Compound Interest LAUNCH (7 MIN) Before How can lines on a graph show the effect of interest rates on savings accounts? During How can you tell what the graph of simple interest looks like? After What would you advise your friend to do? PART 1 (7 MIN) During the Intro When is compound interest like simple interest? Why does the amount of interest increase over time? Javier Says (Screen 2) Use the Javier Says button to point out the difference between simple and compound interest. Why does the amount of interest increase if the rate is constant? PART 2 (7 MIN) During the Intro How is the Compound Interest Formula similar to what you learned about simple interest, for n 1? Javier Says (Screen 1) Use the Javier Says button to engage students in a discussion about financial planning. Contrast taking out the money sooner versus leaving it in the account and letting the interest compound as time passes. What do you observe about the pace at which the balance changes over time? What variables will have the same value for all three parts of the problem? After solving the equation How is using a formula more efficient than calculating each part of a table? KEY CONCEPT (4 MIN) Explain that the number of interest periods is not the number of years. Similarly, the interest rate r refers to the rate for each interest period. The value of r for an interest period of less than a year is the quotient of the annual interest rate and the number of interest periods per year. PART 3 (7 MIN) Before solving the problem How is solving this problem similar to solving problems about balances in bank accounts? How can you find the number of interest periods? Javier Says (Screen 3) Use the Javier Says button to emphasize how interest payments can be a significant additional cost to the original purchase price. Which terms would you prefer when paying off a loan: interest compounded annually or interest compounded monthly? CLOSE AND CHECK (7 MIN) What decisions do you predict you will have to make as an adult that will require an understanding of the mathematics of percent and compound interest? Explain what comparisons you will have to make if you open a bank account or get a loan.
Compound Interest LESSON OBJECTIVE Solve real-world and mathematical problems involving the four operations with rational numbers. FOCUS QUESTION How is compound interest different from simple interest? When do you use each kind? MATH BACKGROUND In this lesson, students investigate a more commonly used and sophisticated form of interest: compound interest. In contrast to simple interest, compound interest is earned on the principal plus the previously accumulated interest. Interest earned in the first year becomes part of the principal when finding the interest for subsequent years. It is assumed in this lesson that no money is added to or withdrawn from the accounts being discussed. The Compound Interest Formula differs from the Simple Interest Formula in a few ways. First, the Compound Interest Formula uses balance instead of interest, so students no longer need to add the interest to the principal as a final step. Also, the Compound Interest Formula involves n, the number of times the interest is applied, instead of t. The use of the exponent n drives students to use calculators more often than they did for simple interest. In the previous lesson, students learned that they can solve for any one of the variables in the Simple Interest Formula if they know the values of the rest of the variables. For the Compound Interest Formula, they can find B or p, but they will not be able to find n or r until they learn about logarithms in high school. Most problems that students have solved before this lesson involve interest that is applied annually. Students should know that interest is often compounded every month and sometimes even more frequently. They will need to divide the annual interest rate by the number of times interest is applied per year to find the rate per interest period. LAUNCH (7 MIN) Objective: Compare simple and compound interest on a graph. Author Intent Students identify which of two curves represents simple interest and compare the balances of two accounts using a graph. While the other curve describes compound interest, students will not learn about compound interest until Part 1. Questions for Understanding Before How can lines on a graph show the effect of interest rates on savings accounts? [Sample answer: They can show by how much and at what rate the balance increases over time.] During How can you tell what the graph of simple interest looks like? How do you know which line shows simple interest? [Sample answer: The graph of simple interest will show the same change in balance from year to year because the interest does not change. You should look for the line that is straight and rises by $2 each year.]
