Enhanced Instructional Transition Guide

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1 0-0 Ehaced Istructioal Trasitio Guide High School Courses Uit Number: /Mathematics Suggested Duratio: 0 days Uit : Sequeces ad Series (0 days) Possible Lesso 0 (0 days) POSSIBLE LESSON 0 (0 DAYS) This lesso is oe approach to teachig the State Stadards associated with this uit. Districts are ecouraged to customize this lesso by supplemetig with districtapproved resources, materials, ad activities to best meet the eeds of learers. The duratio for this lesso is oly a recommedatio, ad districts may modify the time frame to meet studets eeds. To better uderstad how your district is implemetig CSCOPE lessos, please cotact your child s teacher. (For your coveiece, please fid liked the TEA Commissioer s List of State Board of Educatio Approved Istructioal Resources ad Midcycle State Adopted Istructioal Materials.) Lesso Syopsis: Studets develop ad apply the vocabulary ad formulas ivolved with arithmetic ad geometric sequeces ad series. Studets formulate recursive ad explicit formulas to represet sequeces. Studets use these skills to solve problems i a variety of scearios, icludig biomial coefficiets ad coverget series. TEKS: The Texas Essetial Kowledge ad Skills (TEKS) listed below are the stadards adopted by the State Board of Educatio, which are required by Texas law. Ay stadard that has a strike-through (e.g. sample phrase) idicates that portio of the stadard is taught i a previous or subsequet uit. The TEKS are available o the Texas Educatio Agecy website at P. /Kowledge ad Skills. The studet uses sequeces ad series as well as tools ad techology to represet, aalyze, ad solve real-life problems. The studet is expected to: P.A P.B P.C P.D Represet patters usig arithmetic ad geometric sequeces ad series. Use arithmetic, geometric, ad other sequeces ad series to solve real-life problems. Describe limits of sequeces ad apply their properties to ivestigate coverget ad diverget series. Apply sequeces ad series to solve problems icludig sums ad biomial expasio. Performace Idicator(s): page of 90

2 0-0 Ehaced Istructioal Trasitio Guide High School Courses Uit Number: /Mathematics Suggested Duratio: 0 days High School Mathematics Uit PI 0 Aalyze a problem situatio ivolvig sequeces ad series such as the followig: A motocross ramp is supported by 8 evely-spaced wood colums, as show. The first pole is ft tall, the secod is. ft tall, ad the third is ft tall. Write a recursive formula to represet a. Write a explicit formula to fid a, the height of the th colum. Use the explicit formula to evaluate a 8, ad iterpret its meaig i the cotext of the problem. Express the series i sigma otatio. Use the series formula to determie how may total feet of lumber would be required to costruct all 8 support colums? Tomas drew the desig of the spiral below o grid paper to model the ext sectio of his irowork sculpture. Write a recursive formula to represet a. Write a explicit formula to fid a, the legth of the th sectio of iro. Use the explicit formula to evaluate a 9, ad iterpret its meaig i the cotext of the problem. Express the series i sigma otatio. Use the series formula to evaluate S 9, ad iterpret its meaig i the cotext of the problem. Each day, Gradma puts 8 ouces of water i her flower pot. Over the ext hours, 0% of the water evaporates (leavig 60%), at which time she adds aother 8 ouces. Complete the chart to show the amout of water (w ) i the flower pot after days. page of 90

3 0-0 Ehaced Istructioal Trasitio Guide High School Courses Uit Number: /Mathematics Suggested Duratio: 0 days Write the recursive ad explicit formulas to fid w, the amout of water i the flower pot after days. What is the limit of w as? 8? What does this represet i the problem situatio? O ay give school day, Jim has a 70% chace of beig o time (or a 0% chace of beig tardy). For the -day school week ahead, determie each biomial probability. P(o time all days) P(o time exactly days) P(o time exactly days) P(o time at least days) Create a graphic orgaizer for each situatio that icludes ) Table, graph, ad appropriate represetatios ) Formulas ad calculatios ) Justificatio of predictios i terms of the problem situatio Stadard(s): P.A, P.B, P.C, P.D ELPS ELPS.c.C, ELPS.c.H, ELPS.c.G Key Uderstadig(s): Formulas for sequeces ad series i problem situatios ca be determie by aalysis of patters. Patters ca be geeralized ad modeled usig explicitly ad recursively defied fuctios to represet relatioships i terms of the problem situatio. Diverget ad coverget series ca be ivestigated usig the limits ad properties of their respective sequeces. I a coverget series, the ifiite sequece of the partial sums approaches a limit. I a diverget series, the ifiite sequece of the partial sums does ot approach a limit. Miscoceptio(s): Some studets may thik that there is o distictio betwee the term i a sequece (a ) ad its positio i the sequece (), particularly whe usig the related formulas. For example, the fifth term i a sequece (a ) is ot ecessarily a five (eve though = ). For studets, this is particularly troublesome i the otatio for recursively defied sequeces, such as a = a - +. page of 90

4 0-0 Ehaced Istructioal Trasitio Guide High School Courses Uit Number: /Mathematics Suggested Duratio: 0 days Vocabulary of Istructio: amortizatio biomial probability coefficiet commo differece commo ratio coverge diverge iterest rate limit paymet positio probability recursive sequece series term Materials List: graphig calculator ( per studet) graphig calculator with display ( per teacher) Attachmets: All attachmets associated with this lesso are refereced i the body of the lesso. Due to cosideratios for gradig or studet assessmet, attachmets that are coected with Performace Idicators or serve as aswer keys are available i the district site ad are ot accessible o the public website. Patters KEY Patters Arithmetic ad Geometric Sequeces KEY Arithmetic ad Geometric Sequeces Yes We Ca KEY Yes We Ca page of 90

5 0-0 Ehaced Istructioal Trasitio Guide High School Courses Uit Number: /Mathematics Suggested Duratio: 0 days Series KEY Series Sequeces ad Series Problems KEY Sequeces ad Series Problems Goig ad Goig KEY Goig ad Goig Ifiite Series KEY Ifiite Series Repeatig Decimals KEY Repeatig Decimals Meds ad Moey KEY Meds ad Moey From Recursive to Explicit KEY From Recursive to Explicit Amortizatio KEY Amortizatio Aythigs Possible KEY Aythigs Possible page of 90

6 0-0 Ehaced Istructioal Trasitio Guide High School Courses Uit Number: /Mathematics Suggested Duratio: 0 days Permutatios ad Combiatios KEY Permutatios ad Combiatios Combiatios ad Pascals Triagle KEY Combiatios ad Pascals Triagle Biomial Power KEY Biomial Power Biomial Probability KEY Biomial Probability Applicatios of Sequeces ad Series KEY Applicatios of Sequeces ad Series PI Suggested Day Suggested Istructioal Procedures Notes for Teacher Topics: Patters Egage Studets aalyze, exted, ad make predictios usig patters. Istructioal Procedures:. Place studets i pairs. Distribute hadout: Patters to each studet. Refer studets to ATTACHMENTS Teacher Resource: Patters KEY ( per teacher) Teacher Resource: Patters ( per teacher) Hadout: Patters ( per studet) MATERIALS page 6 of 90

7 0-0 Ehaced Istructioal Trasitio Guide High School Courses Uit Number: /Mathematics Suggested Duratio: 0 days Suggested Day Suggested Istructioal Procedures the patters o page. Istruct studets to work with their parter to complete problems 9, leavig the Type colum of the chart blak for ow. Allow studets approximately 0 miutes to complete the table. Moitor ad assess studets to check for uderstadig.. Display teacher resource: Patters, ad facilitate a class discussio of studet results, clarifyig ay miscoceptios. Ask: O problem, what is the first observatio you look for i a patter? Aswers may vary. See if the umbers go up or dow by the same amout each time; etc. How do you verify that the umbers i the patter are goig up or dow by the same amout each time? Aswers may vary. You ca subtract the terms to see if you get the same thig every time; etc. Cosider problem. If there is ot a umber that is added or subtracted each time, what observatio should you look for ext? Aswers may vary. Look to see if you multiply or divide by the same thig each time; etc. Are there some patters where either strategy works? Aswers may vary. Yes, but these two strategies are always a good place to start; etc. Notes for Teacher graphig calculator ( per studet) graphig calculator with display ( per teacher) TEACHER NOTE For the Explaatio sectio of problems 9 o hadout: Patters, studets typically describe a recursive process, such as add 7 each time or multiply by 0.. Whe studets complete problems 7, however, they will be usig a explicit formula.. Usig teacher resource: Patters, facilitate a class discussio to idetify the type of each patter i problems 9 usig A, G, or O. Istruct studets to fill i the Type colum of the chart. Say: I the Type colum, I will ask you to record a letter. The letter will either be a A, a G, or a O. At first, you may ot kow how I am labelig each; however, after several examples, you page 7 of 90

