MY SMALL RESEARCH

MY SMALL RESEARCH

ON THE PSYCHOLOGY OF TEACHING LEARNING MATHEMATICS:

Applying Mathematical Thinking Related to Mathematical Contents in Learning

the Arithmetics Sequence and Series of the 9^th Grade Student of Junior High School

Ratna Ayu Setiyo Siwi

Yogyakarta State University

The psychology of teaching learning mathematics especially mathematical contents guided this small research. The researcher tried to uncover the student attempts in applying mathematical thinking related to the mathematical contents in learning arithmetics sequence and series of the 9^th grade student of Junior High School. The results of the research described that the student attempts to finding relation between variables, express relationship as formulas, and read the meaning of formulas. Mathematical thinking related to the mathematical contents can be compared to mathematical skills as well as estimation or deep mathematical knowledge.

INTRODUCTION

In the National Education Ministerial Decree No. 19 year 2007 about Education Management Standard by Base and Middle Education Unit, quality of study in the school/ Islamic school is developed by purpose so that educative participant reach patterned thinking and free of thinking causing can execute intellectual activity which in the form of thinking, argument, question, study, find, and predict. In other word, the educative participant expected can proceed the informations into knowledges and then using the knowledges to solving problems. According to Kazuyoshi Okubo (2006), A characteristic of school mathematics is that the learning contents in each grade are based on what the children learn in previous year or earlier in the same year, which involves a new stage of learning. Especially to this research, in the National Education Ministerial Decree No. 23 year 2006 about SKL “Graduate Competence Standard” mentioned the competence standard of lesson group in science and technology (include mathematics) for Junior High School is developing logic, ability think and analysis of educative participant. It can be realized by applying mathematical thinking related to mathematical contents. By applying mathematical contents, a student can analized and solved problem such as ideas of fundamental formulas. Above all, the researcher have an opinion that learning arithmetics sequence and series of the 9^th grade student of Junior High School needs application of mathematical thinking related to mathematical contents to increasing student’s ability in case to understanding formulas of this concept.

THEORETICAL FRAMEWORK

In his blog http://powermathematics.blogspot.com, Dr. Marsigit writed Niss’s opinion (in Ernest,1991) about the purpose of mathematics education. He said that it should be enable students to realize, understand, judge, utilize and sometimes also perform the application of mathematics in society, in particular to situations which are of significance to their private, social and professional lives. So, mathematics education has many important roles to solve problems, at present and future. According to Kazuyoshi Okubo (2006), mathematics education has two main purpose, both are following.

1. To enable children to make use what they acquire in the study of mathematics, understand phenomena in their daily life in a mathematical way, and examine and process phenomena through logical thinking.

2. To enable children to enjoy intellectual pleasure through learning mathematics creatively. In this purpose, for a change in approach to allow children to take part in more creative way, have them experience intellectual pleasure and make them feel that they are “creating” mathematics.

Moreover, his matter is also insisted by Shigeo Katagiri (2004) in Marsigit (2006) that mathematical activities can not just be pulled out of a hat, they need to be carefully chosen so that children form concepts, develop skills, learn facts and acquire strategies for investigating and solving problem.

Mathematical Thinking

According to Madihah Khalid (2006), mathematical thinking is the mathematical mode of thought that we use to solve any problem in our daily life including at school. It can be defined as applying mathematical techniques, concepts and processes, either explicitly or implicitly, in the solution of problems. It is, according to Shigeo Katagiri (2006), used during mathematical activities, and is therefore intimately related to the contents and methods of arithmetics and mathematics. According to him, mathematical thinking can be divided into three categories:

1. Mathematical attitudes

2. Mathematical thinking related to mathematical methods

3. Mathematical thinking related to mathematical contents

Mathematical attitudes is mathematical thinking which related to the attitudes, actions, and attempts done by students depend on how interested they are in the problem solving or the lesson. It is considered as the driving force behind the two later categories. Mathematical thinking related to mathematical methods is mathematical thinking which related to mathematical patterns. According to Mason, Burton, and Stacey (1982) in Stacey (2006), it is consisting of specializing (trying special cases, looking at examples), generalizing (looking for patterns and relationships), conjecturing (predicting relationships and results), and convincing (finding and communicating reason why something is true). Mathematical thinking related to mathematical contents is discussed in following sub chapter.

