In the first time when we learn about mathematics, sometimes we want to know about what really mathematics is? What are the functions of mathematics in our life, and etc. We were learning mathematics in every stage of school that is in elementary school, in junior high school, are in senior high school. But, are you know that there are some differences from mathematics in that schools and mathematics in university? Yes, because according Mr. Marsigit there are two types of mathematics. The first is school mathematics and the second is pure mathematics or applied mathematics that we usually get this one in university.
Some or may be almost people in university think that mathematics is very abstract and it means not need to use the actual object. But, the object of it is the abstract idea in your mind. Mathematics as the branch of science actually has a formal and material object. This is thinking thing or abstract ideas in your mind.
The next problem that we found in study mathematics is how to learn it. There are two ways to get the abstract object in mathematics:
1. Abstraction
The means of abstraction here is different from abstract, because abstract means something out of physical object. But, the only one thing that will be the essence of abstractions is value. For example, we know that we can get too much assumptions number 5, but we just need and talk about the value of this number in mathematics, not about the other assumptions. Not about the colors, the taste, the materials, and etc.
2. Idealization
What is an idealization? Idealization is some assumptions or some actual imaginations to think about mathematics. For example, we know that in this world there is no really cute angle, because every goods in this world are consist of molecules that has a very small measure. And in molecule there are some atoms. And in each atom, there are some electrons that have the circle orbits surrounding the centre of atom. So we can conclude that although we can see that the needle is an example of cute angle, but that is actually wrong because the smallest thing in the top of needle is an electron orbit or electron. So we just can see the really cote angle is in our mind.
We are going to talk about what is the definition of mathematics. Mathematics is structure of systems that built with deductively methods. It consists of the definitions, some axioms, theorems, lemmas, formulas, and the role of pattern. And the characteristic of mathematics is logically compound and always be consistent when obey the roles.
According Prof. Kayestesi, mathematics consists of two things, which are:
1. Conjecture
Conjecture is to think or to predict something in our mind.
2. Convince
Convince is to communicate the result of some search to other people. We can communicate it by a lot of ways. For example, by the test, by write it in some journals, or by write it in blog.
The next definition of mathematics is from Prof. Katagiri from
Japan, he said that:
Mathematics is mathematical thinking. Mathematics is not come from books, not from encyclopedia, not from computer, but just from our mind.
There are three components of mathematics thinking:
- Mathematics Attitude
Mathematics Attitude is shows by always has a question. Always positive thinking always be consistent and critical, and always be careful.
- Mathematics Methods
Methods in Mathematics are how to learn Mathematics. There are So many ways to learn it, for example:
Deductions Method
It means explaining something from general to specific items, for example, if we meet someone and know him or her, we a the first sight is just know about the general characteristics, but if we always meet him or her, we can know him until the details characteristics.
Inductions Method
Inductions method means we look something from specific items to general items. For example, we study mathematics in the first time from learn 1+1=2, 2+1=3, 3+1=4, and etc. And after that we learn about something larger from it. That is an example from inductions method.
Syllogism Method
It is the method that taking a conclusion according some
premises or some statements. To be the example is:
Lyra is student.
Every student has an id card.
The conclusion: Lyra has an id card.
Logical Method
It is about mathematics sentence and mathematics prepositions. Mathematics sentence is the sentence that from that we can take a conclusion it is true or not.
And the other methods are:
If then Statement
If and If Then Statement
Direct Proof
Indirect Proof
Incomplete Induction
- Mathematics Content
Mathematics content is the object of mathematics too because in it
there are so many roles and the fixed assumption that used to the source.
Kamis, 17 Februari 2011
My Reflection in Learning English
One day at March 12, 2009 I’ve got an exercise from Mr. Marsigit as my lecturer of English II. The first activity that that must we do is praying before begin that class. So, after praying he asks us to prepare one sheet of paper and a pen. So I was very nervous because I think I am not really has a good preparation for this unplanned exercise. When Mr. Marsigit at that time starting his instruction to answer some questions, I just can preparing my self to be calm and can thinking seriously. Mr. Marsigit starts with number 1, 2, and 3 and continued to latest number. It’s all 50 numbers of questions.
I think that it’s not too difficult questions if I prepared my self before, but I think that my work is not too good because I have not prepared my self before. In that exercise, Mr. Marsigit always remembers us to be honest in every part of job, every part of life. Because he thought that honesty is the most important thing that we must has to be a succeed person. He also asks some students to move to other chair when he looked there are some unwanted activity of the student. I think that’s very good to make us always has self confidence.
When the test was ended we start to make a correction from the result of that test, so finally I’ve got not good mark from this exercise. I was very disappointed and really sad, because from that mark I know that I am not too good in this lecture.
Mr. Marsigit also thought that it’s not too good mark enough from us. Mr. Marsigit told us that we must take a change and use it to be the process if we want to be better from before. We must be always doing the self reflections because the mark is showed that all of student in my class is not too good in English lecture especially in English of mathematics.
After the test, Mr. Marsigit told us a real story about a student from USA that got an international scholarship. Before she get the scholarship, she must pass the TOEFEL test by excellent. She is a student like us that has a good competency and very smart to make challenges to be a change. So we as the student too, we should always has a good competency.
Mr. Marsigit called us like a child of bird that still has no fur, so when we want to fly, we can not do it. Mr. Marsigit compares us with the student from USA. He told that she alike a child of bird that has the fur to make her be able to fly into the sky and reach her dreams. So, from this comparison Mr. Marsigit hopes we can get the fur too to can fly like her. The way to can reach our dreams is we must leave the laziness and we must take a change and challenges. Although in some ways we fail, but we not must to surrender.
The question that appeared after this is how to improve our competent in mathematics, English, attitude and etc. one of that way is by answering “How to use preference?” If we can use the preference by maximal, we can get a lot of materials and it can make us cleverer than before. There are so many ways to use the preference, for example is attended the lecture, using internet access and always search information from other people.
If we want to be a good person, we must always thinking about the greatest and the biggest dream. After we have some great dreams, we must search the way how to make that dreams comes true. The only one answer is we must have no fear factor when we pass some problems.
From the test, the result of the test, from the Mr. Marsigit’s advices, finally I know that I must be strong when passing the hard problems that I find in journey to reach my dreams, and always keep our curiosity with everything and always trying to find the answers of it. It’s all about the reflection in learning mathematics.
I was found from here that life is a struggle to be better and better. So, according me I see that life is the changes and the challenges, and we must take it to can make us better. My second assumption of life is there is just only two results in the end of life, failed and succeed. So what’s your choice?
I think that it’s not too difficult questions if I prepared my self before, but I think that my work is not too good because I have not prepared my self before. In that exercise, Mr. Marsigit always remembers us to be honest in every part of job, every part of life. Because he thought that honesty is the most important thing that we must has to be a succeed person. He also asks some students to move to other chair when he looked there are some unwanted activity of the student. I think that’s very good to make us always has self confidence.
When the test was ended we start to make a correction from the result of that test, so finally I’ve got not good mark from this exercise. I was very disappointed and really sad, because from that mark I know that I am not too good in this lecture.
Mr. Marsigit also thought that it’s not too good mark enough from us. Mr. Marsigit told us that we must take a change and use it to be the process if we want to be better from before. We must be always doing the self reflections because the mark is showed that all of student in my class is not too good in English lecture especially in English of mathematics.
After the test, Mr. Marsigit told us a real story about a student from USA that got an international scholarship. Before she get the scholarship, she must pass the TOEFEL test by excellent. She is a student like us that has a good competency and very smart to make challenges to be a change. So we as the student too, we should always has a good competency.
Mr. Marsigit called us like a child of bird that still has no fur, so when we want to fly, we can not do it. Mr. Marsigit compares us with the student from USA. He told that she alike a child of bird that has the fur to make her be able to fly into the sky and reach her dreams. So, from this comparison Mr. Marsigit hopes we can get the fur too to can fly like her. The way to can reach our dreams is we must leave the laziness and we must take a change and challenges. Although in some ways we fail, but we not must to surrender.
The question that appeared after this is how to improve our competent in mathematics, English, attitude and etc. one of that way is by answering “How to use preference?” If we can use the preference by maximal, we can get a lot of materials and it can make us cleverer than before. There are so many ways to use the preference, for example is attended the lecture, using internet access and always search information from other people.
If we want to be a good person, we must always thinking about the greatest and the biggest dream. After we have some great dreams, we must search the way how to make that dreams comes true. The only one answer is we must have no fear factor when we pass some problems.
From the test, the result of the test, from the Mr. Marsigit’s advices, finally I know that I must be strong when passing the hard problems that I find in journey to reach my dreams, and always keep our curiosity with everything and always trying to find the answers of it. It’s all about the reflection in learning mathematics.
I was found from here that life is a struggle to be better and better. So, according me I see that life is the changes and the challenges, and we must take it to can make us better. My second assumption of life is there is just only two results in the end of life, failed and succeed. So what’s your choice?
REFLECTION OF VIDEOS
VIDEO I
DEAD POET SOCIETY
In this first video, I can see there are so many unique things that appeared from the story. This video told us about a teacher is a class that teaching his student in different way. He wants to make his students having a lot of courage and spirits to face everything in this world.
Firstly, the teacher told about Shakespeare who write something interesting and it make the other people enjoy with it. Sometimes, the teacher also gives some examples about the dialogue in drama. He looks so happy when all of his students in that class interested with what that he said to them.
The teacher also gives some beautiful words to his students. He said that we must always look at thing from the different ways because if we can see for example if we see the problem from different way, exactly we got the solving problem from it. And the second motivation is the teacher want to build the self confidence to his students. He shows that he can get a great spirit when he believes that his self can do it.
The teacher move up to the table in front of the class and the students see him with a lot of curiosity, but at that time the teacher said with the load voice, he said that if we want to get the self confidence, we must has an assumptions that we are the greatest person and we can do everything that we should do. So, after this section, the teacher asks the student to also climb the table one by one. He wants to gives a good motivation to his students to always having the self confidence and always having a good spirits to face every problem.
VIDEO II
THE GREAT SHOW
In the second video, I can see there is a show that shows a child who has a great spirit and he also told a beautiful poem. The poem was said about the spirit, the motivation, the trust, and everything in feels mathematics. We must not only learn mathematics by calculating and remembering the role and formulas, but something deeper from it’s that is we also must know the essentials meaning and the usage the mathematics in our life.
The child makes all of audience in that show was very interested with what that he said. So much applauses and shouts sent to the child. He said his poem by confident and it makes the audiences exactly knows what’s the meaning inside the poem.
The poem can makes the audience get the spirit to love mathematics from the heart. Because of we believe in mathematics so we can be the good mathematicians. So, thank to the child, you can successfully makes the other people believe in their selves that they can do it.
VIDEO III
WHAT’S YOU KOW ABOUT MATH?
In this third video, we can see there are two or may be more than two people singing the rap music and the song tells us about mathematics. In that video, we can see the people who sings that song is the great people who having a big spirit and motivations to find the way how to learn mathematics. Sometimes mathematics makes some people feels confuse and they thing m mathematics is difficult. But, the reality we want or not, we must learn it because mathematics is the most important science in our life.
There are a lot of problems in mathematics because when we want to learn it there’s something too hard to understand the role and formulas. So, in that song, the singers want to help us in study mathematics, and we can learn mathematics by simple way. First, we must having a big spirit to learn it, and the next way is we must try to write some formulas to some papers and put it in the wall of the class or may be our room.
We also sometimes can use the calculator and find there are some unique formulas. And from the song too, we must applied the mathematics to our life if we want to learn it. The trigonometry, logic, integral, exponent, and etc are the material that we must know in mathematics. So make it come true in our life. And finally we can say “I know all about math!”
VIDEO IV
SOLVING THE DIFFERENTIAL EQUATION
In this video, we can learn how to solve the problem in differential equation. For example is:
We want to solve this problem Dy/Dx = 4 x^(2 )
To solve it, we must do, first integrate it (Dy/Dx = 4 x^(2 )) so we can get this formula: ∫▒1 dy= ∫▒〖4x^2 〗 dx
And from this integrating system from whole equation we can get:
y = 4/3 x^(3 )+ C ( y equal four third times x cubic plus c), with C is constant
If we want to know the curve, we must represent it to curve from this equation.
VIDEO V
SOLVING ALGEBRAIC PROBLEMS
We can see the way how to solve some algebraic problems from this video.
For example =
x – 5 = 3 ( x minus five equal 3)
To can solve it, (solve here is we can get the value of x), the steps are:
By adding the right side and left side with 5.
X – 5 +5 = 3+5 (x minus five plus five equal three plus five)
And we can calculate it.
X minus five plus five is equal x + 0 that is x, and in the right side, is three plus five equal eight. So, we can get
X= 8
7 = 4a – 1 ( seven equal four a minus one)
If we want to solve it, that is to get the value of a, we must firstly adding the right and the left side of that problem by 1, we do it to make the right side is only 4a. So, the problem being :
7 + 1= 4a (seven plus one equal four a)
So we can calculate it 8 = 4a
From it, we can see that a is two, it’s come from a = 8/4 =2 (a equal eight fourth equal two)
2/3 x = 8 ( two third x equal eight)
If we want to solve it, first way is we must times the right and the left side by 3/2 (three second) because we want to get the value of x as a variable. So, the pattern is:
3/2 .2/3 x = 3/2 . 8 (three second times two third x equal three second times eight) So, we can get x = 12
The next problem is 5 – 2x = 3x + 1 (five minus two x equal three x plus one)
The first way to can solve it is adding the left and right side by minus 3x, so the pattern will be:
5 – 2x - 3x = 3x + 1 - 3x
(Five minus 2x minus 3x equal five minus 5x, and 3x plus one minus 3x is equal one).
