Teaching & Learning Plans - Project Maths

Teaching & Learning Plans 

Arithmetic 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. Student Learning Tasks – Teacher Input: 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. 

Student Activities – Possible Responses: Gives details of possible student 

reactions and responses and possible misconceptions students may have. 

iii. Teacher’s Support and Actions: Gives details of teacher actions designed to 

support and scaffold student learning. 

iv. 

Assessing the Learning: 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.

Teaching & Learning Plan: 

Leaving Certificate Syllabus 

Aims 

• To understand the concept of arithmetic series 

• To use and manipulate the appropriate formulas 

• To apply the knowledge of arithmetic series in a variety of contexts 

• To deal with combinations of arithmetic sequences and series and 

distinguish between them 

Prior Knowledge 

Students have prior knowledge of: 

• the concept of Patterns 

• basic number systems 

• sequences 

• basic graphs in the co-ordinate plane 

• simultaneous equations with 2 unknowns 

• T n 

as the n th term of a sequence. 

Learning Outcomes 

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

• recognise arithmetic series in a variety of contexts 

• recognise series that are not arithmetic 

• apply their knowledge of arithmetic series in a variety of contexts 

• apply the relevant formulas in both theoretical and practical situations 

• given information about a sequence or series, students should be able to 

derive the first term (a), the common difference (d) ,the nth term (T n 

) and 

the sum of the first n terms (S n 

). 

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

Teaching & Learning Plan: Arithmetic Series 

Catering for Learner Diversity 

In 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 

3.1 Number 

systems 

Students 

working at FL 

should be able 

to 

In addition, 

students working 

at OL should be 

able to 

–– 

find the sum to 

n terms of an 

arithmetic series 

In addition, 

students working 

at HL should be 

able to 


Student Learning Tasks: Teacher 

Input 


Student Activities: Possible 

Responses 

Lesson Interaction 

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 

the10 th week? 

(ii) how much in total will he 

have saved after the first 10 

weeks?» 

• €40, €45, €50, €55, €60, 

€65, €70, €75, €80, €85 

» 

» 

• Answer €85 

• €40 + €45 + €50 + €55 + 

€60 + €65 + €70 + €75 + 

€80 + €85 = €625 

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 10 th 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?» 

Teacher Reflections 

»» 

Which answer is bigger and 

why? 

• The total for the 10 weeks is 

bigger because the savings 

for each week are added. 

» 

» 

» 

» 

» 

» 

» 

»» 

Can they relate this 

difference to the 

question posed? 

»» 



relationship between 

the 10th term and 

the sum of the first 

10 terms?» 

» 

»» 

The amount Seán saved in the 

10 th week is known as the 10 th 

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 10 th 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 that 

when the terms of a 

sequence are added 

together, a series is 

formed? 

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


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?» 


Responses 

• €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 

»» 

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.» 

»» 

Did students arrive at 

the correct answer for 

both questions? » 

»» 

Did any students have 

misconceptions? 


»» 

Does this seem like a 

reasonable answer? Explain. 

• 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. 

»» 

Ask an individual student to 

write the solutions on the 

board and explain what they 

are doing in each step. 

»» 

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 term 

n= Number of terms 

d= Common difference 

S n = 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 T n as well and use the 

opportunity provided by the 

problems on series to revise 

the use of the formula. 

»» 


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 


differences between 

arithmetic sequences 

and series? 


Student Learning Tasks: Teacher Input Student Activities: 

Possible Responses 

»» 

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. 

»» 

Look at Section A: Student Activity 

1 and commence problems 1, 2, 3, 6 

and 7. 


Teacher’s Support and 

Actions 

»» 

On the board, write the 

formula and what each 

letter represents:» 

a = First term 

n= Number of terms 

d= Common difference 

S n = 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. 

»» 

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. 


»» 

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? 

»» 

Are students developing the 

formula for S n ? 

»» 

Are the students clear that 

(i) a 1 , a 2 , a 3 ... constitutes a 

sequence and that a 1 + a 2 + a 3 

+ ... 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 Input 



Responses 


Actions 

Assessing the 

Learning 

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

»» 

Now, I want you all to try 

the interactive quiz called 

“Arithmetic Series” on the 

Students' CD. 

»» 

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?» 