Mathematical Thinking Related to Mathematical Contents

In the Oxford Learner’s Pocket Dictionary, content (plural: contents) can be meant as what is contained in something. Therefore, mathematical contents is the objects (what is contained) in mathematics. According to Madihah Khalid (2006), it includes ideas of sets, units, expressions, operations, algorithms, approximation, fundamental properties and formulas. These can be compared to mathematical skills as well as estimation or deep mathematical knowledge.

According to Kazuyoshi Okubo (2006), A characteristic of school mathematics is that the learning contents in each grade are based on what the children learn in previous year or earlier in the same year, which involves a new stage of learning. Consequently, sufficiently understanding the contents and procedures of the subject is vital for the children to work on problem solving activies. The course of study guidelines mention four areas of learning contents for Junior High School, they are numbers and formulas, graphic figures, functions, and quantitative relations.

LEARNING CONTENTS ABOUT “NUMBERS AND FORMULAS”

Let’s assume that the students are going to learn contents about “numbers and formulas”. It is important to the students: (1) understand the meaning of formula and which formula should be used in a certain situasion, (2) understand the meaning of the procedure previously introduced, if the students learn such ways of learning as the basics then it serves as a great strength when they proceed to next step, (3) understand the manner in which they learned to understand it, (4) in that sense, the teacher sould clearly understand what the students need to learn and how this will be used in future. Therefore, the way to study mathematics is an important part of mathematical thinking.

Mathematical thinking related to mathematical contents is mathematical thinking related to finding relation between variables, express relationship as formulas, and read the meaning of formulas. This is a simple step to learning contents about formulas and then using the problem solving to examine mathematical thinking.

In his blog http://powermathematics.blogspot.com, Dr. Marsigit writed Ebbutt, S. and Straker, A. (1995) said that Mathematics can provide an important set of tools for problems in the main, on paper and in real situations. Students of all ages can develop the skills and processes of problem solving and can initiate their own mathematical problems. Hence, the teacher may help the students learn mathematics by: (1) providing an interesting and stimulating environment in which mathematical problems are likely to occur, (2) suggesting problems themselves and helping students discover and invent their own, (3) helping students to identify what information they need to solve a problem and how to obtain it, (4) encouraging the students to reason logically, to be consistent, to works systematically and to develop recording system, (5) making sure that the students develop and can use mathematical skills and knowledge necessary for solving problems, and (6) helping them to know how and when to use different mathematical tools. In this matter, according to Madihah Khalid (2006), the students are encouraged to use thinking skills and problem solving strategies during mathematics lessons and not just learn mathematical skills and concepts from listening to the teacher. It is featured that if mathematical thinking is not emphasized, the students would end up learning mathematics by rote memorization, without understanding and without the ability to think integently.

According to Kazuyoshi Okubo (2006), a major objective of mathematics education at present is to foster the ability to solve problems. Then, according to Polya in Sri Subarinah (2006), there are four stages be needed in problem solving:

1. The stage of understanding the problem

2. The stage of devising a plan for solution

3. The stage of carrying out the plan for solution

4. The stage of examining the solution

Polya’s four stages seem to have been derived focusing on problem solving as an activity of individuals (Kazuyoshi Okubo, 2006). It is suitable to this research that applying mathematical thinking related to the mathematical contents in learning arithmetics sequence and series of the 9^th grade student of Junior High School, in which the subject of research is only one student. The understanding of mathematical thinking related to mathematical contents about “number and formulas” that the student acquire in each stage based on Polya’s four stages (adapted from Kazuyoshi Okubo, 2006) are following.

