LEARN MATHEMATICS, MAKE IT EASY

GREATEST COMMON FACTOR (GCF)
OBJECTIVES
- Find the greatest common factor (GCF) of numbers
- Find he GCF of terms
- Factor out the GCF
- Factor a four-term expression by grouping
GETTING STARTED
- Product and Factors
15 = 3 x 5
15 is product
13 and 5 are factors
Factoring completely means to have all factors as prime numbers.
another example is 20 = 2 x 2 x 5
FINDING THE GREATEST COMMON FACTOR (GCF) OF NUMBERS
- The GCF of a list of integers is the largest common factor of the integers in the list.
Example 45 = 3 x 3 x 5 = 3^2 x 5
60 = 2^2 x 3 x 5
- To find the GCF, choose prime factors with the smallest exponents and find their product.
the same factors are 3 and 5, then the smallest exponent is 1. So, we can choose 3 and 5 and multiply it.
3 x 5 = 15
15 is the greatest common factor (GCF) of 45 and 60
Once again ...
- To find the GCF, choose prime factors with the smallest exponents and find their product.
36, 60, 108
36 = 2^2 x 3^2
60 = 2^2 x 3 x 5
108 = 2^2 x 3^3
Then,  2^2 x 3 = 12
So, 12 the greatest common factor (GCF) of 36, 60,  dan 108
Another way to solve it ...
2 | 36   60   108
2 | 18   30    54
3 |  9    15    27
      3     5     9
Collect the terms, that are 2 x 2 x 3 = 12
Then, we get the same result.
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THE CHARACTERISTICS OF NUMBER
PROPERTIES OF NUMBERS
- The Reflexive Property of Equality
A number is equal to itself.
symbolicly is A = A
Example 2 same as 2
- The Symmetric Property of Equality
If one value is equal to another, then that second value is the sameas the first.
symbolicly is A = B then B = A
example 3 = x
so, x = 3
- The Transitive Property of Equality
If one value is equal to a second, and the second happens to bethe same as a third, then we can conclude the first value must also equal the third.
symbolicly A = B and B = C, so, A = C
- The Substitution Property
If one value is equal toanother, then the secondvaluecan be used in place of the firstin any algebraic expression dealing with the first value.
- The Additive Property of Equality
We can add equal values to both sides of an equation without changing the validity of the equation.
A = B
A + C = B + C
C + A = C + B
- The Cancellation Law of Additive
A + C = B + C
if we will cancel the substitution, we just substract -C
then A + C - C = B + C - C can be A = B
- The Multiplicative Property of Equality
We can multiply equal values to both sides of an equation without changing the validity of the equation.
A = B
A x C = B x C
C x A = C x B
- The Cancellation Law of Multiplication
A x C = B x C
if we wanna cancel the C, so, we just divide both sides with C and C is not zero
A = B
- The Zero-FactorProperty
If two values that are being multiplied together equal zero, then one of the values, or both of them, must equal zero.
AB = 0
A = 0 or B = 0 or A and B = 0

PROPERTIES OF INEQUALITY
- The Law of Trichotomy
For any two values, only one of the following can be true about these values:
They are equal. The first has a smaller value than the second. The first has a larger value than the second.
Given any numbers A and B
that can be A = B or A < B or A > B
- The Transitive Property of Inequality
If one value is smaller than a second, and the second is less than a third, then we can conclude the first value is smaller than the third.
A < B and B < C so, A < C

PROPERTIES OF ABSOLUTE VALUE
|A| >= 0
|-A| = |A|
|AxB| = |A| x |B|
|A/B| = |A| / |B|
B is not zero

