Comment [AI]: Minor formatting Contextualized Learning Activities (CLAs) changes. For the “other required credits” in the bundle of credits, students in a Specialist High Skills Major program must complete learning activities that are contextualized to the knowledge and skills relevant to the economic sector of the SHSM. Contextualized learning activities (CLAs) address curriculum expectations in these courses. This CLA has been created by teachers for teachers. It has not undergone an approval process by the Ministry of Education. Contact Information Board Rainbow District School Board Development date January 2008 Contact person Keri Endanawas Sheila Giroux Position Mathematics Program Leader Program Coordinator – Numeracy Phone 705-522-2320 705-523-3308 Fax 705-523-3314 E-mail firstname.lastname@example.org email@example.com Specialist High Skills Business Major Course code and course MBF3C – Foundations for College Mathematics title Name of contextualized The Power of Compound Interest learning activity/activities Brief description of In these activities, the students will explore different investment contextualized learning options in simple interest and in compound interest problems. activity/activities Students will identify the main features of each and recognize trends. The activity culminates in exploring the benefits associated with two alternatives for funding a scholarship. Duration 6 hours Overall expectations B1 Compare simple and compound interest, relate compound interest to exponential growth, and solve problems involving compound interest B2 compare services available from financial institutions, and solve problems involving the cost of making purchases on credit Specific expectations B1.1 determine, through investigation using technology, the compound interest for a given investment, using repeated calculations of simple interest, and compare, using a table of values and graphs, the simple and compound interest earned for a given principal (i.e., investment) and a fixed interest rate over time B1.2 determine, through investigation (e.g., using spreadsheets and graphs), and describe the relationship between compound interest and exponential growth B1.3 solve problems, using a scientific calculator, that involve the calculation of the amount, A (also referred to as future value, FV), and the principal P (also referred to as present value, PV) using the ` an compound interest formula in the form A P 1 i [or ` an FV PV 1 i ] B1.4 calculate the total interest earned on an investment or paid on a loan by determining the difference between the amount and the principal [e.g., using I A @P (or I FV @PV )] B1.5 solve problems, using a TVM Solver on a graphing calculator or on a website, that involve the calculation of the interest rate per compounding period, i, or the number of compounding periods, n, in ` an ` an the compound interest formula A P 1 i [or FV PV 1 i ] B2.2 gather and interpret information about investment alternatives (e.g., stocks, mutual funds, real estate, GICs, savings accounts), and compare the alternatives by considering the risk and the rate of return Essential Skills and work Numeracy – Money Math habits Thinking – Job Task Planning and Organizing Oral Communication Catholic graduate expectations (if applicable) Instructional/Assessment Strategies Teacher’s notes There are many considerations in the development of a positive learning environment Processes The seven mathematical processes can be referred to as the ‘actions of math.’ In the revised curriculum, these process expectations have been highlighted in their importance since they support the acquisition and use of mathematical knowledge and skills. They can be mapped to three categories of the Achievement Chart – Thinking, Communication and Application. The fourth category, Knowledge and Understanding, connects to the overall and specific expectations of the course, which can be referred to as the ‘mathematical concepts’. Students apply the mathematical processes as they learn the content for the program. The combination of the mathematical processes and expectations are embedded in the achievement chart as the following: Knowledge and Understanding Concept Understanding Procedural Fluency Thinking Problem Solving Reflecting Reasoning and Proving Application Selecting Tools and Computational Strategies Connecting Communication Communicating Representing Literacy Strategies Mathematics is the most difficult content area material to read because there are more concepts per word, per sentence, and per paragraph than in any other subject; the mixture of words, numerals, letters, symbols, and graphics requires the reader to shift from one type of vocabulary to another. Leading Math Success, Report of the Expert Panel for Mathematical Literacy Gr. 7 – 12 Improved student achievement demands an emphasis on developing literacy competencies linked to mathematics learning. To consolidate understanding, learners need opportunities to share their understanding both in oral as well as written form. Weakness in reading or writing skills provides barriers to success in problem solving. This resource explicitly embeds at least one literacy strategy in every lesson. Starting