# Lesson Study Project in Mathematics, Fall University of Wisconsin Marathon County. Report

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1 Lessn Study Prject in Mathematics, Fall 2008 University f Wiscnsin Marathn Cunty Reprt Date: December Students: MAT 110 (Cllege Algebra) students at UW-Marathn Cunty Team Members: Paul Martin Clare Hemenway Kirthi Premadasa Research Theme (Brad Objectives) T create the ability in students t slve real-wrld prblems. T enhance the team wrking skills f Students. 1. The unit: Applicatins f Lgarithms. 2. The Lessn : The Mathematics f Credit Cards (As an applicatin f expnential equatins) 3. Abut the Unit: Amng the tpics cvered in MAT 110, are expnential functins and lgarithms. The lgarithm cmpnent cntains applicatins f lgarithms which usually invlve the slving f expnential and lgarithmic equatins using lgarithms. Students wh fllw MAT 110 end up taking a diverse selectin f career paths. It was ur bjective t get them t tackle an applicatin f lgarithms which they will find useful regardless f their career path. Fr this we selected a lessn which cntains the analysis f the different payment methdlgies invlved in credit cards. Expnential equatins are encuntered in the analysis when the time taken fr the balance t reach a certain level is calculated and students will use lgarithms t slve these expnential equatins and find the payment perids crrespnding t different payment methds.

2 4. Gals f the lessn. a) Students will learn hw t make basic interest calculatins, when calculating the mnthly payments and the mnthly interest charges n credit card balances. b) Students will learn what amrtizatin means and will learn t fill in an amrtizatin table. c) Students will use lgarithms t slve the expnential equatins encuntered in the calculatins f the payment perids fr different payment methds. d) Students will develp an insight t the advantages and the disadvantages f the different payment methds. 5. Lessn Plan fr research Lessn 1 Delivered by : Paul Martin Date/Time: December 5, 2008, 8.00 a.m. t 9.00 a.m., 23 students participated. Time Activity Anticipated Student Respnses 5 minutes Thanksgiving advert fr Laptp? 5 minutes Ask students the questin? If yu keep paying the minimum payment f a credit card lan, a. What is a minimum payment? b. What d yu think will be the cnsequences? c. When d yu think the lan will be paid ff? 5 minutes Hw abut a specific questin? Steve buys a Laptp wrth \$800 at the last Thanksgiving. Steve decides t pay nly the minimum payment. Hw lng will it take fr Steve t pay ff the lan? Make a guess! 1. Gd thing t d, it will nt effect yur credit scre 2. Yu will never pay it ff 3. Yu will pay it after a lng time 1. 3 years 2. 5 years years 5 minutes Start n activity Sheet Hand the students the amrtizatin wrksheet and ask them t find a. The mnthly interest rate b. The first payment 10 minutes Ask students t wrk ut the first three payments.

3 Guide students thrugh first step? 10 minutes T tell students that an equatin can be derived using this pattern t find the balance after n mnths. Students t eventually end up getting the answer (94 mnths) Give the fllwing specific task. Suppse we want t find the number f mnths it takes fr the lan amunt t reduce t 500. Shw students that we can use the frmula t find it. Guide students t slve the resulting expnential equatin using lgarithms. 5 minutes Discussin n hw it s nt wrth the truble and what can be dne t imprve the situatin 1. Fixed pssible payment fr mnth 2. Pay it all at nce 5 minutes Shw the amrtizatin tables appendix 1 and 2 n screen And shw that fr a fixed mnthly payment f \$ 25, we can reduce the balance t \$500 dllars Please see Appendix I fr the activity sheet fr this Lessn (Lessn 1) 6. Observatins made during the First Delivery f the research lessn.

