English Language Arts 10 – 2



MATHEMATICS 20-1Sequences and SeriesHigh School collaborative venture withEdmonton Christian, Harry Ainlay, J. Percy Page, Jasper Place, Millwoods Christian, Ross Sheppard and W. P. Wagner, M. E LaZerte, McNally, Queen Elizabeth, Strathcona and VictoriaEdm Christian High: Aaron TrimbleHarry Ainlay: Ben LuchkowHarry Ainlay: Darwin HoltHarry Ainlay: Lareina RezewskiHarry Ainlay: Mike ShrimptonJ. Percy Page: Debbie YoungerJasper Place: Matt KatesJasper Place: Sue DvorackMillwoods Christian: Patrick YpmaRoss Sheppard: Patricia ElderRoss Sheppard: Dean WallsW. P. Wagner: Amber SteinhauerM. E. LaZerte: Teena WoudstraQueen Elizabeth: David UnderwoodStrathcona: Christian Digout Victoria: Steven Dyck McNally: Neil PetersonFacilitator: John Scammell (Consulting Services)Editor: Jim Reed (Contracted)2010 - 2011TABLE OF CONTENTSSTAGE 1 DESIRED RESULTSPAGEBig Idea Enduring UnderstandingsEssential Questions445KnowledgeSkills67STAGE 2 ASSESSMENT EVIDENCETransfer Task Arena PlanTeacher Notes for Transfer Task and RubricTransfer TaskRubricPossible Solution8101214STAGE 3 LEARNING PLANSLesson #1 Introduction to Patterns16Lesson #2 Arithmetic Series20Lesson #3 Geometric Sequences23Lesson #4 Geometric Series26Lesson #5 Infinite Geometric Series30 Mathematics 20-1 Sequences and SeriesSTAGE 1 Desired Results Big Idea: The world is full of patterns to be discovered. Students will be able to recognize a pattern and continue modeling the sequence to make predictions for future elements.Implementation note:Post the BIG IDEA in a prominentplace in your classroom and refer to it often. Enduring Understandings:Students will understand …Different types of sequences and series exist.We can use mathematics to model the pattern of the sequence or series.By the end of the unit students should:Use concrete strategies to determine the pattern.Have an idea of where to start in breaking down the sequence/series. Recognize and apply patterns to familiar and unfamiliar situations (predictions).Know that a pattern exists.See patterns in life, application of patterns beyond geometric/arithmetic sequences and series.Make predictions based on an observed pattern.Determine the pattern and identify relevant elements of geometric/arithmetic sequences and series.Investigate or discover patterns and extend them. Essential Questions:Is anything in the universe truly random? Is chaos a pattern?What is the underlying structure in the pattern that allows sequences and series to be expressed mathematically, concretely, symbolically, pictorially, and verbally in different terms depending on the context?Can we recognize that there is universality to patterns that manifest themselves in different contexts in nature?Additional questions to considerIs there an underlying structure/connection that helps us identify that different patterns exist?How are exceptions to the pattern a pattern itself? You have a routine M-F, but your routine is different on Saturday. The exception on Saturday is a pattern in itselfWhen we are looking for patterns, where do we start looking?Can we identify that we have an unconscious sense of patterns (rule of thirds)? Can we learn to recognize and name these patterns?Can we recognize intrinsic/automatic patterns and acknowledge that they are in fact patterns? Knowledge:Enduring UnderstandingSpecific OutcomesDescription ofKnowledgeStudents will understand…Different types of sequences and series exist. *RF9 RF10Students will know …the difference between arithmetic and geometricthe notation of sequences and series (a, d, n, r, tn)the components required to finding the general termthe difference between convergent and divergent geometric series and what leads to convergenceStudents will understand…We can use mathematics to model the pattern of the sequence or series. RF9 RF10Students will know …the difference between arithmetic and geometricthe notation of sequences and series (a, d, n, r, tn)the components required to finding the general termthe difference between convergent and divergent geometric series and what leads to convergence8888I*RF = Relations and Functions Skills: Enduring UnderstandingSpecific OutcomesDescription of SkillsStudents will understand…Different types of sequences and series exist. * RF9 RF10Students will be able to…identify arithmetic and geometric sequencescreate a model for a problem/scenariocalculate any specified parameter for a sequence or series (a, d, n, r, tn, Sn) find the sum of a sequence or the individual terms or a seriesStudents will understand…We can use mathematics to model the pattern of the sequence or series. RF9 RF10Students will be able to…identify arithmetic and geometric sequencescreate a model for a problem/scenariocalculate any specified parameter