points for teachers: Use strategies to develop vocabulary and comprehension skills, including word walls Frayer model concept circles Use strategies relating to the organization of information “inking your thinking” – having students write down their thoughts concept maps anticipation guides Use strategies to help students understand features of textbooks and graphics read problems aloud highlight key words think aloud More details and strategies can be found in Think Literacy: Cross-CurricularApproaches, Mathematics, Grades 10-12, 2005, http://www.curriculum.org/thinkliteracy/library.html Context Over the past several years, there has been remarkable growth in the need to understand and apply basic principles of money management and investment. Technology has been at the front of this surge. Technology allows a larger proportion of the population to access the tools required to make sound financial decisions with respect to their business and personal investments. Entrepreneurs sometimes make investments in local education as part of their social responsibility. They will fund awards, bursaries and scholarships at local educational institutions including secondary schools. In doing so, entrepreneurs often need to examine alternatives in consideration with their cash flow, and assets to allow the maximum return to the school with minimum cost. To do so, entrepreneurs need a sound understanding of simple and/or compound interest. Strategies Lesson 1 - Simple Interest This lesson will be review of previous material for many students. Allow students to work in small groups to solve the examples rather than demonstrate the calculations for students. Provide graph paper, placemat paper, chart paper with markers for each group. Be aware of students who continue to struggle working with percents and provide support when required. Pearson Math 11 (2008) p. 212-216. Additional practice may be assigned for the text as determined by student’s needs. Lesson 2 - Compound vs. Simple Interest Be certain that students are linking the two ideas together. A computer lab is required for this activity but students should work in pairs. Students need to be able to compare and contrast the two types of interest. Pearson Math 11 (2008) p 217-222 MBF3C Support Materials Unit 8 www.oame.on.ca Lesson 3 - The Compound Interest Formula The teacher should ensure that students are connecting the ideas of all three lessons so far in this unit. Pearson Math 11 (2008) p 223-234 MBF3C Support Materials Unit 8 www.oame.on.ca Lesson 4 - The TVM Solver Students use technology in graphing calculators to further explore compound interest, particularly when needing to solve for i or n in the compound interest formula. As a homework activity, students will discover the Rule of 72 (the product of the annual interest rate and the number of years for the investment to double is 72). Emphasis on exponential growth and the effect of different compounding periods on the accumulated amount continues. MBF3C Support Materials Unit 8 www.oame.on.ca Taking Stock in Your Future www.investorED.ca Lesson 5 - Philanthropy Students use the compound interest formula in conjunction with technology to explore the present value of an investment given the accumulated amount. Students will explore two different payout options, make conjectures and make a final argument in writing. This is the summative evaluation of the unit. MBF3C Support Materials Unit 8 www.oame.on.ca Taking Stock in Your Future www.investorED.ca Assessment and Evaluation of Student Achievement Strategies/Tasks Purpose 1. Paper/Pencil Quiz (L1) Diagnostic 2. Peer Assessment (L2) Formative 3. Journal (L2) Formative 4. Checklist (L3) Formative 5. Journal (L3) Formative 6. Gallery Walk (L3) Formative 7. Oral Discussion (L4) Formative 8. Philanthropy Assignment (L5) Summative Assessment tools Achievement Chart Rubric Additional Notes/Comments/Explanations To elevate student interest, the teacher may pull from current events to teach the elements of this unit. These can be incorporated as mathematical examples, investigations or to encourage dialogue and discussion of current investment trends and options. Resources Authentic workplace materials Human resources Print Pearson Math 11 (2008) Video Software Websites www.investorED.ca www.oame.on.ca Other Graphing Calculator Accommodations Through groups set up and identified by the teacher, an entry point to problems for all students can be ensured. The teacher circulates during small group discussion, making notes, and can step in to provide additional support and or review for students in need. The use of graphing calculators, spreadsheets, scientific calculators in addition to tasks requiring little to no technology allows the teacher to differentiate meeting the needs of all learners. In the final task (Day 5), students may wish to use the graphing calculator technology instead of a spreadsheet. Embedding literacy strategies into each day’s activities supports all Mathematics Language Learners in the classroom. The teacher should refer to the word walls whenever possible throughout each day’s lesson to support students’ acquisition of precise mathematical vocabulary. As always, any modifications