4 The team agreed that the Lessn went very well and was well received by the students. The team felt pleasantly surprised at the student's general knwledge abut credit cards. Sme students answered the questins rally but did nt write dwn the answers t the questins in the activity sheet. Mst f the teams did nt teamwrk well. It culd be the seating arrangement. Sme teams finished the amrtizatin activity early and then had nthing t d, s engaged in idle chitchat. Tw f the students in ne team cmpletely ignred the activity and engaged in a persnal cnversatin. It wuld have mre beneficial if the slide with the actual frmula fr the mnthly balance with i=1.5% was advanced befre asking them t determine hw lng it takes t get the balance dwn t \$500. It wuld be mre beneficial if we wrk thrugh the ther payment schemes that students suggested. A graphical representatin f the varius payment plans at the end wuld have helped t drive hme the difference between the fixed vs minimum payment scenaris 7. Observatins made after studying the student wrksheets. Tasks Respnses (5 grups in all) Q1: Hw d Credit Card cmpanies make Q1: (take) yur mney? All grups said interest rates, Tw grups said late charges. Q2: What is a typical interest rate charged n credit card lans? What kind f cmpunding is used? Q2: 18-20%; 12%; 18, 20%; 9-19%; 12-18%. All grups said Mnthly Q3: What is the Minimum Payment n a credit card each mnth? Q4: What are sme cnsequences f yur paying nly the minimum amunt each mnth? Q3: Q4: Abut 10-15\$ 10, 15\$, r 1% One flat rate r 1% f balance. \$15-\$50 depending n card size Typically \$10-20/m. Sme cards it is based n principal balance. End up paying mre, prlngs yur payments giving mre \$ t the cmpany. Higher interest rates, takes lnger t pay ff, if yu get clse t yur limit, yu can t charge any mre until yu pay mre f it ff. Spend mre mney, higher interest rates, fees,

5 pay mre, reprt n credit scre. Yu wuldn t be paying ff yur principal, just interest leading t subsequent fees. Yu retain a balance which the card cmpanies want. Let s cnsider a specific scenari. Michael decides t buy the TV and puts the \$800 n his credit card and makes nly the minimum payment each mnth. Q5 Q5: Make a rugh guess as t hw lng it will take fr Michael t pay ff the lan. 4 years Abut 7.5 years 7 years, 2 years 7-8 years Q6: Lk at the Amrtizatin Wrksheet fr the first mnth s payment. Determine: Q7: Use the frmula t determine hw lng it takes t get the lan balance dwn t \$500. In ther wrds, Slve fr n Q6 All grups wrked the first three rws f the table crrectly. Tw grups wrked thrugh the first five payments. Q7 All grups used lgarithms t calculate the crrect number f mnths (93.77). Q8: Pay mre than minimum, duble r as much as yu can. Pay 4%, pay ff at nce, buy less expensive TV, 50% a mnth.

6 Q8: Can yu suggest a better payment methd? Duble minimum payment, pay ff in full, pay \$50 a mnth. Make higher payments, dn t use credit cards, save \$800 first. Buy it when yu have saved \$ Respnses t a ten-questin student survey n the lessn study a week later. A ttal f 16 students filled ut respnses. Of these, 7 strngly recmmended, 5 recmmended, and 4 recmmended with reservatins that we cnduct mre such lessn studies in the future. (Nne f the respnders chse the ptin t nt recmmend at all r nt sure t recmmend. Fur ther quantitative survey results are listed in the table belw. Math 110 Lessn Study Survey Results (Paul Martin) Q.1. Hw well did the L.S. help yu learn abut hw credit cards wrk? Q.2. Hw did yu like the LS? 2 Q.7. Hw well did the L.S. shw hw math can be used t slve real life prblems? 1 5 Q.8. T what extent did teamwrk help yu learn? 4 Nt at All Nt Very Well OK Quite Well Extremely Well The fllwing is a transcriptin f student respnses t five ther questins in the student survey. On the questin f What did yu like abut Friday s Lessn? Wrking in grups, The idea that it culd be applied in the real wrld I liked being able t talk t and wrk with ther students. Shwn hw math was used in practical life r real life usage. Cunted as 10 pints. It was a learning experience. The fact that if yu just pay the minimum, it was interesting hw lng it wuld take t pay it ff.