PROPERTIES OF NUMBERS
CLOSURE
- The Closure Property of Addition
When you add real numbers to other real numbers, the sum is also real.
Addition is a "closed" operation.
A + B = real numbers
A is real number
B is real number
The result is real
- The Closure Property of Multiplication
When you multiply real numbers to other real numbers, the product is a real number.
Multiplication is a "closed" operation.
A x B = real number
A is real number
B is real number
The result is real
- A Special Note
Substraction operation is not "closed"operation. Example: 3 - 5 = -2
3 is natural number and 5 is natural number, but the result is -2 and its is not a natural number. So, substraction operation is not "closed"operation.
COMMUTATIVE
-The Commutative Property of Addition
It does not matter the order in which numbers are added together.
A + B <--> B + A
-The Commutative Property of Multiplication
It does not matter the order in which numbers are multiplied together.
A x B <--> B x A
ASSOCIATIVITY
-The Associative Property of Addition
When we wish to add three (or more) numbers. It does not matter how we group them together for adding purposes. The parentheses ca be placed as we wish.
(A + B) + C
A + (B + C)
-The Associative Property of Multiplication
When we wish to multiply three (or more) numbers. It does not matter how we group them together for multiplication purposes. The parentheses ca be placed as we wish.
(A x B) x C
A x (B x C)
- By the way ...
It does not valid in substraction and division.
IDENTITY
-The Identity Property of Multiplication
There exists a special number, called the "additive identity", when added to any other number, then that other number will still "keep its identity" and remain the same.
A + 0  = A <--> 0 + A = A (a number plus zero still that number)
-The Identity Property of Multiplication
There exists a special number, called the "multiplicative identity", when multiplied to any other number, then that other number will still "keep its identity" and remain the same.
A x 1  = A <--> 1 x A = A (a number plus zero still that number)
INVERSE
- The Inverse Property of Addition
For every real number, there exists another real number that is called its opposite, such that, when added together, you get the additive identity (the number zero).
A + (-A) = 0
(-A) + A = 0
So, -A is an opposite of A
- The Inverse Property of Multiplication
For every real number, except zero. there exists another real number that is called its multiplicative inverse, or reciprocal, such that, when multiplied together, you get the multiplicative identity (the number one).
A x 1/A = 1
1/A x A = 1
DISTRIBUTIVITY
- The Distributive Law of Multiplication Over Addition
Multipling a number by a sum of numbers is the same asmultiplying each numberinthe sum individually, then adding up our products.
Example:
5 (7 + 3) = 5 (10) = 50
5 (7) + 5 (3) = 35 + 15 = 50
the result is same
So, A (B + C) = AB + AC
(A + B) C = AC + BC
- The Distributive Law of Multiplication Over Subtraction
A (B - C) = AB - AC
- The General Distributive Property
a (b1+b2+b3+...+bn) = ab1 + ab2 + ab3 +...+ abn
2 (1+3+5+7) = 2 + 6 + 10 + 14 = 32
- The Negation Distributive Property
If you negate (or find the opposite) of a sum, just "change the signs" of whatever is inside the parentheses.
-(A+B) = (-A) + (-B) = - A - B
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PRE_CALCULUS
Graph of a Rational Function
Given f(x) = (x+2) / (x-1)
if x=1 then can be 3/0 and the result is can not be define
if we draw in the graph, it will break the graph
So, we must substitute x with number except 1
if x = -3, f(-3) = 1/4
x = -2, f(-2) = 0
x = -1, f(-1) = -1/2
x = 0, f(0) = -2
x = 2, f(2) = 4
x = 3, f(3) = 5/2
Then, the graph is

REFLECTION OF MY LESSON

ANGLE
An angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Initial side of an angle is the ray where measurement of an angle starts.    Terminal side of an angle is the ray where measurement of an angle stops.


Parts of an Angle











The corner point of an angle is called the vertex and the two straight sides are called arms. The angle is the amount of turn between each arm.
Type of Angle     Description    
Acute Angle        an angle that is less than 90°
Right Angle         an angle that is 90° exactly
Obtuse Angle      an angle that is greater than 90° but less than 180°
Straight Angle     an angle that is 180° exactly
Reflex Angle       an angle that is greater than 180°

Positive and Negative Angle
The angle is measured by the amount of rotation from the initial side to the terminal side.  If measured in a counterclockwise direction the measurement is positive (called positive angle).  If measured in a clockwise direction the measurement is negative (called negative angle).  (A negative associated with an angle's measure refers to its "direction" of measurement, clockwise.)