and/or accommodations outlined in a student’s IEP must be adhered to. Simple Interest SHSM Business CLA Day 1 MBF3C Description Materials Students review simple interest and working with percents in preparation for further work Placemat paper with compound interest. Chart paper Markers Graph paper Calculators Assessment Opportunities Minds On… Whole Class – Small Group In groups of 3 or 4, have students discuss when they have ever loaned money or Make sure that a made a financial investment. wide variety of How were the funds re-paid? How much was invested, how much was paid responses are out at the end? presented and What are some advantages to lending money? Are there any accepted. Provide disadvantages? further examples Would you ever lend money to someone with whom you did not have a when required. close personal relationship? Why or why not? Groups record their findings on chart paper and share with class. Teacher facilitates discussion making reference to “Interest”, “Investing”, and “Lending Institutions”. Action! Whole Class – Small Group When working When banks and businesses lend money to individuals and companies, interest is through examples, charged. The bank or business is investing in the person borrowing the money. students should There are different ways of determining the interest charged. work in their small groups on placemat Example: You loan $250 to your brother in January so that he can buy an 8GB mp3 paper. The teacher Player. He plans to repay you in two months. You agree to lend the money but ask should consolidate him to repay $260 for your trouble. with the whole class a) What interest, in dollars, did you charge? after each example. b) What percentage of the principal ($250) is the interest? c) Solution: a) $10 per 2 months $10 fffffffffffffff $250 Definition should be b) added to the 0.04 student’s glossary 4% of terms and the word should be Simple Interest: interest earned only on the principal, using the formula I Prt , added to the where I is the simple interest, P is the principal, r is the annual interest rate as a classroom word decimal, t is the time in years. wall. Note that the interest rate in the example was not an annual rate, it was the rate for 2 1 fffffff two months or fff of a year. 12 6 Example: Your brother does not repay you until the following January. Assume you continue to charge him $10 for each two months that he owes you the money. a) Make a table of values to record the Accumulated Amount now owing every two months. b) Graph your results. c) How much interest did you charge over the course of a year? d) What is the annual interest rate? Solution: Month Total Owing ($) March 260 May 270 July 280 Sept 290 Nov 300 a) Dec 310 Total Ow ing ($) 320 310 300 290 280 270 260 250 240 230 March May July Sept Nov Jan b) I A @P c) Over the course of 1 year, $310 @$250 . So $60 interest was $60 charged over the course of 1 year. 60 fffffffffff d) 250 Therefore, 24% interest was charged over the course of 1 year. 0.24 24% Note that interest is charged 6 times over the course of the year, 4% is charged every 2 months and 6 B 4% 24% . Simple interest is often used for short team investments. When the interest rate is an annual rate, time in months or days must be written as a fraction of 1 year. Example: You invest $4000.00 at your financial institution. How much interest do you earn for each of the following situations: a) 2.25%/a simple interest for 60 days; b) 2.25%/a simple interest for 5 months; c) 3%/a simple interest for 5 months? Solution: 60 a) P $4000.00; i 0.0225; t fffffffffff 365 I Prt 60 $4000.00 B 0.0225 B fffffffffff 365 $14.79 5 fffffff b) P $4000.00; i 0.0225; t 12 I Prt f g ` 5 a fffffff $4000.00 0.0225 12 $37.50 5 c) P $4000.00; i 0.03; t fffffff 12 I Prt 5 $4000.00 B 0.03 B fffffff 12 $50 Note that more interest is earned the longer the investment time and the higher the interest rate. Curriculum Document/Quiz/Marking Scheme Assess students’ ability to apply their understanding of Simple Interest. BLM1 Consolidate Note that the simple interest graph you created showed a linear relationship. When Debrief we looked at the total amount owed, what was the y-intercept? (Ans: Principal or initial amount owing) If we label each month where interest was charged as 1, 2, 3,… what is the slope of the line? (Ans: the product of the two month interest rate and the Principal) Generate an expression for determining the interest earned on a $1200 investment that earns 6.5% simple interest. The time is variable. a) What would the graph of the equation look like? b) What is the y-intercept? the slope? c) What do the y-intercept and the slope represent in this situation? Solution: P $1200; i 0.065; t is a variable I Prt $1200 B 0.065 B t Note that the y- I $78t intercept here is a)The graph of this equation would be linear. different from the b)The y-intercept is 0 and the slope is 78. graph of “Total c)The y-intercept represents the amount of interest earned at the beginning of Amount Owing”. the investment and 78 is the product of the Principal and the annual interest Explain the reason rate. for this difference. Home Activity or Further Classroom Consolidation Concept Practice a) In your own words, describe