8 0-0 Ehaced Istructioal Trasitio Guide High School Courses Uit Number: /Mathematics Suggested Duratio: 0 days Suggested Day Suggested Istructioal Procedures should be able to determie the reasoig. Notes for Teacher. Idetify the type of sequece for the first four problems. Istruct studets to predict the Type for problem. What do the A patters, problems ad, have i commo? Aswers may vary. They are both add/subtract patters; etc. Is problem the same operatio as problems ad? Aswers may vary. No, it is a multiplicatio/divisio patter; etc. What would you put i the Type colum for problem? (G). Istruct studets to complete the Type colum for problems 6 9. Next, istruct studets to complete problems 0. Allow studets time to complete the problems. Moitor ad assess studets to check for uderstadig. Usig teacher resource: Patters, facilitate a class discussio of studet results, clarifyig ay miscoceptios. 6. Refer studets to the top of page. Usig teacher resource: Patters, facilitate a class discussio to summarize the types of sequeces by completig the chart at the top of page. 7. Istruct studets to work idepedetly to complete problems 0 o hadout: Patters. This may be completed as homework, if ecessary. Topics: Recursive formulas Explicit formulas Arithmetic sequeces ATTACHMENTS Teacher Resource: Arithmetic ad Geometric Sequeces KEY ( per teacher) page 8 of 90

9 0-0 Ehaced Istructioal Trasitio Guide High School Courses Uit Number: /Mathematics Suggested Duratio: 0 days Suggested Day Suggested Istructioal Procedures Geometric sequeces Applicatios of arithmetic ad geometric sequeces Explore/Explai Studets defie terms related to sequeces. Studets develop a uderstadig of patters i sequeces by formulatig recursive ad explicit formulas. Studets ivestigate ad compare arithmetic ad geometric sequeces. Notes for Teacher Teacher Resource: Arithmetic ad Geometric Sequeces ( per teacher) Hadout: Arithmetic ad Geometric Sequeces ( per studet) Teacher Resource: Yes, We Ca KEY ( per teacher) Hadout: Yes, We Ca ( per studet) Istructioal Procedures: MATERIALS. Place studets i pairs. Distribute hadout: Arithmetic ad Geometric Sequeces to each studet. Refer studets to the Vocabulary sectio o page. Display teacher resource: Arithmetic ad Geometric Sequeces, ad facilitate a class discussio of the vocabulary of sequeces.. Refer studets to the Arithmetic Sequeces sectio of hadout: Arithmetic ad Geometric Sequeces. Usig teacher resource: Arithmetic ad Geometric Sequeces, facilitate a class discussio of arithmetic sequeces, modelig problems. Istruct studets to work with their parter to complete problems. Allow studets time to complete the problems. Moitor to check for uderstadig. Usig teacher resource: Arithmetic ad Geometric Sequeces, facilitate a class discussio of studet results, clarifyig ay miscoceptios.. Refer studets to the otes o Recursive ad Explicit o page of hadout: Arithmetic ad Geometric Sequeces. Usig teacher resource: Arithmetic ad Geometric Sequeces, facilitate a class discussio of recursive ad explicit formulas, modelig graphig calculator ( per studet) graphig calculator with display ( per teacher) TEACHER NOTE Problem o hadout: Yes, We Ca requires studets to total a list of about 0 umbers, which ca become quite tedious. The tedium of this exercise is meat to drive the eed for the summatio formulas preseted i the subsequet activity. However, for a temporary fix, you ca use the lists i the calculator. Ito L, eter the umbers from to 0. Fill i L usig the explicit fuctio: L = 0000(.0)^(L ). page 9 of 90

10 0-0 Ehaced Istructioal Trasitio Guide High School Courses Uit Number: /Mathematics Suggested Duratio: 0 days Suggested Day Suggested Istructioal Procedures problems ad 7. Istruct studets to work with their parter to complete problems 6 ad 8. Allow studets time to complete the problems. Moitor ad assess studets to check for uderstadig. Usig teacher resource: Arithmetic ad Geometric Sequeces, facilitate a class discussio of studet results, clarifyig ay miscoceptios. Emphasize the differeces betwee the recursive ad explicit formulas. Notes for Teacher Recursive formulas require the terms of a sequece to be developed i order; however, i explicit formulas, you ca fid ay terms i ay order. While a explicit formula is a sigle fuctio or equatio, a recursive defiitio requires two equatios: oe statig the first term, ad a secod defiig the process for gettig the other terms. From the home scree, you ca use the sum commad ( d, [STAT], move over to MATH ad dow to :sum).. Refer studets to the Geometric Sequeces sectio of hadout: Arithmetic ad Geometric Sequeces. Usig teacher resource: Arithmetic ad Geometric Sequeces, facilitate a class discussio of geometric sequeces, modelig problems 9 0. Istruct studets to work with their parter to complete problems. Allow studets time to complete the problems. Moitor ad assess studets to check for uderstadig. Usig teacher resource: Arithmetic ad Geometric Sequeces, facilitate a class discussio of studet results, clarifyig ay miscoceptios.. Refer studets to the otes o Recursive ad Explicit o page of hadout: Arithmetic ad Geometric Sequeces. Usig teacher resource: Arithmetic ad Geometric Sequeces, facilitate a class discussio of recursive ad explicit formulas, modelig problems ad. Istruct studets to work with their parter to complete problems ad 6. Allow studets time to complete the problems. Moitor to check for uderstadig. Usig teacher resource: Arithmetic ad Geometric Sequeces, page 0 of 90

11 0-0 Ehaced Istructioal Trasitio Guide High School Courses Uit Number: /Mathematics Suggested Duratio: 0 days Suggested Day Suggested Istructioal Procedures facilitate a class discussio of studet results, clarifyig ay miscoceptios. 6. Istruct studets to work with their parter to complete problems 7 0 o hadout: Arithmetic ad Geometric Sequeces. Allow studets time to complete the problems. Moitor ad assess studets to check for uderstadig. Usig teacher resource: Arithmetic ad Geometric Sequeces, facilitate a class discussio of studet results, clarifyig ay miscoceptios. 7. Distribute hadout: Yes, We Ca to each studet. Istruct studets to work with their parter to complete the hadout. This may be completed as homework, if ecessary. Notes for Teacher Topics: Series Applicatios of arithmetic ad geometric sequeces ad series Explore/Explai Studets defie series ad apply sigma otatio ad formulas for arithmetic ad geometric series. Studets model problem situatios usig arithmetic ad geometric sequeces. Istructioal Procedures:. Place studets i pairs. Distribute hadout: Series to each studet. Refer studets to page of the hadout. Display teacher resource: Series, ad facilitate a class discussio of series ad sigma otatio, modelig problems. Istruct studets to work with their parter to complete problem. Allow studets time to complete the problem. Moitor ad assess studets to check for uderstadig. Usig teacher ATTACHMENTS Teacher Resource: Series KEY ( per teacher) Teacher Resource: Series ( per teacher) Hadout: Series ( per studet) Teacher Resource: Sequeces ad Series Problems KEY ( per teacher) Hadout: Sequeces ad Series Problems ( per studet) MATERIALS graphig calculator ( per studet) graphig calculator with display ( per page of 90