for a sequence or series (a, d, n, r, tn, Sn) find the sum of a sequence or the individual terms or a seriescalculate the infinite sum of a convergent series* RF = Relations and FunctionsImplementation note:Teachers need to continually askthemselves, if their students are acquiring the knowledge and skills needed for the unit.STAGE 2 Assessment Evidence1 Desired Results Desired Results Arena Plan Teacher NotesThis task is open-ended. Many different student responses are possible.Students could research construction costs, arena designs, and ticketing practices online before beginning. Have students read the whole thing before beginning. Part 3 relies on the answer to Part 2, and students should consider Part 3 as they work on Part 2.Part 3 is difficult to do just with formulas because it mixes both arithmetic and geometric sequences. As a result, students may want to use a spreadsheet to answer part 3.Students should be encouraged to present their proposals to Mr. Dogs in whatever format they like. It could be a verbal presentation, done on a poster, video or PowerPoint. They should consider the best way to present the proposal to Mr. Dogs.Implementation note:Teachers need to consider what performances and products will reveal evidence of understanding?What other evidence will be collected to reflectthe desired results?Glossary arithmetic sequence - A sequence for which the difference between successive terms is constantarithmetic series - The sum of the terms of an arithmetic sequencecommon difference - A constant that is added to each term to produce an arithmetic sequencecommon ratio - A constant that is multiplied to each term to produce a geometric sequenceconvergent sequence – A sequence in which the difference between two consecutive terms is equal to zero when n is largeconvergent series – A series in which the sum is finitedivergent series – A series that is not convergentfirst term – The first value in a/an arithmetic/geometric sequence/series.general term - A function that describes all terms in a sequencegeometric sequence - A sequence in which the ratio of successive terms is constantgeometric series - The sum of the terms of a geometric sequenceinfinite sequence - A sequence that does not end or have a final terminfinite geometric series – A geometric series that does not end or have a final term. An infinite geometric series may be convergent or divergent.sequence - A set or list of numbers arranged in a definite order. A sequence is a function whose domain is a subset of the natural numbers, N, and whose range is a subset of the real numbers, R. The sequence itself shows the range of the function. Arena Plan - Student Assessment TaskScenarioMr. Dogs has asked your company to design a seating plan for a new NHL arena. Currently his team plays in a rink like the one below.Sample ArenaYou must create a proposal to Mr. Dogs that outlines the following information. Support your proposal with appropriate mathematics. Arena Plan - Student Assessment TaskMr. Dogs wants the number of seats in the arena to be between 18 000 and 22 500. One ring of seats all the way around the rink is considered a row, and row 1 is considered to be the row closest to the ice. He wants the number of seats in each row to form an arithmetic sequence, increasing by the same number in each subsequent row. Your task is to decide on the total number of seats in the arena by designing a seating arrangement that has a reasonable number of rows by determining: The number of seats in the first row.The number of rows required.The number of seats by which each row increases.The number of seats in the last row.The total number of seats in the arena. In his current arena, Mr. Dogs charges $6000 per season for seats in rows 1-10, $4000 for season seats in rows 11-20, $3000 for season seats in rows 21-30, and $2000 for season seats in rows 31-40. He thinks that a more fair way to decide on season ticket prices is to use a geometric sequence, and decrease the price in each subsequent row by the same factor based on the price of the row in front of it. For your proposalDetermine a reasonable price per game for each seat in the first row.Determine the factor by which the cost of each seat per game will decrease in each subsequent row from row 1.Determine the price per game of each seat in the last row.There are 41 home games in the regular season. Given that he needs to sell every seat in the arena and generate at least $50 000 000 in revenue, determine the following:The total revenue he will generate by selling all the seats in his rink at the prices you set above. You may have to adjust the prices you set above in order to generate at least $50 000 000 in revenue.Your proposal can take any form, but must be supported by