5 – 5x = 1
The second way is by adding the right and left side by minus 5.
– 5x – 5 = 1 – 5 (five minus five equal one minus five)
-5x = - 4 (negative five x equal negative four)
So, the next trick if we want to find the value of x as a variable, we must dividing the whole equation by negative five ( -5 )
(-5)/(-5) x = (-4)/(-5) (negative five over negative five times x equal negative four over negative five)
So, x is 4/5 (four fifth)
3 – 5(2m – 5) = -2 (three minus five times open bracket two m minus five close bracket equal negative two)
To finish that equation, we must first make this equation being
3 – 10m + 25 = -2 (three minus ten m plus twenty five equal negative two)
And the next is calculating and it being:
-10m + 28 = -2 (negative ten m plus twenty eight equal negative two)
If we want to find the value of m, with the same system like the numbers before, we must adding the whole equation by negative 28
-10m + 28 -28 = -2 -28 (negative ten m plus twenty eight minus twenty eight equal negative two minus twenty eight)
-10m = -30 (negative ten m equal negative thirty)
And we can find the value of m is 3, by dividing the whole equation by negative 10.
1/2 x + 1/4 = 1/3 x + 5/4 (a half x plus a quarter equal one third x plus five fourth)
To can find the value of x, we can do:
First is times the whole equation by 12, it because 12 is the GCD of 2, 3, and 4 from that equation. So, the equation being
12 (1/2 x + 1/4) = 12 (1/3 x +5/4) (twelve times open bracket a half x plus a quarter close bracket equal twelve open bracket one third x plus five fourth close bracket)
6x + 3 = 4x + 15 (six x plus three equal four x plus fifteen)
To continue our work is we can move the 4x from the right side to the left side to make it gather with the 6x, and the next, we can calculate it easily. And we also can move the 3 from left side to the right side to make we can easily calculate it to find the value of x. So, the pattern of that equation will be:
6x – 4x = 15 – 3 (six x minus four x equal fifteen minus three)
2x = 12 (two x equal twelve)
And we can see here if x is 12/2 (twelve over two) = 6
T he next algebraic problem is :
0.35x – 0.2 = 0.15x – 0.1 (oh point three five x minus oh point two equal oh point one five x minus oh point one)
To can find the value of x, we can start the way to answer it by times the whole equation by 100 (one hundred) to make it easily to calculate. So the pattern will be:
100 (0.35x – 0.2) = 100 (0.15x – 0.1) (one hundred times open bracket oh point three five x minus oh point two close bracket equal one hundred times open bracket oh point one five minus oh point one close bracket)
35x – 20 = 15x – 10 (thirty five x minus twenty equal fifteen x minus ten)
With the same way from the upper number, we can see that it being:
35x – 15x = -10 + 20 (thirty five x minus fifteen x equal negative ten plus twenty)
and we just calculate it
20x = 10 (twenty x equal ten)
So, we can find the value of x is a half, it from 10/20 (ten over twenty) = 1/2 (a half)
From the examples, we can learn about the basic way to solve the algebraic problem, it is very important because the basic system is very usable in our process to learn mathematics. So, I wish it can give us some knowledge.
VIDEO VI
LAW OF LOGARITHM
In this video, we showed about some proof from the theorems in logarithm, it will be so important to us because we as the student who learn mathematics.
The first thing that explained here was about the basic principle in logarithm. That is about the pattern of logarithm.
If we have a number with the pattern like this, for example x to the b equals a
x^b = a, we can explain it into the logarithm pattern and it being
〖log〗_x a = b (read: log with the base x from a equals b)
The next are about some law of logarithm with the way how to can find it.
〖log〗_x AB= 〖log〗_xA + 〖log〗_xB (log base x A times B equal log base x A plus log base x B)
If we want to know how to can get that law is we must study this proof:
Assumptions:
〖log〗_xA = L (log base x A equal L) means 〖 x〗^L = A (x to the power of L equal A)
〖log〗_xB = M (log base x B equal M) means x^M = B (x to the power of M equal B)
〖log〗_x AB = N (log base x A times B equal N) means x^N = AB (x to the power of N equal A times B)
From these assumptions, we know that:
〖log〗_x AB = N ⇒ x^N = AB (x to the power of N equal A times B)
x^N = 〖 x〗^L. x^M (x to the power of N equal x to the power of L times x to the power of M)
x^N = x^(L+M) (x to the power of N equal x to the power of L plus M)
So, we can conclude that N = L + M (N equal L plus M)
And from it we can see that 〖log〗_x AB = 〖log〗_xA + 〖log〗_xB (log base x A times B equal log base x A plus log base x B) is true.
〖log〗_x A/B = 〖log〗_xA - 〖log〗_xB (log base x A over B equal log base x A minus log base x B)
With the same way like at number 1 above, we can see how to can get that law. So, let’s we start it.
Assumptions:
〖log〗_xA = L (log base x A equal L) means〖 x〗^L = A ( x to the power of L equal A)
〖log〗_xB = M (log base x B equal M) means x^M = B (x to the power of M equal B)
〖log〗_x A/B = N (log base x A over B equal N) means x^N= A/B (x to the power of N equal A over B)
From these assumptions, we can get:
〖log〗_x A/B = N ⇒ x^N= A/B (x to the power of N equal A over B)
x^N = 〖 x〗^L/(〖 x〗^M ) (x to the power of N equal x to the power of L
over x to the power of M)
x^N = x^(L-M) (x to the power of N equal x to the power of L minus M)
So, we can conclude that: N = L – M ( N equal L minus M)
And from it we can see that 〖log〗_x A/B = 〖log〗_xA - 〖log〗_xB (log base x A over B equal log base x A minus log base x B) is true.
That’s all about that I gave from watching the video I until VI, I hope it will be usable to us and I’m so sorry if there are so many mistakes. And I will very happy if some people want to correct my work.
DEAD POET SOCIETY
In this first video, I can see there are so many unique things that appeared from the story. This video told us about a teacher is a class that teaching his student in different way. He wants to make his students having a lot of courage and spirits to face everything in this world.
Firstly, the teacher told about Shakespeare who write something interesting and it make the other people enjoy with it. Sometimes, the teacher also gives some examples about the dialogue in drama. He looks so happy when all of his students in that class interested with what that he said to them.
The teacher also gives some beautiful words to his students. He said that we must always look at thing from the different ways because if we can see for example if we see the problem from different way, exactly we got the solving problem from it. And the second motivation is the teacher want to build the self confidence to his students. He shows that he can get a great spirit when he believes that his self can do it.
The teacher move up to the table in front of the class and the students see him with a lot of curiosity, but at that time the teacher said with the load voice, he said that if we want to get the self confidence, we must has an assumptions that we are the greatest person and we can do everything that we should do. So, after this section, the teacher asks the student to also climb the table one by one. He wants to gives a good motivation to his students to always having the self confidence and always having a good spirits to face every problem.
VIDEO II
THE GREAT SHOW
In the second video, I can see there is a show that shows a child who has a great spirit and he also told a beautiful poem. The poem was said about the spirit, the motivation, the trust, and everything in feels mathematics. We must not only learn mathematics by calculating and remembering the role and formulas, but something deeper from it’s that is we also must know the essentials meaning and the usage the mathematics in our life.
The child makes all of audience in that show was very interested with what that he said. So much applauses and shouts sent to the child. He said his poem by confident and it makes the audiences exactly knows what’s the meaning inside the poem.
The poem can makes the audience get the spirit to love mathematics from the heart. Because of we believe in mathematics so we can be the good mathematicians. So, thank to the child, you can successfully makes the other people believe in their selves that they can do it.
VIDEO III
WHAT’S YOU KOW ABOUT MATH?
In this third video, we can see there are two or may be more than two people singing the rap music and the song tells us about mathematics. In that video, we can see the people who sings that song is the great people who having a big spirit and motivations to find the way how to learn mathematics. Sometimes mathematics makes some people feels confuse and they thing m mathematics is difficult. But, the reality we want or not, we must learn it because mathematics is the most important science in our life.
There are a lot of problems in mathematics because when we want to learn it there’s something too hard to understand the role and formulas. So, in that song, the singers want to help us in study mathematics, and we can learn mathematics by simple way. First, we must having a big spirit to learn it, and the next way is we must try to write some formulas to some papers and put it in the wall of the class or may be our room.
We also sometimes can use the calculator and find there are some unique formulas. And from the song too, we must applied the mathematics to our life if we want to learn it. The trigonometry, logic, integral, exponent, and etc are the material that we must know in mathematics. So make it come true in our life. And finally we can say “I know all about math!”
VIDEO IV
SOLVING THE DIFFERENTIAL EQUATION
In this video, we can learn how to solve the problem in differential equation. For example is:
We want to solve this problem Dy/Dx = 4 x^(2 )
To solve it, we must do, first integrate it (Dy/Dx = 4 x^(2 )) so we can get this formula: ∫▒1 dy= ∫▒〖4x^2 〗 dx
And from this integrating system from whole equation we can get:
y = 4/3 x^(3 )+ C ( y equal four third times x cubic plus c), with C is constant
If we want to know the curve, we must represent it to curve from this equation.
VIDEO V
SOLVING ALGEBRAIC PROBLEMS
We can see the way how to solve some algebraic problems from this video.
For example =
x – 5 = 3 ( x minus five equal 3)
To can solve it, (solve here is we can get the value of x), the steps are:
By adding the right side and left side with 5.
X – 5 +5 = 3+5 (x minus five plus five equal three plus five)
And we can calculate it.
X minus five plus five is equal x + 0 that is x, and in the right side, is three plus five equal eight. So, we can get
X= 8
7 = 4a – 1 ( seven equal four a minus one)
If we want to solve it, that is to get the value of a, we must firstly adding the right and the left side of that problem by 1, we do it to make the right side is only 4a. So, the problem being :
7 + 1= 4a (seven plus one equal four a)
So we can calculate it 8 = 4a
From it, we can see that a is two, it’s come from a = 8/4 =2 (a equal eight fourth equal two)
2/3 x = 8 ( two third x equal eight)
If we want to solve it, first way is we must times the right and the left side by 3/2 (three second) because we want to get the value of x as a variable. So, the pattern is:
3/2 .2/3 x = 3/2 . 8 (three second times two third x equal three second times eight) So, we can get x = 12
The next problem is 5 – 2x = 3x + 1 (five minus two x equal three x plus one)
The first way to can solve it is adding the left and right side by minus 3x, so the pattern will be:
5 – 2x - 3x = 3x + 1 - 3x
(Five minus 2x minus 3x equal five minus 5x, and 3x plus one minus 3x is equal one).
5 – 5x = 1
The second way is by adding the right and left side by minus 5.
– 5x – 5 = 1 – 5 (five minus five equal one minus five)
-5x = - 4 (negative five x equal negative four)
So, the next trick if we want to find the value of x as a variable, we must dividing the whole equation by negative five ( -5 )
(-5)/(-5) x = (-4)/(-5) (negative five over negative five times x equal negative four over negative five)
So, x is 4/5 (four fifth)
3 – 5(2m – 5) = -2 (three minus five times open bracket two m minus five close bracket equal negative two)
To finish that equation, we must first make this equation being
3 – 10m + 25 = -2 (three minus ten m plus twenty five equal negative two)
And the next is calculating and it being:
-10m + 28 = -2 (negative ten m plus twenty eight equal negative two)
If we want to find the value of m, with the same system like the numbers before, we must adding the whole equation by negative 28
-10m + 28 -28 = -2 -28 (negative ten m plus twenty eight minus twenty eight equal negative two minus twenty eight)
-10m = -30 (negative ten m equal negative thirty)
And we can find the value of m is 3, by dividing the whole equation by negative 10.
1/2 x + 1/4 = 1/3 x + 5/4 (a half x plus a quarter equal one third x plus five fourth)
To can find the value of x, we can do:
First is times the whole equation by 12, it because 12 is the GCD of 2, 3, and 4 from that equation. So, the equation being
12 (1/2 x + 1/4) = 12 (1/3 x +5/4) (twelve times open bracket a half x plus a quarter close bracket equal twelve open bracket one third x plus five fourth close bracket)
6x + 3 = 4x + 15 (six x plus three equal four x plus fifteen)
To continue our work is we can move the 4x from the right side to the left side to make it gather with the 6x, and the next, we can calculate it easily. And we also can move the 3 from left side to the right side to make we can easily calculate it to find the value of x. So, the pattern of that equation will be:
6x – 4x = 15 – 3 (six x minus four x equal fifteen minus three)
2x = 12 (two x equal twelve)
And we can see here if x is 12/2 (twelve over two) = 6
T he next algebraic problem is :
0.35x – 0.2 = 0.15x – 0.1 (oh point three five x minus oh point two equal oh point one five x minus oh point one)
To can find the value of x, we can start the way to answer it by times the whole equation by 100 (one hundred) to make it easily to calculate. So the pattern will be:
100 (0.35x – 0.2) = 100 (0.15x – 0.1) (one hundred times open bracket oh point three five x minus oh point two close bracket equal one hundred times open bracket oh point one five minus oh point one close bracket)
35x – 20 = 15x – 10 (thirty five x minus twenty equal fifteen x minus ten)
With the same way from the upper number, we can see that it being:
35x – 15x = -10 + 20 (thirty five x minus fifteen x equal negative ten plus twenty)
and we just calculate it
20x = 10 (twenty x equal ten)
So, we can find the value of x is a half, it from 10/20 (ten over twenty) = 1/2 (a half)
From the examples, we can learn about the basic way to solve the algebraic problem, it is very important because the basic system is very usable in our process to learn mathematics. So, I wish it can give us some knowledge.