Simple Interest. b) How is Simple Interest calculated? Differentiated Student responses Exploration c) Create an “interesting” Simple Interest example to be solved by a classmate will vary. tomorrow. You must do the solution to your problem as homework. Keep It Simple! Name: Date: Josie holds the following bonds in her investment portfolio. Each bond earns simple interest at the given rate. A $1200 Ontario Savings Bond earning 6.7% per year A $5000 Canada Savings Bond earning 5.9% per year A $5000 corporate bond earning 8.25% per year Each year, Josie must report the interest earned on these bonds on her tax return. She has the Ontario Savings Bond for the entire year, the Canada Savings Bond for 10 months and the corporate bond for 95 days. a) What is the total interest she must report on her tax return? Show all of your calculations. (9 marks) b) Graph the line showing the total interest earned on the Ontario Savings Bond for each of the twelve months in the year. Determine the y-intercept and the slope and state their meaning in the context of the question. (5 marks) Solution: a) For the Ontario Savings Bond (2 marks): P $1200; i 0.067; t 1 I Prt $1200 B 0.067 B 1 $80.40 For the Canada Savings Bond (3 marks): 10 P $500; i 0.059; t fffffff 12 I Prt 10 $500 B 0.059 B fffffff 12 $24.58 For the Corporate Savings Bond (3 marks): 95 P $5000; i 0.0825; t fffffffffff 365 I Prt 95 $5000 B 0.0825 B fffffffffff 365 $107.36 On her tax return, Josie must report $80.40 $24.58 107.36 $212.34 (1 mark) b) Month 1 2 3 4 5 6 7 8 9 10 11 12 Number Interest 6.70 13.40 20.10 26.80 33.50 40.20 46.90 53.60 60.30 67.00 73.70 80.40 SHSM Business CLA – BLM1.1 Earned ($) Total Interest Earned 100 Interest Earned ($) 80 60 40 20 0 1 2 3 4 5 6 7 8 9 10 11 12 Month Number The y-intercept is 0 and this is the total interest earned during the year on January 1. rise $80.40 @$6.70 The slope is ffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ; this represents to total interest earned on the investment each ` a run 12 @1 months $6.70 per month month. Compound vs. Simple Interest SHSM Business CLA Day 2 MBF3C Description Materials A first look at compound interest. Students will use the Simple Interest formula Computer lab with repeatedly along with a spreadsheet to calculate the Total Interest earned with simple enough stations for compounding periods. students to work in pairs Computer projector Assessment Opportunities Minds On… Whole Class – Pairs In pairs student solve each other’s problems that were created as part of the homework. If there is a pair without problems to be solved, encourage them to create one problem together and solve it as a pair. Administer Quiz (BLM1.1) if not completed yesterday and take up solutions making noteworthy points. Think-Pair-Share Students discuss You are a millionaire at age 60!!! Your grandparents invested $2500 when you are the reasonableness born in a savings bond. of this statement in pairs. Each pair will share their perspective and rationale with the class. Action! Whole Class – Small Groups Build on mp3 player question from yesterday…. Example: Suppose in March when your brother cannot repay you for the mp3 player loan, you decide that you don’t want to be without the money for much longer. You Students work on tell him that either he has to pay the interest every two months ($10) or you will now this in pairs. The consider the Principal amount to be $250 + $10 = $260 and charge the 4% interest for teacher takes up the next two months. He cannot pay the $10 so you charge interest on $260. This the solution using a continues for the entire year. computer projection of one pair’s work. a) Make a spreadsheet showing the Compounding Period, Total Principal and It is important to Interest Charged and Total Interest Charged. circulate so that b) Graph Total Interest against Compounding Period using the spreadsheet students have a functions. good understanding of the calculations Solution: to be carried out. The teacher may wish to show a table started and ask the students how each entry in the second row is obtained. d rio Pe ed l arg ing ipa est nd Ch nc er ou Pr i Int st mp ere tal tal Co To To Int 1 $250 $10 $10 2 $260 $10.40 $20.40 3 $270 $10.82 $31.22 4 $281 $11.25 $42.46 5 $292 $11.70 $54.16 a) 6 $304 $12.17 $66.33 Total Interest $70 $60 Total Interest Charged ($) $50 $40 $30 $20 Definition – add to $10 word wall and glossary. $0 1 2 3 4 5 6 Encourage students Com pounding Period to use a table of b) differences to predict the shape of Compounding Period: The time interval at which interest is added to the principal. the graph. Teacher debriefs Small Group Discussion: the discussion and Why does the interest charged increase every two months? shows a projection At the end of the year, how much additional interest is charged by not of a graph with the paying the $10 out each time? linear and What are the advantages of paying the interest as it accrues? Why might exponential graphs someone not choose to do this? on the same chart. What shape is the resulting graph? Compare this to yesterday’s graph. Answers: The interest charged increases because each time the interest is rolled into the Principal amount and essentially he has “borrowed” more money. The additional interest charged is $66.33 - $60.00 = $6.33. If you pay the interest as it accrues then you save money and end up paying less total interest at the end. Someone may choose not to do this because they don’t have enough money. You might want to do this if it is an Add to word wall investment rather than a loan as you will earn more total interest over the and glossary. life of the investment. The resulting graph is exponential. Yesterday’s was linear. Exponential curves grow more rapidly than the linear graphs…bad for a loan but good for an investment. The kind of interest that works this way is called Compound Interest. Students work in pairs at computer Have students work in their groups to create a word-wall card for Compound Interest. stations. Example: a) Suppose you invested $1500 in a Canada Savings Bond for seven years. The simple interest rate is 6.35%/a. Make a spreadsheet showing the total interest earned at the end of each year. Calculate the first differences. b) Suppose you invested $1500 in a Canada Savings Bond for seven years. The compound interest rate is 6.35%/a. Make a spreadsheet showing the total interest earned at the end of each year. Calculate the first differences. Note that the First c) Plot Total Interest against Year for each situation on the same chart. Differences for Simple Interest are Solution: constant – the Year Total Int First Diff graph is linear. 1 $95.25 2 $190.50 $95.25 3 $285.75 $95.25 The First 4 $381.00 $95.25 Differences for 5 $476.25 $95.25 Compound Interest 6 $571.50 $95.25 are not constant. The total interest a) 7 $666.75 $95.25 changes by an Year Total Int First Diff increasing amount 1 $95.25 each year – the 2 $196.55 $101.30 graph is not linear. 3 $304.28 $107.73 Ask students to 4 $418.85 $114.57 calculate further 5 $540.70 $121.85 differences to 6 $670.28 $129.58 identify the shape 7 $808.10 $137.81 as exponential. b) Interest $800.00 Total Interest $600.00 Simple $400.00 Compound $200.00 $0.00 1 2 3 4 5 6 Year c) Consolidate The points for simple interest lie on a straight line. Debrief The points for compound interest lie on a curve. Simple interest illustrates linear growth. Compound interest illustrates exponential growth. In the first year of the example, the simple interest and the compound interest are the same. Money grows more rapidly when interest is compounded. Revisit the Minds On questions – A millionaire at age 60!! Pairs revisit their If interest is earned monthly, how many times will the bond earn interest responses to the over the lifespan? Minds On question Estimate the interest rate that would result in this Accumulated Amount. posed at the Explore on your calculator or using a graph. beginning of class. Is this situation realistic? Home Activity or Further Classroom Consolidation Peer Assessment of Part a) of Homework in small groups (3 – 4) Concept Practice a) Explain the difference between simple interest and compound interest. Include tables and graphs in your explanation. Reflective Journal is collected for Part B and teacher provides written feedback. b) Which bond earns more interest? Justify your answer. A simple interest bond that earns 3.7% per year for three years Reflection A compound interest bond that earns 3.6% per year for three years Solution: a) See Consolidate and Debrief. b) Suppose $1000 is invested. I Prt A simple interest bond earns $1000 B 0.037 B 3 $111.00 A compound interest bond earns… I Prt …in year 1 $1000 B 0.36 B 1 $36.00 I Prt …in year 2 $1036 B 0.36 B 1 $37.30 I Prt …in year 3 $1073.30 B 0.036 B 1 $38.64 …total interest earned over 3 years: $36.00 $37.30 $38.64 $111.94 The compound interest bond earns more. Peer Assessment – Different Interests Indicate Y/N in the left hand column beside each item. Staple this page to the back of your classmate’s work. Remember to be positive in your written feedback. This activity provides you and your classmates an opportunity to improve your work and understanding. Is simple interest well defined? (refer to the word-wall card) Is compound interest well defined? (refer to the word-wall card) Based on the written description, can you tell what the difference is between simple interest and compound interest? Does the description use correct mathematical vocabulary (e.g. interest, principal, present value, compound, amount, compounding)? Are there tables included to show the difference between simple interest and compound interest? Are there graphs included to show the difference between simple interest and compound interest? Are there any recommendations you could provide to your peer? Remember to use positive wording. Reviewer’s signature: _________________________________ The Compound Interest Formula SHSM Business CLA Day 3 MBF3C Description Materials Chart paper ` an Students are introduced to the compound interest formula, A P 1 i , through Markers Tape investigation. Blank word wall cards Assessment Opportunities Minds On… In Pairs Thinking back to the mp3 player example (first example) from yesterday, encourage Teacher circulates students to express the entries under Total Principal (the Accumulated Amount) as a and prompts product of initial Principal ($250) and some other