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Teaching & Learning PlansArithmetic Series

Leaving Certificate Syllabus

The Teaching & Learning Plans are structured as follows:

Aims outline what the lesson, or series of lessons, hopes to achieve.

Prior Knowledge points to relevant knowledge students may already have and also to knowledge which may be necessary in order to support them in accessing this new topic.

Learning Outcomes outline what a student will be able to do, know and understand having completed the topic.

Relationship to Syllabus refers to the relevant section of either the Junior and/or Leaving Certificate Syllabus.

Resources Required lists the resources which will be needed in the teaching and learning of a particular topic.

Introducing the topic (in some plans only) outlines an approach to introducing the topic.

Lesson Interaction is set out under four sub-headings:

i. StudentLearningTasks–TeacherInput:This section focuses on possible lines of inquiry and gives details of the key student tasks and teacher questions which move the lesson forward.

ii. StudentActivities–PossibleResponses:Gives details of possible student reactions and responses and possible misconceptions students may have.

iii. Teacher’sSupportandActions:Gives details of teacher actions designed to support and scaffold student learning.

iv. AssessingtheLearning:Suggests questions a teacher might ask to evaluate whether the goals/learning outcomes are being/have been achieved. This evaluation will inform and direct the teaching and learning activities of the next class(es).

Student Activities linked to the lesson(s) are provided at the end of each plan.

© Project Maths Development Team 2011 www.projectmaths.ie 1

Teaching & Learning Plan: Leaving Certificate Syllabus

Aims• Tounderstandtheconceptofarithmeticseries

• Touseandmanipulatetheappropriateformulas

• Toapplytheknowledgeofarithmeticseriesinavarietyofcontexts

• Todealwithcombinationsofarithmeticsequencesandseriesanddistinguishbetweenthem

Prior Knowledge Students have prior knowledge of:

• theconceptofPatterns

• basicnumbersystems

• sequences

• basicgraphsintheco-ordinateplane

• simultaneousequationswith2unknowns

• Tnasthenthtermofasequence.

Learning OutcomesAs a result of studying this topic, students will be able to:

• recognisearithmeticseriesinavarietyofcontexts

• recogniseseriesthatarenotarithmetic

• applytheirknowledgeofarithmeticseriesinavarietyofcontexts

• applytherelevantformulasinboththeoreticalandpracticalsituations

• giveninformationaboutasequenceorseries,studentsshouldbeabletoderivethefirstterm(a),thecommondifference(d),thenthterm(Tn)andthesumofthefirstnterms(Sn).

Teaching & Learning Plan: Arithmetic Series


Catering for Learner DiversityIn class, the needs of all students, whatever their level of ability are equally important. In daily classroom teaching, teachers can cater for different abilities by providing students with different activities and assignments graded according to levels of difficulty so that students can work on exercises that match their progress in learning. Less able students, may engage with the activities in a relatively straightforward way while the more able students should engage in more open-ended and challenging activities. In this Teaching and Learning Plan, for example, teachers can provide students with different applications of arithmetic series and with appropriate amounts and styles of support.

In interacting with the whole class, teachers can make adjustments to suit the needs of students. For example, derive the formula for an arithmetic series in order to gain a greater understanding of the topic.

Apart from whole-class teaching, teachers can utilise pair and group work to encourage peer interaction and to facilitate discussion. The use of different grouping arrangements in these lessons should help ensure that the needs of all students are met and that students are encouraged to verbalise their mathematics openly and to share their learning.

Relationship to Leaving Certificate Syllabus

Students learn about

Students working at FL should be able to

In addition, students working at OL should be able to

In addition, students working at HL should be able to

3.1 Numbersystems

– findthesumtontermsofanarithmeticseries


© Project Maths Development Team 2011 www.projectmaths.ie KEY: » next step • student answer/response 3

Teacher Reflections

Lesson InteractionStudent Learning Tasks: Teacher Input

Student Activities: Possible Responses

Teacher’s Support and Actions Assessing the Learning

Section A: To find the sum of first n terms of an arithmetic series»» Try»this»question»with»or»

without»your»calculator.»“If»Seán»saves»€40»for»the»first»week»and»increases»this»amount»by»€5»per»week»each»week»thereafter»(i)»how»much»will»he»save»in»the10th»week?(ii)»how»much»in»total»will»he»have»saved»after»the»first»10»weeks?»