12 0-0 Ehaced Istructioal Trasitio Guide High School Courses Uit Number: /Mathematics Suggested Duratio: 0 days Suggested Day Suggested Istructioal Procedures resource: Series, facilitate a class discussio of studet results, clarifyig ay miscoceptios.. Refer studets to the top of page o hadout: Series. Usig teacher resource: Series, facilitate a class discussio of shortcuts for totalig terms i a sequece, particularly if the sequeces are arithmetic or geometric. Go over the summatio formulas at the top of page, ad model problems 6.. Istruct studets to work with their parter o problems 7 0 o hadout: Series. Allow studets time to complete the problems. Moitor ad assess studets to check for uderstadig. Usig teacher resource: Series, facilitate a class discussio of studet results, clarifyig ay miscoceptios.. Distribute hadout: Sequeces ad Series Problems to each studet. Istruct studets to work with their parter to complete the problems. This may be completed as homework, if ecessary. Notes for Teacher teacher) TEACHER NOTE As time allows, explai why the summatio formulas work. The arithmetic formula, for example, comes from the fact that the sum of the first ad last terms i the series is the same as the sum of the secod ad extto-last terms, etc. Pairig terms i this way creates pairs that have the same sum (a +a ), thus geeratig the formula provided. Similar proofs or demostratios are available i various resources for the summatio formula for terms i a geometric series. TEACHER NOTE As a extesio, itroduce studets to Karl Friedrich Gauss (777-8). I school, whe his teacher gave the problem of summig the itegers from to 00 to keep the studets busy, Gauss immediately wrote dow the correct aswer of 00 o his slate. page of 90

13 0-0 Ehaced Istructioal Trasitio Guide High School Courses Uit Number: /Mathematics Suggested Duratio: 0 days Suggested Day Suggested Istructioal Procedures Notes for Teacher Gauss's presumed method was to realize that pairwise additio of terms from opposite eds of the list yielded idetical itermediate sums: + 00 = 0, + 99 = 0, + 98 = 0, ad so o, for a total sum of 0 0 = 00. Topics: ATTACHMENTS Ifiite series Applicatios of ifiite series Explore/Explai Studets defie ifiite series ad apply sigma otatio ad formulas for ifiite series. Studets model problem situatios usig ifiite series, icludig repeatig decimals. Istructioal Procedures:. Place studets i pairs. Distribute hadout: Goig ad Goig to each studet. Istruct studets to work with their parter, read the calculator istructios at the top of page to eter sigma otatio ito the calculator, ad complete problems 0. Allow studets time to complete the problems. Moitor ad assess studets to check for uderstadig. Facilitate a class discussio of studet results, clarifyig ay miscoceptios.. Distribute hadout: Ifiite Series to each studet. Refer studets to the Covergece Teacher Resource: Goig ad Goig KEY ( per teacher) Hadout: Goig ad Goig ( per studet) Teacher Resource: Ifiite Series KEY ( per teacher) Teacher Resource: Ifiite Series ( per teacher) Hadout: Ifiite Series ( per studet) Teacher Resource: Repeatig Decimals KEY ( per teacher) Hadout: Repeatig Decimals ( per studet) MATERIALS page of 90

14 0-0 Ehaced Istructioal Trasitio Guide High School Courses Uit Number: /Mathematics Suggested Duratio: 0 days Suggested Day Suggested Istructioal Procedures of a Sequece sectio. Display teacher resource: Ifiite Series, ad facilitate a class discussio of covergig ad divergig sequeces. Istruct studets to work with their parter to complete problems. Allow studets time to complete the problems. Moitor ad assess studets to check for uderstadig. Usig teacher resource: Ifiite Series, facilitate a class discussio of studet results, clarifyig ay miscoceptios.. Refer studets to the Covergece of a Series sectio. Usig teacher resource: Ifiite Series, facilitate a class discussio of covergig series. Istruct studets to work with their parter to complete problems 7. Allow studets time to complete the problems. Moitor ad assess studets to check for uderstadig. Usig teacher resource: Ifiite Series, facilitate a class discussio of studet results, clarifyig ay miscoceptios.. Refer studets to the Note ad Ifiite Geometric Series sectios. Usig teacher resource: Ifiite Series, facilitate a class discussio to summarize covergig series ad to defie the characteristics of a ifiite geometric series, modelig problems 8 ad. Istruct studets to work with their parter to complete the remaiig problems o hadout: Ifiite Series. Allow studets time to complete the problems. Moitor ad assess studets to check for uderstadig. Usig teacher resource: Ifiite Series, facilitate a class discussio of studet results, clarifyig ay miscoceptios.. For additioal practice, distribute hadout: Repeatig Decimals to each studet. Istruct studets to work idepedetly to complete the hadout. This may be completed for homework, if ecessary. Notes for Teacher graphig calculator ( per studet) graphig calculator with display ( per teacher) TEACHER NOTE Whe the calculator is i sequece mode, the variable key will give the small used i sequeces. However, calculatios also work usig the ALPHA N method. TEACHER NOTE Whe studets take Calculus II i college (or AP Calculus BC i high school), they sped some eergy discussig the differece betwee a sequece that coverges ad a series that coverges. Specifically, if a ifiite series of the form Σa coverges, the the related sequece a must also coverge, ad it must coverge to 0 (or a 0). However, just because the terms of a sequece approach 0, this does ot mea that the related ifiite series will coverge. The famous example is called the harmoic series: page of 90

15 0-0 Ehaced Istructioal Trasitio Guide High School Courses Uit Number: /Mathematics Suggested Duratio: 0 days Suggested Day Suggested Istructioal Procedures Notes for Teacher While the sequece a = coverges (a 0), the ifiite series diverges (or, the totals keep icreasig without limit). (See problem 6 o hadout: Goig ad Goig.) - 6 Topics: Applicatios of recursive ad explicit formulas Amortizatio Explore/Explai Studets model ad solve problems usig recursive ad explicit formulas. Studets apply amortizatio formulas to determie variables o loas. Istructioal Procedures: Day. Place studets o pairs. Distribute hadout: Meds ad Moey to each studet. Facilitate a class discussio of the problem situatio o page ad the problem situatio o page. Istruct studets to work with their parter to complete problems 8. Allow studets time to complete the problems. Moitor ad assess studets to check for uderstadig. Facilitate a class discussio of studet results, clarifyig ay miscoceptios.. Distribute hadout: From Recursive to Explicit to each studet. This hadout makes ATTACHMENTS Teacher Resource: Meds ad Moey KEY ( per teacher) Hadout: Meds ad Moey ( per studet) Teacher Resource: From Recursive to Explicit KEY ( per teacher) Teacher Resource: From Recursive to Explicit ( per teacher) Hadout: From Recursive to Explicit ( per studet) Teacher Resource: Amortizatio KEY ( per teacher) Teacher Resource: Amortizatio ( per teacher) Hadout: Amortizatio ( per studet) MATERIALS page of 90

16 0-0 Ehaced Istructioal Trasitio Guide High School Courses Uit Number: /Mathematics Suggested Duratio: 0 days Suggested Day Suggested Istructioal Procedures referece to the problem sets about doses of medicie ad paymets o a credit card from the previous hadout; therefore, studets eed to keep hadout: Meds ad Moey for referece. Istruct studets to work with their parter to complete the hadout. This may be completed as homework, if ecessary. Day 6. Display teacher resource: From Recursive to Explicit, ad facilitate a class discussio to debrief hadout: From Recursive to Explicit. Ask: Notes for Teacher graphig calculator ( per studet) graphig calculator with display ( per teacher) TEACHER NOTE Referece materials for the various types of graphig calculators should have istructios for use of the TVM solver. This is sometimes located i Applicatios uder Fiace. For problem 9, what is the first umber listed i the problem? ($9,000) Would you call that a A, P, or r? Why? (A; It is the amout that is origially owed.) What is the paymet P? ($00) What is r? (The r is 7.%, or 0.07.) Look at the explicit formula i the box. The first value i the equatio is. What is the value of? (( ) = 6,000) Next, what is the value of A? (9,000 6,000 =,000) How would you evaluate ( + )? Aswers may vary. We ca use ( + ), or divide ad add to get (.006); etc. What does the explicit formula for B look like? (B = 6,000,000(.006).) page 6 of 90