1. The Stage of Understanding the Problem

To understanding the problem or question, the student must consider the phenomenon mathematically. In order to establish a mathematical formula necessary to answer the question, the student must understand the following:

a. The mathematical conditions related to the question

b. What must be found

c. Remembering previously learned related questions

d. Establishing a formula by selecting division as a procedure, which was previously learned

The teacher should extend guidance to help them develop the ability to practice using a mathematical way of thinking such as that mentioned above. The teacher should also use ingenuity to present a concrete phenomenon and a concrete situation related to the question shown to them.

2. The Stage of Devising a Plan for Solution

The ability to translate such a problem into a formula depends on how much they understand the of this fundamental formula or other formulas from past lessons. Then, they can finding relationship between formulas. When the student try to find the answer to a question, it is important to have an estimate of the answer and the ability to estimate which procedure should be use to find the answer. Having an estimated answer is important in the sense that they have their own viewpoint and respond using their own way of thinking.

3. The Stage of Carrying Out the Plan for Solution

It is expected that the student can find the correct procedure for finding formula. Choosing the correct procedure means to have the ability to foresee the process that leads to the result. By considering which procedure should be used to find the result before taking the steps to solve the problem, one can reasonably and efficiently find the solution. Then, the student doing the selected procedure that the student assumed correct.

4. The Stage of Examining the Solution

As described by previous stage, each student can think using his/ her own way of thinking. There are various procedures for finding the solution. The student can deepen their understanding by exchanging their opinions among themselves, finding similarities and differences and correlating them. It is important to have them examine and understand which procedure is more intelligible and easier to understand, or have them find a more general means of solving the problem, if necessary (which procedure would seem to be most suitable for future development). When one ascertains that one’s way of thinking is correct, or one shows that it is correct and conveys it to others, it is important to think logically, that is, inductively, analogously and deductively.

RESEARCH METHOD

The main aim of the research is uncovering the student attempts in applying mathematical thinking related to the mathematical contents in learning arithmetics sequence and series of the 9^th grade student of Junior High School. The subject of research is only one student, she named herself “Chako”. Because of in the National Education Ministerial Decree No. 19 year 2007 was mentioned that one of education management standard by base and middle education unit is the educative participant expected can proceed the informations into knowledges and then using the knowledges to solving problems, then one matter to indicating student’s success in studying mathematics is the student has ability to finding mathematics concepts, definitions, patterns, and relationships so that can use knowledges in the process of problem solving. The instrument used for collecting data consists of observation and interview. Observation done participatively. Interview done directly, it means the researcher collects the data from the student directly. Because of the topic about arithmetics sequence and series has not studied yet, the design of the research included: preparation, implementation, and reflection.

LESSON PLAN

Day and Date : Sunday, December 20^th 2009

Time : 08.30-10.00

Subject : Chako

Grade : 9^th of Junior High School

Competency Standard : Understanding the arithmetics sequence and series and using both in

the process of problem solving

Basic Competencies : 1. Determining the n^th term of arithmetics sequence

2. Determining the sum of the first n terms of arithmetics series

3. Solving problem related to the arithmetics sequence and series

Teaching Scenario : 1. Apperception

2. Developing concepts

3. Reflection

4. Conclusion and closing

PART 1

Aim : Determining the n^th term of arithmetics sequence

1. Introduction : Topic Comprehension

a. The teacher writes some numbers pattern on the whiteboard in which all of them described the arithmetics sequence.

b. The teacher lets the student observes given numbers pattern, what is she thinking about them?

c. The teacher informs that each number set is called sequence number and each number of sequence is called sequence term. n^th sequence term is symbolized by U_n. Then, the teacher lets the student defines sequence number and sequence term, and symbolized each number of given sequence number.

d. The teacher informs in each sequence number, the differences between each two term in series is always constant number, it is called balance (beda, Indonesia) and it is symbolized by b.

2. Activity 1 : Understanding the Problem

a. The teacher lets the student defines the arithmetics sequence.

b. The teacher lets the student identify the terms of arithmetics sequence.

c. The teacher lets the student defines the concept of n^th term of arithmetics sequence.