»» Which»answer»is»bigger»and»why?

•» €40,»€45,»€50,»€55,»€60,»€65,»€70,»€75,»€80,»€85»»

•» Answer»€85

•» €40»+»€45»+»€50»+»€55»+»€60»+»€65»+»€70»+»€75»+»€80»+»€85»=»€625

•» The»total»for»the»10»weeks»is»bigger»because»the»savings»for»each»week»are»added.

Note: As»the»formula»will»not»yet»be»introduced»the»emphasis»at»this»stage»is»on»getting»the»students»to»understand»the»difference»in»how»much»Seán»will»save»on»the»10th»week»and»how»much»in»total»will»he»have»saved»after»the»first»10»weeks?

»» Can»students»successfully»complete»the»question?»

»» Do»the»students»understand»the»difference»in»phrasing»of»parts»(i)»and»(ii)»of»the»question?»

»» Can»they»relate»this»difference»to»the»question»posed?

»» The»amount»Seán»saved»in»the»10th»week»is»known»as»the»10th»term»of»the»series»and»the»total»amount»he»had»saved»in»the»first»10»weeks»is»known»as»the»sum»of»the»first»ten»terms»(or»the»partial»sum)»of»the»series.»

»» Can»anyone»tell»me»if»there»is»a»relationship»between»arithmetic»sequences»and»series»and»if»so»what»is»it?

»»»»»»»

•» There»is»a»relationship.»An»arithmetic»series»is»formed»when»the»terms»of»an»arithmetic»sequence»are»added»together.

»» Write»on»the»board»the»10th»term»of»the»series»(€85)»and»the»sum»of»the»first»ten»terms»of»the»series:»€40»+»€45»+»€50»+»€55»+»€60»+»€65»+»€70»+»€75»+»€80»+»€85»=»€625.

»» Explain»that»when»the»series»is»written»in»the»form»€40»+»€45»+»€50»+»€55»+»€60»+»€65»+»€70»+»€75»+»€80»+»€85»is»referred»to»as»the»unevaluated»sum.»

»» Explain»that»€40,»€45,»€50,»€55,»€60,»€65,»€70,»€75,»€80,»€85»…»constitutes»a»sequence»whereas»€40»+»€45»+»€50»+»€55»+»€60»+»€65»+»€70»+»€75»+»€80»+»€85»is»the»corresponding»series.

»» Do»the»students»understand»the»relationship»between»the»10th»term»and»the»sum»of»the»first»10»terms?»»

»» Do»the»students»understand»that»when»the»terms»of»a»sequence»are»added»together,»a»series»is»formed?



Teacher Reflections

Student Learning Tasks: Teacher Input



»» Emily»earned»€20,000»in»her»first»year»of»employment»and»got»an»annual»increase»of»€4.000,»thereafter.»How»much»will»she»earn»in»the»eigth»year»of»her»employment»and»how»much»will»her»total»earnings»be»in»the»first»eight»years?»

»» Does»this»seem»like»a»reasonable»answer?»Explain.

•» €20,000»+»€4,000»+»€4,000»+»€4,000»+»€4,000»+»€4,000»+»€4,000»+»€4,000»=»€48,000

•» Her»total»earnings»will»be»€20,000»+»€24,000»+»€28,000»+»€32,000»+»€36,000»+»€40,000»+»€44,000»+»€48,000»=»€272,000

•» Yes.»Without»any»increase»in»salary»Emily»would»earn»€20,000»×»8»=»€160,000.»With»the»increase»each»year,»the»total»should»be»greater»than»€160,000.

