Simple Interest LAUNCH (7 MIN) Before Why do banks pay interest to their customers? During What is the relationship between Line 1 and the deposit amount? After Which of the two lines shows a proportional relationship? KEY CONCEPT (4 MIN) Point out that I is not the new balance of the account but the interest earned on principal. To find the new balance, you need to add the principal to the interest: B p I How is finding the product pr similar to using the percent equation? Javier Says (Simple Interest (I)) Use the Javier Says button to address the Focus Question, Why is simple interest called simple? PART 1 (7 MIN) Javier Says (Screen 1) Use the Javier Says button to discuss what happens to your money when you place it in a savings account versus a checking account. What do you estimate is the interest earned? Explain. After solving the problem How can you use the percent equation to prove that I is proportional to p? PART 2 (7 MIN) During the Intro What is the relationship between the new account balance and the simple interest earned? Javier Says (Screen 2) Use the Javier Says button to support students understanding of the concept of proportional relationships. What pattern do you see in the table? Why can t you find the New Account Balance using the percent equation? While solving the equation How will you use the data in the Time column to complete the table? PART 3 (7 MIN) Why can t you just compare the interest for each account? While solving the problem What must you do once you find the amount of interest each person earns? After solving the problem Suppose their money were kept in the bank for 18 mo. What value can you use for time? CLOSE AND CHECK (8 MIN) Describe a real-life situation from this lesson in which it is convenient to use percents. What is another time period in which interest might be paid?
Simple Interest LESSON OBJECTIVES 1. Recognize and represent proportional relationships between quantities. 2. Use proportional relationships to solve multi-step ratio and percent problem involving simple interest. FOCUS QUESTION Why is simple interest called simple? When would you use simple interest? MATH BACKGROUND A key idea is that simple interest is found on the principal only and is not recalculated at the end of each year. For example, the simple interest earned on an account with a principal of $100 and an interest rate of 4% is always $4 per year even though the balance increases each year. Some problems in this lesson involve fractions of years. It is important that students understand the variable t in the formula for simple interest is the time in years, so they need to convert any time to those units. You may need to review how to rename parts of years as fractions. For example, 3 months 1 4 year. While the formula for simple interest is straightforward and helpful to students to understand how percents apply to real-world situations, simple interest is not typically used. Some examples of accounts that involve simple interest are certificates of deposit (CDs) and short loans of 30 90 days. Use this lesson as a precursor to compound interest, which is the focus of the next lesson. An understanding of interest as a percent increase will prepare students for their work in the lessons on percent change and markups/markdowns. LAUNCH (7 MIN) Objective: Identify simple interest on a graph. Author Intent Students pair bank offers with lines on a graph. They recognize one offer as a constant and the other as a variable. This problem demonstrates how a constant interest rate does not mean that the interest earned is constant. The interest earned is dependent on the principal. Questions for Understanding Before Why do banks pay interest to their customers? [Sample answer: When people deposit money into bank accounts they are, in essence, loaning money to the bank that the bank will invest. Interest is what the banks pay customers for the use of their money.] How can banks afford to pay customers interest? [Sample answer: Banks invest your money and generally make more than they pay you in interest.] During What is the relationship between Line 1 and the deposit amount? Between Line 2 and the deposit amount? [The value of Offer 1 increases as the deposit amount increases. The value of Offer 2 is constant and does not change as the deposit amount increases.]