VIDEO VI
LAW OF LOGARITHM
In this video, we showed about some proof from the theorems in logarithm, it will be so important to us because we as the student who learn mathematics.
The first thing that explained here was about the basic principle in logarithm. That is about the pattern of logarithm.
If we have a number with the pattern like this, for example x to the b equals a
x^b = a, we can explain it into the logarithm pattern and it being
〖log〗_x a = b (read: log with the base x from a equals b)
The next are about some law of logarithm with the way how to can find it.
〖log〗_x AB= 〖log〗_xA + 〖log〗_xB (log base x A times B equal log base x A plus log base x B)
If we want to know how to can get that law is we must study this proof:
Assumptions:
〖log〗_xA = L (log base x A equal L) means 〖 x〗^L = A (x to the power of L equal A)
〖log〗_xB = M (log base x B equal M) means x^M = B (x to the power of M equal B)
〖log〗_x AB = N (log base x A times B equal N) means x^N = AB (x to the power of N equal A times B)
From these assumptions, we know that:
〖log〗_x AB = N ⇒ x^N = AB (x to the power of N equal A times B)
x^N = 〖 x〗^L. x^M (x to the power of N equal x to the power of L times x to the power of M)
x^N = x^(L+M) (x to the power of N equal x to the power of L plus M)
So, we can conclude that N = L + M (N equal L plus M)
And from it we can see that 〖log〗_x AB = 〖log〗_xA + 〖log〗_xB (log base x A times B equal log base x A plus log base x B) is true.
〖log〗_x A/B = 〖log〗_xA - 〖log〗_xB (log base x A over B equal log base x A minus log base x B)
With the same way like at number 1 above, we can see how to can get that law. So, let’s we start it.
Assumptions:
〖log〗_xA = L (log base x A equal L) means〖 x〗^L = A ( x to the power of L equal A)
〖log〗_xB = M (log base x B equal M) means x^M = B (x to the power of M equal B)
〖log〗_x A/B = N (log base x A over B equal N) means x^N= A/B (x to the power of N equal A over B)
From these assumptions, we can get:
〖log〗_x A/B = N ⇒ x^N= A/B (x to the power of N equal A over B)
x^N = 〖 x〗^L/(〖 x〗^M ) (x to the power of N equal x to the power of L
over x to the power of M)
x^N = x^(L-M) (x to the power of N equal x to the power of L minus M)
So, we can conclude that: N = L – M ( N equal L minus M)
And from it we can see that 〖log〗_x A/B = 〖log〗_xA - 〖log〗_xB (log base x A over B equal log base x A minus log base x B) is true.
That’s all about that I gave from watching the video I until VI, I hope it will be usable to us and I’m so sorry if there are so many mistakes. And I will very happy if some people want to correct my work.
THE PROOF
I.
How to proof that the square root of two is irrational number?
To proof that the square root of two is irrational number is we can use the reductio ad absurdum. It can show that the square root of two is irrational number by make an assumption that the square root of two is not the irrational number, or it’s called rational number.
First, the definition of rational number is the number that can explain by comparing two number x and y that’s x over y, and x and y are two numbers which relatively prime. And from this statement we can make an assumption that square root of two equal x over y. And the next step is we can get x equal square root of two times y. So, if we squared the two side of this equation, we can get x square equal two times y square. And from this equation, we can make a conclusion that x square always even number because x is a number that made from two times y square and it for y equal all number. So from it we also can get that x is even number too.
The next step is we make the second assumption that x equal two times z. So we from this statement we can get two times z equal square root of two times y. if we squared these two side of this equation, we can see that four times z square equal two times y square. And if we dividing these side of this equation by two, we can get two times z square equal y square. From this equation, we can identify that y square is even number because it’s made from two times z square for z is all number, and we also know that y is even number too.
Finally from these steps, we found that x and y are always even number. So we can’t called square root of two is rational number because both of x and y is always not relatively prime number. So, square root of two is irrational number.
How to show or indicate that the sum angle of triangle is equal to one hundred and eighty degree?
The steps to show that the sum angle of triangle is equal to one hundred and eighty degree is we must draw a triangle which having x, y and z angle. And firstly we know that the sum angle of the straight line is one hundred and eighty degree too. So we can get that the sum angle of straight line is equal x angle, y angle, and z angle in triangle. To make it clearer, we can see this picture of the triangle. We can make some parallel lines to help us imagine the angles in triangle.
z
z
x y x
How to get the value of phi?
To get the value of phi we can do these steps that are we must take some measure from some circle about the diameter. And after this part of the job, we also must find the measure of each perimeter of circles. We can make some assumptions about it for example for the first circle we call it a circle, and having a diameter and also having a perimeter. And the second circle called b circle and also having b diameter and b perimeter too. And if we take some measure from a lot of circle we can called its c circle, d circle, and etc.
We can find the value of phi by comparing the perimeter a with diameter a, perimeter b with diameter b, perimeter c with diameter c, and etc.
Phi = Pa/Da + Pb/Db + Pc/Dc + … dividing by the sum of comparisons.
From this comparison, we get the mean from each comparison, and finally we find phi is 3.14 or 22/7 (twenty seventh).
Explain how you are able to find out the area of region bounded by the graph of y equal x square and y equal x plus two?
To know the region of the wanted area is we must find out the axis of x by make a substitution process to the equalities. We know that y equal x square and y equal x plus two, so we can operate the equation by y equal y, so we can see here we get x square equal x plus two too. From this equation we get x square minus x minus two equal null. So the next is we get x minus two in bracket times x plus one in bracket equal null. Because the value of x minus two in bracket times x plus one in bracket is same with get x square minus x minus two. And the last is we know that e equal two and x equal negative one and it being the axis.
The second step to know the area is we can use the integral, because the area is equal integral with lower boundary x equal negative one and the upper boundary is x equal two of x plus two in bracket minus x square dx. So, the next is we must finish this define integral, that is integral with lower boundary x equal negative one and the upper boundary is x equal two of x plus two minus x square in bracket dx, equal negative one third times x cube plus a half times x square plus two times x from x equal negative one until x equal two.
To finish this equation we must substitution the value of x equal two into this formula minus the value of x equal negative one that also substitution in this formula. So the value is negative one third times two cube plus a half times two square plus two times two in bracket minus negative one third times negative one cube plus a half times negative one square plus two times negative one in bracket.
We continue to finish it by algebraic roles as usually we did. So we get negative eight third plus two plus four in bracket minus one third plus a half minus two in bracket equal negative nine third minus a half plus eight equal five minus a half equal four and a half. And finally we get the area of the region that bounded by the graph of y equal x square and y equal x plus two is four and a half.
Explain how you’re able to determine the intersection points between the circle x square plus y square equal twenty and y equal x plus one!
To find the intersection points of the equation, we can use the substitution way. The way is: we substituting y equal x plus one to equation x square plus y square equal twenty. So we get x square plus x plus one in bracket square equal twenty. And then we get x square plus x square plus two times x plus one equal twenty. Two times x square plus two times x minus nineteen equal zero. And to find the value of x, we use the abc formula.
So, x one and two is negative b plus minus square root of b square minus four times a times c all over two times a. So we get negative two plus minus four minus four times two times negative nineteen in bracket to the power of a half all over two times two. And later we get negative two plus minus square root of one hundred and fifty six all over four. And from this equation, we know that x one is negative two plus square root of one hundred and fifty six all over four, and then x two equal negative two minus square root of one hundred and fifty six all over four.
And by substituting x one and x two to y equal x plus one, we can get the value of y one and y two. So, y one equal two plus square root of one hundred and fifty six all over four, and y two equal two minus square root of one hundred and fifty six all over four. And finally we know that the intersection points are x one point y one that is negative two plus square root of one hundred and fifty six all over four point two plus square root of one hundred and fifty six all over four, and x two point y two equal negative two minus square root of one hundred and fifty six all over four point equal two minus square root of one hundred and fifty six all over four.
II.
SIMILARITY OF TRIANGLES
Definition of similar triangles is triangles whose corresponding angles are congruent and whose corresponding sides are in proportion. We can say triangle is also planar polygonal figures. Two or more triangles are similar when and only when:
(i)Corresponding angles are equal (have the same measure).
(ii)Corresponding sides are proportional.
There are some theorems about the similarity of triangles. The theorems are always used if we found some problems about similarity of triangle, so the theorems are very important to know. These theorems are:
The following theorems are valid in Euclidean geometry:
Theorem AA (angle-angle)
If one triangle has a pair of angles that are congruent to a pair of angles in another triangle, then the two triangles are similar.
Theorem SAS (side-angle-side)
If the pairs of sides of a triangle are proportional to the pairs of sides in another triangle and if the angles included by the side-pairs are congruent, then the triangles are similar.
Theorem SSS (side-side-side)
If the sides of a triangle are proportional to the sides of another triangle, so the triangles are similar.
The applications of the similarity of triangles are it’s used to measure how long and how wide some area, it also used to make we easier to determine which triangle that similar each other, and etc.
The example problems about the similarity of triangles are:
In the triangle ABC shown below, A accent C accent is parallel to AC. Find the length y of BC accent and the length x of A accent A!
The picture is a triangle that having A, B and C angle. And there is a line that parallel with line AC and it lies and made two intersection points with line AB in A accent, and with the line BC in point C accent. And we can make assumption that A A accent is x and C C accent is y.
The picture also shown that the length of AC is twenty two centimeters, the length of C C accent is fifteen centimeters, and the length of A accent B is thirty centimeters.
Solution of problem one:
BA is a transversal that intersects the two parallel lines A accent C accent and AC, hence the corresponding angles BA accent C accent and BAC are congruent. BC is also a transversal to the two parallel lines A accent C accent and AC and therefore angles BC accent A accent and BCA are congruent. These two triangles have two congruent angles are therefore similar and the lengths of their sides are proportional. Let us separate the two triangles as shown below.
Picture:
Shown two triangles that are ABC triangle and A accent B C accent triangle. So the next we call ABC triangle is first triangle and the other triangle is second triangle. So, AB equals thirty plus x, AC equal twenty two, and BC equal y plus fifteen. And the second triangle is A accent C equal fourteen, A accent B equal thirty, and B C accent equal y.
Because we know that the angle of A in first triangle is having the same measure with the A accent angle in the second triangle, we now use the proportionality of the lengths of the side to write equations that help in solving for x and y, that is:
Thirty plus x in bracket over thirty equal twenty two over fourteen equal y plus fifteen in bracket over y
An equation in x may be written as follows. Thirty plus x in bracket over thirty equal twenty two over fourteen.
Solve the above for x. That’s four hundred and twenty plus fourteen x equal six hundred and sixty. So, fourteen times x equal six hundred and sixty minus four hundred and twenty. And we get fourteen times x equal two hundred and fourty to get the value of x is by dividing two hundred and fourty by fourteen.
So, x equal seventeen point one (rounded to one decimal place).
An equation in y may be written as follows. Twenty two over fourteen equal y plus fifteen in bracket over y. And solve the above for y to obtain, twenty two times y equal fourteen times y plus two hundred and ten. So, twenty times y minus fourteen times y equal two hundred and ten. So, eight y equals two hundred and ten, and we get y equal two hundred ten over eight, y equal twenty six point two five.
Problem two:
ABC is a right triangle. AM is perpendicular from vertex A to the hypotenuse BC of the triangle. How many similar triangles are there?
Answer:
Picture 1: triangle ABC, angle A is right angle and angle AMC equal angle AMB are right angle too. AB equal a, AC equal b, BM equal y, BM equal x, and AM equal h.
Picture 2: triangle AMB which has right angle in angle AMB, BM equal x, MA equal h, and AB equal a.
Picture 3: triangle AMC which has angle M as right angle AM equal h, MC equal y, and AC equal b.
Solution to Problem 5:
Consider triangles ABC and MBA. They have two corresponding congruent angles: the right angle and angle B. They are similar. Also triangles ABC and MAC have two congruent angles: the right angle and angle C. Therefore there are three similar triangles: ABC, MBA and MAC.
ARITHMETIC SERIES
The definition of arithmetic series is the sum of an arithmetic sequence. To make it clearer we should see this example:
A series such as three plus seven plus eleven plus fifteen plus etc until plus ninety nine or ten plus twenty plus thirty plus etc until one thousand. Which is has a constant difference between terms. The first term is a1 (a one), the common difference is d, and the number of terms is n. The sum of an arithmetic series is found by multiplying the number of terms times the average of the first and last terms.