»» Delay»telling»students»the»algorithm.»Allow»students»to»explore»this»for»themselves.»

Note: for»less»able»students,»the»teacher»can»select»smaller»numbers,»but»students»will»appreciate»the»difference»more»if»working»with»larger»figures.»

»» Ask»an»individual»student»to»write»the»solutions»on»the»board»and»explain»what»they»are»doing»in»each»step.»

»» Did»students»arrive»at»the»correct»answer»for»both»questions?»»

»» Did»any»students»have»misconceptions?

»» If»we»need»to»calculate»Emily’s»earnings»over»thirty»years»we»will»need»a»less»time»consuming»and»more»robust»technique.»»

»» The»formula»for»the»sum»of»the»first»n»terms»of»an»arithmetic»series»is»»»

(See»Tables»and»Formulae»booklet).»a»=»First»termn=»Number»of»termsd=»Common»differenceSn»=»Sum»of»the»first»n»terms

»» Write»the»formula»»»

»and»the»meaning»of»the»terms»on»the»board.»

»» The»proof»of»this»formula»can»be»seen»in»Appendix A»and»it»is»recommended»that»where»appropriate»a»class»should»be»shown»how»it»is»derived.»

»» Write»down»the»formula»for»Tn»as»well»and»use»the»opportunity»provided»by»the»problems»on»series»to»revise»the»use»of»the»formula.

»» Do»the»students»understand»that»a,»d»and»n»have»the»same»meaning»as»they»had»for»an»arithmetic»sequence?»

»» Can»students»explain»what»each»of»the»terms»mean?»

»» Do»all»students»understand»the»differences»between»arithmetic»sequences»and»series?



Teacher Reflections

Student Learning Tasks: Teacher Input Student Activities: Possible Responses

Teacher’s Support and Actions

Assessing the Learning

»» Now»I»want»you»to»work»in»pairs.»One»student»is»to»be»called»student»A»and»the»other»student»in»the»group»student»B.»

»» Student»A,»write»down»an»arithmetic»series»containing»at»least»ten»terms»and»inform»student»B»of»the»first»number,»the»common»difference»and»the»number»of»terms»in»the»series.»

»» Student»B»must»now»find»the»partial»sum»(the»sum»to»n»terms)»of»the»chosen»series»using»the»formula.»Student»A»must»find»the»sum»using»their»calculator.»Then»compare»your»answers.

»» On»the»board,»write»the»formula»and»what»each»letter»represents:»a»=»First»termn=»Number»of»termsd=»Common»differenceSn»=»Sum»of»the»first»n»terms

»» Walk»around»the»room»and»check»that»all»students»are»engaged»in»creating»their»own»problem.»

»» When»some»groups»have»completed»the»exercise»they»can»be»asked»to»reverse»the»roles.

»» Can»students»verbally»express»their»own»everyday»examples»of»arithmetic»sequence»and»series?»

»» Did»all»students»devise»their»own»problem?»

»» Were»all»students»able»to»apply»the»formula»correctly»to»the»problem?»

»» Were»the»students’»misconceptions»addressed»as»a»result»of»doing»the»problems?

»» Look»at»Section A: Student Activity 1»and»commence»problems»1,»2,»3,»6»and»7.

»» As»students»progress»through»this»activity»sheet»the»lesson»can»be»stopped»from»time»to»time»and»discussions»developed»in»relation»to»the»questions.»The»content»of»this»discussion»will»depend»on»what»the»teacher»has»seen»in»the»students’»answers.»

»» Further»questions»from»the»Section A: Student Activity 1»can»be»given»for»homework.

»» Are»students»developing»the»formula»for»Sn?

»» Are»the»students»clear»that»(i)»a1,»a2,»a3...»constitutes»a»sequence»and»that»a1»+»a2»+»a3»+»...»is»the»corresponding»series?

»» (ii)»»is»the»general»term»of»an»arithmetic»sequence»and»that»»

gives»the»partial»sum»(the»sum»to»n»terms)»of»an»arithmetic»series?