17 0-0 Ehaced Istructioal Trasitio Guide High School Courses Uit Number: /Mathematics Suggested Duratio: 0 days Suggested Day Suggested Istructioal Procedures Notes for Teacher. Related to these formulas to fid the balace due o a loa, there are also formulas that ca be used to fid the umber of moths or the paymets required to pay off a loa with a certai aual iterest rate. These formulas are provided at the top of the ext activity. Place studets i pairs. Distribute hadout: Amortizatio to each studet.. Display teacher resource: Amortizatio, ad facilitate a class discussio of amortizatio, modelig problem. Ask: After readig problem, decide whether you are asked to determie the mothly paymet or the umber of moths. (The mothly paymet.) Look at the formula, ad determie what variables must be used. (You eed to kow A, r, ad.) What is $80,000? (This is the amout borrowed (A).) What is the 6%? (That s the aual iterest rate (r).) What should be used for? (0, because it s supposed to be i moths.) 6. Istruct studets to work with their parter to complete problems. Allow studets time to complete the problems. Moitor ad assess studets to check for uderstadig. Usig teacher resource: Amortizatio, facilitate a class discussio of studet results, clarifyig ay miscoceptios. 7. Demostrate for studets how the graphig calculator ca use a TVM solver to calculate solutios for amortizatio problems with the push of a butto. Model problem usig the TVM solver o the graphig calculator. page 7 of 90

18 0-0 Ehaced Istructioal Trasitio Guide High School Courses Uit Number: /Mathematics Suggested Duratio: 0 days Suggested Day Suggested Istructioal Procedures 8. Istruct studets to work idepedetly to complete problems 6 0 usig the TVM solver o the graphig calculator. Notes for Teacher 7 Topics: Factorials Permutatios Combiatios Pascal s Triagle Explore/Explai Studets evaluate factorial expressios ad determie probabilities of permutatios ad combiatios. Studets apply combiatios to ivestigate Pascal s Triagle. Istructioal Procedures:. Place studets i pairs. Distribute hadout: Aythig s Possible to each studet. Istruct studets to work with their parter to complete problems. Allow studets to complete the problems. Moitor ad assess studets to check for uderstadig. Facilitate a class discussio of studet results, clarifyig ay miscoceptios.. Istruct studets to work with their parter to complete problem. Allow studets to complete the problems. Moitor ad assess studets to check for uderstadig. Facilitate a class discussio of studet results, clarifyig ay miscoceptios.. Distribute hadout: Permutatios ad Combiatios to each studet. Refer studets ATTACHMENTS Teacher Resource: Aythig s Possible KEY ( per teacher) Hadout: Aythig s Possible ( per studet) Teacher Resource: Permutatios ad Combiatios KEY ( per teacher) Hadout: Permutatios ad Combiatios ( per studet) Teacher Resource: Combiatios ad Pascal s Triagle KEY ( per teacher) Hadout: Combiatios ad Pascal s Triagle ( per studet) MATERIALS graphig calculator ( per studet) graphig calculator with display ( per teacher) TEACHER NOTE page 8 of 90

19 0-0 Ehaced Istructioal Trasitio Guide High School Courses Uit Number: /Mathematics Suggested Duratio: 0 days Suggested Day Suggested Istructioal Procedures to the Factorial sectio. Display teacher resource: Permutatios ad Combiatios, ad facilitate a class discussio of factorials. Istruct studets to work with their parter to complete problems 6. Allow studets to complete the problems. Moitor ad assess studets to check for uderstadig. Facilitate a class discussio of studet results, clarifyig ay miscoceptios.. Refer studets to the Permutatios sectio o hadout: Permutatios ad Combiatios. Usig teacher resource: Permutatios ad Combiatios, facilitate a class discussio of the meaig of permutatios (where order matters). Go over formulas for determiig permutatios with repeats ad without repeats. Ask: I the previous activity, which situatios were permutatios the two digit umbers or the hads of cards? Why? (The two-digit umbers.) Aswers may vary. Because order matters. is differet from ; etc. Which situatios were permutatios the radio statio playig sogs or the lady playig CD s? Why? (Both were permutatios.) Aswers may vary. I both situatios, order mattered; etc.. Istruct studets to work with their parter to complete problems 7. Allow studets to complete the problems. Moitor ad assess studets to check for uderstadig. Usig teacher resource: Permutatios ad Combiatios, facilitate a class discussio of studet results, clarifyig ay miscoceptios. 6. Refer studets to the Combiatios sectio o hadout: Permutatios ad Combiatios. Usig teacher resource: Permutatios ad Combiatios, facilitate a Notes for Teacher After usig the formula to determie the umber of permutatios, studets should start to recogize that the terms i the factorial expressios cacel each other out. A shortcut for evaluatig these expressios could be writte as: TEACHER NOTE As studets use combiatio otatio, they may otice that, if a + b = c, the For example, sice + =, Or, sice + 6 = 7, page 9 of 90

20 0-0 Ehaced Istructioal Trasitio Guide High School Courses Uit Number: /Mathematics Suggested Duratio: 0 days Suggested Day Suggested Istructioal Procedures class discussio of the meaig of combiatios. Go over the formula for determiig combiatios without repeats. Ask: Notes for Teacher I the previous activity, were ay of the situatios combiatios the twodigit umbers, or the hads of cards? Why? (The hads of cards.) Aswers may vary. Because order did ot matter. Gettig & would be the same as & ; etc. 7. Istruct studets to work with their parter to complete problems 6. Allow studets to complete the problems. Moitor ad assess studets to check for uderstadig. Usig teacher resource: Permutatios ad Combiatios, facilitate a class discussio of studet results, clarifyig ay miscoceptios. 8. For additioal practice o combiatio otatio, distribute hadout: Combiatios ad Pascal s Triagle to each studet. Istruct studets to work idepedetly to complete the hadout. This may be completed as homework, if ecessary. 8-9 Topics: Biomial probability Elaborate Studets use biomial expasio to determie biomial probability. Istructioal Procedures: Day 8 ATTACHMENTS Teacher Resource: Biomial Power KEY ( per teacher) Teacher Resource: Biomial Power ( per teacher) Hadout: Biomial Power ( per studet) Teacher Resource: Biomial page 0 of 90

21 0-0 Ehaced Istructioal Trasitio Guide High School Courses Uit Number: /Mathematics Suggested Duratio: 0 days Suggested Day Suggested Istructioal Procedures. Place studets i pairs. Distribute hadout: Biomial Power to each studet.. Istruct studets to work idepedetly to complete problems, expadig the expressios (x + y) ad (x + y) by multiplyig polyomials. As studets fiish, have them compare aswers with their parter. Facilitate a class discussio of studet results. Make sure studets recogize the biomial coefficiets from Pascal s triagle i the expaded polyomials, as well as the patters i the expoets. To check for uderstadig, istruct studets to work with their parter to complete problems 6 by writig the expasios for (x + y) ad (x + y) 6 usig oly Pascal s triagle as a guide.. Refer studets to page of hadout: Biomial Power. Display teacher resource: Biomial Power, ad facilitate a class discussio of the Biomial Theorem at the top of page. Ask: Why do you thik this is called the biomial theorem? Aswers may vary. Because you have a biomial to a power. There are oly two terms iside the paretheses; etc. How do you get the coefficiets of the expaded polyomial? Aswers may vary. The coefficiets come from the combiatio umbers, like C r. These values will also fall o a specific row i Pascal s triagle; etc. What patter arises i the expoets? (It starts with a to the highest power. The, with each term, the power of a decreases oe, ad power of b i the term icreases oe.) Ad so o. Notes for Teacher Probability KEY ( per teacher) Teacher Resource: Biomial Probability ( per teacher) Hadout: Biomial Probability ( per studet) MATERIALS graphig calculator ( per studet) graphig calculator with display ( per teacher) TEACHER NOTE Whe writte i sigma otatio, the idex variable starts o zero (ot, as i may of the previous problems). This meas that, i the expaded expressio for (a + b), there are actually + terms. TEACHER NOTE I computig each probability i a situatio such as the girl-boy birth order problem o hadout: Biomial Probability, compare the results to the expaded biomial. page of 90