value. Encourage students to think students with about the operation they are performing each time to come up with the revised appropriate expression. questions as ` a required. $260 $250 1.04 ` a ` a` a ` a2 Teacher uses $270.40 $260 1.04 $250 1.04 1.04 $250 1.04 checklist to assess ` a ` a3 student $281.22 $270.40 1.04 $250 1.04 understanding of ` a4 the introduction to $292.46 $250 1.04 compound interest from the previous Have students conjecture a formula based on the values of the principal, P, the day. interest rate, i, and the number of times interest is charged, n to calculate the Accumulated Amount of the loan, A. Selecting Tools and Computational Strategies/Observation/ Checklist: Observe students, watching their calculator use and patterning skills. Action! Whole Class – In Pairs The compound interest formula allows us to quickly make compound interest calculations without the need for a chart. ` an A P 1 i A is the total Accumulated Amount of investment (or future value) P is the Principal i is the interest rate as a decimal n is the number of compounding periods Example: Suppose you invest $1000 at 6.37% compounded annually for 15 years. a) Calculate the Accumulated Amount at maturity. b) Calculate the interest earned. Solution: a) A = ? P = $1000 i = 0.0637 n = 15 ` an A P 1 i ` a15 $1000 1 0.0637 ` a15 $1000 1.0637 $2525.15 The Accumulated Amount of the investment is $2525.15 b) To calculate the interest earned, calculate the difference between the Accumulated Amount at maturity and the principal invested. I A @P $2525.15 @$1000.00 $1525.15 The interest earned is $1525.15 Interest can be compounded at different intervals each year. Compounding could be: Annually Through mediating Semi-annually respectful class Quarterly discussion, the Bi-weekly teacher can ensure Semi-monthly that each word wall Weekly card is complete Daily and accurate. Assign each group of four one word. As a group, they create a word-wall card for each compounding period. A reporter from each group will present and provide an overview of the group’s word wall card to the whole class before posting it in the classroom. When interest is not compounded annually then in the formula, i is the interest rate per compounding period and n is the number of times interest is compounded. To calculate i, divide the annual interest rate by the number of times interest is earned/charged per year. Example: Calculate the Accumulated Amount of an investment of $600 that is i is calculated by invested for 7 years at 5.5%/a compounded quarterly. dividing the annual interest rate by the Solution: A = ? number of P = $600 compounding i 0.055 D 2 periods in one year. 0.0275 n is calculated by n 7B 4 multiplying the ` 28an number of years A P 1 i and the number of ` a28 compounding $600 1 0.0275 periods in one year. $1282.46 The Accumulated Amount at maturity is $1282.46 Example: You are saving for a spring break vacation in two years’ time. Your bank will pay you 3.6%/a compounded monthly. How much do you need to invest now to have $2500 to pay for your vacation? 0.036 fffffffffffffffff Solution: A $2500; P ?; i 0.003; n 2 B 12 24 ` an 12 A P 1 i ` a24 $2500 P 1 0.003 ` a24 $2500 P 1.003 $2500 ffffffffffffffffffffff 24 P 1.003 $2326.58 P You have to invest $2326.58. Consolidate Small Groups Students work in Debrief If you have a choice between two investments with the same interest rate except one small groups and offers simple interest and the alternative offers compound interest, which should you record their choose? Justify your answer and support your position with an example. response on chart paper. Once all responses have been posted, the class goes on a gallery walk, each individual records their thoughts about the different responses and these are discussed in the small groups. Home Activity or Further Classroom Consolidation Application Journal Entry: x Allow students to You have learned that the equation y ab describes exponential growth. Explain discuss the x how the compound interest formula is related to y ab . Include an example in question in small your explanation. groups before they begin their independent responses. 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Na me Ca n use ex po ne nt b Assessment Checklist utt Un on de rs t Calculator use and patterning an ds r ep ea te d Ad mu ds lt 1 to ipli ca inte ti on res t ra t Ide e nti fie s th eP r in cip a l Ex pr e s ses in ter est as Re a de cog c im niz an e s th ep att er n The TVM Solver SHSM Business CLA Day 4 MBF3C Description Materials Students will use the TVM (Time Value Money) Solver to calculate different amounts in TI-83Plus or TI- compound interest problems. 84Plus calculators Assessment Opportunities Minds On… Small Groups With the assistance of the graphing calculator (not the TVM Solver application), students work on the following problem… $500 is invested at 4%/a compounded quarterly. How long would it take to grow to $600. Solution 1: Using a table. d Perio ding po un i pal unt es t Prin c Com Am o Inter 0 $500.00 0.01 $500.00 1 $500.00 0.01 $505.00 2 $500.00 0.01 $510.05 3 $500.00 0.01 $515.15 4 $500.00 0.01 $520.30 5 $500.00 0.01 $525.51 6 $500.00 0.01 $530.76 7 $500.00 0.01 $536.07 8 $500.00 0.01 $541.43 9 $500.00 0.01 $546.84 10 $500.00 0.01 $552.31 11 $500.00 0.01 $557.83 12 $500.00 0.01 $563.41 13 $500.00 0.01 $569.05 14 $500.00 0.01 $574.74 15 $500.00 0.01 $580.48 16 $500.00 0.01 $586.29 17 $500.00 0.01 $592.15 18 $500.00 0.01 $598.07 19 $500.00 0.01 $604.05 Therefore, it will take 19 compounding periods or 19/4=4.75 years (4years, 9months). To grow to $600.00. Solution 2: Using a graph and the trace function on the graphing calculator. 