Teacher Reflections





Section B: To further develop the concept of Sn of an arithmetic series

»» Now,»I»want»you»all»to»try»the»interactive»quiz»called»“Arithmetic»Series”»on»the»Students'»CD.

»» If»computers»are»not»available,»hard»copies»of»the»quiz»can»be»printed»for»the»students.

»» Are»students»getting»the»majority»of»answers»correct?

»» We»know»that»the»sum»of»the»first»eight»terms»of»an»arithmetic»series»is»80»and»that»the»sum»of»the»first»sixteen»terms»is»288.»How»can»we»represent»this»information»using»algebra?»

»» Now,»what»types»of»equations»do»we»have»and»what»can»we»do»with»them?»

»» What»information»does»this»give»us?

»»

•» We»have»Simultaneous»Equations»and»we»can»solve»them.2a»+»15d»=»362a»+»»»7d»=»20»»»»»»»»»»8d»=»16»»»»»»»»»»»»d»=»2

2a+15(2)»=»362a»=»6a»=»3

•» The»first»term»is»3»and»the»common»difference»is»2.

»» Write»the»question»and»the»students’»responses»on»the»board»and»discuss»each»stage»as»it»progresses.»

Note: Delay»giving»the»procedure.

»» Can»students»develop»the»equations?»

»» Can»students»solve»the»equations?»

»» Do»students»understand»the»meaning»of»the»solution»of»these»equations?»

»» Are»the»students»certain»that»the»8th»term»is»given»by»»»

and»that»the»sum»to»eight»terms»is»given»by»»»»



Teacher Reflections

Student Learning Tasks: Teacher Input Student Activities: Possible Responses



»» Do»questions»1-9»in»Section B: Student Activity 2.

»» Compare»your»answers»around»the»class»and»have»a»discussion»if»the»answers»do»not»all»agree.

»» Teacher»distributes»Section B: Student Activity 2.

Note: Select»questions»depending»on»the»ability»and»progress»of»the»students.»

»» Encourage»students»to»explain»their»reasoning.»

»» Allocate»the»more»difficult»questions»to»the»students»who»have»made»most»progress»to»date.

»» Are»students»able»to»complete»the»given»question?»

»» Are»any»misconceptions»realised»and»rectified?



Teacher Reflections





»» A»runner»is»already»running»1km»per»day»and»decides»to»increase»this»amount»by»0.1km»per»day»starting»on»the»1st»August.How»far»does»he»run»on»the»10th»August?»

»» Why»is»a»equal»to»1.1?»»»

»» If»you»used»the»formula»»

to»calculate»this,»what»value»was»a»(the»first»term)?

»» Find»the»total»distance»he»ran»up»to»and»including»10th»August.

»» As»in»earlier»examples»you»need»to»read»the»question»carefully»to»establish,»the»value»of»the»first»term.»

»» Now»do»questions»10»and»11»in»Section B: Student Activity 2.

»» »»»»»»

»

•» Because»the»1km»was»the»distance»he»was»running»before»the»1st»August»and»was»not»the»first»term.»1.1»was»the»first»term.»»

•» 1.1»»»»

•» »»

»» Write»the»problem»on»the»board.»

»» Give»students»time»to»explore»possibilities»and»to»discuss»what»is»happening.»

»» Encourage»students»to»explain»their»reasoning.»

»» Walk»around»the»room»and»check»the»students’»responses»to»the»questions»on»the»activity»sheet.

»» Do»the»students»recognise»that»1.1»is»the»first»term?»

»» Do»students»recognise»the»need»to»read»all»questions»carefully?



Teacher Reflections




Reflection»» Now»to»recap,»what»do»you»

understand»by»the»word»‘arithmetic’»sequence?»»

»» Can»you»tell»me»anything»you»know»about»a»sequence?»»»»»»»»

»» What»can»you»tell»be»about»a»series?»»»»

»» How»is»an»arithmetic»series»formed?