The difference between the consecutive terms of the sequence is constant. The constant difference is called common difference. Hence the sequence is arithmetic.
If we have an arithmetic series for example:
1+2+3+…+n, (one plus two plus three plus etc until plus n)
We can write it into this formula or the sum identity is:
(sigma k from k equal one until k equal n equal a half times n times open bracket n plus one close bracket)
Then, if we have an arithmetic series that can be shown like this:
a one plus a two plus a three plus etc until a n, we can write it into this formula:
S n equal a half times a one plus a n in bracket, and we can see here that S n is same with sigma k from k equal one until k equal n equal a half times n times open bracket n plus one close bracket.
Sum = n/2(2a + (n-1)b)
Summary equal n over two times two times a plus open bracket n minus one close bracket times b in bracket.
The next formula is a n equal a one plus n minus one in bracket times d, which are a n is the n-th series, a one is first series, n is the sum of whole series, and d is difference value from that series, or d is a two minus a one equal a three minus a two, etc.
an = a1 + (n – 1)d
The application of arithmetic series is very large; we can easier to know the sum of some series for example is to predict the sum or value of data.
Example problem 1: three plus seven plus eleven plus fifteen plus etc until plus ninety nine which has a one equal three, and d equal four.
3 + 7 + 11 + 15 + ••• + 99 has a1 = 3 and d = 4. To find n, use the explicit formula for an arithmetic sequence.
We solve three plus n minus one in bracket times four equal ninety nine. So we start to calculate it, three plus four n minus four equal ninety nine. So, four times n minus one equal ninety nine. Four times n equal one hundred, so n equal one hundred over four equal twenty five.
3 + (n – 1)• 4 = 99
n = 25
So, the sum is twenty five times three plus ninety nine in bracket over two equal one thousand two hundred seventy five.
or: sum is twenty five over two times open bracket two times three plus twenty five minus one in bracket times four close bracket equal one thousand two hundred seventy five.
Example problem 2: Find the 35th partial sum of a n equal a half times n plus one.
an = (1/2)n + 1
The thirty fifth partial sum of this sequence is the sum of the first thirty-five terms. The first few terms of the sequence are:
a one equal a half times one plus one equal three second
a two equal a half times two plus one equal two
a three equal a half three plus one equal five second
a1 = (1/2)(1) + 1 = 3/2
a2 = (1/2)(2) + 1 = 2
a3 = (1/2)(3) + 1 = 5/2
The terms have a common difference d equal a half, so this is indeed an arithmetic sequence. The last term in the partial sum will be a thirty five equal a one plus thirty five minus one in bracket times d equal three second plus thirty four times a half equal thirty seven second.
a35 = a1 + (35 – 1)(d) = 3/2 + (34)(1/2) = 37/2.
Then, plugging into the formula, the 35th partial sum is:
n over two in bracket times a one plus a n in bracket equal thirty five second times three second plus thirty seven second in bracket equal thirty five second times fourty second equal three hundred and fifty.
COMPOSITE NUMBER
The most important thing that we should know before we learn about the composite number is we must know the definition of composite number. So, a composite number is a natural greater than 1 that is divisible by a number other than itself and 1 (in other words, it has more than the two factors of 1 and itself).
The integer fourteen is a composite number because it can be factored as two times seven.
And to make we learn about it deeper, we must also know about the prime number, the definition of prime number is a natural number greater than 1 that has only itself and 1 as factors.
One thing as important as the definition above is about the fundamental theorem that usually used in this material, as shown below:
Every composite number can be expressed as a product of prime numbers in one and only one way.
The prime factorization of a whole number is the number written as the productof its prime factors.
And also one method used to find the prime factorization of a composite number is called a factor tree.
For example, find the factor from twenty!
The answer is: we must make a factor tree from twenty. That is twenty equal two times ten, and in the picture of factor tree is two and ten being the branch of twenty, and it continued until twenty can be expressed just as a product of prime number. So, the next is ten equal two times five, here we find that two, two, and five are all prime number being the factors of twenty.
We also must know about the relatively prime numbers, and the definition of relatively prime number is pairs of numbers that have 1 as their greatest common divisor. For example, the greatest common divisor of seven and twenty five is 1. Thus, seven and twenty five are relatively prime numbers.
To find the greatest common divisor of two or more numbers,
Write the prime factorization of each number.
Select each prime factor with the smallest exponent that is common to each of the prime factorizations.
Form the product of the numbers from step two. The greatest common divisor is the product of these factors.
Example: Find the greatest common divisor of two hundred and sixteen and two hundred and thirty four.
Solution: Step 1. Write the prime factorization of each number.
The picture of factor tree is explain about two hundred and sixteen equal two times one hundred and eighteen, and one hundred and eighteen equal two times fifty and four, and fifty and four equal two times twenty seven, and twenty seven equal three times nine, and nine equal three times three.
And the second factor tree is explain that two hundred and thirty four equal two times one hundred and seventeen, one hundred and seventeen equal three times thirty nine, and thirty nine equal three times thirteen.
The factor tree at the left indicates that two hundred and sixteen equal two to the power of three times three to the power of three.
The factor tree at the right indicates that two hundred and thirty four equal two times three to the power of two times thirteen.
Step 2: Select each prime factor with the smallest exponent that is common to each of the prime factorizations.
Which exponent is appropriate for two and three? We choose the smallest exponent; for two we take two to the power of one and for three we take three square.
Step 3: Form the product of the numbers from step three. The greatest common divisor is the product of these factors. Greatest common divisor equal two times three square equal two times nine equal eighteen. Thus, the greatest common factor for two hundred and sixteen and two hundred and thirty four is eighteen.
The least common multiple (LCM) of two or more natural numbers is the smallest natural number that is divisible by all of the numbers.
To find the least common multiple using prime factorization of two or more numbers:
Write the prime factorization of each number.
Select every prime factor that occurs raised to the greatest power to which it occurs, in these factorizations.
Form the product of the numbers from step 2. The least common multiple is the product of these factors.
Example: Find the least common multiple of one hundred forty four and three hundred.
Solution: Step one. Write the prime factorization of each number.
One hundred and forty four equal two to the power four times three square.
Three hundred equal two square times three times five square.
Step 2: Select every prime factor that occurs, raised to the greatest power to which it occurs, in these factorizations.
One hundred and forty four equal two to the power of four times three square.
Three hundred equal two square times three times five square.
Step 3: Form the product of the numbers from step 2. The least common multiple is the product of these factors.
LCM equal two to the power of four times three square times five square equal sixteen times nine times twenty five equal three thousand and six hundred.
Hence, the LCM of one hundred and forty four and three hundred is three thousand and six hundred. Thus, the smallest natural number divisible by one hundred and fourty four and three hundred is three thousand and six hundred.
Properties:
All even numbers greater than two are composite numbers.
The smallest composite number is four.
Every composite number can be written as the product of (not necessarily distinct) primes. (Fundamental theorem of arithmetic)
Also, n minus one in bracket factorial equivalent with zero modulo n for all composite numbers, n more than five. (Wilson's theorem)
Kinds of composite numbers:
One way to classify composite numbers is by counting the number of prime factors. A composite number with two prime factors is a semi prime or 2-almost prime (the factors need not be distinct; hence squares of primes are included). A composite number with three distinct prime factors is a sphenic number. In some applications, it is necessary to differentiate between composite numbers with an odd number of distinct prime factors and those with an even number of distinct prime factors.
Some theorems about composite number are:
Theorem1)
If C is an odd number composed of two prime factors a and b, then the number of consecutive odd integers required to add up to C, is equal to the smaller of the two prime factors of C.
Theorem2)
The larger of the two prime factors of C is the average to the first and last number in the consecutive sum of odd numbers that add up to C.
Theorem3)
If C is an odd number that is composed of two prime factors a and C, then there exists at least one non-arbitrary perfect square that can be used to factor C through a finite number of algebraic operations.
Theorem4)
If C is an odd number that is composed of two prime factors a and b, then the non-arbitrary perfect square that can be used to factor C via finite number of algebraic operations is the first perfect square acquired by adding a consecutive number of odd integers starting with the number 1 to C and is equal to the average of the prime factors of C quantity squared.
Example:
Solution: Find the sum of the first one hundred even positive numbers.
The sum of the first one even positive numbers is two or one times one plus one equal one times two.
The sum of the first two even positive numbers is two plus four equal six or two times two plus one in bracket equals two times three.
The sum of the first three even positive numbers is two plus four plus six equal twelve or three times open bracket three plus one close bracket equal three times four.
The sum of the first four even positive numbers is two plus four plus six plus eight equal twenty or four times open bracket four plus one close bracket equal four times five.
Look for a pattern :
The sum of the first one hundred even positive numbers is two plus four plus six plus eight plus etc equal x or one hundred times one hundred plus one in bracket equal one hundred times one hundred and one or ten thousand and one hundred.
And finally this is the end of my fifth task and I wish it can be used well.
Best Regards….
How to proof that the square root of two is irrational number?
To proof that the square root of two is irrational number is we can use the reductio ad absurdum. It can show that the square root of two is irrational number by make an assumption that the square root of two is not the irrational number, or it’s called rational number.
First, the definition of rational number is the number that can explain by comparing two number x and y that’s x over y, and x and y are two numbers which relatively prime. And from this statement we can make an assumption that square root of two equal x over y. And the next step is we can get x equal square root of two times y. So, if we squared the two side of this equation, we can get x square equal two times y square. And from this equation, we can make a conclusion that x square always even number because x is a number that made from two times y square and it for y equal all number. So from it we also can get that x is even number too.
The next step is we make the second assumption that x equal two times z. So we from this statement we can get two times z equal square root of two times y. if we squared these two side of this equation, we can see that four times z square equal two times y square. And if we dividing these side of this equation by two, we can get two times z square equal y square. From this equation, we can identify that y square is even number because it’s made from two times z square for z is all number, and we also know that y is even number too.
Finally from these steps, we found that x and y are always even number. So we can’t called square root of two is rational number because both of x and y is always not relatively prime number. So, square root of two is irrational number.
How to show or indicate that the sum angle of triangle is equal to one hundred and eighty degree?
The steps to show that the sum angle of triangle is equal to one hundred and eighty degree is we must draw a triangle which having x, y and z angle. And firstly we know that the sum angle of the straight line is one hundred and eighty degree too. So we can get that the sum angle of straight line is equal x angle, y angle, and z angle in triangle. To make it clearer, we can see this picture of the triangle. We can make some parallel lines to help us imagine the angles in triangle.
z
z
x y x
How to get the value of phi?
To get the value of phi we can do these steps that are we must take some measure from some circle about the diameter. And after this part of the job, we also must find the measure of each perimeter of circles. We can make some assumptions about it for example for the first circle we call it a circle, and having a diameter and also having a perimeter. And the second circle called b circle and also having b diameter and b perimeter too. And if we take some measure from a lot of circle we can called its c circle, d circle, and etc.
We can find the value of phi by comparing the perimeter a with diameter a, perimeter b with diameter b, perimeter c with diameter c, and etc.
Phi = Pa/Da + Pb/Db + Pc/Dc + … dividing by the sum of comparisons.
From this comparison, we get the mean from each comparison, and finally we find phi is 3.14 or 22/7 (twenty seventh).
Explain how you are able to find out the area of region bounded by the graph of y equal x square and y equal x plus two?
To know the region of the wanted area is we must find out the axis of x by make a substitution process to the equalities. We know that y equal x square and y equal x plus two, so we can operate the equation by y equal y, so we can see here we get x square equal x plus two too. From this equation we get x square minus x minus two equal null. So the next is we get x minus two in bracket times x plus one in bracket equal null. Because the value of x minus two in bracket times x plus one in bracket is same with get x square minus x minus two. And the last is we know that e equal two and x equal negative one and it being the axis.
The second step to know the area is we can use the integral, because the area is equal integral with lower boundary x equal negative one and the upper boundary is x equal two of x plus two in bracket minus x square dx. So, the next is we must finish this define integral, that is integral with lower boundary x equal negative one and the upper boundary is x equal two of x plus two minus x square in bracket dx, equal negative one third times x cube plus a half times x square plus two times x from x equal negative one until x equal two.
To finish this equation we must substitution the value of x equal two into this formula minus the value of x equal negative one that also substitution in this formula. So the value is negative one third times two cube plus a half times two square plus two times two in bracket minus negative one third times negative one cube plus a half times negative one square plus two times negative one in bracket.
We continue to finish it by algebraic roles as usually we did. So we get negative eight third plus two plus four in bracket minus one third plus a half minus two in bracket equal negative nine third minus a half plus eight equal five minus a half equal four and a half. And finally we get the area of the region that bounded by the graph of y equal x square and y equal x plus two is four and a half.
Explain how you’re able to determine the intersection points between the circle x square plus y square equal twenty and y equal x plus one!
To find the intersection points of the equation, we can use the substitution way. The way is: we substituting y equal x plus one to equation x square plus y square equal twenty. So we get x square plus x plus one in bracket square equal twenty. And then we get x square plus x square plus two times x plus one equal twenty. Two times x square plus two times x minus nineteen equal zero. And to find the value of x, we use the abc formula.