» 

» 

»» 

If computers are not 

available, hard copies of the 

quiz can be printed for the 

students. 

»» 

Write the question and 

the students’ responses on 

the board and discuss each 

stage as it progresses.» 

Note: Delay giving the 

procedure. 

»» 

Are students 

getting the 

majority of 

answers correct? 

»» 

Can students 

develop the 

equations?» 

»» 

Can students solve 

the equations?» 

»» 

Do students 


meaning of the 

solution of these 

equations?» 


»» 

What information does 

this give us? 

• We have Simultaneous Equations 

and we can solve them. 

2a + 15d = 36 

2a + 7d = 20 

8d = 16 

d = 2 

2a+15(2) = 36 

2a = 6 

a = 3 

»» 

Are the students 

certain that the 8 th 

term is given by » 

» 

and that the sum 

to eight terms is 

given by » 

» 

• The first term is 3 and the common 

difference is 2. 



Student Learning Tasks: Teacher Input 


Responses 


Actions 

Assessing the 


»» 

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.» 

»» 

Are students able to 

complete the given 

question?» 

» » Are any 

misconceptions 

realised and 

rectified? 


»» 

Encourage students 

to explain their 

reasoning.» 

»» 

Allocate the more 

difficult questions 

to the students who 

have made most 

progress to date. 




Teacher Input 

»» 

A runner is already running 

1km per day and decides 

to increase this amount by 

0.1km per day starting on 

the 1 st August. 

How far does he run on the 

10 th August? 

» 


Responses 

»» 

» 

» 

» 

» 

» 

» 


Actions 

»» 

Write the problem on the 

board.» 


»» 

Why is a equal to 1.1? 

» 

» 

» 

»» 

If you used the formula» 

» 

to calculate this, what value 

was a (the first term)? 

» 

• Because the 1km was the 

distance he was running before 

the 1 st August and was not the 

first term. 1.1 was the first term. » 

• 1.1 

» 

» 

» 

» 

»» 

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. 

»» 


recognise that 1.1 is the 

first term?» 

» » Do students recognise 

the need to read all 

questions carefully? 


»» 

Find the total distance he 

ran up to and including 10 th 

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. 



Student Learning Tasks: Teacher 

Input 

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?» 

» 

» 

» 

Student Activities: 

Possible Responses 

• 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 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 a 1 , 

a 2 , a 3 ... 

»» 

Write a number of series on 

the board so that the students 

are clear that a series is the 

unevaluated sum a 1 + a 2 + a 3 + ... 

»» 

Make sure that the students 

recognise that the general term of 

an arithmetic sequence is also the 

general term of the series. 

Assessing the 


»» 

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? 


»» 

How is an arithmetic series 

formed? 

• It is formed when the 

terms of the arithmetic 

sequence are added 

together. 



Appendix A 

Proof of Formula for arithmetic series 

(Students will not be required to prove this formula.) 

S n = T 1 + T 2 + T 3 + T 4 + ............................................... T n-1 + T n 

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

Note S n can also be written as T n + T n-1 ...............+T 4 + T 3 + T 2 + T 1 

Writing S n in reverse: 

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

Adding (1) and (2) 

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

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

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

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

S n = 

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

S n = {a + a + (n-1)d} = {first term + n th 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? 

6 

2 

2. Find the total amount of metal required to continue 

this shape with 20 sides. The lengths of the sides are in 

metres. 

5 

A 

4 

1 

3 

7 

G 

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. 



Section A: Student Activity 1 (continued) 

8. Find an expression for S n 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 30 th 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 S n 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, S 4 = 26 and S 6 = 57, find T n . 

2. The terms of an arithmetic sequence are given by the formula T n = 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 S 1 of an arithmetic series is -3 and S 2 of the same series is 3. 

a. Find the common difference. 

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

c. When S n 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 5 th 

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 

S n = n 2 − 15n. 

a. Find the first term and the common difference. 

b. Find T n the n th 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 T 20 , the 20 th term of the equivalent sequence. 

c. Find S 20 , 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 S n 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 1 st 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 10 th 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 1 st 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 1 st 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 

1 st February, ten years after she bought the first car? 

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

Teaching & Learning Plans - Project Maths

Create successful ePaper yourself

Delete template?

Save as template?