Which line indicates the higher savings balance? [For the first two years, Line 2 has the higher balance. At exactly two years, they have the same balance. After two years, Line 1 has a higher balance.] After What would you advise your friend to do? [Sample answer: If my friend is saving money for a long period of time, I would advise my friend to take the other plan (not the simple interest plan) because the line increases at a much faster rate after two years.] Solution Notes Students may know that simple interest is a proportional relationship and that Line 2 shows a proportional relationship because it is a straight line. Other students may calculate the interest per year and find the line that shows 2% simple interest. For example, they can look for the account that has a balance of $102 after 1 year. Students may mention that it is important to ask your friend how long the money will be in the account before choosing the type of interest. The decision is based on whether the money will be earning interest for more than 2 years. Connect Your Learning Move to the Connect Your Learning screen. Discuss how the graph of the straight line in the Launch shows a constant rate of change, which means the account earns interest based only on principal. Listen for students to identify this as simple interest, which they learned about in the previous lesson. The other graph does not have a constant rate of change because it is not a straight line. Let students know that it shows a new kind of interest they will study in this lesson. PART 1 (8 MIN) Objective: Solve compound interest problems involving the four operations. ELL Support Beginning As you work through the Intro with students, pause to display the Spanish text and play the vocabulary words in both Spanish and English where appropriate. Click through each layer of the Annual Compound Interest box and point out the use of color in the example. Discuss how color and visuals can enhance understanding of concepts as students follow the changes in compound interest and balance even though the rate is constant. Intermediate Do the activity for Beginning learners. Assess their understanding by asking basic questions, such as: On what is simple interest calculated? On what is compound interest calculated? When are compound interest and simple interest equal? How do you calculate compound interest? Revisit any concepts that are causing students difficulty. Advanced Work through the Intro with students. Discuss the main concepts to assess students understanding. Then revisit the parts of the Key Concept that students do not fully understand. Play the vocabulary words in English and then in Spanish if necessary. Alternatively, pair students so that those that understand parts of the Key Concept may explain those parts to students who are having difficulty.
Author Intent Compound Interest continued Students are shown how to calculate compound interest year by year and the major difference between simple interest and compound interest. They complete a table to find the final balance of a savings account that earns interest compounded annually. This prepares students to learn a formula for compound interest in Part 2. Instructional Design In the Intro, you can call on students to click on each check box and reveal the steps for finding compound interest from year to year. Note that the first year is the same as simple interest. Get students to notice that the balance increases more than it would for simple interest because you earn interest on all previous interest (as well as the principal). On Screen 2, let students volunteer to find the interest and final balance for each year in the table. Have them complete the table on the whiteboard and show all necessary work. Check that students understand that the first row of the table represents the time that the principal is first deposited and that no interest has been earned yet. Questions for Understanding During the Intro When is compound interest like simple interest? [only for the first interest period] Why does the amount of interest increase over time? [Sample answer: The principal is greater every time you earn interest, so the percent of a greater number will be greater.] Javier Says (Screen 2) Use the Javier Says button to point out the difference between simple and compound interest. You might want to explain that in an account with simple interest, you earn the same amount of money in interest each time period. In an account with compound interest, you earn a different amount of interest each time period. As an account with compound interest grows, the amount of money earned in interest grows. Why does the amount of interest increase if the rate is constant? [After the first year, compound interest is calculated based on the balance in the account, not the original principal. The rate is a percent of the whole, and the whole is greater every year.] Why is there zero interest in year 0? [When you first deposit money, no interest has been earned yet. For this account, the first time interest is earned is after 1 year.] After solving the problem Suppose there are two accounts with the same rate, one offering simple interest and the other compound. Which account should you choose? [Compound interest; since the rate is the same, you will earn more interest if it is compounded.] Solution Notes The provided solution is animated to help you efficiently show the completed table. Point out the parallels in the numbers as the table is filled in. Students can see that completing this short table requires several computations. This method can seem inefficient if you need to find the balance after several years. Help students realize that they would benefit from a more efficient way to find the balance of an account with compound interest.