mathematics.Glossary arithmetic sequence - A sequence for which the difference between successive terms is constantarithmetic series - The sum of the terms of an arithmetic sequencecommon difference - A constant that is added to each term to produce an arithmetic sequencecommon ratio - A constant that is multiplied to each term to produce a geometric sequenceconvergent sequence – A sequence in which the difference between two consecutive terms is equal to zero when n is largeconvergent series – A series in which the sum is finitedivergent series – A series that is not convergentfirst term – The first value in a/an arithmetic/geometric sequence/series.general term - A function that describes all terms in a sequencegeometric sequence - A sequence in which the ratio of successive terms is constantgeometric series - The sum of the terms of a geometric sequenceinfinite sequence - A sequence that does not end or have a final terminfinite geometric series – A geometric series that does not end or have a final term. An infinite geometric series may be convergent or divergent.sequence - A set or list of numbers arranged in a definite order. A sequence is a function whose domain is a subset of the natural numbers, N, and whose range is a subset of the real numbers, R. The sequence itself shows the range of the function. AssessmentMathematics 20-1Sequences and SeriesRubricLevelExcellentProficientAdequateLimitedInsufficientCriteria4321BlankMath ContentPart 1All required elements are present and correctAll required elements are present but may contain minor errorsSome required elements are missing, or contain major errorsMost required elements are missing or incorrectNo score is awarded as there is no evidence givenMath ContentPart 2All required elements are present and correctAll required elements are present but may contain minor errorsSome required elements are missing, or contain major errorsMost required elements are missing or incorrectNo score is awarded as there is no evidence givenMath Content Part 3All required elements are present and correctAll required elements are present but may contain minor errorsSome required elements are missing, or contain major errorsMost required elements are missing or incorrectNo score is awarded as there is no evidence givenPresents DataPresentation of data is clear, precise and accuratePresentation of data is complete and unambiguousPresentation of data is simplistic and plausiblePresentation of data is vague and inaccuratePresentation of data is incomprehensibleExplains ChoicesProvides insightful explanationsProvides logical explanationsProvides explanations that are complete but vagueProvides explanations that are incomplete or confusing.No explanation is providedWhen work is judged to be limited or insufficient, the teacher makes decisions.Possible Solution to Arena PlanA solution such as this could be presented in many different ways:Number of SeatsWe propose to have 460 seats in row 1, and increase the number of seats by 4 in each subsequent row. If we have 40 rows, the total number of seats in the arena will be 21 520, as shown below. Ticket PriceWe propose that the ticket price per game for seats in row 1 should be $400. Each subsequent row should receive an 8% decrease in this price, making the ticket price per game in row 40 a very reasonable $15.48. Total RevenueBased on our proposed model, Mr. Dogs can expect a total revenue of $98?868?825.80. A spreadsheet is useful in determining the total revenue based on the above model. RowSeats$/GmRow's Revenue/GMGamesRow's Revenue/Season1460$400.00$184,000.0041$7,544,000.002464$368.00$170,752.0041$7,000,832.003468$338.56$158,446.0841$6,496,289.284472$311.48$147,016.2941$6,027,668.075476$286.56$136,401.2241$5,592,450.006480$263.63$126,543.6541$5,188,289.757484$242.54$117,390.3341$4,813,003.468488$223.14$108,891.6641$4,464,557.929492$205.29$101,001.4741$4,141,060.4410496$188.86$93,676.8141$3,840,749.3911500$173.76$86,877.6941$3,561,985.3212504$159.85$80,566.9041$3,303,242.7113508$147.07$74,709.8141$3,063,102.2114512$135.30$69,274.2341$2,840,243.4315516$124.48$64,230.2041$2,633,438.2116520$114.52$59,549.8641$2,441,544.2617524$105.36$55,207.3041$2,263,499.3418528$96.93$51,178.4341$2,098,315.7319532$89.17$47,440.8641$1,945,075.0920536$82.04$43,973.7541$1,802,923.7421540$75.48$40,757.7641$1,671,068.1222544$69.44$37,774.8941$1,548,770.6923548$63.88$35,008.4441$1,435,346.0224552$58.77$32,442.8641$1,330,157.1525556$54.07$30,063.7141$1,232,612.3026560$49.75$27,857.6041$1,142,161.6127564$45.77$25,812.0641$1,058,294.3128568$42.10$23,915.5141$980,535.9529572$38.74$22,157.2241$908,445.8430576$35.64$20,527.1941$841,614.7231580$32.79$19,016.1641$779,662.5332584$30.16$17,615.5241$722,236.3533588$27.75$16,317.2841$669,008.5234592$25.53$15,114.0241$619,674.8335596$23.49$13,998.8541$573,952.8836600$21.61$12,965.3841$531,580.5237604$19.88$12,007.6741$492,314.4438608$18.29$11,120.2141$455,928.8139612$16.83$10,297.9041$422,214.0840616$15.48$9,535.9941$390,975.75STAGE 