3. Activity 2 : Devising a Plan for Solution

a. The student attempts define the terms of arithmetics sequence.

b. The student learned that the next term is sum of previous term and balance (b).

4. Activity 3 : Carrying Out the Plan for Solution

The student learned that the n^th term of arithmetics sequence is U_n = U_n-1 + b.

5. Activity 4 : Examining the Solution

a. The student presented that U_n = a + (n - 1) b

in which

U_n is n^th term

a is the first term

n is number of term

b is balance (beda, Indonesia)

b. The student needed to have clarification whether her formula was correct?

PART 2

Aim : Determining the sum of the first n terms of arithmetics series

1. Introduction : Topic Comprehension

a. The teacher informs that if U₁, U₂, U₃, ..., U_n are sequence number then the sum of them is called number series. It is symbolyzed by S_n. Then, the teacher lets the student defines number series and symbolyzed it.

b. The teacher lets the student defines S_n in which n = 1, 2, 3, ..., n in related to the terms of sequence number. Then, the teacher lets the student symbolyzed them.

2. Activity 1 : Understanding the Problem

a. The teacher lets the student defines the arithmetics series (based on the previous informations).

b. The teacher informs the sum of the first n terms of arithmetics series (or symbolized by S_n) can be determined by sum of two arithmetics series.

c. The teacher lets the student defines this concept of sum of the first n terms of arithmetics series.

3. Activity 2 : Devising a Plan for Solution

a. The student attempts define the sum of arithmetics series.

b. The student learned that the sum of the first n terms of arithmetics series (or symbolized by S_n) can be determined by sum of two arithmetics series.

4. Activity 3 : Carrying Out the Plan for Solution

The student learned that the sum of the first n terms of arithmetics series is

S_n = n [2a + (n - 1) b] / 2.

5. Activity 4 : Examining the Solution

a. The student presented that S_n = n [2a + (n - 1) b] / 2

in which

a is the first term

n is number of term

b is balance (beda, Indonesia)

b. The student needed to have clarification whether her formula was correct?

PART 3

Aim : Solving problem related to the arithmetics sequence and series

1. Problem/ : The first term of arithmetics series is 3, the n^th term is 41, and its balance is

Question 3, find the sum of the first n terms of arithmetics series!

2. Activity 1 : Understanding the Problem

a. The student attempts to understanding the relation between variables (the relations between the first term of arithmetics series, the n^th term, and its balance) and what must be found (the sum of the first n terms of arithmetics series).

b. The student remembers previously learned related question

c. The student establishes a formula by selecting division as procedure which as previously learned.

3. Activity 2 : Devising a Plan for Solution

a. The student learned that the sum of the first n terms of arithmetics series (or symbolized by S_n) can be determined by following formula:

S_n = n [2a + (n - 1) b] / 2

b. The student learned that the first term is U₁ = a and the n^th term is U_n = a + (n - 1) b.

4. Activity 3 : Carrying Out the Plan for Solution

a. The student learned that the n^th term is U_n = a + (n - 1) b, so that number of term is

n = (U_n – a + b) / b

b. The student learned that

S_n = n [2a + (n - 1) b] / 2

So this formula can be expressed by

S_n = n [a + (a + (n - 1) b)] / 2

Therefore, because of U₁ = a and U_n = a + (n - 1) b, then

S_n = n (U₁ + U_n) / 2

According to this formulas, the student attempts to solving the problem by put U₁ = a = 3, U_n = 41, and b = 2 into the formulas.

n = (U_n – a + b) / b = (41 – 3 + 2) / 2 = 20

and

S_n = n (U₁ + U_n) / 2 = 20 (3 + 41) / 2 = 440

So, the sum of the first n terms of arithmetics series is 440.

5. Activity 4 : Examining the Solution

a. The student presented that the sum of the first n terms of arithmetics series is 440.

b. The student needed to have clarification whether her formula was correct?