22 0-0 Ehaced Istructioal Trasitio Guide High School Courses Uit Number: /Mathematics Suggested Duratio: 0 days Suggested Day Suggested Istructioal Procedures. Usig teacher resource: Biomial Power, facilitate a class discussio, modelig sample problem, (x + ). Studets may have questios about why the etire (x) is raised to a power (istead of just the variable x). Emphasize that, i this biomial, the a represets x ad the b represets.. Istruct studets to work with their parter to complete problems 6 o hadout: Biomial Power. This may be completed as homework, if ecessary. Notes for Teacher G + G B + GB + B A iterpretatio of this polyomial is that there is way to get girls (G ), ways to get girls ad a boy (G B), ways to get girl ad boys (GB ), ad way to get boys (B ). Day 9 6. Usig teacher resource: Biomial Power, facilitate a class discussio to debrief problems 6 o hadout: Biomial Power to check for uderstadig. 7. Place studets i pairs. Distribute hadout: Biomial Probability to each studet. Refer studets to the problems o page. Display teacher resource: Biomial Probability, ad facilitate a class discussio of problems ad. Ask: Are these items permutatios or combiatios? Why? (permutatios; order matters) Ca optios be repeated? Explai. (yes) Aswers may vary. You ca have more tha oe girl or boy, ad more tha oe head or tail; etc. What formula ca be used to determie that there are 8 possibilities i problem ad 6 i problem? Aswers may vary. Sice the formula is () r, for problem, = 8, ad for problem, page of 90

23 0-0 Ehaced Istructioal Trasitio Guide High School Courses Uit Number: /Mathematics Suggested Duratio: 0 days Suggested Day = 6; etc. Suggested Istructioal Procedures Notes for Teacher 8. Istruct studets to work with their parter to complete problems o hadout: Biomial Probability. Allow studets time to complete the problems. Moitor ad assess studets to check for uderstadig. Usig teacher resource: Biomial Probability, facilitate a class discussio of teacher results, clarifyig ay miscoceptios. 9. Refer studets to the top of page o hadout: Biomial Probability. Usig teacher resource: Biomial Probability, facilitate a class discussio of the biomial probability formula, modelig problem. Ask: Why do you thik these are called biomial probability problems? Aswers may vary. Because there are oly two possible outcomes like heads or tails, boy or girl, make it or miss it; etc. What do these problems ad formulas have i commo with the expasios of the biomials to a power? Aswers may vary. They both use the combiatio umbers from Pascal s triagle. Also, they both have two values raised to powers; etc. 0. Istruct studets to work with their parter to complete problems o hadout: Biomial Probability. This may be completed as homework, if ecessary. 0 Evaluate ATTACHMENTS page of 90

24 0-0 Ehaced Istructioal Trasitio Guide High School Courses Uit Number: /Mathematics Suggested Duratio: 0 days Suggested Day Suggested Istructioal Procedures Istructioal Procedures:. Assess studet uderstadig of related cocepts ad processes by usig the Performace Idicator(s) aliged to this lesso. Performace Idicator(s): High School Mathematics Uit PI 0 Aalyze a problem situatio ivolvig sequeces ad series such as the followig: A motocross ramp is supported by 8 evely-spaced wood colums, as show. Notes for Teacher Teacher Resource (optioal): Applicatios of Sequeces ad Series KEY ( per teacher) Hadout (optioal): Applicatios of Sequeces ad Series PI ( per studet) MATERIALS graphig calculator ( per studet) TEACHER NOTE As a optioal assessmet tool, use hadout (optioal): Applicatios of Sequeces ad Series PI. The first pole is ft tall, the secod is. ft tall, ad the third is ft tall. Write a recursive formula to represet a. Write a explicit formula to fid a, the height of the th colum. Use the explicit formula to evaluate a 8, ad iterpret its meaig i the cotext of the problem. Express the series i sigma otatio. Use the series formula to determie how may total feet of lumber would be required to costruct all 8 support colums? Tomas drew the desig of the spiral below o grid paper to model the ext sectio of his irowork sculpture. page of 90

25 0-0 Ehaced Istructioal Trasitio Guide High School Courses Uit Number: /Mathematics Suggested Duratio: 0 days Suggested Day Suggested Istructioal Procedures Notes for Teacher Write a recursive formula to represet a. Write a explicit formula to fid a, the legth of the th sectio of iro. Use the explicit formula to evaluate a 9, ad iterpret its meaig i the cotext of the problem. Express the series i sigma otatio. Use the series formula to evaluate S 9, ad iterpret its meaig i the cotext of the problem. Each day, Gradma puts 8 ouces of water i her flower pot. Over the ext hours, 0% of the water evaporates (leavig 60%), at which time she adds aother 8 ouces. Complete the chart to show the amout of water (w ) i the flower pot after days. Write the recursive ad explicit formulas to fid w, the amout of water i the flower pot after days. What is the limit of w as? What does this represet i the problem situatio? page of 90

26 0-0 Ehaced Istructioal Trasitio Guide High School Courses Uit Number: /Mathematics Suggested Duratio: 0 days Suggested Day Suggested Istructioal Procedures Notes for Teacher O ay give school day, Jim has a 70% chace of beig o time (or a 0% chace of beig tardy). For the -day school week ahead, determie each biomial probability. P(o time all days) P(o time exactly days) P(o time exactly days) P(o time at least days) Create a graphic orgaizer for each situatio that icludes ) Table, graph, ad appropriate represetatios ) Formulas ad calculatios ) Justificatio of predictios i terms of the problem situatio Stadard(s): P.A, P.B, P.C, P.D ELPS ELPS.c.C, ELPS.c.H, ELPS.c.G 0/0/ page 6 of 90

27 Patters KEY Uit: Lesso: 0 Write dow the ext (fifth) term i each patter, ad explai how you determied the aswer. Predict the 0 th term i the patter. Leave the type colum blak (util istructed by your teacher). First term Secod term Third term Fourth term Fifth term Explaatio Teth term Type ) 6,, 0, 7, Add 7 each time 79 A ) 6, 8,,, Multiply by or 0. = 0.0 ),, 7,, 9 Subtract - A ),, 9, 6, Square umbers; or, add odd umbers G 00 O ),, 6, 6, 6 Multiply by 6, G 6) 00, 89, 78, 67, 6 Subtract A 7),,, 7 0 Add.. A 8) 6,, 6,, 8 9),, 6,, 0 Multiply by. (or, take half ad add it back) Multiply by,,,, 6, etc. 6 G,68,800 O 0) First term Secod term Third term Fourth term Fifth Term How may dots would be i the 0 th term? Type: O ) First term Secod term Third term Fourth term Fifth Term How may squares would be i the 0 th term? Type: A ) First term Secod term Third term Fourth term Fifth Term What would the 0 th term look like? Type: O 0, TESCCC 9/ page of

28 Patters KEY Uit: Lesso: 0 With your teachers help, label the type of each of the patter (# ) as either A, G, or O. Describe each type i the space below. A Type Arithmetic Descriptio Add/subtract patters, where the terms have a commo differece G Geometric Multiply/divide patters, where the terms have a commo ratio O Other Aythig else that does ot fit the two previous descriptios Use each expressio to geerate a sequece of umbers by evaluatig at =,,, ad. Idetify the type of sequece usig the descriptios above (A, G, or O). Expressio = = = = Type ) Arithmetic ) Arithmetic ) 6) ( ) 6 0 Other Other 7) 6 8 Geometric BONUS! Complete these other patters (just for fu). Patter Next three terms Explaatio 8),,,,, 8,,,,, 89,,, Each term is the sum of the two previous terms (Explai to studets that this is a special sequece called the Fiboacci sequece) 9) J, F, M, A, M, J, J, A, S, O, N, D 0) O, T, T, F, F, S, S, E, N, T, E, T, T, First letters i the moths (Jauary, February ) First letters i the coutig umbers (Oe, Two, ) 0, TESCCC 9/ page of