•» Arithmetic»means»that»any»term»can»be»obtained»by»adding»a»fixed»number»to»the»preceding»term.»

•» A»sequence»is»a»group/array»of»numbers»that»are»in»some»form»of»pattern.»Arithmetic»sequences»have»a»common»difference»i.e.»»»

»

•» If»the»sequence»is»for»example»1,»3,»5,»9,…»»then»the»series»is»1»+»3»+»5»+»7»+»9»+»…»

•» It»is»formed»when»the»terms»of»the»arithmetic»sequence»are»added»together.

»» It»is»important»that»this»section»is»concluded»on»a»positive»note»and»that»students»can»see»that»arithmetic»sequences»and»series»are»relevant»and»are»encountered»in»everyday»life.»

»» Write»a»number»of»sequences»on»the»board»so»that»student»are»clear»that»sequences»take»the»form»a1,»a2,»a3...

»» Write»a»number»of»series»on»the»board»so»that»the»students»are»clear»that»a»series»is»the»unevaluated»sum»a1»+»a2»+»a3»+»...»

»» Make»sure»that»the»students»recognise»that»the»general»term»of»an»arithmetic»sequence»is»also»the»general»term»of»the»series.

»» Are»the»students»confident»in»their»knowledge»of»this»topic?»

»» Can»students»effectively»articulate»the»concept?»

»» Can»students»explain»the»properties»of»arithmetic»sequences»and»series?



Appendix A

Proof of Formula for arithmetic series(Students will not be required to prove this formula.)

Sn = T1 + T2 + T3 + T4 + ............................................... Tn-1 + Tn

Sn = a + a + d + a + 2d + a + 3d +..........a + (n-2)d + a + (n-1)d (1)

Note Sn can also be written as Tn+ Tn-1...............+T4 + T3 + T2 + T1

Writing Sn in reverse:

Sn = a + (n-1)d + a + (n-2)d ..................... a + 3d + a + 2d + a + d +a (2)

Adding (1) and (2)

Sn = a + a + d + a + 2d + a + 3d +..........a + (n-2)d + a + (n-1)d (1)

Sn = a + (n-1)d + a + (n-2)d ..................... a + 3d + a + 2d + a + d +a (2)

2 Sn = {2a + (n-1)d} + {2a + (n-1)d} + {2a + (n-1)d}.....+{2a + (n-1)d} + {2a + (n-1)d}

2 Sn = n{2a + (n-1)d}

Sn = {2a + (n-1)d} Formula as per tables but note

Sn = {a + a + (n-1)d} = {first term + nth term}



Section A: Student Activity 1(Calculations must be shown in all cases.)

1. A craftsman uses 100 beads on the first day of the month and this amount increases by 15 beads each day thereafter. If he works 24 days in the month, how many beads will he need to order in advance to have a month’s supply?

2. Find the total amount of metal required to continue this shape with 20 sides. The lengths of the sides are in metres.

3. A factory produced 10, 13, 16 and 19 items per week in the first four weeks of the year. If this pattern continues how many items will this factory produce in the last week of the year and how many items will the factory produce in total in a complete year of business (52 weeks in the year)?

4. If James saves €40 during the first week of January and increases this amount by €5 per week every week for the following ten weeks, how much will he save in total?

5. A woman has a starting salary of €20,000 and gets an annual increase of €2,000 per year thereafter. How much will she earn in total during her working life, if she retires after working for 40 years?

6. Your new employer offers you a choice of 2 salary packages. Package A has a starting salary of €12,000 per year with an annual increase of €2,000. Package B has a starting salary of €20,000 and an annual increase of €1,500. Assuming you plan to remain in the firm for ten years which is the best package and by how much? Illustrate your reasoning with the help of calculations.

7. In a cinema, there are 140 seats in the front row, 135 in the second and 130 in the third row. This pattern continues until the last row. If the last row has 45 seats, how many rows are there in the cinema? Calculate the total number of seats in the cinema.