Simple Interest continued After Which of the two lines shows a proportional relationship? [The diagonal line shows that the two quantities have a proportional relationship.] Solution Notes Invite students to brainstorm a list of reasons, other than their initial offers, for choosing one bank over another. Answers may include fees, interest rates, the minimum initial deposit, hours of operation, number of locations, and other investment services available. Students may compare the two offers to determine which has a greater value. They should notice that Line 2 has a greater value for smaller deposit amounts, the intersection of the two lines represents the deposit amount where the values are equal, and Line 1 has a greater value for all larger deposit amounts. Connect Your Learning Move to the Connect Your Learning screen. Use the Launch to help clarify the meaning of fixed percent. Explain that the value is not fixed itself, but the percent of the whole is fixed. Use the Focus Question to begin a conversation about interest. If necessary, you can prompt students to list reasons why people put their money in banks, such as to keep it safe or to gain interest. Explain that there are different kinds of interest, and how in this lesson, students will study a relatively uncomplicated kind. KEY CONCEPT (4 MIN) ELL Support Beginning While listening to the definitions in the Key Concept, have students write down words that they do not understand. Make a list on the board of these words and write simple definitions for students to copy. Discuss the definition of each word to reinforce understanding. Intermediate Have students make a list of academic words they do not understand as they listen to the definitions in the Key Concept. Have them look up the words in a dictionary or the online glossary and write definitions for each of the words. Advanced Have students make a list of academic words as in the activity for Intermediate learners. Then have students who have listed different words work together to define the words. For example, if one student lists one word and a second student lists a different word, they should be able to define each other s words. Teaching Tips for the Key Concept You can call on students to click on each check box to show the various parts of the Simple Interest Formula. Make sure students understand the meaning of interest and define each variable as it appears. When all layers are showing, students can see the relationship between the parts of the formula. Point out that I is not the new balance of the account but the interest earned on principal p at interest rate r over t years. To find the new balance, you need to add the principal to the interest: B p I. You might compare the Simple Interest Formula to the percent equation. Discuss how the equation is in the form y mx, where m r t.
Questions for Understanding Simple Interest continued How is finding the product pr similar to using the percent equation? [Sample answer: p is the whole, and r is the percent, so the product is a part of the whole. It is the interest you earn in a year.] Why can you multiply by t in the formula to find the interest for t years? [Sample answer: The interest is the same every year. Once you know the interest for one year, you can multiply by the number of years to find the total interest.] Javier Says (Simple Interest (I)) Use the Javier Says button to address the Focus Question, Why is simple interest called simple? Using the example provided, point out that if the money is taken out after 1 year, you earn $2 in interest. If it's taken out after 2 years, you earn $4 in interest, and so on. What is the pattern in the relationship between the number of years the money is in the bank and the amount of interest earned? [Sample answer: The interest earned is double the number of years the money is in the bank.] PART 1 (7 MIN) Objective: Identify parts of a proportional relationship represented by the simple interest formula. Author Intent Students use the Simple Interest Formula they learned in the Key Concept. They apply the formula to calculate interest earned given principal, rate, and time. This problem is designed to help students become comfortable with the formula before they find the balance of an account by adding the interest to the principal in Part 2. Instructional Design You can call on students to drag tiles to complete the Words to Equation organizer. Once the class agrees on where the variables belong in the equation, have students identify the values of each variable from the problem statement and place the tiles accordingly. They will need to solve the equation to find the interest, I. Questions for Understanding Javier Says (Screen 1) Use the Javier Says button to discuss what happens to your money when you place it in a savings account versus a checking account. Focus on the idea that banks are businesses that make money with your money. They use depositors funds to make loans to other people and businesses, and they invest your money in outside funds, hoping to earn more interest than the bank pays you. Which quantity is not given in the problem statement? [I] How is the rate, r, different from the interest, I? [Sample answer: The rate tells you the percent of the principal that makes up the interest. The interest is an actual quantity that you can calculate using the rate and principal.] What do you estimate is the interest earned? Explain. [Sample answer: Since 6 5 30, then you need to find 30% of $400. You can either multiply 10% by 3, or you estimate that it s a little less than 1 of 400.] 3