29 Patters Uit: Lesso: 0 Write dow the ext (fifth) term i each patter, ad explai how you determied the aswer. Predict the 0 th term i the patter. Leave the type colum blak (util istructed by your teacher). First term Secod term Third term Fourth term Fifth term ) 6,, 0, 7, Add 7 each time ) 6, 8,,, ),, 7,, ),, 9, 6, ),, 6, 6, 6) 00, 89, 78, 67, 7),,, 7 8) 6,, 6,, 9),, 6,, Explaatio Teth term Type 0) First term Secod term Third term Fourth term Fifth Term How may dots would be i the 0 th term? Type: ) First term Secod term Third term Fourth term Fifth Term How may squares would be i the 0 th term? Type: ) First term Secod term Third term Fourth term Fifth Term What would the 0 th term look like? Type: 0, TESCCC /9/ page of

30 Patters Uit: Lesso: 0 With your teachers help, label the type of each of the patter (# -) as either A, G, or O. Describe each type i the space below. Type Descriptio A G O Use each expressio to geerate a sequece of umbers by evaluatig at =,,, ad. Idetify the type of sequece usig the descriptios above (A, G, or O). ) 8 Expressio = = = = Type ) 0 ) 6) ( ) 7) BONUS! Complete these other patters (just for fu). Patter Next three terms Explaatio 8),,,,, 8,,,,,,,... 9) J, F, M, A, M, J, J, A, S,,, 0) O, T, T, F, F, S, S, E, N, T,,,... 0, TESCCC /9/ page of

31 Arithmetic ad Geometric Sequeces KEY Uit: Lesso: 0 Vocabulary Word Defiitio Symbols Example Sequece A list of umbers, usually i a patter Brackets, commas {,, 8, 6,, } Term A umber i a sequece Letters with subscripts a =, a =, a = 8, a = 6, a =, Positio Recursive Formula A term s ordial locatio i a sequece A process for fidig cosecutive terms i a sequece based o the previous terms A variable (usually ) I this case, a a a = (first), = (secod), = (third) a a () 6 6 a 6() = 8 7 a 8() = 6 8 Explicit Formula A formula for fidig ay term i a sequece based o its positio i the sequece I this case, a a 0 =,0 0 a = 6,8 Arithmetic Sequeces I a arithmetic sequece, the terms have a commo differece (d). Fid the commo differece for each arithmetic sequece. ) {, 9,, 7,, } d ) {8,, 68, 99, } d 67 ) {8,, 8,, -, } d - ) {8, 8, 9,, } d - Recursive: Give a, a a d A arithmetic sequece ca be defied usig either of the Explicit: a a ( ) formulas show here. d Use these formulas to complete the chart for each arithmetic sequece. ) 6) 7) 8) Recursive Formula Sequece (first five terms) Explicit Formula a a a {, 9, 6, 87,, } a = + ( ) a 08 a a {08, 96, 8, 7, 60, } a = 08 ( ) a = 6 a = (a ) + 8 a = 0 a = (a ) 9 {6,,, 0, 8, } 6 8( ) {0,,, -7, -6, } a = 0 9( ) a 0, TESCCC /9/ page of

32 Arithmetic ad Geometric Sequeces KEY Uit: Lesso: 0 Geometric Sequeces I a geometric sequece, the terms have a commo ratio (r). Fid the commo ratio for each geometric sequece. 9) {,, 7,, 9 } r 6 ) {, -0, 6, -.8, } r ) {,, 9, 68.6, } r. ) {7, 9,,,, } r Recursive: Give a, a a r A geometric sequece ca be defied usig either of the Explicit: a formulas show here. a ( r ) Use these formulas to complete the chart for each geometric sequece. ) ) ) 6) Recursive Formula Sequece (first five terms) Explicit Formula a 6 a a {6, 8,,,, } a = 6(½) a 7 a a {7,, 6, 89, 67, } a = 7() a = 00 a = (a ) 0.6 a = 8 a = (a ) ( / ) {00, 60, 6,.6,.96, } a 00(0.6) {8,, 6,, 6, } a = 8( / ) Mixed Sequeces 7) A arithmetic sequece has a 9 ad d = -8. A) Write the first five terms of the sequece. 9) A geometric sequece has a ad r =. A) Write the first five terms of the sequece. {9, 86, 78, 70, 6, } {, 0, 0, 0, 0, } B) Write the explicit formula for a, ad use it to fid the 0 th term. B) Write the explicit formula for a, ad use it to fid the 0 th term. a = 9 8( ), a 0 = 9 8(9) = -8 a = (), a 0 = () 9 =,906,0 8) A arithmetic sequece has a 9 ad a ) A geometric sequece has a ad a. A) What is d, the commo differece? A) What is r, the commo ratio? 0 = 9 + d(6 ), d = 6. = (r), r =.7 B) Fid the 0 th term ( a 0). B) Fid the 7 th term ( a 7 ). a 0 = (9) = 6.8 a 7 = ( ) 6 = 08 0, TESCCC /9/ page of

33 Arithmetic ad Geometric Sequeces Uit: Lesso: 0 Vocabulary Word Defiitio Symbols Example Sequece {,, 8, 6,, } Term a =, a =, a = 8, a = 6, a =, Positio Recursive Formula A process for fidig cosecutive terms i a sequece based o the previous terms I this case, a a a = (first), = (secod), = (third) a a () 6 6 a 7 a 8 Explicit Formula A formula for fidig ay term i a sequece based o its positio i the sequece I this case, a a 0 a Arithmetic Sequeces I a arithmetic sequece, the terms have a commo differece (d). Fid the commo differece for each arithmetic sequece. ) {, 9,, 7,, } d ) {8,, 68, 99, } d ) {8,, 8,, -, } d ) {8, 8, 9,, } d Recursive: Give a, a a d A arithmetic sequece ca be defied usig either of the Explicit: a a ( ) formulas show here. d Use these formulas to complete the chart for each arithmetic sequece. ) 6) Recursive Formula Sequece (first five terms) Explicit Formula a a a a 08 a a 7) 6 8( ) a 8) {0,,, -7, -6, } 0, TESCCC /9/ page of

34 Arithmetic ad Geometric Sequeces Uit: Lesso: 0 Geometric Sequeces I a geometric sequece, the terms have a commo ratio (r). Fid the commo ratio for each geometric sequece. 9) {,, 7,, 9 } r ) {, -0, 6, -.8, } r 0) {,, 9, 68.6, } r ) {7, 9,,,, } r Recursive: Give a, a a r A geometric sequece ca be defied usig either of the Explicit: a formulas show here. a ( r ) Use these formulas to complete the chart for each geometric sequece. ) ) Recursive Formula Sequece (first five terms) Explicit Formula a 6 a a a 7 a a ) a 00(0.6) 6) {8,, 6,, 6, } Mixed Sequeces 7) A arithmetic sequece has a 9 ad d = -8. A) Write the first five terms of the sequece. B) Write the explicit formula for a, ad use it to fid the 0 th term. 9) A geometric sequece has a ad r =. A) Write the first five terms of the sequece. B) Write the explicit formula for a, ad use it to fid the 0 th term. 8) A arithmetic sequece has a 9 ad a ) A geometric sequece has a ad a. A) What is d, the commo differece? A) What is r, the commo ratio? B) Fid the 0 th term ( a 0). B) Fid the 7 th term ( a 7 ). 0, TESCCC /9/ page of

35 Yes, We Ca KEY Alexader is i his first year of work at a grocery store. He has bee put i charge of makig a display by stackig cas of soup. ) The top row (a ) is to have cas, the secod row (a ) eeds, ad so forth i a arithmetic sequece (d = ). I stackig the cas, however, Alexader has to begi with the bottom row. A) To start, Alexader thiks he ca fit either 6 or 7 cas o the bottom row. Which should he choose? Why? 7 cas. With a = ad d =, the sequece will have oly odd umbers (ad will ot cotai 6 as a term). a a a Uit: Lesso: 0 Top Row B) Usig your aswer from part (A) with the equatio a a d( ), determie how may rows of soup cas will be i Alexader s display. a = + ( ) 8 rows of cas 7 = + ( ) = 8 C) How may total cas of soup will be eeded to make this display? = 60 cas ) Alexader does such a good job with the display of soup cas that the maager offers him a fulltime job at the grocery store with a ice aual salary ad a guarateed % raise each year. A) If the first year s salary is $0,000, how much would Alexader make the followig year? The year after that? Secod year: $0,000(.0) = $,00 Third year: $,00(.0) = $,07 B) Write a explicit formula of the form a a ( r ) that projects Alexader s salary i the th year of employmet at the grocery store. a = 0,000(.0) C) If Alexader worked at the grocery store for 0 years, what would be his aual salary the? a 0 = 0,000(.0) 9 a 0 $7,808. D) Over these 0 years, how much total salary would Alexader ear? Would it be more or less tha $ millio? $0,000 + $,00 + $, $7,808. $99,979 This is slightly less tha $ millio. 0, TESCCC /9/ page of