So, x one and two is negative b plus minus square root of b square minus four times a times c all over two times a. So we get negative two plus minus four minus four times two times negative nineteen in bracket to the power of a half all over two times two. And later we get negative two plus minus square root of one hundred and fifty six all over four. And from this equation, we know that x one is negative two plus square root of one hundred and fifty six all over four, and then x two equal negative two minus square root of one hundred and fifty six all over four.
And by substituting x one and x two to y equal x plus one, we can get the value of y one and y two. So, y one equal two plus square root of one hundred and fifty six all over four, and y two equal two minus square root of one hundred and fifty six all over four. And finally we know that the intersection points are x one point y one that is negative two plus square root of one hundred and fifty six all over four point two plus square root of one hundred and fifty six all over four, and x two point y two equal negative two minus square root of one hundred and fifty six all over four point equal two minus square root of one hundred and fifty six all over four.
II.
SIMILARITY OF TRIANGLES
Definition of similar triangles is triangles whose corresponding angles are congruent and whose corresponding sides are in proportion. We can say triangle is also planar polygonal figures. Two or more triangles are similar when and only when:
(i)Corresponding angles are equal (have the same measure).
(ii)Corresponding sides are proportional.
There are some theorems about the similarity of triangles. The theorems are always used if we found some problems about similarity of triangle, so the theorems are very important to know. These theorems are:
The following theorems are valid in Euclidean geometry:
Theorem AA (angle-angle)
If one triangle has a pair of angles that are congruent to a pair of angles in another triangle, then the two triangles are similar.
Theorem SAS (side-angle-side)
If the pairs of sides of a triangle are proportional to the pairs of sides in another triangle and if the angles included by the side-pairs are congruent, then the triangles are similar.
Theorem SSS (side-side-side)
If the sides of a triangle are proportional to the sides of another triangle, so the triangles are similar.
The applications of the similarity of triangles are it’s used to measure how long and how wide some area, it also used to make we easier to determine which triangle that similar each other, and etc.
The example problems about the similarity of triangles are:
In the triangle ABC shown below, A accent C accent is parallel to AC. Find the length y of BC accent and the length x of A accent A!
The picture is a triangle that having A, B and C angle. And there is a line that parallel with line AC and it lies and made two intersection points with line AB in A accent, and with the line BC in point C accent. And we can make assumption that A A accent is x and C C accent is y.
The picture also shown that the length of AC is twenty two centimeters, the length of C C accent is fifteen centimeters, and the length of A accent B is thirty centimeters.
Solution of problem one:
BA is a transversal that intersects the two parallel lines A accent C accent and AC, hence the corresponding angles BA accent C accent and BAC are congruent. BC is also a transversal to the two parallel lines A accent C accent and AC and therefore angles BC accent A accent and BCA are congruent. These two triangles have two congruent angles are therefore similar and the lengths of their sides are proportional. Let us separate the two triangles as shown below.
Picture:
Shown two triangles that are ABC triangle and A accent B C accent triangle. So the next we call ABC triangle is first triangle and the other triangle is second triangle. So, AB equals thirty plus x, AC equal twenty two, and BC equal y plus fifteen. And the second triangle is A accent C equal fourteen, A accent B equal thirty, and B C accent equal y.
Because we know that the angle of A in first triangle is having the same measure with the A accent angle in the second triangle, we now use the proportionality of the lengths of the side to write equations that help in solving for x and y, that is:
Thirty plus x in bracket over thirty equal twenty two over fourteen equal y plus fifteen in bracket over y
An equation in x may be written as follows. Thirty plus x in bracket over thirty equal twenty two over fourteen.
Solve the above for x. That’s four hundred and twenty plus fourteen x equal six hundred and sixty. So, fourteen times x equal six hundred and sixty minus four hundred and twenty. And we get fourteen times x equal two hundred and fourty to get the value of x is by dividing two hundred and fourty by fourteen.
So, x equal seventeen point one (rounded to one decimal place).
An equation in y may be written as follows. Twenty two over fourteen equal y plus fifteen in bracket over y. And solve the above for y to obtain, twenty two times y equal fourteen times y plus two hundred and ten. So, twenty times y minus fourteen times y equal two hundred and ten. So, eight y equals two hundred and ten, and we get y equal two hundred ten over eight, y equal twenty six point two five.
Problem two:
ABC is a right triangle. AM is perpendicular from vertex A to the hypotenuse BC of the triangle. How many similar triangles are there?
Answer:
Picture 1: triangle ABC, angle A is right angle and angle AMC equal angle AMB are right angle too. AB equal a, AC equal b, BM equal y, BM equal x, and AM equal h.
Picture 2: triangle AMB which has right angle in angle AMB, BM equal x, MA equal h, and AB equal a.
Picture 3: triangle AMC which has angle M as right angle AM equal h, MC equal y, and AC equal b.
Solution to Problem 5:
Consider triangles ABC and MBA. They have two corresponding congruent angles: the right angle and angle B. They are similar. Also triangles ABC and MAC have two congruent angles: the right angle and angle C. Therefore there are three similar triangles: ABC, MBA and MAC.
ARITHMETIC SERIES
The definition of arithmetic series is the sum of an arithmetic sequence. To make it clearer we should see this example:
A series such as three plus seven plus eleven plus fifteen plus etc until plus ninety nine or ten plus twenty plus thirty plus etc until one thousand. Which is has a constant difference between terms. The first term is a1 (a one), the common difference is d, and the number of terms is n. The sum of an arithmetic series is found by multiplying the number of terms times the average of the first and last terms.
The difference between the consecutive terms of the sequence is constant. The constant difference is called common difference. Hence the sequence is arithmetic.
If we have an arithmetic series for example:
1+2+3+…+n, (one plus two plus three plus etc until plus n)
We can write it into this formula or the sum identity is:
(sigma k from k equal one until k equal n equal a half times n times open bracket n plus one close bracket)
Then, if we have an arithmetic series that can be shown like this:
a one plus a two plus a three plus etc until a n, we can write it into this formula:
S n equal a half times a one plus a n in bracket, and we can see here that S n is same with sigma k from k equal one until k equal n equal a half times n times open bracket n plus one close bracket.
Sum = n/2(2a + (n-1)b)
Summary equal n over two times two times a plus open bracket n minus one close bracket times b in bracket.
The next formula is a n equal a one plus n minus one in bracket times d, which are a n is the n-th series, a one is first series, n is the sum of whole series, and d is difference value from that series, or d is a two minus a one equal a three minus a two, etc.
an = a1 + (n – 1)d
The application of arithmetic series is very large; we can easier to know the sum of some series for example is to predict the sum or value of data.
Example problem 1: three plus seven plus eleven plus fifteen plus etc until plus ninety nine which has a one equal three, and d equal four.
3 + 7 + 11 + 15 + ••• + 99 has a1 = 3 and d = 4. To find n, use the explicit formula for an arithmetic sequence.
We solve three plus n minus one in bracket times four equal ninety nine. So we start to calculate it, three plus four n minus four equal ninety nine. So, four times n minus one equal ninety nine. Four times n equal one hundred, so n equal one hundred over four equal twenty five.
3 + (n – 1)• 4 = 99
n = 25
So, the sum is twenty five times three plus ninety nine in bracket over two equal one thousand two hundred seventy five.
or: sum is twenty five over two times open bracket two times three plus twenty five minus one in bracket times four close bracket equal one thousand two hundred seventy five.
Example problem 2: Find the 35th partial sum of a n equal a half times n plus one.
an = (1/2)n + 1
The thirty fifth partial sum of this sequence is the sum of the first thirty-five terms. The first few terms of the sequence are:
a one equal a half times one plus one equal three second
a two equal a half times two plus one equal two
a three equal a half three plus one equal five second
a1 = (1/2)(1) + 1 = 3/2
a2 = (1/2)(2) + 1 = 2
a3 = (1/2)(3) + 1 = 5/2
The terms have a common difference d equal a half, so this is indeed an arithmetic sequence. The last term in the partial sum will be a thirty five equal a one plus thirty five minus one in bracket times d equal three second plus thirty four times a half equal thirty seven second.
a35 = a1 + (35 – 1)(d) = 3/2 + (34)(1/2) = 37/2.
Then, plugging into the formula, the 35th partial sum is:
n over two in bracket times a one plus a n in bracket equal thirty five second times three second plus thirty seven second in bracket equal thirty five second times fourty second equal three hundred and fifty.
COMPOSITE NUMBER
The most important thing that we should know before we learn about the composite number is we must know the definition of composite number. So, a composite number is a natural greater than 1 that is divisible by a number other than itself and 1 (in other words, it has more than the two factors of 1 and itself).
The integer fourteen is a composite number because it can be factored as two times seven.
And to make we learn about it deeper, we must also know about the prime number, the definition of prime number is a natural number greater than 1 that has only itself and 1 as factors.
One thing as important as the definition above is about the fundamental theorem that usually used in this material, as shown below:
Every composite number can be expressed as a product of prime numbers in one and only one way.
The prime factorization of a whole number is the number written as the productof its prime factors.
And also one method used to find the prime factorization of a composite number is called a factor tree.
For example, find the factor from twenty!
The answer is: we must make a factor tree from twenty. That is twenty equal two times ten, and in the picture of factor tree is two and ten being the branch of twenty, and it continued until twenty can be expressed just as a product of prime number. So, the next is ten equal two times five, here we find that two, two, and five are all prime number being the factors of twenty.
We also must know about the relatively prime numbers, and the definition of relatively prime number is pairs of numbers that have 1 as their greatest common divisor. For example, the greatest common divisor of seven and twenty five is 1. Thus, seven and twenty five are relatively prime numbers.
To find the greatest common divisor of two or more numbers,
Write the prime factorization of each number.
Select each prime factor with the smallest exponent that is common to each of the prime factorizations.
Form the product of the numbers from step two. The greatest common divisor is the product of these factors.
Example: Find the greatest common divisor of two hundred and sixteen and two hundred and thirty four.
Solution: Step 1. Write the prime factorization of each number.
The picture of factor tree is explain about two hundred and sixteen equal two times one hundred and eighteen, and one hundred and eighteen equal two times fifty and four, and fifty and four equal two times twenty seven, and twenty seven equal three times nine, and nine equal three times three.
And the second factor tree is explain that two hundred and thirty four equal two times one hundred and seventeen, one hundred and seventeen equal three times thirty nine, and thirty nine equal three times thirteen.
The factor tree at the left indicates that two hundred and sixteen equal two to the power of three times three to the power of three.
The factor tree at the right indicates that two hundred and thirty four equal two times three to the power of two times thirteen.
Step 2: Select each prime factor with the smallest exponent that is common to each of the prime factorizations.
Which exponent is appropriate for two and three? We choose the smallest exponent; for two we take two to the power of one and for three we take three square.
Step 3: Form the product of the numbers from step three. The greatest common divisor is the product of these factors. Greatest common divisor equal two times three square equal two times nine equal eighteen. Thus, the greatest common factor for two hundred and sixteen and two hundred and thirty four is eighteen.
The least common multiple (LCM) of two or more natural numbers is the smallest natural number that is divisible by all of the numbers.
To find the least common multiple using prime factorization of two or more numbers:
Write the prime factorization of each number.
Select every prime factor that occurs raised to the greatest power to which it occurs, in these factorizations.
Form the product of the numbers from step 2. The least common multiple is the product of these factors.
Example: Find the least common multiple of one hundred forty four and three hundred.
Solution: Step one. Write the prime factorization of each number.
One hundred and forty four equal two to the power four times three square.
Three hundred equal two square times three times five square.
Step 2: Select every prime factor that occurs, raised to the greatest power to which it occurs, in these factorizations.
One hundred and forty four equal two to the power of four times three square.
Three hundred equal two square times three times five square.
Step 3: Form the product of the numbers from step 2. The least common multiple is the product of these factors.
LCM equal two to the power of four times three square times five square equal sixteen times nine times twenty five equal three thousand and six hundred.
Hence, the LCM of one hundred and forty four and three hundred is three thousand and six hundred. Thus, the smallest natural number divisible by one hundred and fourty four and three hundred is three thousand and six hundred.
Properties:
All even numbers greater than two are composite numbers.
The smallest composite number is four.
Every composite number can be written as the product of (not necessarily distinct) primes. (Fundamental theorem of arithmetic)
Also, n minus one in bracket factorial equivalent with zero modulo n for all composite numbers, n more than five. (Wilson's theorem)
Kinds of composite numbers:
One way to classify composite numbers is by counting the number of prime factors. A composite number with two prime factors is a semi prime or 2-almost prime (the factors need not be distinct; hence squares of primes are included). A composite number with three distinct prime factors is a sphenic number. In some applications, it is necessary to differentiate between composite numbers with an odd number of distinct prime factors and those with an even number of distinct prime factors.
Some theorems about composite number are:
Theorem1)
If C is an odd number composed of two prime factors a and b, then the number of consecutive odd integers required to add up to C, is equal to the smaller of the two prime factors of C.
Theorem2)
The larger of the two prime factors of C is the average to the first and last number in the consecutive sum of odd numbers that add up to C.