G

A

6

7

1

2

3

4

5



Section A: Student Activity 1 (continued)

8. Find an expression for Sn for the arithmetic Series 2+4+6+8+….

9. How many terms of the arithmetic series 1 + 3 + 5 +.... are required to give a sum in excess of 600?

10. Kayla got her new mobile phone on the first of April. She sent 1 text that day, 3 texts the next day and 5 texts the next day. If this pattern continues how many texts will she send on 30th April and how many texts in total will she send in the month of April that year? (April has 30 days.) If each text message costs 13 cent how much will she spend sending texts in April?

11. Is it possible for an arithmetic series to have a first term and a common difference that are both non-zero and have a partial sum of zero? If so, give an example and explain the circumstance that causes this to happen.

12. A water tank containing 377 litres of water develops a leak. On the first day the tank leaks 5 litres of water and this increases by 4 litres each day thereafter. Show that the amount of water that leaks each day follows an arithmetic progression and apply the Sn formula to determine how long it takes for the tank to empty. Show your calculations.

13. A bricklayer has 400 bricks and wants to build a wall following the pattern below. How many layers high will the wall be if he plans to use all his bricks?



Section B: Student Activity 2(Calculations must be shown in all cases.)

1. For a given arithmetic series, S4 = 26 and S6 = 57, find Tn.

2. The terms of an arithmetic sequence are given by the formula Tn = 25−4n.

a. Find the first three terms of the sequence. What is the value of d, the common difference?

b. Find the first negative term of the sequence.

c. For what value of n is the sum of the first n terms of the series equal to 30?

3. Given that S1 of an arithmetic series is -3 and S2 of the same series is 3.

a. Find the common difference.

b. Find the 20th term of the equivalent sequence.

c. When Sn of this series is equal to 75, what value has n?

4. Jonathan saved a certain amount for one year and increased this amount by a regular amount each year thereafter. If the total amount he saved in the first 8 years of this savings plan is €1,690 and he saved €220 in the 5th year. Find how much he saved in the first year and by how much did he increase his savings each year.

5. The sum of the first n terms of an arithmetic series is given by

Sn = n2 − 15n.

a. Find the first term and the common difference.

b. Find Tn the nth term of the equivalent sequence.

c. When is the series equal to -50?

6. The first 4 terms of arithmetic series is -3 + 4 + 11 + 18 +...

a. Find d, the common difference.

b. Find T20, the 20th term of the equivalent sequence.

c. Find S20, the sum of the first twenty terms of the series.



Section B: Student Activity 2 (continued)

7. Three consecutive terms of an arithmetic sequence are x + 3, 4x + 1 and 6x + 1. Find the value of x. Find an expression for Sn the equivalent series.

8. If a household uses 1kg of sugar each week and decideds to reduce this amount by 10g per week. How much sugar per week would the household be using at the end of the year (52 weeks in the year)? What is the total amount of sugar this household would use that year?

9. Prove that the formula for the sum of the first n Natural Numbers (N) is

N = {1, 2, 3, 4, ...}

10. Emer purchases a new car every year on 1st January. She purchased her first car in 2001 and it cost €20,000. Each year after that the cost of her new car increases by €3,000.

a. How much did she spend on her 10th car?

b. How much did she spend on the car she purchased in 2011?

c. Why were the previous two answers not the same?

d. How much did she spend, in total, on her first ten cars?

e. By 1st February 2011, how much would she have spent on cars, assuming that she bought no cars other than those in the pattern mentioned in this question?

11. Emer purchases a new car every second year on 1st January. If the first car she purchases costs €20,000 and each time she changes the cost increasses by €6,000

a. how much will she have spent in total in buying the cars on 1st February, ten years after she bought the first car?

b. how much will she spend in total on her first ten cars?

Documents

Arithmetic Series - Mr. Loisel's Classroom - Homemrloiselsclassroom.weebly.com/uploads/9/0/4/4/... · apply their knowledge of arithmetic series in a variety of contexts • apply