ANALYSIS OF DATA

1. Topic Comprehension

Part 1 : Determining the n^th term of arithmetics sequence

It is aimed to informing studied topic because the topic about arithmetics sequence and series has not studied yet by the student. So, the student is coming to know about this topic.

The first, the teacher writes numbers pattern on the whiteboard:

(1) 2, 4, 6, 8, ...

(2) 1, 3, 5, 7, ...

(3) 3, 6, 9, 12, ...

The student observed given numbers pattern. The teacher guides the student in order to understanding similarities between them. The student defined “the differences between two number in series of (1) is 2, the differences between two number in series of (2) is 2, and the differences between two number in series of (3) is 3”.

The second, the teacher informs the differences is called balance (beda, Indonesia) in which symbolized by b, and its value is always constant. The teacher guides the student understands each numbers pattern in (1), (2), and (3) are following a certain pattern. The teacher informs each number set is called sequence number and each number of sequence is called sequence term or usually called term. The n^th sequence term is symbolized by U_n. The student defined “sequence number is a numbers set in which each numbers are following a certain pattern” and “sequence term is each numbers in sequence number”. Therefore, the student defined “in sequence number of (1), U₁ = 2, U₂ = 4, U₃ = 6, U₄ = 8”.

Part 2 : Determining the sum of the first n terms of arithmetics series

The teacher informs that if U₁, U₂, U₃, ..., U_n are sequence number then the sum of them is called number series. It is symbolyzed by S_n. Then, the student defined number series as U₁ + U₂ + U₃ + ... + U_n. the student determined S_n in which n = 1, 2, 3, ..., n. It means the student defines S_n as result generalized of S₁ (sum of the first 1 term), S₂ (sum of the first 2 terms), S₃ (sum of the first 3 terms), ... . Therefore, S_n is sum of the first n terms. Then, symbolyzed them as follows: S₁ = U₁, S₂ = U₁ + U₂, S₃ = U₁ + U₂ + U₃, ..., S_n = U₁ + U₂ + U₃ + ... + U_n. The teacher informs this is a arithmetics series, then the teacher lets the student defines it.

2. Understanding the Problem

Part 1 : Determining the n^th term of arithmetics sequence

After the student defined, the teacher informs its sequence is called the arithmetics sequence. According to the teacher information, the student defined arithmetics sequence based on related to balance as “a sequence number in which its differences between two term is constant”. The teacher guides based on this concept, sequence number of (1) can be written as follows.

U₁ = 2, U₂ = 4 = 2 + 2, U₃ = 6 = 4 + 2, U₄ = 8 = 6 + 2.

The teacher lets the student defines the concept of n^th term of arithmetics sequence. Previously, the teacher informs the n^th term of arithmetics sequence (or symbolized by U_n) can be determined by takes example by U₁, U₂, U₃, ..., U_n are the terms of arithmetics sequence with first term is a and balance is b. Understanding this concept, observe previous information about sequence number of (1).

Part 2 : Determining the sum of the first n terms of arithmetics series

The student defined the arithmetics series as “ the sum of all terms of arithmetics series”.

Teacher informs the sum of the first n terms of arithmetics series (or symbolized by S_n) can be determined by sum of two arithmetics series. The teacher lets the student defines this concept of sum of the first n terms of arithmetics series.

3. Devising a Plan for Solution

Part 1 : Determining the n^th term of arithmetics sequence

After getting information from the teacher, the student defined in the sequence number of (1), U₁ = a = 2 and b = U₂ – U₁ = U₃ – U₂ = U₄ – U₃ = 2. From here, the student generalized this concept into U₁ = a and b = U₂ – U₁ = U₃ – U₂ = U₄ – U₃ = ... = U_n – U_n-1.

The student attempts define the terms of arithmetics sequence (U₁, U₂, U₃, ..., U_n) in a and b formulas. By using previous formula, the student expresses the relationship as formulas, such as because of b = U₂ – U₁ then U₂ = U₁ + b, because of b = U₃ – U₂ then U₃ = U₂ + b, therefore, because of b = U_n – U_n-1 then U_n = U_n-1 + b.