3 Learning PlansLesson 1Introduction to PatternsSTAGE 1BIG IDEA: The world is full of patterns to be discovered. Students will be able to recognize a pattern and continue modeling the sequence to make predictions for future elements.ENDURING UNDERSTANDINGS:Students will understand …We can use mathematics to model the pattern of the sequence or series.ESSENTIAL QUESTIONS:Is anything in the universe truly random? Is chaos a pattern?What is the underlying structure in the pattern that allows sequences and series to be expressed mathematically, concretely, symbolically, pictorially, and verbally in different terms depending on the context?Can we recognize that there is universality to patterns that manifest themselves in different contexts in nature?KNOWLEDGE:Students will know …the components required to finding the general termSKILLS:Students will be able to …identify arithmetic and geometric sequencescreate a model for a problem/scenariocalculate any specified parameter for a sequence or series (a, d, n, r, tn, Sn)Implementation note:Each lesson is a conceptual unit and is not intended to be taught on a one lesson per block basis. Each represents a concept to be covered and can take anywhere from part of a class to several classes to complete.Lesson SummaryUse patterns to generate the general term of an arithmetic sequence and examine each parameter. Lesson PlanThinkWhy do we need/where do we see sequences and series?Explore PatternsGive examples of arithmetic, geometric and other sequences. Ask students to find the next two terms and determine a rule for the pattern.HookWhat animal are you (your birth year - )? What does your animal tell you about your personality traits, and is it accurate?What about other family members?LessonHave students explore and discover the general term by doing the following:Have students choose an animal in the Chinese calendar.Students will list the first 6 terms for the years of the animal they choose.Students will create a formula for their sequence (not necessarily using the parameters from the general term).Begin to lead the discussion towards the general term, using common parameters.Introduce students to these parameters:a, n and d, find tn and the formula for the general term.A variety of examples involving arithmetic sequences should be given to students (including word problems).2, 5, 8, 11, 14, ...find the 100th term and the general term.9, 2, -5, -12, -18, ...find the 200th term and the general term.-55 in the sequence 26, 23, 20, ... is which term number?A sequence is defined by tn = 3n - 2. Find a and d.In an arithmetic sequence, the 5th term is 20 and the 9th term is 36. Find the common difference, the first term, the general term and the 47th term.You are going to train for a marathon over the summer holidays. The first week you will run 5 km. Each additional week you run another 2 km. How many kilometres do you run in week 8. Extension: What is the total distance you will run at the end of eight weeks? Going Beyond Group Project: Create your own zodiac calendar with your own animals and year span.ResourcesMath 20-1 (McGraw-Hill: sec 1.1) SupportingAssessment Exit slip – some examples of possible exit slipsGive a real-life example of an arithmetic sequence.Given a formula for the general term of an arithmetic sequence, find a parameter (a,?d,?n,?t,?tn).Glossaryarithmetic sequence - A sequence for which the difference between successive terms is constantcommon difference - A constant that is added to each term to produce an arithmetic sequencefirst term – The first value in a/an arithmetic/geometric sequence/series.general term - A function that describes all terms in a sequencesequence - A set or list of numbers arranged in a definite order. A sequence is a function whose domain is a subset of the natural numbers, N, and whose range is a subset of the real numbers, R. The sequence itself shows the range of the function.OtherLesson 2Arithmetic SeriesSTAGE 1BIG IDEA: The world is full of patterns to be discovered. Students will be able to recognize a pattern and continue modeling the sequence to make predictions for future elements.ENDURING UNDERSTANDINGS:Students will understand …Different types of sequences and series exist.We can use mathematics to model the pattern of the sequence or series.ESSENTIAL QUESTIONS:Is anything in the universe truly random? Is chaos a pattern?What is the underlying structure in the pattern that allows sequences and series to be expressed mathematically, concretely, symbolically, pictorially, and