7 Hw lng it takes t pay ff a balance. The lessn was easy t fllw and wrking in grups was fun. The interactin with the instructrs and classmates. A change in math class, peple gave their pinins, grup activity. The grup wrk and hw the website was shwn. We talked as a whle. Gt everyne invlved. I learned abut credit cards withut the pain f a bring lecture it was hands n. Hw we were able t wrk in grups t learn abut credit cards and hw we learned hw they wrked. It taught us abut credit cards and hw they wrk in a fun interesting manner. Nthing. On the questin f What did yu dislike abut Friday s Lessn? Nthing ther than that I didn t put much int lking nice that day and ended up getting filmed, (g figure). Didn t use actual credit card statements minus persnal infrmatin. Being filmed. There was sme repetitin. It tk the entire class perid and nne f it was n the exam. Nthing, it was gd. There wasn t anything I didn t like. I thught it was a well taught lessn. We talked abut credit. I feel like the camera caught me n a bad day. There was a camera. The camera caught name n a bad day. At my life right nw, I am nt respnsible enugh t have a credit card s I did nt pay attentin at all. On the questin f What suggestins d yu have fr imprving Friday s Lessn? Use actual credit card bill minus persnal infrmatin. Dn t film it. Have student d 3 simple equatins. Making it mre fun. Maybe having an activity r a game. Yu culd have mre examples abut credit cards. Tell us ahead f time and make it a bit mre rganized. On the questin f What made Friday s Lessn interesting r nt interesting? It made yu think abut the current financial crisis. Learning hw lng it takes t pay fr things using credit cards was very interesting. Set up a lessn that it was yur wn credit card being used and that yu want t check t make sure the cmpany did it crrectly. T see hw credit card cmpanies rape yu.

8 It applies directly t life. The numbers made it interesting. Hw lng it takes t pay ff a balance. It was interesting t see hw lng it might take t pay f a televisin. The infrmatin n minimum payments. We were talking abut smething we might need in the near future. I thught that everything was interesting. The internet page we had. (This was an amrtizatin applet.) The fact that it was a life lessn made it interesting. Being in grups and having t wrk in made it mre interesting. Learned different ways f credit card usage. All f the prfessrs. On the questin f What did yu learn in Friday s Lessn? Hw t determine hw much smething really csts yu if yu charge it. I learned that I will never buy an \$800 TV with a credit card unless I can pay mre than the minimum payment each mnth. That different labels are used that mean exactly the same thing, n wnder peple get cnfused. Hw credit card cmpanies rb yu f yur mney if yu dn t pay mre than the minimum payment. Hw credit cards wrk. Nt t pay just the minimum. Credit cards can be dangerus if nt used prperly. The imprtance f making mre than the minimum payment. That making minimum payments increases the actual cst f an item. Frm a practical standpint, saving fr a majr purchase is wiser than using a credit card and making minimum payments. Hw t calculate credit card interest and the payments. Hw t cmpute and lk at the credit card bills and hw much interest is being charged. Interest. I learned a lt. We learned hw t calculate interest. Hw t calculate the interest n yur credit card. That I still am nt respnsible t have a credit card.

9 9. Lessn Plan fr research Lessn 2 Lessn Plan (Research Lessn 2) Instructr: Kirthi Premadasa Class: MAT 110 (Sectin 3) Date: December 10, a.m. t 9.50 a.m. (Changes made t the Lessn plan are in Red) Time Guide t Instructr Activity Anticipated Student Respnses 5 minutes Thanksgiving advert fr Laptp? 5 minutes Be sure t tell students t write dwn their grup answers n their sheets. Make sure the grups are mre fcused arund ne table. Ask students the questin? If yu keep paying the minimum payment f a credit card lan, a. What is a minimum payment? b. What d yu think will be the cnsequences? c. When d yu think the lan will be paid ff? 1. Gd thing t d, it will nt effect yur credit scre 2. Yu will never pay it ff 3. Yu will pay it after a lng time 5 minutes Hw abut a specific questin? Steve buys a Laptp wrth \$800 at the last Thanksgiving. Steve decides t pay nly the minimum payment. Hw lng will it take fr Steve t pay ff the lan? Make a guess! 1. 3 years 2. 5 years years 5 minutes Hand the students the amrtizatin wrksheet and ask them t find a. The mnthly interest rate b. The first payment