Simple Interest continued After solving the problem How can you use the percent equation to prove that I is proportional to p? [Sample answer: You can rewrite the Simple Interest Formula to look like the percent equation: part percent The number 6 in the formula represents 6 years. How do you think you would represent 6 months? [Sample answer: 1 ; 6 months is half a year.] 2 Solution Notes This problem involves the Words to Equation organizer to help students translate the problem statement into the Simple Interest Formula. You can check your answer against the provided solution to see whether students translated the problem into the formula correctly. Although students could place p, r, and t in any order by the Associative Property of Multiplication, emphasize that the formula specifies an order that is useful to remember. Differentiated Instruction For struggling students: Students may need help choosing the correct tile to represent the rate r. Point out that 5% is not equal to 5, but 0.05. Provide a hundreds chart and have students shade 5% of it. Note that the total chart is equal to 1. For advanced students: Have students write another equation that relates balance to principal, rate, and time. They should realize that they need to add the principal to the interest, which can be written as Error Prevention Students may write the percent as a fraction or a decimal. Tell them that the rate is most commonly written as a decimal. Watch for students who convert the percent incorrectly, no matter which form they choose. For example, in the Got It, students may have trouble writing 1.2% as a decimal. Got It Notes If you show answer choices, consider the following possible student errors: Students who choose A may not notice that the problem asks about two years. If students have trouble writing the rate as a decimal in order to calculate the interest, they may choose B. Students who select D have swapped the meanings of p and r. Got It 2 Notes Help students connect this problem to the previous lesson. Point out that when the value of t is 1 year, you can write the formula without t, and there exists a proportional relationship between a part, a whole, and a constant of proportionality. The provided solution uses color to distinguish the different variables as the steps clearly transform the Simple Interest Formula into the percent equation. PART 2 (7 MIN) Objective: Find simple interest and account balances. ELL Support On the Student Companion page for the Part 2 Got It, there are three tasks for students to complete and discuss: Choose a row from the table. Explain what the entries in Columns 1, 2, and 3 mean for the row that you selected.
Take turns until you have reviewed all the rows in the table. Simple Interest continued Beginning and Intermediate Be sure that students have a common understanding of what makes up a row in the table vs. a column in the table. Consider numbering the rows to make it easy for students to reference. Then have students choose one of the rows and complete the second task, paying particular attention to the use of the column headings and how information is related across the cells in that row. Advanced Have students complete the first two tasks in pairs. Then have them discuss the challenges of describing the information in the row. Was there a column of information that was more challenging to describe than the others? How does understanding the information in the table better prepare you to answer the problem? Author Intent Students calculate simple interest for various times using the same principal and interest rate. They find the new account balance after each year and identify that the balance changes by the same amount every year. Computing simple interest over a period of years prepares students to compare simple and compound interest in the next lesson. Instructional Design In the Intro, have students define the word balance in their own words. You can compare this problem to the previous lesson, in which students found the tax/tip as a percent of the cost of food and added the quantities together to find a total. On Screen 2, call on students to fill in the table one row at a time. They may realize that you can fill in the entire Simple Interest Earned column immediately because every entry is the same. You may need to discuss what certificates of deposit (CDs) are and why people buy them. Explain that the rate for a CD is usually higher than that of a savings account because you are not allowed to withdraw the money out of a CD without a penalty. Questions for Understanding During the Intro What is the relationship between the new account balance and the simple interest earned? [The new account balance is the sum of the principal plus the interest earned.] Using the variables in the Simple Interest Formula, write another formula for B, the balance of an account. [B p I] Javier Says (Screen 2) Use the Javier Says button to support students understanding of the concept of proportional relationships by studying the table. Some students may be able to recall from the previous topic, Proportional Relationships, how to use their knowledge to find interest earned in 10 years. What pattern do you see in the table between the Time column and Simple Interest Earned column? [Sample answer: The Simple Interest earned is the Year multiplied by 87.5, the product of the principal and rate.] Describe a proportional relationship in the table. [Sample answer: The ratio of the time and the simple interest earned is the same for every row of the table.] Why can t you find the New Account Balance using the percent equation? [Sample answer: The new balance is neither the part nor the whole. You need to add the previous balance to the interest earned to find the New Account Balance.]