In other word, the student learned that the next term is sum of previous term and balance (b).

Part 2 : Determining the sum of the first n terms of arithmetics series

The student attempts defined that S_n is the sum of the first n terms of arithmetics series, so that:

S_n = U₁ + U₂ + U₃ + ... + U_n-1 + U_n

Because of

U₁ = a

U₂ = a + b

U₃ = a + 2b

U_n-2 =a + (n - 3) b

U_n-1 =a + (n - 2) b

U_n = a + (n - 1) b

Therefore,

S_n = a + (a + b) + (a + 2b) + ... + [a + (n - 3) b] + [a + (n - 2) b] + [a + (n - 1) b]

4. Carrying Out the Plan for Solution

Part 1 : Determining the n^th term of arithmetics sequence

Student defined that:

The first term is U₁ = a = a (according to understanding the problem)

The second term is U₂ = U₁ + b = a + b

The third term is U₃ = U₂ + b = a + 2b

The fourth term is U₄ = U₃ + b = a + 3b

Therefore, the n^th term is U_n = U_n-1 + b. Then appear a question, what is the express U_n in a and b formula because the express of U_n-1 is known.

Then, based on the last formulas, the student defined that:

U₁ = a = a + (1 - 1) b

U₂ = a + b = a + (2 - 1) b

U₃ = a + 2b = a + (3 - 1) b

U₄ = a + 3b = a + (4 - 1) b

Therefore,

U_n = a + (n - 1) b

Part 2 : Determining the sum of the first n terms of arithmetics series

The student learned that the sum of the first n terms of arithmetics series (or symbolized by S_n) can be determined by sum of two arithmetics series, so that:

S_n = a + (a + b) + (a + 2b) + ... + [a + (n - 3) b] + [a + (n - 2) b] + [a + (n - 1) b]

S_n = a + (a + b) + (a + 2b) + ... + [a + (n - 3) b] + [a + (n - 2) b] + [a + (n - 1) b]

_______________________________________________________________ +

This is written as follows.

S_n = a + (a + b) + (a + 2b) + ... + [a + (n - 3) b] + [a + (n - 2) b] + [a + (n - 1) b]

S_n = [a + (n - 1) b] + [a + (n - 2) b] + [a + (n - 3) b] + ... + (a + 2b) + (a + b) + a

____________________________________________________________________________+

2S_n = [2a + (n - 1) b] + [2a + (n - 1) b] + [2a + (n - 1) b] + ... + [2a + (n - 1) b] + [2a + (n - 1) b]

(n terms)

2S_n = n [2a + (n - 1) b]

S_n = n [2a + (n - 1) b] / 2

So, the sum of the first n terms of arithmetics series is S_n = n [2a + (n - 1) b] / 2.

5. Examining the Solution

Part 1 : Determining the n^th term of arithmetics sequence

The student presented her formula:

U_n = a + (n - 1) b

in which

U_n is n^th term

a is the first term

n is number of term

b is balance (beda, Indonesia)

So, the n^th term of arithmetics sequence is U_n = a + (n - 1) b.

Part 2 : Determining the sum of the first n terms of arithmetics series

Student presented that S_n = n [2a + (n - 1) b] / 2

DISCUSSION (STUDENT’S REFLECTION)

Researcher : How do you find the n^th term of arithmetics sequence formula?

Student : By generalized the terms of this sequence.

Researcher : What do you mean?

Student : The first term is a and the balance is b in which b = U₂ – U₁ = U₃ – U₂ = U₄ – U₃ = ... = U_n – U_n-1, so we can conclused that the next term is sum of previous term and balance. Then, we can expressed them in a and b formulas. The last, by generalized them, we found the formula.

Researcher : Then, how about the sum of the first n terms of arithmetics series formula?