10 5-7 minutes Sme teams will finish early, and if s, ask thse teams t wrk n activity 2 1. Ask students t wrk ut the first three payments. Guide students thrugh first step? 2 If a team finishes early early, have them wrk ut anther activity. The first few mnths fr say a \$25 fixed payment. 10 minutes Advance t the slide with the actual frmula fr the mnthly balance with i=1.5% befre asking them t determine hw lng it takes t get the balance dwn t \$500. T tell students that an equatin can be derived using this pattern t find the balance after n mnths. Give the fllwing specific task. Suppse we want t find the number f mnths it takes fr the lan amunt t reduce t 500. Students t eventually end up getting the answer (94 mnths) Shw students that we can use the frmula t find it. Guide students t slve the resulting expnential equatin using lgarithms. 5 minutes Discussin n hw it s nt wrth the truble and 1. Fixed pssible

11 5 minutes. what can be dne t imprve the situatin Shw the amrtizatin tables appendix 1 and 2 n screen And shw that fr a fixed mnthly payment f \$ 25, we can reduce the balance t \$500 dllars payment fr mnth 2. Pay 4% f balance. 3. Buy cheaper TV. 2. Please see Appendix II fr the activity sheet fr this Lessn (Lessn 2). Please see Appendix IV fr the student wrksheets fr Lessn Observatins made during Research Lessn 2 The team agreed that the verall lessn went very well. Mst students fund the wrding f Questin Q3 f the wrksheet which was wrded as "What is the Minimum Payment n a credit card each mnth?" was cnfusing and that many students felt that what was asking was a literal interpretatin. It was felt that the instructr gave a little t much guidance t the students t d the tasks where cmputatin was needed and felt that specially wrking as a grup, the students culd have perfrmed the cmputatinal tasks with less guidance frm the instructr. It was felt that it wuld have been better if the questin Q5, "Make a rugh guess as t hw lng it will take fr Michael t pay ff the lan" seemed like a dead end questin. It wuld have been better if it was fllwed by asking the teams fr the reasning behind their guesses The team agreed that the teams team-wrked much better than last time but still there were lapses (the circular seating arrangement in sme teams culd have caused the better team perfrmance this time). It was felt that the ideal team size shuld nt exceed 4 and that the arrangement shuld have mre than 3 students in a straight line. The team bserved that in spite f all teams finishing the tasks very mre r less n time, casual talk still happened within sme teams.

12 11. Observatins made after studying the student wrksheets (Lessn 2) Tasks Q1: Hw d Credit Card cmpanies make (take) yur mney? Q2: What is a typical interest rate charged n credit card lans? What kind f cmpunding is used? Q3: What is the Minimum Payment n a credit card each mnth? Q4: What are sme cnsequences f yur paying nly the minimum amunt each mnth? Respnses (6 grups in all) Q1: All grups said interest rates, Three grups said late charges, and ne grup said annual fees. Q2:Three grups said arund 8 t 9%. Tw grups said 18% and 21% All grups said Mnthly Q3: percentage f bill percentage f bill, r flat payment, whichever was smaller. 1/3 f ttal 20% lwest payment takes yu t pay per mnth but larger payments Let s cnsider a specific scenari. Michael decides t buy the TV and puts the \$800 n his credit card and makes nly the minimum payment each mnth. Q4: Q5 Builds interest. Lnger t pay ff Hurts Credit scre Desn t hurt credit scre Q5: Make a rugh guess as t hw lng it will take fr Michael t pay ff the lan. Q6: Lk at the Amrtizatin Wrksheet fr the first mnth s payment. Determine: (If yu finish ding the first three lines quickly, d the table again, this time assuming that the payment is flat \$25 payment, instead f 2% payment) Q7: Q6 5 years (three respnses) 1-2 years 3.5 years 3 years All grups wrked the first three rws f the table crrect. Q7 All grups used lgarithms t calculate the crrect number f mnths (93.77). (Sme students initially tried t take the lgarithm withut islating the expnential

13 Use the frmula t determine hw lng it takes t get the lan balance dwn t \$500. In ther wrds, Slve fr n Q8: Can yu suggest a better payment methd? but later crrect themselves) Q8: Cash (2 grups) Get a prmtin t have a zer interest rate. Fixed payment (mst cmmn respnse) Pay mre than minimum payment r fixed payment (als a cmmn respnse) Pay it ff when yu can r pay the mst yu can each mnth. Pay as ften as pssible 12. Future Plans A spring 2009 delivery f a 3 rd Research Lessn is planned.