36 Yes, We Ca Alexader is i his first year of work at a grocery store. He has bee put i charge of makig a display by stackig cas of soup. ) The top row (a ) is to have cas, the secod row (a ) eeds, ad so forth i a arithmetic sequece (d = ). I stackig the cas, however, Alexader has to begi with the bottom row. A) To start, Alexader thiks he ca fit either 6 or 7 cas o the bottom row. Which should he choose? Why? a a a Uit: Lesso: 0 Top Row B) Usig your aswer from part (A) with the equatio a a d( ), determie how may rows of soup cas will be i Alexader s display. C) How may total cas of soup will be eeded to make this display? ) Alexader does such a good job with the display of soup cas that the maager offers him a fulltime job at the grocery store with a ice aual salary ad a guarateed % raise each year. A) If the first year s salary is $0,000, how much would Alexader make the followig year? The year after that? B) Write a explicit formula of the form a a ( r ) that projects Alexader s salary i the th year of employmet at the grocery store. C) If Alexader worked at the grocery store for 0 years, what would be his aual salary the? D) Over these 0 years, how much total salary would Alexader ear? Would it be more or less tha $ millio? 0, TESCCC /9/ page of

37 Series KEY Uit: Lesso: 0 A series is simply the sum (or total) of the terms i a sequece. Sequece: {,, 7, 9,,,,, 7} Series: The otatio S (a capital letter S with a umber as subscript) is ofte used to describe a series. S is read S, sub, ad meas the sum of the first terms of the give sequece. ) For the sequece a = {,, 7, 9, } (or, a ), evaluate each of the followig series (also called partial sums ). S + = 8 S = S = S = 80 8 S = 60 8 Sigma Notatio There is special otatio to describe totalig terms i a sequece. The otatio uses the Greek letter sigma (), which i mathematics idicates to fid a sum. 8 upper boud (ed) formula The sum of +, for values of from to 8. How it s read lower boud (start) idex variable = = 80 How it is (,,,,, 6, 7, 8 ) evaluated Geerate a sequece usig the umbers from to 8 for i the formula +. The fid the total of all the terms. What it meas ) For the sequece a 8, evaluate each of the followig series by writig out the terms. ) For the sequece a 8, evaluate each of the followig series by writig out the terms. S Sigma otatio Series Total S = S 6 = S 8 = = = = (-) + (-6) 6 S Sigma otatio Series Total S = S = S 7 = = = = , TESCCC /9/ page of

38 Series KEY Uit: Lesso: 0 Series Shortcuts For certai types of sequeces, there are formulas to determie the sums of their terms. Arithmetic Series: S ak ( a a ) Geometric Series: S k a a r r k k Here, a = first term, a = last ( th ) term, ad = umber of terms i the series Here, a = first term, r = commo ratio, ad = umber of terms i the series Use these formulas to evaluate the series below. I doig so, idetify the values that would be used for the variables i each formula. (See the example.) Series Formula Variable Values Aswer S ( a a ) a Ex = 0, a = 0, S (0 0) ad =.(0) 00 (Arithmetic) a r ) S a =, r =, r ad = 7 ( 7 )/( ) = 09 ) S ( a a ) a = 8, a = 0, ad = 9 (9/)(8 + 0) = 6 6) 8( ) S ( a a ) 7) 0(.) 6 a r S a =, a = 9, ad = a = 0, r =., r ad = 6 (/)( + 9) = 88 0(. 6 )/(.) = Use the explicit formulas for arithmetic ad geometric sequeces a a d( ) ad to help write the followig series usig sigma otatio ad evaluate. a a, ( r ) 8) ) ) a = 9, d = -6 a = 9 6( ) = a =, d = 9 a = + 9( ) Here, solvig whe a = 9 yields = 6 a = 80, r =. a = 80(.) = 9 6( ) = 6 9( ) = (6/)(+9) =, 80(.) = 80(. )/(.) = 0 0, TESCCC /9/ page of

39 Series Uit: Lesso: 0 A series is simply the sum (or total) of the terms i a sequece. Sequece: {,, 7, 9,,,,, 7} Series: The otatio S (a capital letter S with a umber as subscript) is ofte used to describe a series. S is read S, sub, ad meas the sum of the first terms of the give sequece. ) For the sequece a = {,, 7, 9, } (or, a ), evaluate each of the followig series (also called partial sums ). S + = 8 S = S S 8 S 8 Sigma Notatio There is special otatio to describe totalig terms i a sequece. The otatio uses the Greek letter sigma (), which i mathematics idicates to fid a sum. 8 upper boud (ed) formula The sum of +, for values of from to 8. How it s read lower boud (start) idex variable = = 80 How it is (,,,,, 6, 7, 8 ) evaluated Geerate a sequece usig the umbers from to 8 for i the formula +. The fid the total of all the terms. What it meas ) For the sequece a 8, evaluate each of the followig series by writig out the terms. S Sigma otatio Series Total S = S 6 = 6 8 = 8 = S 8 = ) For the sequece a 8, evaluate each of the followig series by writig out the terms. S Sigma otatio Series Total S = S = 8 = S 7 = 0, TESCCC /9/ page of

40 Series Uit: Lesso: 0 Series Shortcuts For certai types of sequeces, there are formulas to determie the sums of their terms. Arithmetic Series: S ak ( a a ) Geometric Series: S k a a r r k k Here, a = first term, a = last ( th ) term, ad = umber of terms i the series Here, a = first term, r = commo ratio, ad = umber of terms i the series Use these formulas to evaluate the series below. I doig so, idetify the values that would be used for the variables i each formula. (See the example.) Series Formula Variable Values Aswer S ( a a ) a Ex = 0, a = 0, S (0 0) ad =.(0) 00 (Arithmetic) ) ) ) 8( ) 7) 0(.) 6 Use the explicit formulas for arithmetic ad geometric sequeces ( a a d( ) ad respectively) to help write the followig series usig sigma otatio. The evaluate. a a, ( r ) 8) ) ) , TESCCC /9/ page of

41 a (8 cm) Sequeces ad Series Problems KEY Uit: Lesso: 0 ) I a amphitheater, there are 0 seats i the frot row, seats i the secod row, 6 seats i the third row, ad so o, i a arithmetic sequece. A) Write a explicit formula to fid a, the umber of seats i the th row. a = 0 + ( ) B) The last row i the amphitheater has 8 seats i it. How may rows of seats are there? 8 = 0 + ( ) There are rows of seats = Stage Back Stage rows Rows of Seats C) Write a expressio usig sigma otatio that represets the total umber of seats i the amphitheater. 0 + ( ) D) Evaluate this expressio. How may total seats are there? (/)(0 + 8) =,, seats ) To make a spiral desig, a art studet draws a segmet 0 cm i legth. The she repeats the followig two steps: Rotate the paper 90 Draw aother segmet that is 80% of the legth of the previous oe. A) Write a explicit formula to fid a, the legth of the th segmet. a = 0(0.8) a (0 cm) a (6. cm) B) Use the formula to approximate the legth of the 9 th segmet (a 9 ). a 9 = 0(0.8) 8 = The segmet is about.7 cm log C) Write a expressio usig sigma otatio that represets the combied legths of the first 9 segmets i this spiral. 9 0(0.8) D) How far would the artist s pecil move o paper to create this desig? 0( )/( 0.8) =.896 About. cm 0, TESCCC /9/ page of