Student : It can be determined by sum of two arithmetics series.

Researcher : How?

Student : Instance, we have two arithmetics series. We can sum both, one series written from first term to last term but the other written from last term to first term. Then, we have a sum of two series. Therefore, the sum of one series is half of it.

Researcher : So, do you wish to say that the formula be found by expressing relationship between previously fundamental formulas?

Student : I think like that.

CONCLUSION

In this research, tried to uncover the student attempts in applying mathematical thinking related to the mathematical contents in learning arithmetics sequence and series of the 9^th grade student of Junior High School. The results of the research described that the student attempts to finding relation between variables, express relationship as formulas, and read the meaning of formulas. In other word, the formula be found by expressing relationship between previously fundamental formulas. The research stated that mathematical thinking related to mathematical content can be known by indicating student’s success in studying mathematics. It means the student has ability to finding mathematics concepts, definitions, patterns, and relationships so that can use knowledges in the process of problem solving. The schema of teaching learning activities are following.

1. Topic Comprehension

a. It is aimed to informing studied topic because the topic about arithmetics sequence and series has not studied yet by the student.

b. Observed given some numbers pattern which described the arithmetics sequence.

c. Defined sequence number and sequence term, and symbolized each number of given sequence number.

d. Defined number series and symbolyzed it.

e. Defined S_n in which n = 1, 2, 3, ..., n in related to the terms of sequence number and symbolyzed them.

2. Understanding the Problem

a. Defined the arithmetics sequence.

b. Identified the terms of arithmetics sequence.

c. Defined the concept of n^th term of arithmetics sequence.

d. Defines the arithmetics series.

e. Got notices to define the concept of sum of the first n terms of arithmetics series.

f. In problem solving related to the arithmetics sequence and series:

(1) attempted understand the relation between variables

(2) remembered previously learned related question

(3) established a formula by selecting division as procedure which as previously learned.

3. Devising a Plan for Solution

a. Defined the terms of arithmetics sequence.

b. Learned that the next term is sum of previous term and balace (b).

c. Defined the sum of arithmetics series.

d. Learned that the sum of the first n terms of arithmetics series (or symbolized by s_n) can be determined by sum of two arithmetics series.

e. In problem solving related to the arithmetics sequence and series:

(1) learned that the sum of the first n terms of arithmetics series is S_n = n [2a + (n - 1) b] / 2

(2) learned that the first term is U₁ = a and the n^th term is U_n = a + (n - 1) b.

4. Carrying Out the Plan for Solution

a. Learned that the n^th term of arithmetics sequence is U_n = U_n-1 + b.

b. Learned that the sum of the first n terms of arithmetics series is S_n = n [2a + (n - 1) b] / 2

a. In problem solving related to the arithmetics sequence and series:

(1) learned that number of term is

n = (U_n – a + b) / b

(2) learned that

S_n = n (U₁ + U_n) / 2

5. Examining the Solution

a. Presented that U_n = a + (n - 1) b

b. Presented that S_n = n [2a + (n - 1) b] / 2

c. In problem solving related to the arithmetics sequence and series:

Presented that the sum of the first n terms of arithmetics series is 440

Note:

Problem/ question related to the arithmetics sequence and series:

The first term of arithmetics series is 3, the n^th term is 41, and its balance is 3, find the sum of the first n terms of arithmetics series!

REFERENCES

Badan Standar Nasional Pendidikan. 2006. Peraturan Menteri Pendidikan Nasional No. 23 Tahun 2006 tentang Standar Kompetensi Lulusan (SKL) (The National Education Ministerial Decree No. 23 year 2006 about SKL “Graduate Competence Standard”).

Badan Standar Nasional Pendidikan. 2007. Peraturan Menteri Pendidikan Nasional No. 19 Tahun 2007 tentang Standar Pengelolaan Pendidikan oleh Satuan Pendidikan Dasar dan Menengah (The National Education Ministerial Decree No. 19 year 2007 about Education Management Standard by Base and Middle Education Unit).

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