14 Appendix I Activity Sheet fr Research Lessn 1 Please nte that the questins Q1-Q7 fr this activity were put n the prjectr slides fr students t see Credit Card Activity fr Math 110 Grup Member Names,,,,. This activity is being filmed and may be used fr instructinal purpses t enhance teaching. We d like each grup member t fill in the release infrmatin in the table belw. If yu wuld rather nt sign, that is k t and we will nt shw any vide with yu in it. I, the undersigned hereby give Clare Hemenway, Paul Martin, Kirthi Premadasa and the UW-Clleges permissin t use vide f me in this class fr educatinal purpses. Signature Print Name Address Tel. # Date Use the spaces belw t recrd grup discussin/respnses t the questins that will cme up during this activity. Q1: Q2: Q3: Q4:

15 Q5: Credit Card Amrtizatin Table Fr Lan Size: \$800. Annual Interest rate : 18% Minimum Payment: 2% PAYMENT SCHEDULE Mnth Minimum Payment Interest Paid Principal Paid Remaining Balance Q6: Q7: Results fr Yur suggestins in Q7:

16 Appendix II Revised Activity Sheet fr Lessn 2 Credit Card Activity fr Math 110 Research Lessn 2 (PLEASE SUBMIT ONLY ONE WORKSHEET PER GROUP) Grup Member Names,,,,. This activity is being filmed and may be used fr instructinal purpses t enhance teaching. We d like each grup member t fill in the release infrmatin in the table belw. If yu wuld rather nt sign, that is k t and we will nt shw any vide with yu in it. I, the undersigned hereby give Paul Martin, Kirthi Premadasa and the UW-Clleges permissin t use vide f me in this class fr educatinal purpses. Signature Print Name Address Tel. # Date Use the spaces belw t recrd grup discussin/respnses t the questins that will cme up during this activity. Q1: Hw d Credit Card cmpanies make (take) yur mney? Q2: What is a typical interest rate charged n credit card lans? What kind f cmpunding is used?

17 Q3: What is the Minimum Payment n a credit card each mnth? Q4: What are sme cnsequences f yur paying nly the minimum amunt each mnth? Let s cnsider a specific scenari. Michael decides t buy the TV and puts the \$800 n his credit card and makes nly the minimum payment each mnth. Q5: Make a rugh guess as t hw lng it will take fr Michael t pay ff the lan.

18 Q6: Lk at the Amrtizatin Wrksheet fr the first mnth s payment. Determine: 1. The mnthly interest rate = 2. The minimum payment. 3. The amunt f principal paid. 4. The amunt f interest paid. 5. The remaining balance. Credit Card Amrtizatin Table Fr Lan Size: \$800. Annual Interest rate : 18% Minimum Payment: 2% PAYMENT SCHEDULE Mnth Minimum Payment Interest Paid Principal Paid Remaining Balance

19 1 2 3 (If yu finish ding the first three lines quickly, d the table again, this time assuming that the payment is flat \$25 payment, instead f 2% payment)

20 Q7: Use the frmula t determine hw lng it takes t get the lan balance dwn t \$500. In ther wrds, Slve fr n Step 1: Plug in the mnthly interest rate fr i. Step 2: Islate the expnent. Step 3: Take lgarithms n bth sides Step 4: Slve fr n.

21 The number f mnths = The number f years+ Q8: Can yu suggest a better payment methd? Results fr Yur suggestins in Q8:

22 Appendix IV

23 Sample Student Wrk (Lessn 1)

24

25 Sample Student Wrk (Lessn 2)

26

27

28

29

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