Theorem3)
If C is an odd number that is composed of two prime factors a and C, then there exists at least one non-arbitrary perfect square that can be used to factor C through a finite number of algebraic operations.
Theorem4)
If C is an odd number that is composed of two prime factors a and b, then the non-arbitrary perfect square that can be used to factor C via finite number of algebraic operations is the first perfect square acquired by adding a consecutive number of odd integers starting with the number 1 to C and is equal to the average of the prime factors of C quantity squared.
Example:
Solution: Find the sum of the first one hundred even positive numbers.
The sum of the first one even positive numbers is two or one times one plus one equal one times two.
The sum of the first two even positive numbers is two plus four equal six or two times two plus one in bracket equals two times three.
The sum of the first three even positive numbers is two plus four plus six equal twelve or three times open bracket three plus one close bracket equal three times four.
The sum of the first four even positive numbers is two plus four plus six plus eight equal twenty or four times open bracket four plus one close bracket equal four times five.
Look for a pattern :
The sum of the first one hundred even positive numbers is two plus four plus six plus eight plus etc equal x or one hundred times one hundred plus one in bracket equal one hundred times one hundred and one or ten thousand and one hundred.
And finally this is the end of my fifth task and I wish it can be used well.
Best Regards….
Mathematical Thinking and Scientific Work
This whole article will explain about the correlation from mathematical thinking and scientific work. Mathematical thinking is thinking by consistent and always paying attention about the principle and the procedure when we solve some problems especially in mathematics. And if we do a fault or a step that it is not consistent with the assumptions, so the whole procedure is failed. As we know that mathematics consists of two ways which are pure mathematics and applied mathematics.
The school mathematics is the mathematics that learned in some grades of school, for example mathematics for junior high school or may be mathematics in senior high school. In this case, the school mathematics as the teacher’s task to explain to the student is having a pattern, solving problems, having some investigations of formulas, and also has the mathematics communications. And then the pure mathematics is sometimes interpreted by formal mathematics, and the characteristic of formal mathematics that is axiomatic mathematics. In formal mathematics sometimes each mathematicians are having a different thinking about mathematics too.
Mathematics established by deductive method consist that method itself, the definitions, some theorems, the axioms, rules, and procedure to proof, and etc. And then our first question is how to start establish mathematics as a system?
First we should have an assumption. Usually assumption is a concept of definition. All people can establish mathematics if they have a strong ground in mathematics, but to make we have a strong ground in mathematics is not too simple. So it is our tasks to always improve our ability in mathematics by always develop the mathematical thinking. To make the strong foundation in mathematics we also must paying attention with the definitions, the axioms, rules, and the procedure.
Second we must know the object of mathematics. And the object is your own idea in your mind. And then the second problem is how to get mathematical object? We can get it from the whole thing in the world, but be careful because some things in the world just being the example not the really object in mathematics. So, the examples of example in mathematics are maybe paper, pen chair, pyramid, and etc. The most difficult problem in here is how to make the example in mathematics being the object of mathematics? And it is what that did by Meta mathematics or thinking about mathematics. To make it can be the real object there are two ways.
The characteristics of real object are can be touched or can be tasted by us. And difference of real object and the mathematical object is mathematical object is abstract object, it can not be touched and can not be tasted, we can not manipulate it with sensory tools.
To make it can be the real object there are two ways. The first way is the idealization method. Idealization method is the method that we must always think the object is perfect because in mathematics sometimes we need the perfect object, for example the line that absolutely straight, the circle that absolutely circle, absolutely plane, and etc. The second way is by abstraction method. The abstraction method is just learning the one or some characteristics from all of characteristics that owned by that real object. For example the characteristics of a paper are it is light, flimsy, maybe the color is white and etc. But in mathematics we just learn the shape and the size of that paper. We just concern with the shape, for example it is rectangular paper, it is circle, it is triangular paper and etc. Well, the next characteristic that used in mathematics is size because in mathematics we just use the size or the value of something real. So the most important characteristics in abstraction method are the shape and the size.
Mathematical thinking is thinking by consistent and always according the early agreement. Consistent is always holding the principles of procedure. If we make a step that it is not consistent with the agreement, so the whole procedure is failed. And then mathematical thinking is always thinking by logic. Logic is consisting of two parts; they are daily logic and formal logic. Logical mathematics used very large in the world, for example comparison (seven more than four, and three less than ten, etc). The essential thing in logical mathematics is relationship, for example is nine equal ten minus one, four is three plus one, twenty five is five square, and etc. Mathematics operations are arithmetic operation for example addition, subtraction, to the power, square root, the radix, and etc. In mathematics we also must know about the “if then statement”. “If then statement” is usually write in if a then b, and the ‘a’ and ‘b’ are some simple statements or prepositions in mathematics. The sentence in mathematics can be the close sentence and open sentence. We also can see the truth table to understand what the relations of those statements are.
The table of truth is like that shown below:
If a is true and b is true, so a or b is true, a and b is true.
If a is true and b is false, so a or b is true, a and b is false.
If a is false and b is true, so a or b is true, a and b is false.
If a is false and b is false, so a or b is false, a and b is false.
How to get conclusion of some sentences? We must have some premises; mega premise (major premise), or minor premises.
In a procedure or to get the conclusion of some sentences sometimes we also find the antithesis, thesis, and hypothesis. The thesis in mathematics is a sentence that showing the truth, and then the antithesis is sentence that it becoming the contradiction sentence from the thesis. The hypothesis is a temporary assumption to make we can solve the mathematical problems by easier. And to proof the hypothesis we need the tool called conjecture.
The relationship between mathematical thinking and scientific work
Before we talk about the relationship between the mathematical thinking and scientific work, it can be wiser if we know that the scientific work has some special characters. Those characters are:
First, scientific work is impersonal work or it not related with personality or the maker. It not too much and it fulfilling standard definition, and also it is clear enough to can be able read by all people.
Second, scientific work has the standard of criteria; it can be just for local consumption, national, or maybe international standard. It also fulfilling the writing rule and it must become objective work.
The examples of scientific work are a report, paper, proposal, note, text book or presentation and etc. For scientific paper is consisting of abstractions, an introduction, and the background, discussion, conclusion, recommendation, ways of head, and preference. And also it must fulfill the ethical code that is far away from plagiarism and we must mention the sources and the references.
So, the relation of mathematical thinking and the scientific work is, if a mathematician makes some researches, and it proofed true, it can become the scientific work too. Because in the fact mathematics also as a science knowledge. And it can be used to all people to make them richer with new knowledge in mathematics. And we must always believe that if we want to be a good mathematician, please do not always want the other people helps us in mathematics, but we must think that mathematics is us and just we and ourselves that can make we become the good mathematician like what Mr. Marsigit said.
It is the end of this sixth article in my blog, I wish it is good for us and we can use it by maximal.
Regards…
The school mathematics is the mathematics that learned in some grades of school, for example mathematics for junior high school or may be mathematics in senior high school. In this case, the school mathematics as the teacher’s task to explain to the student is having a pattern, solving problems, having some investigations of formulas, and also has the mathematics communications. And then the pure mathematics is sometimes interpreted by formal mathematics, and the characteristic of formal mathematics that is axiomatic mathematics. In formal mathematics sometimes each mathematicians are having a different thinking about mathematics too.
Mathematics established by deductive method consist that method itself, the definitions, some theorems, the axioms, rules, and procedure to proof, and etc. And then our first question is how to start establish mathematics as a system?
First we should have an assumption. Usually assumption is a concept of definition. All people can establish mathematics if they have a strong ground in mathematics, but to make we have a strong ground in mathematics is not too simple. So it is our tasks to always improve our ability in mathematics by always develop the mathematical thinking. To make the strong foundation in mathematics we also must paying attention with the definitions, the axioms, rules, and the procedure.
Second we must know the object of mathematics. And the object is your own idea in your mind. And then the second problem is how to get mathematical object? We can get it from the whole thing in the world, but be careful because some things in the world just being the example not the really object in mathematics. So, the examples of example in mathematics are maybe paper, pen chair, pyramid, and etc. The most difficult problem in here is how to make the example in mathematics being the object of mathematics? And it is what that did by Meta mathematics or thinking about mathematics. To make it can be the real object there are two ways.
The characteristics of real object are can be touched or can be tasted by us. And difference of real object and the mathematical object is mathematical object is abstract object, it can not be touched and can not be tasted, we can not manipulate it with sensory tools.
To make it can be the real object there are two ways. The first way is the idealization method. Idealization method is the method that we must always think the object is perfect because in mathematics sometimes we need the perfect object, for example the line that absolutely straight, the circle that absolutely circle, absolutely plane, and etc. The second way is by abstraction method. The abstraction method is just learning the one or some characteristics from all of characteristics that owned by that real object. For example the characteristics of a paper are it is light, flimsy, maybe the color is white and etc. But in mathematics we just learn the shape and the size of that paper. We just concern with the shape, for example it is rectangular paper, it is circle, it is triangular paper and etc. Well, the next characteristic that used in mathematics is size because in mathematics we just use the size or the value of something real. So the most important characteristics in abstraction method are the shape and the size.
Mathematical thinking is thinking by consistent and always according the early agreement. Consistent is always holding the principles of procedure. If we make a step that it is not consistent with the agreement, so the whole procedure is failed. And then mathematical thinking is always thinking by logic. Logic is consisting of two parts; they are daily logic and formal logic. Logical mathematics used very large in the world, for example comparison (seven more than four, and three less than ten, etc). The essential thing in logical mathematics is relationship, for example is nine equal ten minus one, four is three plus one, twenty five is five square, and etc. Mathematics operations are arithmetic operation for example addition, subtraction, to the power, square root, the radix, and etc. In mathematics we also must know about the “if then statement”. “If then statement” is usually write in if a then b, and the ‘a’ and ‘b’ are some simple statements or prepositions in mathematics. The sentence in mathematics can be the close sentence and open sentence. We also can see the truth table to understand what the relations of those statements are.
The table of truth is like that shown below:
If a is true and b is true, so a or b is true, a and b is true.
If a is true and b is false, so a or b is true, a and b is false.
If a is false and b is true, so a or b is true, a and b is false.
If a is false and b is false, so a or b is false, a and b is false.
How to get conclusion of some sentences? We must have some premises; mega premise (major premise), or minor premises.
In a procedure or to get the conclusion of some sentences sometimes we also find the antithesis, thesis, and hypothesis. The thesis in mathematics is a sentence that showing the truth, and then the antithesis is sentence that it becoming the contradiction sentence from the thesis. The hypothesis is a temporary assumption to make we can solve the mathematical problems by easier. And to proof the hypothesis we need the tool called conjecture.
The relationship between mathematical thinking and scientific work
Before we talk about the relationship between the mathematical thinking and scientific work, it can be wiser if we know that the scientific work has some special characters. Those characters are:
First, scientific work is impersonal work or it not related with personality or the maker. It not too much and it fulfilling standard definition, and also it is clear enough to can be able read by all people.
Second, scientific work has the standard of criteria; it can be just for local consumption, national, or maybe international standard. It also fulfilling the writing rule and it must become objective work.
The examples of scientific work are a report, paper, proposal, note, text book or presentation and etc. For scientific paper is consisting of abstractions, an introduction, and the background, discussion, conclusion, recommendation, ways of head, and preference. And also it must fulfill the ethical code that is far away from plagiarism and we must mention the sources and the references.
So, the relation of mathematical thinking and the scientific work is, if a mathematician makes some researches, and it proofed true, it can become the scientific work too. Because in the fact mathematics also as a science knowledge. And it can be used to all people to make them richer with new knowledge in mathematics. And we must always believe that if we want to be a good mathematician, please do not always want the other people helps us in mathematics, but we must think that mathematics is us and just we and ourselves that can make we become the good mathematician like what Mr. Marsigit said.
It is the end of this sixth article in my blog, I wish it is good for us and we can use it by maximal.
Regards…
THE PSYCOLOGICAL PHENOMENON
HOW TO UNCOVER THE PSYCOLOGICAL PHENOMENON
We as human that lives in our environment of course have a lot of problems and chance to can always exist. We have a lot of reason to survive and defend our selves in the world. But sometimes, the problems occur when we try to face this reality. We just walk in our line as good as we can, but we don’t know why we still do the mistake and it make we get a problem as the effect. From this phenomenon we can learn how to solve the problem wisely.
World is full of traumatic things. In here, traumatic is the phenomenon or something not explained and something happens and we don’t know what the aim of the phenomenon is. The traumatic thing makes we feel trauma and scare about something. Sometimes the thing that make we get traumatic feeling is caused by strange behavior of someone, because of the implicit motif, and because of the natural activity that we don’t know.
The problem occurs because there’s a communication problem and we life in the world is just how to can make a good adaptation. We can make a good adaptation by create a good communication in every part of our conversation, no matter with whom. Because from a good communication we can share our idea and we also can get new ideas from the others, so we get our knowledge growing larger than before. The communication problem sometimes happens in to the people that have too low level competent and then too high competent, or because of there is a big and fast changing of something, for example the job mutation or the job pressure.
The central communication principle is sender and receiver. But if sometimes there are constraints that make the communication getting worse. The constraints cause the miss understanding and miss understanding cause the bad communication.