verbally in different terms depending on the context?Can we recognize that there is universality to patterns that manifest themselves in different contexts in nature?KNOWLEDGE:Students will know …the notation of sequences and series (a, d, n, r, t, tn, Sn)the components required to finding the general termSKILLS:Students will be able to …create a model for a problem/scenariocalculate any specified parameter for a sequence or series (a, d, n, tn, Sn)find the sum of a sequence or the individual terms or a seriesLesson SummaryUse a pattern to determine the parameters of arithmetic series.Find the sum of a series. Lesson PlanHookVisit the following website and discuss the structures in pictures. Discuss how to determine the number of cans in each picture. students generate three or four different finite arithmetic sequences and find the sum of those sequences.Teacher Notes: The text resource uses t1 instead of a in the general term and sum formula.Introduce the difference between a finite and infinite sequence.Some students will create short sequences and manually add up the terms.Encourage students to then look for patterns in finding the sum to develop their own formula.A prompt could indicate that the last term of the sequence is tn.It may also help to prompt students to examine a relationship between the number of terms, the first term and the last term of the sequence in finding the sum of an arithmetic sequence.Provide a few sequences and the formulae (tn?=?a + (n?-?1)d, Sn?=? and Sn?=?. Given:n, a and d, find the sumn, a and tn, find the sumSn, a and n find parameter dSn, a and n find tnSn, a and d find parameter nSn, n and d find parameter aSn, a and tn find parameter nSome examples can include: 2, 5, 8, 11, 14, ...find the sum of the first 100 terms.9, 2, -5, -12, -18, ...find the sum of the first 200 terms.Find the sum of the first 15 terms of the sequence defined by tn = 3n-2. Relate back to the Canstruction website and have students design a symmetrical shape using soup cans and calculate how many cans would be required to build their shape. Teacher Notes: For students who finish early, they could work on developing a formula for the sum of an arithmetic series.Some possible shapes students could design are pyramids, cones, football, etc …For this to work, the number of cans on each level of the shape does have to work out to be an arithmetic sequence.Students will share their designs with the class. Going BeyondExample #2 on p. 26 McGraw-Hill Ryerson Pre-Calculus 11 TextbookResourcesMath 20-1 (McGraw-Hill: sec 1.1, 1.2) Supporting If you wish to relate an arithmetric sequence to a linear function, consider this applet. You may want to use graphing calculators to show the general term of an arithmetic sequence (tn?=?a + (n?-?1)d) as a transformation of the linear function (tn?=?a?+?nd). Students may need coaching to realize that what they know about y = mx + b applies to tn?=?a?+?nd and ultimately to tn?=?a + (n?-?1)d.Source: Glossaryarithmetic sequence - A sequence for which the difference between successive terms is constantarithmetic series - The sum of the terms of an arithmetic sequencecommon difference - A constant that is added to each term to produce an arithmetic sequencefirst term – The first value in a/an arithmetic/geometric sequence/series.general term - A function that describes all terms in a sequencesequence - A set or list of numbers arranged in a definite order. A sequence is a function whose domain is a subset of the natural numbers, N, and whose range is a subset of the real numbers, R. The sequence itself shows the range of the function.finite sequence – A sequence that has a specific number of terms.infinite sequence – A sequence that has an unlimited number of terms.OtherLesson 3Geometric SequenceSTAGE 1BIG IDEA: . The world is full of patterns to be discovered. Students will be able to recognize a pattern and continue modeling the sequence to make predictions for future elements.ENDURING UNDERSTANDINGS:Students will understand …Different types of sequences and series exist.We can use mathematics to model the pattern of the sequence or series.ESSENTIAL QUESTIONS:Is anything in the universe truly random? Is chaos a pattern?What is the underlying structure in the pattern that allows sequences and series to be expressed mathematically, concretely, symbolically, pictorially, and verbally in different terms depending on the context?Can we recognize that there is universality to patterns that manifest themselves in different contexts in nature?KNOWLEDGE:Students will know …the difference between arithmetic and geometricthe notation of sequences and series (a, d, n, r, t, tn)the components required to finding the general termthe difference between convergent