Angles are usually measured in degrees (denoted degrees) and radians (denoted rad, or without a unit). It can be representated in quadrant (x-y graph).
 

Standard Position
 
An angle is in standard position if its vertex is located at the origin and one ray is on the positive x-axis.  The ray on the x-axis is called the initial side and the other ray is called the terminal side.  If the terminal side of an angle lies "on" the axes (such as 0º, 90º, 180º, 270º, 360º ), it is called a quadrantal angle.  The angle shown at the right is referred to as a Quadrant II angle since its terminal side lies in Quadrant II.    
Special angle : 0°, 30°, 45°, 60°, and 90°




DEGREES AND RADIANS
Degrees
We can measure Angles in Degrees.
There are 360 degrees in one Full Rotation (one complete circle around). A Full Circle is 360°.
One complete turn (360°) is equal to 2π radians, so 180° is equal to π radians, or equivalently, the degree is a mathematical constant: 1° = π⁄180.
The Degree Symbol: °
We use a little circle ° following the number to mean degrees. For example 90° means 90 degrees.

 Radians
We can measure Angles in Radians.
1 Radian is about 57.2958 degrees. Radian is a pure measure based on the Radius of the circle.
Radians and Degrees
    There are π radians in a half circle
    And also 180° in a half circle
So π radians = 180°
So, 1 radian = 180°/π = 57.2958° (approximately)

Degrees     Radians         Radians
                  (exact)           (approx)
0°              0                    0
30°            π/6                 0.524
45°            π/4                 0.785
60°            π/3                 1.047
90°            π/2                 1.571
180°          π                    3.142
270°          3π/2               4.712
360°          2π                  6.283

How to convert

360°               =    2π
360° / 2          =    2π / 2
180° / 180°    =    π / 180°
1°                   =    π / 180°
and
2π radian           =    360°
2π radian / 2π    =    360° / 2π
1 radians            =    180° / π

Example
120°     = ... radians
120°     = 120° x π/180 radians
             = 2/3 π

11/12 π rad = ... °
11/12 π       = 180°/π x 11/12 π
                   = 165°



MULTIPLYING EXPONENTS
There are some rules to solve exponent problems:
Rule 1        
If the bases are different but the exponents are the same, then we can combine them.
Example    3^5 x 4^5 = (3 x 4)^5 = 125 = 248832

Rule 2        
Dividing different bases can’t be simplified unless the exponents are equal. So, we can divide bases, then count its exponent.
Example    6^3 / 3^3 = (6/3)^3 = 3^3 = 27

Rule 3        
We just multiply the exponents.
Example    (23)^2 = (2 x 2 x 2)^2 = 8^2 = 64 = 2^6
        (23)^2 = (2)^(3x2) = 2^6 = 64

Rule 4        
When the bases are the same, then, just adding the exponents.
Example    2^3 x 2^5 = 2^(3+5) = 2^8 = 256

Rule 5        
For division with same bases, then subtract exponents.
Example    4^5 / 4^3 = 4^(5-3) = 4^2 = 16

MULTI DIVISION MATH
How to calculate 26 x 31 ?
Standard algorithm for multiplying
26 x 31 = ...
  26
  31  x
  26
78    +
806

We can calculate with other methods:
# Partial Product Method
26 x 31 = ...
   26
   31  x
     6       = 1 x 6
   12       = 20 x 1
180        = 30 x 6
600  +    = 200 x 3
806

# 26 x 31 = ...
26 = 20 + 5 + 1
then, (20×31) + (5×31) + (1×31) = 620 + 155 + 31 = 806

# Lattice Method
26 x 31 = ...

So, 26 x 31 = 806

Standard algorithm for division
Long Division
133 : 6 = ...
     22 Remain 1
6 /133
     12     -
       13
       12   -
         1
So, 133 : 6 = 22 1/6

# The other methods
133 : 6 = ...
6 x 10    =   60
6 x 20    = 120
6 x 1      =     6
6 x 21    = 126
6 x 1      =     6
6 x 22    = 132
6×22+1  = 133
So, 133 : 6 = 22 Remain 1