The education is an important thing in the human life. And then in the education process there are a lot of communication that happens among teacher and students. So if the teacher wants to make a good class, she must make a good communication first. A good communication starts with a good responsible to each other. Start by hearing and the response is a good speech or a good conversation. The most terrible mistake that did by the teacher is makes the students feel trauma. A trauma in here is all about not logic things or does the activity without the real cause. The bad teacher just give the concept for the student without give how to can get the concept, and how to apply the whole concept for our real life.
For the mathematics teacher, it’s wrong to just directly give the definition, theorems, lemmas, and also formulas to the student without explain first how to get that concept from real problem in our daily life. The student will be more excited if they are needed in the process of find the concept for some real problem. It’s very important to make them really understand about the whole material in the class and always remember the concept. To revitalize the teaching model, we must make a big changing to the learning process.
The pure science particularly mathematics begin from the assumption and the real problem in the world. Assumption is the undefined term and it’s become the main meaning that we use later in our effort to develop this branch of science. Begin from the assumption and after that we can take a definition, and then make an axiom, and theorems. Theorem in mathematics is much needed so it’s make theorem become an important thing and it can be used in every time, and the truth of it was legal in every where in the world. So, from the statement we must do the test from axioms to be the theorem.
This whole world is full of phenomenon. This whole world always pushes us to can face and solve the problems every day, every second. So we as human that want to keep our existence in this world must be a strong person. The characteristic from phenomenon are degree and measure. Everything in the world is having degree or quality, and then also having the measure or dimension. So to can solve the problem, someone must be can understand what’s the characteristic of that problem, for example the reason why the problems occur, the negative thing that maybe occur when the problem doesn’t be solved, the positive thing that will be happened if the problem solved, how to solve the problem, and etc.
The theory of mind teaches us that in our head (our brain) there are some parts. They are concept, perception, and the last is sensation. Concept consists of two parts, a priory concept and empirical concept. A priory concept is a recursion concept, usually we get the concept from the fact that we get in mathematics term. For example x1= 1+5, x2=2+10, x3=3+15, so we can guess that xi=i+5i. The empirical concept is the concept that we take from real problem in the daily life and we transform it into some reminder in our brain, or maybe can be a theorem or predictions.
Perception is all about the opinion from someone. Perception makes we have a main way to see some problems for each sight. Perception is a central thought that we use to determine some materials or phenomenon in our life. The perception built by the process of internalization knowledge or a new idea from the environment. We absorb the knowledge and new ideas from the world, and later it rearranges into a good basic our own ideas. So when we use that kind of ideas arrangement if we look some phenomenon in the daily life, it called we use our perception.
The third part in our mind is sensation. Sensation is the feeling that usually appears in our mind suddenly after we get a stimulus from the outside. The stimulus can be like some activities, phenomenon, ideas, rule, faith, and etc. we usually get the sensation appears suddenly because this whole world also full of stimulus. Every one have their own sensation to feel some phenomenon, it’s because every people have their own perception have the different level of knowledge and different way to understanding something.
In recent days, in the fact we usually find some of teachers are having the wrong process to teach his/her student. The fact brings us to can see that the reality the teacher assume that the teaching and learning process is the process of transfer of knowledge. In addition, this reality builds an assumption that the student is the empty vessel that knows nothing and then the teacher is the subject that full of knowledge. This fact makes a bad impact for the teaching and learning process because it very fragile to can build the traumatic feeling of the student and they can feel scare of their own teacher.
The best structure of a process is there are introduction, teaching and learning process, and then the last is closing remark. If every study process in the class is have that three aspects, it means we can make a good condition in teaching and learning. So the aim for the study process can be reached.
We hope there is a good revitalizing process of study process in our future.
We as human that lives in our environment of course have a lot of problems and chance to can always exist. We have a lot of reason to survive and defend our selves in the world. But sometimes, the problems occur when we try to face this reality. We just walk in our line as good as we can, but we don’t know why we still do the mistake and it make we get a problem as the effect. From this phenomenon we can learn how to solve the problem wisely.
World is full of traumatic things. In here, traumatic is the phenomenon or something not explained and something happens and we don’t know what the aim of the phenomenon is. The traumatic thing makes we feel trauma and scare about something. Sometimes the thing that make we get traumatic feeling is caused by strange behavior of someone, because of the implicit motif, and because of the natural activity that we don’t know.
The problem occurs because there’s a communication problem and we life in the world is just how to can make a good adaptation. We can make a good adaptation by create a good communication in every part of our conversation, no matter with whom. Because from a good communication we can share our idea and we also can get new ideas from the others, so we get our knowledge growing larger than before. The communication problem sometimes happens in to the people that have too low level competent and then too high competent, or because of there is a big and fast changing of something, for example the job mutation or the job pressure.
The central communication principle is sender and receiver. But if sometimes there are constraints that make the communication getting worse. The constraints cause the miss understanding and miss understanding cause the bad communication.
The education is an important thing in the human life. And then in the education process there are a lot of communication that happens among teacher and students. So if the teacher wants to make a good class, she must make a good communication first. A good communication starts with a good responsible to each other. Start by hearing and the response is a good speech or a good conversation. The most terrible mistake that did by the teacher is makes the students feel trauma. A trauma in here is all about not logic things or does the activity without the real cause. The bad teacher just give the concept for the student without give how to can get the concept, and how to apply the whole concept for our real life.
For the mathematics teacher, it’s wrong to just directly give the definition, theorems, lemmas, and also formulas to the student without explain first how to get that concept from real problem in our daily life. The student will be more excited if they are needed in the process of find the concept for some real problem. It’s very important to make them really understand about the whole material in the class and always remember the concept. To revitalize the teaching model, we must make a big changing to the learning process.
The pure science particularly mathematics begin from the assumption and the real problem in the world. Assumption is the undefined term and it’s become the main meaning that we use later in our effort to develop this branch of science. Begin from the assumption and after that we can take a definition, and then make an axiom, and theorems. Theorem in mathematics is much needed so it’s make theorem become an important thing and it can be used in every time, and the truth of it was legal in every where in the world. So, from the statement we must do the test from axioms to be the theorem.
This whole world is full of phenomenon. This whole world always pushes us to can face and solve the problems every day, every second. So we as human that want to keep our existence in this world must be a strong person. The characteristic from phenomenon are degree and measure. Everything in the world is having degree or quality, and then also having the measure or dimension. So to can solve the problem, someone must be can understand what’s the characteristic of that problem, for example the reason why the problems occur, the negative thing that maybe occur when the problem doesn’t be solved, the positive thing that will be happened if the problem solved, how to solve the problem, and etc.
The theory of mind teaches us that in our head (our brain) there are some parts. They are concept, perception, and the last is sensation. Concept consists of two parts, a priory concept and empirical concept. A priory concept is a recursion concept, usually we get the concept from the fact that we get in mathematics term. For example x1= 1+5, x2=2+10, x3=3+15, so we can guess that xi=i+5i. The empirical concept is the concept that we take from real problem in the daily life and we transform it into some reminder in our brain, or maybe can be a theorem or predictions.
Perception is all about the opinion from someone. Perception makes we have a main way to see some problems for each sight. Perception is a central thought that we use to determine some materials or phenomenon in our life. The perception built by the process of internalization knowledge or a new idea from the environment. We absorb the knowledge and new ideas from the world, and later it rearranges into a good basic our own ideas. So when we use that kind of ideas arrangement if we look some phenomenon in the daily life, it called we use our perception.
The third part in our mind is sensation. Sensation is the feeling that usually appears in our mind suddenly after we get a stimulus from the outside. The stimulus can be like some activities, phenomenon, ideas, rule, faith, and etc. we usually get the sensation appears suddenly because this whole world also full of stimulus. Every one have their own sensation to feel some phenomenon, it’s because every people have their own perception have the different level of knowledge and different way to understanding something.
In recent days, in the fact we usually find some of teachers are having the wrong process to teach his/her student. The fact brings us to can see that the reality the teacher assume that the teaching and learning process is the process of transfer of knowledge. In addition, this reality builds an assumption that the student is the empty vessel that knows nothing and then the teacher is the subject that full of knowledge. This fact makes a bad impact for the teaching and learning process because it very fragile to can build the traumatic feeling of the student and they can feel scare of their own teacher.
The best structure of a process is there are introduction, teaching and learning process, and then the last is closing remark. If every study process in the class is have that three aspects, it means we can make a good condition in teaching and learning. So the aim for the study process can be reached.
We hope there is a good revitalizing process of study process in our future.
SMALL RESEARCH
MY SMALL RESEARCH OF MATHEMATICAL THINKING
The small research is the research to complete the task of the lecture of psychology. This research also aimed to be the way to show the real mathematical phenomenon in the frame work of school mathematics. The knowledge of real phenomenon of school mathematics is used to helps us study about the best way to make a good conditions of teaching and learning mathematics in the school. So it’s much needed for the teacher and everyone who concern about the education, especially mathematical education.
According to Katagiri as the expert of mathematics from Japan, mathematics consists of three things.
1. Mathematics Attitude
Mathematics Attitude is shown by always has a question. Always keep the positive thinking and always be consistent and critical, and always be careful to learn, see and understand mathematics as the important science.
2. Mathematics Methods
Methods in Mathematics are how to learn Mathematics. There are So many ways to learn it, for example:
Deductions Method
It means explaining something from general to specific items, for example, if we meet someone and know him or her, we a the first sight is just know about the general characteristics, but if we always meet him or her, we can know him until the details characteristics.
Inductions Method
Inductions method means we look something from specific items to general items. For example, we study mathematics in the first time from learn 1+1=2, 2+1=3, 3+1=4, and etc. And after that we learn about something larger from it. That is an example from inductions method.
Syllogism Method
It is the method that taking a conclusion according some
Premises or some statements. To be the example is:
Lyra is student.
Every student has an id card.
The conclusion: Lyra has an id card.
Logical Method
It is about mathematics sentence and mathematics prepositions. Mathematics sentence is the sentence that from that we can take a conclusion it is true or not.
(And the other mathematical methods).
3. Mathematics Content
Mathematics content is the object of mathematics because we can find there are
many roles and the process to understanding and to solve the mathematical problems.
To build the mathematical concept and theorem, we must collect the mathematical evidences first. The mathematical evidences are very close to our life in the world. Everything is can be inferred into mathematics. For example the economical terms is always related from the linear equation system. The mathematical evidences are not always same for everyone. Of course mathematical evidences for the expert are different from mathematical evidences for the student when learning mathematics. The expert see more complicated real evidences than the student because their necessary and aim to learn mathematics maybe different, and then they have the different level of mathematical knowledge too.
Now we will just concern about the student’s sight of mathematical evidences. Student usually thinks that mathematical evidences in our life are not too clear. It’s because their knowledge of mathematics is not deep enough. They see mathematics just like a counting science and it’s very complicated. They don’t have the awareness that mathematics is very close with our life, and factually the mathematical evidences are taken from the real phenomenon of life. The student finds the mathematical evidences are simpler than the expert. They usually find the evidences after they study about mathematical material first. It’s not too good for the mathematical building in their mind because it can make they think that mathematics application is not as many as what they hope before.
The mathematical evidences can be collected by student from the events of some facts. For the student, the mathematical evidence is same with the mathematical object. Mathematical object become a very important thing because we get the data and we do the analyzing from that object and also it usually shows the quantitative data. In other word, mathematical object as the object in common become the center of the mathematical topic. Mathematical object have some mathematical problems, and it’s needed for us to investigate it.
A bit different from the mathematical object, the mathematics educational object is the object that shows the quantitative also qualitative data. The example of mathematics educational object is the mathematical reactions or mathematical attitude by the student. Student’s mathematical thinking become important because we can analyze what is the problem that immediately occurred in the student’s mind when the face the mathematical problems. The aim from this small research is to uncover the difficulties of student learning mathematics, and from that we can make some correction about the mathematics education to create the good quality of mathematics education and to recover mathematical thinking. After this part, we can see the facts that I collected from the junior high school of Bantul 2, located in Yogyakarta.
To collect the data of this small research, I observe the student thinking of mathematics by ask them to fill a kind of blanket and answer some questions there, for example about mathematical difficulties, mathematical motivation, and also the method that used by the teacher in the class, so from this activity I got the qualitative data.
The questions are:
1. Is mathematics difficult?
2. If mathematics is difficult, what is the reason?
a. The complicated material and problems of mathematics
b. Communication problems
3. What is the most difficult subject?
a. Natural science (consists of biology and physics)
b. Social science
c. English
4. What is the method that used by the teacher to do the mathematical teaching?
a. Lecture (teacher explanation)
b. Discussion
c. Both
5. Is the teacher uses the mathematical model?
6. What is the most difficult material in mathematics according to you?
a. Exponent
b. Line analysis
c. Probability
d. Linear equation system
7. What is the simplest material in mathematics according to you?
a. Statistics
b. Algebra
c. Linear equation system
d. Probability
e. The set material
f. Dimensional figure
8. Are you continuing learn mathematics in home?
9. Are you motivated to learn mathematics?
10. If you motivated to learn mathematics, who is your big motivator?
a. Your own self
b. Parent
c. Friend
The respondents consist of seven students who become the student in the junior high school at fifth semester, IX grade in the same class. The data will be shown below.