and divergent geometric series and what leads to convergenceSKILLS:Students will be able to …identify arithmetic and geometric sequencescreate a model for a problem/scenariocalculate any specified parameter for a sequence or series (a, d, n, r, tn, Sn)find the sum of a sequence or the individual terms or a seriescalculate the infinite sum of a convergent seriesLesson SummaryDevelop understanding/formula of the general term of a geometric sequence. Lesson PlanIntroductionWhen Spider Man was bitten, the radioactive spider injected 1 mg of venom into his body. The venom concentration doubles every hour. How many mg were in his blood stream eight hours later? A car brand new costs $25 000. Each year, on average, the car is worth 80% of the previous year. In what year is it worth half of its original value? LessonDetermine the common ratio for a given geometric sequence.Find specific terms given the general term.Given:n, r, tn, find parameter aa, n, tn, find parmeter r Going BeyondExample #3, page 36 (method 2) McGraw-Hill Ryerson Pre-Calculus 11 TextbookResourcesMath 20-1 (McGraw-Hill: sec 1.3) Supporting Consider showing the trailer of the first Spiderman movie. you wish to show the general term of a geometric sequence as a transformation of the exponential function, consider:source slipAssessed on unit exam Glossarycommon ratio - A constant that is multiplied to each term to produce a geometric sequencefirst term – The first value in a/an arithmetic/geometric sequence/series.general term - A function that describes all terms in a sequencegeometric sequence - A sequence in which the ratio of successive terms is constantOtherLesson 4Geometric SeriesSTAGE 1BIG IDEA: . The world is full of patterns to be discovered. Students will be able to recognize a pattern and continue modeling the sequence to make predictions for future elements.ENDURING UNDERSTANDINGS:Students will understand …Different types of sequences and series exist.We can use mathematics to model the pattern of the sequence or series.ESSENTIAL QUESTIONS:Is anything in the universe truly random? Is chaos a pattern?What is the underlying structure in the pattern that allows sequences and series to be expressed mathematically, concretely, symbolically, pictorially, and verbally in different terms depending on the context?Can we recognize that there is universality to patterns that manifest themselves in different contexts in nature?KNOWLEDGE:Students will know …the notation of sequences and series (a, d, n, r, t, tn)the components required to finding the general termSKILLS:Students will be able to …calculate any specified parameter for a sequence or series (a, d, n, r, tn, Sn) find the sum of a sequence or the individual terms or a seriescalculate the infinite sum of a convergent seriesLesson SummaryDevelop an understanding and uses for the formulae of a geometric series. Lesson PlanIntroduction ActivityYou and your parent agree on a payment plan for you to do your household chores for the next 16 weeks. Your parents, thinking they are so smart, agreed to pay you one penny on the first week and keep doubling the payment each week for 4 months (16 weeks). By the end of the 16 weeks what is the total amount of money your parents have paid you to do your chores.Investigating FractalsRefer to Page 46 of the MGH-Ryerson textbook on Investigating Fractals.Or refer to Applied Math 30 resources for fractal activitiesLessonDevelop the formula for a geometric series in stages. Sum = 64 - 1= 63: Sum of 6 iterations = and Sum of n iterations = Sum = 2 + 4 + 8 + 16 + 32 + 64 = 126Try : , but 63 is 126/2.Sum of 6 iterations = and Sum of n iterations = Sum = 2 + 6 + 18 + 54 + 162 + 486 = 728 (each term is 3 times the previous term)Try : , but 728 is 1456/2 and 3 – 1 = 2Sum of 6 iterations = Sum of n iterations = Sn?=?Test the first 2 examples:1 + 2 + 4 + 8 + 16 + 32 = 63Sn?=? = 63 ?2 + 6 + 18 + 54 + 162 + 486 = 728Sn?=? = 728 ?Give geometric series and have students come up with the sums when given:a, r and na, r, and tnGiven Sn, students must determine the parameters:a, given r, nr, given a, ntn, given a, r Going BeyondResourcesMath 20-1 (McGraw-Hill: sec 1.4) Supporting If you wish to show the general term of a geometric sequence as a transformation of the exponential function and a geometric series as the sum of the underlying sequence:source: Slip Glossaryarithmetic sequence - A sequence for which the difference between successive terms is constantarithmetic series - The sum of the terms of an arithmetic sequencecommon difference - A constant that is added to each term to produce an arithmetic sequencefirst term – The first value in a/an arithmetic/geometric sequence/series.general term - A function that describes all terms in a sequencegeometric