133 : 6 = ...
6 × 10 = 60
6 × 10 = 60
6 × 1   = 6
6 × 1   = 6
(10 + 10 + 1 + 1) = 22
133 = 6 × 22 + 1
So, 133 : 6 = 22 Remain 1




QUADRATICS FORM
Standard form for a quadratic equation:
y = ax2 + bx + c
A quadratic equation is a polynomial function of degree 2, where a, b, and c are all real numbers and a ≠ 0.
Quadratic equations are square. The rate of slope is changing (never same at every point).
Example    f(x) = x2 + 4x + 3
 

Linear equation
y = mx + b
m = slope
b = y-intercept
The gradient is constant and it is in straight line.
Example    y = 3x + 1

Mathematics Song

Prime Number

Hey my friends, hey my friends

I have a something new

There are 2, 3, and 5

Do you know what are they?

Its name is prime number

And then has two factors

Number 1 and itself

They are its factors

(Song: "Pelangi")

THE SOUNDTRACK OF LIFE

Mathematics can we practice from a song. It is also more fun! Believe it! Why? Because we can express our emotion and our expression. Do you know? From a song, we can show our creativity. That is that Mr. Marsigit give to his students include me. We were given some mathematics songs.

Song: Do you know about math

It is tell us what mathematics is. There are many chapters in mathematics. For example : multiply, memorizing pi, take limits, to simplify, graphing utility, trigonometry, limits, logarithm, exponential, etc. From that song we know that learn mathematics is fun.

Song: Perimeter

From this song we can learn how to measure perimeter of a shape. It can be circle, triangle, squares, rectangles, parallelograms, trapezoidal, kite, etc.

Song: Mean, median, and mode

From this song we learn about statistic. We can practice how to find median, mean, and mode. Mean is the average of the data. To find mean, it is to add up the numbers in the data, then divide the total by the number of items. Then median is the middle value, to find it, we must look arrange the numbers in order from the lowest to the highest values. The middle number is the median for an odd number of items. The average of two middle numbers is the median for an even number of items. Mode is the value that often arise, to determine it just find the number that repeats most often.  If there is no number that repeats most often, there is no mode.

Song: The math song

It is song from Bruno Mars with title The Lazy Song. That song is made with fun style. So, it can interest student. We can practice more easy.

Then some videos from Mr. Marsigit. That video show about derivative, integer, and angle.

Derivative is slope of tangent line at point (x, f(x)) ((x+h),f(x+h)), h is change of x. Slope = (y2-y1)/(x2-x1). Definition of derivative F’(x)= limh→0 (f(x+h)-f(x))/h. So, derivative is a slope. Derivative is limit a slope, limit change. Then, try find f’(2) from f(x)= 4×2-8x+3. Substitute f(2) and f (2+h)to the function. Then, simplify and factory.

Integer is whole number, not fractional or decimal. Whole numbers that greater than zero are called positive integers, It is on the right of zero in the number line. Whole numbers less than zero are called negative integers, it is on the left of zero in the number line. In the integer, there are place number, such as unit place, tens place, hundred place, thousand place, etc.

Angle. There are several kinds of angles: acute angle, the angle whose measure is less than 90⁰. The angle brackets, which the measure is 90^0. Obtuse angle, the angle whose measure is greater than 90⁰. Straight angle, the angle whose measure is180⁰. Two angles that complement each other is total of the measure is 90⁰. Two angles that supplement each other is total of the measure is 180⁰. Vertical angles are two angles that the places are opposite each other. Angle bisector is a ray that splits an angle became two congruent angles. Perpendicular lines are intersecting lines that from right angle.

That’s all…. So, we can say that practice mathematics is fun. Sing your own song and show your expression. Be creative student!

TEACH MATHEMATICS, TEACH OUR LIFE

Stadium General

Monday, March 12th 2012, 1 p.m.

By Prof. Mohan Chinnappan, Ph.D

Last Monday, I had to join a stadium general at Seminar Room floor 2, MIPA. It was also task from my lecturer, Mr. Marsigit. There was presented someone from Australia that was Prof. Mohan Chinnappan, Ph.D. I must make a report after I attended the stadium general. During the stadium general, I try to listen carefully and write that I can understand. It was very pleasant because from him I can know about education from other country especially about mathematics education.