0.04 A $600; P $500; i fffffffffffff 0.01; n ? 4 ` an A P 1 i ` an A 500 1.01 ` ax Graph: y 500 1.01 It will take 19 compounding periods or 19/4 = 4.75 years. Action! Teacher-led Discussion There is a built-in function on the graphing calculator, TVM Solver, that allows us to Use the overhead easily determine any of the variables. projection screen to demonstrate use of On the TI-83Plus enter the key sequence [APPS]   the TVM Solver Define the screen variables: N is the number of years I% is the interest rate as a percent (not a decimal) PV is the Present Value or Principal PMT is the payment amount FV is the Future Value or Accumulated Amount P/Y is the number of payments per year N = 5 (years) C/Y is the number of compounding periods per year I% = ? PV = 5300 Example: You invest $5300 in a fund where the interest is compounded semi- PMT = 0 (we are annually. At the end of 5 years your investment is worth $6500. What was the not making regular interest rate of your investment? payments to the investment) Solution: FV = -6500 (negative because we may withdraw from the account) P/Y = 1 (no regular payments) C/Y = 2 (compounded semi- annually) After entering all of the information place the cursor The interest rate was 4.12%. beside I% and key in [ALPHA] [ENTER] to solve. Students work on this problem in pairs. Remind students that more frequent compounding results in a greater amount of interest Example: For how many years did you invest $5300 at 4.12%/a compounded earned. monthly, if the value now is $6511? Solution: The money was invested for 5 years. Consolidate Small Groups Debrief What difference would consumers notice in a loan that carries a 5%/a interest rate compounded annually versus the same interest rate compounded monthly? Include an example and calculations to support your explanation. The teacher may want to carry this out as a placemat activity. Each group divides chart paper into the same number of sections as the number of members of the group. Small Group Each person works independently in their chart paper area. A rectangle in the centre Discussions: is reserved for the group’s final thoughts. One member from each placemat group will Placemat. share the group’s response with the whole class. Think Literacy: The teacher assesses for understanding by listening. Cross- CurricularApproach es, Mathematics, Grades 10-12, pp 66-71 Home Activity or Further Classroom Consolidation Concept Practice 1. You need $4500 to buy a car. You find a savings investment that pays 3.75% compounded monthly. How much would you need to invest now if you wait 2 years to buy your car? 2. Approximately how long would it take for a $2500 investment to double if it earns 9.5%/a compounded a) annually? b) quarterly? c) monthly? d) daily? 3. How long would it take for a $1000 investment to double is interest is compounded annually at a) 4%/a? b) 6%/a? c) 8%/a? d) 12%/a? Do you notice any pattern in your answers that you could use to predict the doubling time for the same investment at 9%/a? Assign other practice/reflection questions to suit the needs of the students. Solutions: 1. N = 24 I% = 3.75% PV = ? PMT = 0 FV = 4500 P/Y = 1 C/Y = 12 PV = $1832.13 I would need to invest $1832.13 to have enough money to purchase the car in two years. 2. N=? I% = 9.5 PV = 2500 PMT = 0 FV = -5000 P/Y = 1 a) C/Y = 1 N = 7.64 It will take 7.64 years (or 7 years, 8 months) for the investment to double. b) C/Y = 4 N = 7.38 It will take 7.38 years (or 7 years, 5 months) for the investment to double. c) C/Y = 12 N = 7.32 It will take 7.32 years ( or 7 years, 4 months) for the investment to double. d) C/Y = 365 N = 7.29 It will take 7.29 years (or 7 years, 3.5 months) for the investment to double. 3. N=? I% = variable PV = 1000 PMT = 0 FV = -2000 P/Y = 1 C/Y = 1 a) I% = 4 N = 18 It will take 18 years for the investment to double. b) I% = 6 N = 12 It will take 12 years for the investment to double. c) I% = 8 N = 9 It will take 9 years for the investment to double. d) I% = 12 N = 6 It will take 6 years for the investment to double. The product of the interest rate and the number of years to double is always 72. This means that for an investment at 9%, it will take 72 D 9 8 years for the investment to double in value. Group Assessment Group Number ________ Strengths: Misconceptions: # # Philanthropy SHSM Business CLA Day 5 MBF3C Description Materials Computer lab with Students explore the Present Value of money by considering different options for access to 1 providing a scholarship at a local secondary school. Students will work in pairs to explore computer per pair. the activity thus providing support and entry points to all students. Each student will submit an individual response to the activity to be graded as a summative task. Students