Financial Mathemetics


 Agnes Miles
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1 Fnancal Mathemetcs 15
2 Mathematcs Grade 12 Teacher Gude Fnancal Maths Seres Overvew In ths seres we am to show how Mathematcs can be used to support personal fnancal decsons. In ths seres we jon Tebogo, a Grade 12 learner, who s makng plans for the future. We consder a savngs plan, a student loan, the cost of buyng a home and strateges for buyng a car. Currculum Lnks The lessons n ths seres lnk to the followng Learnng Outcomes and Assessment Standards of the Natonal Currculum Statement: Learnng Outcome 1: Number and Number Relatonshps Assessment Standard: A P 1+ (b) Apply knowledge of geometrc seres to solvng annuty, bond repayment and snkng fund problems, wth or wthout the use of the formulae: (a) Calculate the value of n n the formula: ( ) n F v x n [ 1 ( 1+ ) ] and P v x n [ 1 ( 1+ ) ] Assessment Standard Crtcally analyse nvestment and loan optons and make nformed decsons as to the best opton(s) (ncludng pyramd and mcrolenders schemes). E Educatonal Approach There are four lessons n ths seres. Apart from teachng the Mathematcal processes nvolved n the context of fnance we have take ths opportunty to stress the mportance of good fnancal plannng. We have tred to emphasze the mportance of savng and the dangers of debt. These are real ssues that many young people have to face when they leave school. We suggest that ths seres provdes an opportunty to dscuss mportance ssues, such as poverty and wealth, too. We have not just used the formulae n ths seres but have ntroduced the concepts of an annuty and a loan by compound nterest from frst prncples and have shown how each month s contrbuton forms a geometrc seres. It would be useful to revse calculatons of geometrc seres and fndng the sum of a geometrc seres before startng ths secton. In the teacher vdeo gude, we have also shown how to derve the future value and present value formulae. In addton, we use logarthms to solve the number of repayments requred for the compound nterest formula and then apply ths to the present value formula, when calculatng a home loan. To get the full beneft of the lessons, your learners need to engage actvely wth the concepts presented. So, when you prevew the vdeos, thnk about how to ntroduce each lesson and what follow up actvtes wll be useful. Also watch out for places n the vdeo where you could pause to have a class dscusson, or ask learners to complete an actvty or solve a problem posed n the vdeo. In the vdeo lessons, we use a pcture of a pause button where we thnk you mght want to stop the tape to have a class dscusson, or ask learners to complete an actvty or solve a problem posed n the vdeo. 16
3 Mathematcs Grade 12 Teacher Gude Fnancal Maths Seres at a Glance Lesson Ttle Lesson Outcomes By the end of ths lesson, the learner should be able to: 1. Savng for future studes apply knowledge of geometrc seres to solve future value annuty problems solve future value annuty problems by usng the future value annuty formula: F n x[( 1+ ) 1] 2. Evaluatng loan optons crtcally examne and analyse varous loan optons usng the Present value formula: P x[(1 (1 + ) n ] 3. Housng loans solve dfferent home loan problems usng the formula: P x[(1 (1 + ) n ] 4. A snkng fund explan the terms nflaton and deprecaton solve problems that requre you to establsh a snkng fund 17
4 Mathematcs Grade 12 Teacher Gude Fnancal Maths G Teachng Gudelnes Lesson 1: Savng for future studes In ths lesson we jon Tebogo as he plans for ways to pay for hs future studes. Ths s an deal opportunty to dscuss the future wth your learners and to alert them to make adequate preparatons. You could make them aware of career opportuntes that nvolve Mathematcs and gude to makng applcatons for regstraton and bursares. The approach taken of seeng what happens to a sngle R700 when t s nvested for nne months may be dfferent from other books. However, your learners wll develop a very clear dea that regular savngs over a longer tme gves the most beneft. We would encourage your learners to fll n the values n a table. Ths data can be used to calculate the values of the other nvestment amount for dfferent months. Please note that ths approach does not gve the balance n the annuty after each month drectly. However, the values from the table added together gve the future value of the annuty. We show that these terms form a geometrc seres and use the sum formula to calculate the future value. We confrm these calculatons usng the future value formula. In the teacher vdeo gude, we show how the future value formula can be derved from a dfferent geometrc seres. You may want to extend your more able learners by askng them to do ths dervaton too. In ths task learners need to use the future value formula to fnd exactly how much Tebogo needs to save each month so that he can have enough money to pay for hs fees. Lesson 2: Evaluatng loan optons Ths lesson examnes dfferent loan optons. We suggest that you dscuss the problem of debt wth your learners. May school leavers do not manage ther fnances well and often spend more than they can afford. Easy credt s often not the most sensble opton to take. The approach used n ths lesson s to start wth an affordable repayment n mnd and then work backwards to fnd the present value of the loan. We show that loan repayments also form a geometrc seres and use the sum formula to calculate the present value of a loan. You may wsh to pont out that when payng off a loan, fnancal nsttutons actually reverse the geometrc seres. Ths means that your frst repayment s mostly nterest and very lttle of the captal s reduced. There are many opportuntes for learners to partcpate n ths lesson. We would encourage you to pause the vdeo so that they can complete the calculaton and then check them when you restart the vdeo. The task n ths lesson requres learners to compare a bank student loan to a NSFAS student loan. NSFAS offer loans lnked to nflaton but n ths task we have gnored nflaton and mantaned a constant rate. You may wsh to extend some of your more able learners by changng the rate of nflaton too. Encourage your able learners who are need fnancal assstance to apply for NSFAS student loans. 18
5 Mathematcs Grade 12 Teacher Gude Fnancal Maths Lesson 3: Housng loans In ths lesson Tebogo and Deyasha talk about hs plans for the future. The purchase of a house s probably the sngle bggest nvestment most people make. But buyng property s a good fnancal strategy because the value often ncreases over tme. Tebogo goes through the process of plannng for a home loan. He chooses a small apartment, decdes what he wll be able to afford n repayments and then calculates how long t wll take hm to pay off or amortze a home loan. To do ths we use logarthms. We frst show how to calculate the number of terms n n the compound nterest formula and then extend ths process to the present value formula. We show that t does not make sense to choose a longer perod to pay off a home loan. We suggest that you could gve learners addtonal problems on home loans to solve ncludng cases where the repayment exceeds the mnmum requrements. Learners are requred to use the present value formula to calculate the repayment of a home loan over twenty years. They are then told that the nterest ncreases by 0,5%. They need to calculate the new repayment and the dfferent ths small change n nterest rate makes over twenty years. Lesson 4: A Snkng fund IIn the future Tebogo would lke to own a car. In ths lesson we nvestgate how a snkng fund can help save you money n the long term. We show that a car deprecates n value, unlke a house and so a snkng fund can be used to help you buy a replacement vehcle wthout havng to take out a loan The task renforces the learners understandng of a nflaton, deprecaton and the advantages of establshng a snkng fund to replace machnery. 19
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