42 Sequeces ad Series Problems KEY Uit: Lesso: 0 ) Aother art studet makes a desig that starts with a isosceles right triagle with a base (hypoteuse) of 6 cm ad a height of cm. More triagles are created i a patter where the leg of oe triagle serves as the base (hypoteuse) of the ext. A) Let a represet the area (i cm ) of the th triagle made i this desig. Give that a = 9, fid a. (See picture.) a =. a (9 cm ) a a a B) Explai how you kow that this sequece (a ) must be geometric (ad ot arithmetic). Sample: The first two terms are 9 ad.. If this sequece were arithmetic, the d would be -., ad the third term would be 0 (followed by egative terms), which could ot describe area. Istead, each triagle is half the area of the oe precedig it (r = 0.). C) Write a explicit formula to fid a. a = 9(0.) D) Fid the total area of all 8 triagles that are used to create this desig. 9( 0. 8 )/( 0.) = About 7.9 cm ) Imagie that your rich gradparets wat to give you some last-miute moey for college, but wat you to choose betwee oe of two differet paymet plas: The Get-Rich-Quick Pla The Pey-Saver Pla Day : They give you $0 Day : They give you pey ($0.0) Day : They give you $70 Day : They give you cets ($0.0) Day : They give you $90 Day : They give you cets ($0.0) ad so o, i a arithmetic patter ad so o, i a geometric patter for weeks (8 days) for weeks ( days) A) Write explicit formulas to fid the amouts you would get o each day () for both the Get- Rich-Quick Pla ad the Pey-Saver Pla. Get-Rich-Quick: g = 0 + 0( ), Pey-Saver: p = 0.0() B) Demostrate that you would actually get more moey o the fial day of the Pey-Saver Pla tha you would get i all weeks (total) of the Get-Rich-Quick Pla. Pey-Saver, o Day : Get-Rich-Quick, over all 8 days: p = 0.0() 0 = 0,8.76 g 8 = 0 + 0(7) = 90 Or, o Day you d get $0,8.76 S 8 = (8/)(0 + 90) = $8,960 O Day of the Pey-Saver Pla, you would get $,.76 more tha you would get i all 8 days (combied) of the other pla. 0, TESCCC /9/ page of

43 a (8 cm) Sequeces ad Series Problems Uit: Lesso: 0 ) I a amphitheater, there are 0 seats i the frot row, seats i the secod row, 6 seats i the third row, ad so o, i a arithmetic sequece. A) Write a explicit formula to fid a, the umber of seats i the th row. B) The last row i the amphitheater has 8 seats i it. How may rows of seats are there? Stage Back Stage rows Rows of Seats C) Write a expressio usig sigma otatio that represets the total umber of seats i the amphitheater. D) Evaluate this expressio. How may total seats are there? ) To make a spiral desig, a art studet draws a segmet 0 cm i legth. The she repeats the followig two steps: Rotate the paper 90 Draw aother segmet that is 80% of the legth of the previous oe. A) Write a explicit formula to fid a, the legth of the th segmet. a (0 cm) a (6. cm) B) Use the formula to approximate the legth of the 9 th segmet (a 9 ). C) Write a expressio usig sigma otatio that represets the combied legths of the first 9 segmets i this spiral. D) How far would the artist s pecil move o paper to create this desig? 0, TESCCC /9/ page of

44 Sequeces ad Series Problems Uit: Lesso: 0 ) Aother art studet makes a desig that starts with a isosceles right triagle with a base (hypoteuse) of 6 cm ad a height of cm. More triagles are created i a patter where the leg of oe triagle serves as the base (hypoteuse) of the ext. A) Let a represet the area (i cm ) of the th triagle made i this desig. Give that a = 9, fid a. (See picture.) B) Explai how you kow that this sequece (a ) must be geometric (ad ot arithmetic). a (9 cm ) a a a C) Write a explicit formula to fid a. D) Fid the total area of all 8 triagles that are used to create this desig. ) Imagie that your rich gradparets wat to give you some last-miute moey for college, but wat you to choose betwee oe of two differet paymet plas: The Get-Rich-Quick Pla The Pey-Saver Pla Day : They give you $0 Day : They give you pey ($0.0) Day : They give you $70 Day : They give you cets ($0.0) Day : They give you $90 Day : They give you cets ($0.0) ad so o, i a arithmetic patter ad so o, i a geometric patter for weeks (8 days) for weeks ( days) A) Write explicit formulas to fid the amouts you would get o each day () for both the Get- Rich-Quick Pla ad the Pey-Saver Pla. B) Demostrate that you would actually get more moey o the fial day of the Pey-Saver Pla tha you would get i all weeks (total) of the Get-Rich-Quick Pla. 0, TESCCC /9/ page of

45 Goig ad Goig KEY Uit: Lesso: 0 A series writte i sigma otatio ca be evaluated i a graphig calculator usig the sum ad sequece commads, as show below ad to the right. The ca be etered by puttig the calculator i sequece mode before begiig ad usig the variable key or by usig ALPHA N. Both methods will work. Whe the sum commad is left out, the calculator ca be used to geerate the sequece itself. Use the calculator s sum ad sequece commads to evaluate each of the followig. ) a 8 {,, 9, 7, } A) 8 B) 8 C) 8 D) 8 0 0,900 6,800 ) a 0 {00, 9, 90, 8, } A) 0 0 B) 0 0, TESCCC /9/ page of 00 0 C) D) ,70 ) a 0 (.) {,., 7.8, 0.76, } A) 0(.) 0 B) 0(.) 0 C) 0(.) 00 D) 0(.) ,969,078, ) {,,,,...} a A) 8 B) 9 0 C) 9 0 D) ) a (0.7) {8,., 0., 7.97, } A) (0.7) 0 B) (0.7) 0 C) (0.7) 80 D) (0.7) ) a,,,,...} { A) B) C) D)

46 Goig ad Goig KEY Uit: Lesso: 0 7) Items # ad (show below, from the previous page) have a special property that the other series do ot have. What is it? For each, the sums have a limit. Or, o matter how x x may terms you total up, the sum ever icreases ) 9 ) (0.7) past a certai amout. For #, that limit is.. For #, the limit is 7. 8) Look at the explicit formulas as well as the terms i the sequeces for items # ad (show below). What do they have i commo? Both formulas are expoetial (or, they create ) 9 a geometric sequeces). {,,,,...} 9 The sequeces are both decreasig (meaig that ) a (0.7) {8,., 0., } the terms get smaller for greater values of ), but the terms will ever reach 0. 9) Compare the explicit formulas ad sequeces for items # ad (above) with those of # (show below). How is # like the other two? How is it differet? Like # ad, the formulas are expoetial (or, they ) a 0 (.) {,., 7.8, } create geometric sequeces). However, i #, the terms i the sequece icrease for greater values of (because the ratio r > ). 0) Compare the explicit formulas ad sequeces for items # ad (above) with those of #6 (show below). How is #6 like the other two? How is it differet? Like # ad, the sequece is decreasig 6) a {,,,,...} (meaig that the terms get smaller for greater values of, but the terms will ever reach 0. However, i #6, the explicit formula is ot expoetial (ad so the sequece is ot geometric). 0, TESCCC /9/ page of

47 Goig ad Goig Uit: Lesso: 0 A series writte i sigma otatio ca be evaluated i a graphig calculator usig the sum ad sequece commads, as show below ad to the right. The ca be etered by puttig the calculator i sequece mode before begiig ad usig the variable key or by usig ALPHA N. Both methods will work. Whe the sum commad is left out, the calculator ca be used to geerate the sequece itself. Use the calculator s sum ad sequece commads to evaluate each of the followig. ) a 8 {,, 9, 7, } A) 8 B) 8 C) 8 D) ) a 0 {00, 9, 90, 8, } A) 0 0 B) 0 0 C) 0 00 D) 0 ) a 0 (.) {,., 7.8, 0.76, } A) 0(.) 0 B) 0(.) 0 C) 0(.) 00 D) 0(.) ) {,,,,...} a A) 8 B) 9 0 C) 9 0 D) 9 ) a (0.7) {8,., 0., 7.97, } A) (0.7) 0 B) (0.7) 0 C) (0.7) 80 D) (0.7) 6) a,,,,...} A) { 0 B) 00 C) 00 D) 0, TESCCC /9/ page of