Result 1
Mathematics is difficult: 1
Mathematics is not difficult: 6
Result 2
The reason why mathematics difficult:
Because of the difficult and complicated material of mathematics: 1
So much material in mathematics: 0
Problem of teacher communication: 0
Result 3
The most difficult subject in the school
Natural science: 6
Social science: 4
English: 2
Result 4
The method that used by the teacher in the mathematics class
Lecturing: 0
Discussion: 1
Both: 6
Result 5
Is the teacher uses mathematical model?
Yes: 6
No: 0
Result 6
The most difficult material in mathematics;
Exponent: 3
Line equation: 4
Probability: 1
Linear equation system: 1
Result 7
The simplest material in mathematics;
Statistics: 4
Algebra: 1
Linear equation system: 1
Probability: 2
The set material: 1
Dimensional figure: 3
Result 8
They continue to learn mathematics in the home:
Yes: 2
No: 4
Result 9
They motivated to learn mathematics;
Yes: 6
No: 0
Result 10
The biggest motivation of learning mathematics;
Their own self: 4
Parent: 2
Friend: 1
The result of the data in the table tells us that the students basically have no problem with the mathematical material, and their understanding is good enough. It shown from the answer of the first question is mostly “no”. From the seven answer, we get 6 “yes” and 1 “no”, so 6 respondents said that mathematics is not difficult, and 1 respondent said that mathematics is difficult. And then the problem of student learning mathematics is because of the material and role is complicated. From the second respondent that stated that mathematics is difficult we get the reason why. Mathematics is quite difficult for her (as her statement) because mathematics has a complicated role and material.
Go to the next analysis from third question, we can see from the result that the most difficult subject in junior high school is natural science (6 arguments), and then second is social science (4arguments) and the last is English language (2 arguments). No one said that mathematics is difficult. Although the second respondent think that mathematics is a difficult science, but she stated that natural science is the most difficult subject for her.
From the fourth question, we can conclude that the teacher use the mixed method from lecturing method and the discussion method to explain the student about the mathematical material. It’s quite effective because the student need a lot of new interesting way to learn mathematics. The creativity is much needed in the mathematical teaching and learning because it can be the power of the teacher to can always find some mathematical model to help them teaching. The teacher in the ninth grade of junior high school of Bantul 2 uses some models to explain the mathematical concept in the school. It’s a good conditions to make the mathematical education involve, and then it’s much helps the student to understand and absorb the knowledge as good as they can.
The respondents as the student who still learn mathematics state that there is some most difficult material in mathematics. The examples most difficult mathematics materials are exponent, line analysis, probability and the linear equation system. From the result, we know that the most difficult material in mathematics is line analysis in the line equation. This material maybe difficult because of it’s a bit hard to explain the material by use model. From the facts, we also know that the simplest material of mathematics according to the respondent is statistic. Statistic is a branch of mathematics that concern about the data processing, in the case analyzing every conclusion that we will find in the data. The student feel enjoy and think that statistics is easy because in the statistics there are a lot of real problem in daily life that can become the mathematics problems, so the understanding and the study process can be more real than if they study about some abstract material like the line and line equation, graph, Cartesian axis, and others.
Students as the respondent not always continue to learn mathematics again in their hum. It’s shown by the answer of the 8th question, almost of them stated not continue than continue. The mathematical understanding can be better if they always keep the spirit and the curiosity to learn mathematics. From the facts, we can conclude that everything that they get in mathematics lesson in the school is not the maximal value. They still can improve and make it better by study and continue the mathematical pattern everywhere. Related to the nature of school mathematics by Ebutt and Straher in 1995, they denoted that school mathematics is the pattern, problem solving, investigation and communication. So to make student can learn mathematics good, they must apply the statement, to understand the mathematics pattern by always make the investigation and finally find the problem solving of mathematics, and communicate it to the others.
The student motivated to learn mathematics mostly because of their own self. It’s better than they motivated by the other because if their self can motivating them to learning mathematics deeply, they will always find the new spirit and they are free from the other influence. It’s very important because the spirit of learning mathematics is much needed to make them have a lot curiosity of mathematics and it’s can make them continue to learn mathematics again and again. And finally they can get a lot of information by learn mathematics and they change their opinion about mathematics, because in reality mathematics is much helps us in our life.
References: http://powermathematics.blogspot.com
Wikipedia.com (philosophy of power)
The small research is the research to complete the task of the lecture of psychology. This research also aimed to be the way to show the real mathematical phenomenon in the frame work of school mathematics. The knowledge of real phenomenon of school mathematics is used to helps us study about the best way to make a good conditions of teaching and learning mathematics in the school. So it’s much needed for the teacher and everyone who concern about the education, especially mathematical education.
According to Katagiri as the expert of mathematics from Japan, mathematics consists of three things.
1. Mathematics Attitude
Mathematics Attitude is shown by always has a question. Always keep the positive thinking and always be consistent and critical, and always be careful to learn, see and understand mathematics as the important science.
2. Mathematics Methods
Methods in Mathematics are how to learn Mathematics. There are So many ways to learn it, for example:
Deductions Method
It means explaining something from general to specific items, for example, if we meet someone and know him or her, we a the first sight is just know about the general characteristics, but if we always meet him or her, we can know him until the details characteristics.
Inductions Method
Inductions method means we look something from specific items to general items. For example, we study mathematics in the first time from learn 1+1=2, 2+1=3, 3+1=4, and etc. And after that we learn about something larger from it. That is an example from inductions method.
Syllogism Method
It is the method that taking a conclusion according some
Premises or some statements. To be the example is:
Lyra is student.
Every student has an id card.
The conclusion: Lyra has an id card.
Logical Method
It is about mathematics sentence and mathematics prepositions. Mathematics sentence is the sentence that from that we can take a conclusion it is true or not.
(And the other mathematical methods).
3. Mathematics Content
Mathematics content is the object of mathematics because we can find there are
many roles and the process to understanding and to solve the mathematical problems.
To build the mathematical concept and theorem, we must collect the mathematical evidences first. The mathematical evidences are very close to our life in the world. Everything is can be inferred into mathematics. For example the economical terms is always related from the linear equation system. The mathematical evidences are not always same for everyone. Of course mathematical evidences for the expert are different from mathematical evidences for the student when learning mathematics. The expert see more complicated real evidences than the student because their necessary and aim to learn mathematics maybe different, and then they have the different level of mathematical knowledge too.
Now we will just concern about the student’s sight of mathematical evidences. Student usually thinks that mathematical evidences in our life are not too clear. It’s because their knowledge of mathematics is not deep enough. They see mathematics just like a counting science and it’s very complicated. They don’t have the awareness that mathematics is very close with our life, and factually the mathematical evidences are taken from the real phenomenon of life. The student finds the mathematical evidences are simpler than the expert. They usually find the evidences after they study about mathematical material first. It’s not too good for the mathematical building in their mind because it can make they think that mathematics application is not as many as what they hope before.
The mathematical evidences can be collected by student from the events of some facts. For the student, the mathematical evidence is same with the mathematical object. Mathematical object become a very important thing because we get the data and we do the analyzing from that object and also it usually shows the quantitative data. In other word, mathematical object as the object in common become the center of the mathematical topic. Mathematical object have some mathematical problems, and it’s needed for us to investigate it.
A bit different from the mathematical object, the mathematics educational object is the object that shows the quantitative also qualitative data. The example of mathematics educational object is the mathematical reactions or mathematical attitude by the student. Student’s mathematical thinking become important because we can analyze what is the problem that immediately occurred in the student’s mind when the face the mathematical problems. The aim from this small research is to uncover the difficulties of student learning mathematics, and from that we can make some correction about the mathematics education to create the good quality of mathematics education and to recover mathematical thinking. After this part, we can see the facts that I collected from the junior high school of Bantul 2, located in Yogyakarta.
To collect the data of this small research, I observe the student thinking of mathematics by ask them to fill a kind of blanket and answer some questions there, for example about mathematical difficulties, mathematical motivation, and also the method that used by the teacher in the class, so from this activity I got the qualitative data.
The questions are:
1. Is mathematics difficult?
2. If mathematics is difficult, what is the reason?
a. The complicated material and problems of mathematics
b. Communication problems
3. What is the most difficult subject?
a. Natural science (consists of biology and physics)
b. Social science
c. English
4. What is the method that used by the teacher to do the mathematical teaching?
a. Lecture (teacher explanation)
b. Discussion
c. Both
5. Is the teacher uses the mathematical model?
6. What is the most difficult material in mathematics according to you?
a. Exponent
b. Line analysis
c. Probability
d. Linear equation system
7. What is the simplest material in mathematics according to you?
a. Statistics
b. Algebra
c. Linear equation system
d. Probability
e. The set material
f. Dimensional figure
8. Are you continuing learn mathematics in home?
9. Are you motivated to learn mathematics?
10. If you motivated to learn mathematics, who is your big motivator?
a. Your own self
b. Parent
c. Friend
The respondents consist of seven students who become the student in the junior high school at fifth semester, IX grade in the same class. The data will be shown below.
Result 1
Mathematics is difficult: 1
Mathematics is not difficult: 6
Result 2
The reason why mathematics difficult:
Because of the difficult and complicated material of mathematics: 1
So much material in mathematics: 0
Problem of teacher communication: 0
Result 3
The most difficult subject in the school
Natural science: 6
Social science: 4
English: 2
Result 4
The method that used by the teacher in the mathematics class
Lecturing: 0
Discussion: 1
Both: 6
Result 5
Is the teacher uses mathematical model?
Yes: 6
No: 0
Result 6
The most difficult material in mathematics;
Exponent: 3
Line equation: 4
Probability: 1
Linear equation system: 1
Result 7
The simplest material in mathematics;
Statistics: 4
Algebra: 1
Linear equation system: 1
Probability: 2
The set material: 1
Dimensional figure: 3
Result 8
They continue to learn mathematics in the home:
Yes: 2
No: 4
Result 9
They motivated to learn mathematics;
Yes: 6
No: 0
Result 10
The biggest motivation of learning mathematics;
Their own self: 4
Parent: 2
Friend: 1
The result of the data in the table tells us that the students basically have no problem with the mathematical material, and their understanding is good enough. It shown from the answer of the first question is mostly “no”. From the seven answer, we get 6 “yes” and 1 “no”, so 6 respondents said that mathematics is not difficult, and 1 respondent said that mathematics is difficult. And then the problem of student learning mathematics is because of the material and role is complicated. From the second respondent that stated that mathematics is difficult we get the reason why. Mathematics is quite difficult for her (as her statement) because mathematics has a complicated role and material.
Go to the next analysis from third question, we can see from the result that the most difficult subject in junior high school is natural science (6 arguments), and then second is social science (4arguments) and the last is English language (2 arguments). No one said that mathematics is difficult. Although the second respondent think that mathematics is a difficult science, but she stated that natural science is the most difficult subject for her.
From the fourth question, we can conclude that the teacher use the mixed method from lecturing method and the discussion method to explain the student about the mathematical material. It’s quite effective because the student need a lot of new interesting way to learn mathematics. The creativity is much needed in the mathematical teaching and learning because it can be the power of the teacher to can always find some mathematical model to help them teaching. The teacher in the ninth grade of junior high school of Bantul 2 uses some models to explain the mathematical concept in the school. It’s a good conditions to make the mathematical education involve, and then it’s much helps the student to understand and absorb the knowledge as good as they can.
The respondents as the student who still learn mathematics state that there is some most difficult material in mathematics. The examples most difficult mathematics materials are exponent, line analysis, probability and the linear equation system. From the result, we know that the most difficult material in mathematics is line analysis in the line equation. This material maybe difficult because of it’s a bit hard to explain the material by use model. From the facts, we also know that the simplest material of mathematics according to the respondent is statistic. Statistic is a branch of mathematics that concern about the data processing, in the case analyzing every conclusion that we will find in the data. The student feel enjoy and think that statistics is easy because in the statistics there are a lot of real problem in daily life that can become the mathematics problems, so the understanding and the study process can be more real than if they study about some abstract material like the line and line equation, graph, Cartesian axis, and others.
Students as the respondent not always continue to learn mathematics again in their hum. It’s shown by the answer of the 8th question, almost of them stated not continue than continue. The mathematical understanding can be better if they always keep the spirit and the curiosity to learn mathematics. From the facts, we can conclude that everything that they get in mathematics lesson in the school is not the maximal value. They still can improve and make it better by study and continue the mathematical pattern everywhere. Related to the nature of school mathematics by Ebutt and Straher in 1995, they denoted that school mathematics is the pattern, problem solving, investigation and communication. So to make student can learn mathematics good, they must apply the statement, to understand the mathematics pattern by always make the investigation and finally find the problem solving of mathematics, and communicate it to the others.
The student motivated to learn mathematics mostly because of their own self. It’s better than they motivated by the other because if their self can motivating them to learning mathematics deeply, they will always find the new spirit and they are free from the other influence. It’s very important because the spirit of learning mathematics is much needed to make them have a lot curiosity of mathematics and it’s can make them continue to learn mathematics again and again. And finally they can get a lot of information by learn mathematics and they change their opinion about mathematics, because in reality mathematics is much helps us in our life.
References: http://powermathematics.blogspot.com
Wikipedia.com (philosophy of power)
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