Many things that Prof. Mohan Chinnappan, Ph.D. explained to participant. First, about mathematical thinking like the theory of Shigeo Katagiri. It include 3 matters.

Mathematical thinking consist of:

1.      Mathematical attitude

2.      Mathematical method

3.      Mathematical content

Then, an analogy between mathematics and tailor. Before the tailor sew the cloth, he make draw what will he sew. He does it in order to make easy when he sew the cloth. So, there is no mistake and his job will finished well. We can say that the sailor make a pattern. From that pattern, he can work well. Now, like the sailor, mathematics also have pattern. That pattern has function when we learn mathematics. So, pattern and relationship in mathematics include:

1.      Problem solving

2.      Communication

3.      Investigation

The next explanation is about religion knowledge and mathematics knowledge. There is a question. What is the relationship between natural knowledge and religion knowledge? To answer it, Prof. Mohan Chinnappan, Ph.D. give example of some matter.

1.      Relationship between praying and mathematics. Before and finish the lesson we are praying.

2.      Zakat is depend on our property. To count how much we must give zakat, we can use mathematical operation. Every kind property that we have, it is has different calculation. So, we need mathematics.

3.      We can calculate inheritance.

4.      Zero = notion of nothing. It use for nothing. Decimal system become more complete.

5.      Basis number  system: 0, 1,2, 3, . . ., 9. We can not change the position of Al-Qur’an.

From that explanation, we can state that there is close connection between mathematics and religion.

Then, it is about culture in Indonesia. What is the culture of Indonesia in mathematics knowledge? Mathematics can be interpreted in Indonesian culture. From our culture, we can know how mathematics is used. When count the day dead in Java like “pasaran” days. For example is Pon, Wage, Kliwon, 7 days, 100 days, 1000 days, etc. It is just use modulo five.

The other is the construction of temple. Geometry construction is five steps and around the temple there is relief. Ramayana: 1 à earth

                                                   2 à couple

                                                   0 à the symbol of loss

There is also important about lesson study to teach. Lesson study include teachers as model, observers, and resource persons. Lesson study is based on students  learning activities.

In lesson study, we need to do three actions. First, detailed planning. Include material approach, how to present to the students, and the estimated constraints. Second, doing. Present the lesson, observing the activity of students, when students begin to learn, when students are getting bored, and what activities that support student learning and not support the learning. We must pay attention our student in the class and need to understand situation in the class. Third, reflection. In reflection, we do evaluation. From activities in the class, we can conclude many things. So, we can know if we do a mistake or there are something that we must be increased or decreased. Teacher must have strategy to teach his/her students. The lesson study can help teacher to solve the problem and make the teacher ready when face his/her students. So, we should involve lesson study in the class room.

The last session is discussion. There are some question from the participants.

- So many subject in the school because some society need many knowledge. So they must study more. To understand more knowledge, we need learn more subject. It is different with Indonesia. In Indonesia, the subjects are more than in Australia.

- In Australia, people can choose 5 subjects since 12 years old. For example science, mathematics, language, etc. It is also different. In Indonesia, students learn all subjects since in elementary school. Then in senior high school, there are three majors and also kind of school like SMA, SMK, etc.

- How to teaching mathematics in Australia? Use ICT, smart board, Internet to download and solve problem, and use laptop, etc. From Prof. Mohan Chinnappan, Ph.D. answer, we know that facilities in Australia are more good than in Indonesia.

- How to cope with children with skills below average?  

Based on the facilities on the school. Government also support the school. Teacher creates worksheets that correspond to student ability. Learning to do and learning to be.

- Character of education in Indonesia and Australia is different.

Australia: secular country and liberal democracy.

Indonesia: there is religion school for example Islamic school, Christian school. But public school do not support religion as many as in religion school.

For the closing, there are some conclusions:

- We need to be rational to the subjects and what kind of activity.

- Developing of teacher professionalism.

- Understand national curriculum and theory in mathematics.

- Attention to our culture and we have characteristics because our culture is important.

- How to manage the different competence in mathematics.