Got It Notes Compound Interest continued As students set up their tables, you may wish to guide them to insert a column at the left that identifies the starting balance that they need to use to calculate interest for each period. Students should realize that this quantity is the final balance of the previous row. Got It 2 Notes Point out that when the rates are the same, compound interest will always result in a higher balance because the principal changes for each interest period. Remind students that to earn the full benefits of compound interest, you cannot withdraw money from your account. PART 2 (7 MIN) Objective: Find interest compounded annually using the compound interest formula. Author Intent Students learn the Compound Interest Formula and apply the formula to find account balances. Once students understand why the formula works, they can use this efficient method instead of finding the interest for each year. Instructional Design In the Intro, introduce the formula for compound interest and help students understand the meaning of each variable. Pay special attention to B and n, which were not in the Simple Interest Formula. You may wish to go back to Part 1 of this lesson to confirm that using the formula gives the same final balance as using the table. On Screen 2, you can call on three students to find the account balances after each of the three times. Questions for Understanding During the Intro How is the Compound Interest Formula similar to what you learned about simple interest, for n 1? [Sample answer: If n 1, then B p(1 r). By the Distributive Property, B p pr. In the first year (t 1), simple interest I pr. As in the previous lesson, the balance is the principal plus the interest earned.] Javier Says (Screen 1) Use the Javier Says button to engage students in a discussion about financial planning. Contrast taking out the money sooner versus leaving it in the account and letting the interest compound as time passes. Help students see that if you can leave money in the account for a significant number of years, the balance grows a lot more in the final years compared to the early years. What do you observe about the pace at which the balance in the account changes over time? [Sample answer: The longer money is in the account, the greater will be the difference between interest earned from year to year.] What else might affect your decision for how long to leave money in an account? [Sample answer: You should consider when you will need the money and how much less the money will be worth due to inflation over time.] What variables will have the same value for all three parts of the problem? [p and r]
After solving the problem How is using a formula more efficient than calculating each part of a table? [Sample answer: Using a formula means you have fewer calculations to do, and therefore, fewer chances to make mistakes.] Solution Notes You can help students use the same method for each of the three balances with a Think-Write organizer. Once you have found the balance after 10 years, you can use the same Think column to find the other balances. Emphasize that you cannot use the balance after 10 years to find the balance after 20 years. You need to start over by substituting into the formula, but you only need to solve the formula for B to find the balance, which is easier than completing a table. To evaluate the exponents in this problem, you can open the Calculator tool as students solve the equations on the whiteboard. Differentiated Instruction For struggling students: As needed, review how to express percents as decimals and how to round money amounts to the nearest cent. When interest is not an exact amount of money, students should always round to the nearest cent. Have students make a chart showing rounded amounts of money for 5 different percents. For advanced students: Have students research actual interest rates given at banks. Give students the names of various banks in the area and have them look for the rates on money market (savings) accounts. Then, have them calculate how much money would be in each account for a principal of $1,000 that earned interest over a 10-year period. Error Prevention Watch for students who perform operations out of order. If students are using a calculator, remind them to use parentheses to make sure they raise the correct quantity to the exponent. Got It Notes This problem asks students to work backwards to identify the principal given the compound interest rate and the final balance. Show students that you can solve for p as well as B, but that they do not yet know how to solve for n or r. If you show answer choices, tell students that one effective way to solve the problem is to substitute each answer choice into the formula and see if the equation is true. Students who realize that the principal must be less than the final balance can use estimation to eliminate C and D. If they recognize that choice A is far too small compared to the final balance, they know the answer is B. If students choose C, they may be assuming that the problem asks for the final balance. Ask them to read the question part of the problem statement aloud. KEY CONCEPT (4 MIN) Teaching Tips for the Key Concept Previously in this lesson, students applied a formula to find balances when interest was compounded annually. Use this Key Concept as a review of the parts of the Compound Interest Formula. You can have students click on each check box to point out one of the variables in the formula.
To prepare for Part 3, have students focus on n and r. For n, explain that the number of interest periods is not the number of years. For example, if interest is compounded monthly for two years, the number of interest periods is 12 2, or 24. Similarly, the interest rate r refers to the rate for each interest period. The value of r for an interest period of less than a year is the quotient of the annual interest rate and the number of interest periods per year. You might note to students that the formula is not in the form y mx, so compound interest does not describe a proportional relationship. Stress that once students identify the values of each part of the Compound Interest Formula, they still need to follow the order of operations. PART 3 (7 MIN) Objective: Find interest compounded other than annually using the compound interest formula. Author Intent Students calculate compound interest for an interest period of less than 1 year. They need to find the values of n and r in the Compound Interest Formula. This problem involves a real-world situation in which money is owed instead of the balance of an account. Instructional Design This problem uses a blank Know-Need-Plan organizer to help students understand how this problem is more involved than the previous one. Call on students to fill in each part of the organizer. Before students tackle the questions, point out that balance in the case of a loan is the amount of money the borrower owes. The problem assumes that the loan is paid back all at once after 5 years even though most loans are paid back in installments. Questions for Understanding Before solving the problem How is solving this problem similar to solving problems about balances in bank accounts? [Sample answer: In both cases, one party owes money to another party, and the balance increases each period by a percentage (interest).] How can you find the number of interest periods? [You can find the number of periods in 1 year and multiply by the number of years.] What is the interest rate for each four-month period? How did you find it? [0.02, or 2%; you divide the annual interest rate, 6%, by the number of interest periods per year, 3.] After solving the problem What does the borrower pay for the loan altogether? How do you know? [$346; you can subtract the final balance from the original loan.] How could you reduce the interest you pay on this loan? [Sample answers: You can pay some of the balance off early, such as once a month.] How does solving this problem help you understand how to pay off a credit card? [Sample answer: Paying for purchases with a credit card can result in paying much more than the cost of the items over time due to interest fees. You want to plan to pay off your credit card as soon as is reasonable to avoid penalties and excessive interest.]
