of-current-balance growth.” No special setup is needed.
Opening
DisplayA savings account starts with $1,000. In Account A, the bank adds exactly $100 at the end of each year. Ask students to predict the balance after years 1, 2, and 3. Students should produce $1,100, $1,200, and $1,300. Ask, “How much is added each year?” and “Would this make a straight-line graph?”
ConnectExplain that their reasoning is correct for a fixed-dollar increase: the same amount is added each period, so the balance rises in equal steps. Then introduce Account B: the bank adds 10% of the current balance each year. Ask students to predict whether the dollar amount added in year 2 will be less than, equal to, or greater than the amount added in year 1. Have students commit with a hand signal and briefly explain to a partner. Do not calculate yet. Emphasize that the bank is applying the same rate, but not necessarily adding the same number of dollars.
Direct instruction
ModelFor Account B, calculate the first four years aloud. Year 0: $1,000. Year 1: 10% of $1,000 is $100, so the balance is $1,100. Year 2: 10% of $1,100 is $110, so the balance is $1,210. Year 3: 10% of $1,210 is $121, so the balance is $1,331. Year 4: 10% of $1,331 is $133.10, so the balance is $1,464.10.
Think aloud“I do not keep adding $100, because the interest is calculated on the new balance. The common mistake is to treat a percentage rate like a fixed dollar deposit.” Display the comparison: fixed-dollar balances $1,000, $1,100, $1,200, $1,300, $1,400; compounded balances $1,000, $1,100, $1,210, $1,331, $1,464.10. Point out that the compounded graph begins at the same place and has the same first increase, then rises more sharply because each interest amount is added to the balance.
DefineCompounding means that earned interest remains in the account and earns interest in later periods. The calculation shortcut is A = P(1 + r)^t, where P is the initial deposit, r is the interest rate written as a decimal, t is the number of periods, and A is the final balance. Show that $1,000 at 10% for 4 years gives $1,000(1.10)^4 = $1,464.10.
Check for understandingAsk students to calculate or estimate the year 5 interest on the compounded account. The correct response is $146.41, not $100. Students explain why the added amount increased.
Guided practice
Worked exampleSolve $1,000 invested at 6% for 5, 10, and 20 years. Model the calculator entry and record the results: $1,000(1.06)^5 = $1,338.23; $1,000(1.06)^10 = $1,790.85; $1,000(1.06)^20 = $3,207.14. Compare these with simple fixed-dollar growth that adds $60 each year: $1,300, $1,600, and $2,200. Ask, “Why does the difference become larger at 20 years?” Require students to refer to interest earning interest, not merely to say that the account had more time.
ScaffoldGive students the organizer “P = __, r = __ as a decimal, t = __, expression = __, A = __, explanation = __.” Model filling in the first row, and allow students to use the displayed formula and a calculator. For students who confuse 6% with 6, have them write 6% = 0.06 before entering the expression.
Independent work or discussion
Students solve and explain these problems. They must show the expression, final balance rounded to the nearest cent, and one sentence explaining how compounding affects the result.
- Compare $500 invested at 5% for 2 years and 10 years. Use A = 500(1.05)^t. Results: $551.25 after 2 years and $814.45 after 10 years.
- A student deposits $1,200 at 4% and leaves it untouched. Find the balance after 15 years using A = 1,200(1.04)^15. Result: $2,161.13.
- Explain why $1,000 at 8% for 10 years does not equal $1,000 plus ten separate $80 additions. Calculate both values: compound growth gives $2,158.92, while fixed-dollar growth gives $1,800.
CirculateAsk, “What is the base for the next interest calculation?” and “Is your rate entered as a decimal?” Listen for students who still add the original interest each year. Have those students return to the year-by-year $1,000 example and identify the changing interest amounts.
ExtensionStudents determine when a $1,000 deposit at 6% first exceeds $2,000. They test whole-number years with a calculator and justify the answer: it exceeds $2,000 after 12 years because $1,000(1.06)^12 = $2,012.20, while after 11 years it is $1,898.99.
Closing
Exit ticketA $2,000 deposit earns 5% annually and is left untouched. Calculate the balance after 1 year and after 10 years using A = 2,000(1.05)^t. Then write two sentences: one comparing the dollar increase in year 1 with the later growth, and one explaining why the balance does not follow a straight line. Expected results are $2,100 after 1 year and $3,257.79 after 10 years. Collect responses to check that students connect the increasing interest amount to interest being earned on the growing balance.