Simple Interest continued While solving the problem How will you use the data in the Time column to complete the table? [Sample answer: The principal and rate are constant for this problem. The time in the Simple Interest Formula is the only quantity that changes when calculating interest for each row.] Solution Notes Consider using the Coordinate Grapher tool to plot New Account Balance vs. Time and help students identify a proportional relationship. Make sure students remember that the principal is listed above the table. They should use that same value every time they find the interest and the balance. Error Prevention Watch for students who write an incorrect decimal for 3.5%. If necessary, review how to convert a percent to a fraction and/or decimal. Students may want to use the new account balance as the principal for the next row. Explain that the principal is the same for each situation but that they will learn another type of interest that builds on the new account balance in the next lesson. Got It Notes This problem supplies the data table from the Example. Although the principal has changed, students can use number sense to solve this problem. Ask them how the principal compares to that of the Example (it is double). If students understand that they can double every amount in the table, they may look for the first row when the account earns at least $250, which would translate to more than $500 in interest for an account with double the principal. Other students may subtract and realize that $500 of interest is 10% of the principal. They can locate the row of the table that shows at least 10% interest on $2,500, which is 3 years. Got It 2 Notes Unlike the previous problems, students know I and are solving for t. They should recognize that they can solve for any variable in the formula given the other values. They may use the Simple Interest Formula and solve for t or rewrite the formula in terms of t before substituting values. PART 3 (7 MIN) Objective: Compare simple interest rates. Author Intent Students compute simple interest for a fraction of a year. They understand the Simple Interest Formula is in terms of years and convert time from months to years. Working with shorter periods of interest prepares students to work with compound interest. Instructional Design On Screen 1, explain that the time t in the formula is always in years and that the rate r is an annual percent. Use the example to demonstrate how to convert the time when presented in terms of months. Move to Screen 2. Call on two students to each work with one of the two accounts shown. You can have each student solve their problem separately or have both students complete one step at a time.
Questions for Understanding Simple Interest continued Why can t you just compare the interest for each account? [The principals are also different.] While solving the problem Should you express 8 months as a fraction or a decimal? [Sample answer: a fraction; 8 is a repeating decimal.] 12 What must you do once you find the amount of interest each person earns to solve this problem? [Sample answer: Add the interest to the principal and then compare total amounts in each account.] After solving the problem Would you get the same answer if you rounded 8 to the decimal 0.67? Explain. 12 [Yes; the problem does not ask for the exact balance of each account. It asks which account has a higher balance.] Suppose their money were to be kept in the bank for 18 months. What value can you use to represent time? [Sample answers: 1 1 2, 1.5, 3 2 ] Solution Notes Ask students how you can apply the Associative Property of Multiplication to solve the problem more efficiently. They may mention that you can use mental math to multiply Alex s principal of $3,000 by 2 3 Got It Notes first and then multiply by 0.025. Students can see that they need to apply the same formula, this time rewriting the time as an improper fraction, to solve the problem. Guide them to see that reducing the fraction to 4 3 can make the computation easier. You may wish to elicit from students that since 4% is proportionally much greater than 2.5%, the trend is that Mia s account will continue to grow faster than Alex s will. Got It 2 Notes As in Part 2, this problem applies the Simple Interest Formula to find yet another unknown quantity: the principal. Students may ask why they should round their answer. Explain that when banks calculate interest, they round to the nearest cent. The exact interest may not be $5.51. Show that several balances, including $245 and $244.75, result in $5.51 in interest after 9 months. CLOSE AND CHECK (8 MIN) Focus Question Sample Answer Simple interest might be called simple because it is based on the principal only. This makes the interest the same amount every year. You would use simple interest when you have a financial account that applies simple interest to the account. Focus Question Notes Listen for students to describe loaning money to a sibling (or borrowing money from a sibling!) as a situation in which they might use simple interest.
Simple Interest continued You may wish to explain that simple interest is generally used when the loan period is less than a year. For example, if a corporation borrows a sum from another corporation for just a day or two, the loan is often paid back with simple interest. You may also want to discuss with students how most of the loans and mortgages they encounter in life will not use simple interest. In the next lesson, they will look at a kind of interest that is commonly used when an individual borrows a sum for more than a year. Essential Question Connection In this lesson, students see that percents are a convenient way to describe simple interest, which directly addresses the Essential Question: When is it most convenient to use percents? Throughout the lesson, students compared various bank offers and determined which account had more money at the end of a given time period. This supports the portion of the Essential Question that describes comparisons with percents as helpful when making plans and decisions. Describe a real-life situation from this lesson in which it is convenient to use percents. [Sample answers: when determining how much money is in your account; when deciding whether to put your money in a CD; when comparing bank offers] In Parts 1 and 2, interest is paid every year. In Part 3, interest is paid every month. What is another time period in which interest might be paid? [Sample answers: daily, weekly]