Javier Says (Screen 3) Use the Javier Says button to emphasize how interest payments can be a significant additional cost to the original purchase price. If you don't pay off a credit card bill quickly, you end up paying a lot more than you planned. Help students recognize the relationship between the number of interest periods and amount owed. Suppose the interest were compounded quarterly (4 times a year). Without doing any computations, how would the amount owed compare to the amount owed for an account compounded monthly and an account compounded yearly? [The quarterly amount would fall between.] Which terms would you prefer when paying off a loan: interest compounded annually or interest compounded monthly? [annually; the loan would be less.] Which terms would you prefer when loaning your money to the bank: interest compounded annually or interest compounded monthly? [monthly; the balance would be higher.] What is the relationship between number of interest periods and amount owed? [The amount owed increases as the number of interest periods increases.] Solution Notes Show students that the balance is greater than it would be if the interest were compounded annually. In that case, B 1,000(1 0.06)⁵ 1,338.23. Differentiated Instruction For struggling students: To help students find the rate per period, ask them what the interest rate would be if the interest were compounded annually. Now ask students how many times they earn interest in one year, which is 3. Have them break the interest rate into 3 parts by making a number line indicating times of the year and describe the percent interest earned in each part of the year. For advanced students: Have students estimate the monthly payments on the loan b(1 + r) from the Example by using the formula P =, where P is the monthly payments, n b is the balance owed, r is the interest rate, and n is the number of months left. Students will have to use this formula in iterations, so have them calculate only the first 3 payments. Error Prevention Students may think that n represents the number of interest periods in 1 year. Explain that n is the total number, which is the number per year times the number of years. In this problem, there are 3 periods per year times 5 years, so n 15. Got It Notes If you show answer choices, consider the following possible student errors: Students who choose A may need help finding the interest rate and number of interest periods. If students choose B, they are adding the percent to the principal. Students who select C are finding interest compounded annually. Got It 2 Notes Have students explain the relationship between the number of interest periods and increases in account balances. Have them choose which terms they would prefer when paying off a loan: interest compounded annually or interest compounded monthly.
CLOSE AND CHECK (7 MIN) Focus Question Sample Answer Simple interest is calculated based on the principal of an account, and does not change per year. Compound interest is based on the balance of an account, and changes as the balance changes. You use each kind based on the financial account you have. Focus Question Notes Listen for students to reference situations in this lesson as real-life instances when they may use compound interest: computing how much will be in an account after a certain number of years, finding the total balance owed on a car loan, and determining how an unpaid credit card balance grows. Essential Question Connection Percents are central to compound interest so this lesson directly addresses the Essential Question: When is it most convenient to use percents? Connect to the Essential Question idea that comparisons are central to plans, predictions, and decisions. Make sure students see that of two accounts with all things equal except the kind of interest, the account with compound interest grows faster than the account with simple interest. What decisions do you predict you will have to make as an adult that will require an understanding of the mathematics of percent and compound interest? [Sample answer: if you ever buy a car or a home and need a loan or mortgage; if you own a business and need to borrow money to make improvements or buy inventory; if you get a loan to pay for college; if you open a bank account; if you have a credit card] Explain what kinds of comparisons you will have to make if you open a bank account or get a loan. [Sample answer: If you open a bank account you will have to compare account packages offered by various banks and the different packages offered by each bank. If you get a loan, you will need to compare loans of various lengths and interest rates.]