may decide to write a complete report to the company’s board of directors, make an oral presentation, or make an electronic presentation to the board of directors. Assessment Opportunities Minds On… Think-Pair-Share The Scholarship? Students work in pairs and make an The company you work for would like to provide a graduation scholarship at a local initial prediction secondary school. They would like the award to be given each year for the next ten providing their (10) years valued at $500.00. Currently they have $5000.00 ready to be put aside for reasoning. This the award. part of the activity Should the finance department issue a cheque to the school for $5000.00 should be done immediately? What other options are there? without access to a Provide advice in a written or oral format as part of the opening piece to spreadsheet or a your final presentation? graphing calculator. Action! In Pairs If the company decides to issue one payment for all ten years to the school, the school would receive $5000.00. If the school invests this money at current interest rates (use the web to locate these and provide sources), determine the future value of the investment assuming the interest is compounded monthly. Note: the first $500 is issued immediately, the second is invested for 12 months, the third for 24 months, etc. Given the total future value of this initial investment, if the school wanted to award the entire future value, how much should each award be worth? Note that changing the award amount will change how much is invested and thus change the total interest. Suppose the company issues $500 per year to the school and places the remaining amount in a GIC (use the web to determine current rates and provide sources). How much does the company need to put aside today? That is, they pay out $500 now and invest enough to be worth $500 in 12 months and $500 in 24 months and so on… Consolidate Individually Debrief List the advantages and disadvantages of each option. Given this information, make a complete presentation to the board of directors providing details of your analysis and a final, justified recommendation. Home Activity or Further Classroom Consolidation Complete activity for submission and grading. Curriculum Document/Achievement Chart/Evaluation Rubric January 2008 Specialist High Skills Major (SHSM) 31 Evaluation Rubric – Philanthropy Criteria Level 1 Level 2 Level 3 Level 4 - Solve problems The student solves The student solves The student solves The student uses related to simple compound more complex complex compound compound interest to compound interest interest problems to compound interest interest problems to explore possibilities determine the problems that may accurately determine beyond the lump sum amount given the require the present value of and payment plan present value, the rearrangement of the the payment plan options presented. interest rate and the compound interest option. (e.g. explore the compounding formula. (e.g. solve possibility of period. for present value accepting a lump sum given accumulated to invest and amount, interest rate withdraw amounts at and compounding various intervals of period). time). - Compare services The student makes The student makes The student The student explores available from limited mention of some reasonable incorporates the investment financial the different mention of the difference between opportunities not institutions. investments different investments investing in a GIC mentioned here as presented here. presented and compounded monthly part of their final incorporates them and investing at prime recommendation. loosely into their compounded annually OR recommendation. into their The student makes a recommendation. good argument for or against one method of investment in their final recommendation. - Apply reasoning The student uses The student uses The student uses The student makes skills to make limited mathematics to justify mathematical thorough use of mathematical mathematical conclusion and reasoning to mathematical conjectures, and reasoning to employs some formulate any concepts presented in justify conclusions. formulate mathematical conjectures and then this unit to formulate conjectures. reasoning to carries out supporting conjectures and to Conjectures seem formulate conjectures. work to justify support their position. to be mostly conclusions. The student then instinctive with proceeds to use the limited tools and concepts consideration given mentioned to verify to the concepts (or deny) the explored in this unit. conjecture and make a final conclusion. - Communicate The student makes The student makes The student uses The student makes visually and in limited use of some use of precise mathematical thorough use of all of writing. mathematical mathematical language, charts, the mathematical language, charts, language, charts, graphs, and tables of communication tools graphs, and tables graphs, and tables to values to clearly mentioned and in their work. The communicate their communicate the incorporates them mathematical message. message in their seamlessly into their reasoning is not conjectures and final work. clearly recommendation. communicated to the audience.