sequence - A sequence in which the ratio of successive terms is constantgeometric series - The sum of the terms of a geometric sequenceOtherLesson 5Infinite Geometric SeriesSTAGE 1BIG IDEA: The world is full of patterns to be discovered. Students will be able to recognize a pattern and continue modeling the sequence to make predictions for future elements.ENDURING UNDERSTANDINGS:Students will understand …Different types of sequences and series exist.We can use mathematics to model the pattern of the sequence or series.ESSENTIAL QUESTIONS:Is anything in the universe truly random? Is chaos a pattern?What is the underlying structure in the pattern that allows sequences and series to be expressed mathematically, concretely, symbolically, pictorially, and verbally in different terms depending on the context?Can we recognize that there is universality to patterns that manifest themselves in different contexts in nature?KNOWLEDGE:Students will know …the notation of sequences and series (a, d, n, r, t, tn)the components required to finding the general termSKILLS:Students will be able to …create a model for a problem/scenariocalculate any specified parameter for a sequence or series (a, d, n, r, t, tn, Sn) find the sum of a sequence or the individual terms or a seriescalculate the infinite sum of a convergent seriesLesson SummaryUnderstand the conditions necessary to determine the sum of an infinite geometric series.Explain why a geometric series is convergent or divergent Lesson PlanHookEach student will need a blank white sheet of paper. Have students colour one half of the sheet and label it . Students will then colour half of the remaining white space, labeling it . Continue this for , , , and .Ex.Step 1Step 2Step 3Explain to students that they have just modelled a geometric series. Can they write out the first few terms? What is the general equation?Solution:Consider asking students to calculate S5 using the information on their coloured sheet as well as by the formula. What would happen if we continued colouring half of the remaining white space an infinite number of times? What sum are we approaching? How does our colour sheet help us check the last answer?Discuss with students the idea that the sum gets closer and closer to 1. Because the series approaches one value, we say that it is convergent. Consider having students graph so they can see an example of a convergent graph.What would happen if we coloured twice as much each time? + 1 + 2 + 4 + 8 + …Does this series approach one particular sum?Explain that series that do not approach a particular sum are called divergent. Consider having students graph so they can see an example of a divergent graph. See if students can determine the factors, which determine whether a series is convergent (| r | < 1) and divergent (| r | > 1). Note: If students have not been introduced to absolute value notation, consider writing |?r?|?<?1 as -1 < r < 1. Consider coming back to this example when students study absolute value. Graphing |?r?|?<?1 and discussing -1 < r < 1 may help students remember the meaning of absolute value.Introduction ActivityFind the sum of the following series0.3 + 0.03 + 0.003 + 0.0003 +….2 + 4 + 8 + 16 +….Discuss why you can find the sum of the first example but not the second?Distinguish between divergent and convergent series.LessonMath 20-1 (McGraw-Hill, page 60): Convergent Series, and Divergent Series (with or without graphing calculator).Analyze a geometric sequence to determine whether or not it has a sum.Find the sum of a convergent series.Math 20-1 (McGraw-Hill, page 60): Infinite Geometric Series,See Math 20-1 (McGraw-Hill, page 64), #11).For what values of x will this series become convergent?Note: At this point students may be able to follow the steps of the examples using , but have trouble applying what they know to new examples. After students complete Math 20-1 (McGraw-Hill, page 64), #11), consider returning to . Rewrite the series, using the first term as a common factor. 0.3(1 + 0.1 + 0.01 + 0.001 +….)Now 1 becomes t1 and t2 is r in the new series. Help students understand that when the first term is factored out becomes , where t1 = 1 and t2 = r. Going BeyondResourcesMath 20-1 (McGraw-Hill: sec 1.5) Supporting Assessment Glossaryconvergent series – A series in which the sum is finite, where r is between -1 and +1divergent series – A series that is not convergent, where r is greater than or equal to 1 or less than or equal to -1.infinite sequence - A sequence that does not end or have a final terminfinite geometric series – A geometric series that does not end or have a final term. An infinite geometric series may be convergent or divergent.Other ................
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