MY EXERCISE

NAME : NURAIDA LUTFI HASTUTI

NIM : 11301241031

PRODI : MATHEMATICS EDUCATION 2011

EXERCISE 1 (PAGE 47,MARSIGIT’S BOOK, MATHEMATICS FOR JUNIOR HIGH SCHOOL YEAR IX)

1.        ∆ABC and ∆CDA are congruent. The fulfilled requirements are …

 AB=DC, BC=DA, AC=AC (coincides).

The answer is side, side, side.

2.        Consider the following figure.

Given, <CAD=<DBE, ∆AEC are congruent because they fulfill the requirements angle, side, angle. <CAD=<DBE=45°, AE=EB, <AEC=<DEB because they are opposite to each other.

3.        ∆KLO and ∆KMN are congruent because they fulfill the requirements angle, side, angle.

Because <KOL=<KNM=100°, KO=KN, <OKL=<NKM (coincides)

4.        ∆GHI and ∆XYZ are congruent.

The value of m is 25° because m=<GIH=<XZY. <XZY=180°-95°-60° and the answer is 25°.

5.        –

6.        AB || CD. The Length of AB is 18 cm.

Because ∆ECD and ∆EAB are similar, so

AB/CD = AE/CE

AB/12 = 12/3

       AB = 18

7.        The length of AC is 7 cm.

Because ∆ABC and ∆DEF are similar, so

AC/DF = BC/EF

AC/14 = 8/16

AC = 7

8.        Consider the following figure.

Because both triangle are similar (angle, side, angle), so

KL/EF = KM/FG

KL/4 = 9/6

KL = 6

The answer is 6 cm.

9.        For the following figure, let AB=18 cm, PQ=6cm, PR=5cm, <ABC=<PQR, and <BCA=<QRP. The length of BC is 15 cm.

Because both triangle are similar (angle, side, angle), so

BC/QR = AB/PQ

BC/5 = 18/6

BC = 15

10.    In the following figure, let the length of CD = 9 cm, CE = 6 cm, and BC = 12 cm. the length of AC is 12 cm.

∆ADC similar ∆BEC (similar angle, angle, side), so

AC/BC = DC/EC

AB/12 = 9/6

AC = 12

11.    The proportion between the length of two similar sides is 2:3. If the diagonal length of the small rectangle is 30 cm, so the length of the diagonal of the bigger rectangle is  45 cm. Solution : if x is the length of the diagonal of the bigger rectangle. We can state that 30/x = 2/3 then x = 45

12.    A flagpole with 5 m high is at a distance of 12 m from a tower and collinear with the shadow of the tower. The length of the flagpole’s shadow is 3 m. The height of the tower is 25 m.

Solution : The height of a tower is comparable with the height of the flagpole, then the sum of shadow of tower and flagpole is comparable with the shadow of flagpole. If x is the height of the tower then x/5 = 15/3 then x = 25.

13.    A painting is put on a triplex. The size of the triplex is 30 cm x 50 cm. we found that, on the top, left, and right of the painting there are still the remains of the triplex which is uncovered. If the painting is similar with triplex, so the area of uncovered triplex is 360, 5 cm area.

Solution: The size of painting is 50 cm – 7 cm = 43 cm, 30 cm – 3.5 cm = 26,5 cm. the area of painting is 43 x 26.5 = 1139.5. The area of triplex is 50 x 30 = 1500. Then the area that uncovered is 1500 – 1139.5 = 360.5 cm area.

14.    ∆ABC and ∆DBE are similar. The correct statement is AC/DE = CB/EB.

Because they are similar so, their sides can be compared, then

AC/DE = CB/EB = AB/DB

15.    ∆ABC and ∆ADE are similar. Let, AB = 7 cm, AD = 5 cm, and DE = 6 cm. the length of BC is 8.40 cm

because they are similar so, BC/DE = AB/AD = AC/AE

to find BC then

BC/DE = AB/AD

BC/6 = 7/5

BC = 8.40

16.    ∆ABC and ∆DEF are congruent, the length of the sides of  ∆DEF  are DE = 3 cm, EF = 2.5 cm, and FD = 2 cm.

Solution: because they are congruent so it has 3 pairs side that have same size (length). They are AB=DE, EF=BC, FD=AC.

17.    If ∆ABC and ∆BAD are congruent, so the values of x and y are 45° and 1 cm.

solution: <DAB = <ABC

          (3x – 90)° = 45°

                      3x = 135°

                        x = 45°

then AD = BC

      y + 1 = 2

            y = 1 cm

18.    ∆ABC and ∆PQR are congruent. The length  of PQ is 4 cm.

solution: they are congruent side, side, side

so first we must find x

because PR=AC, QR=BC, PQ=AB, then we can use AC = PR

2x – 2 = 2

x = 2

then QR = AB

QR = 3x – 2

QR = 4

19.    See the figure below. Let AC = 2 cm, BC = 4.5cm. if DE = 1 cm, so the length of BE is 2.25 cm.

solution: ∆ABC and ∆DEB are similar, so

BE/BC=DE/AC=BD/BA

To find BE, it use

BE/BC = DE/AC

BE/4.5 = ½

BE = 2.25

20.    Given the parallelogram PQRS, if the area of parallelogram is 8 cm area, so the length of QR is 3 cm.

solution: the area of parallelogram is PQ multiply the height.

A = PQ . 2

8 = (2x + 2) . 2

x = 1

after we find x, then we can find QR by substitute x to QR

QR = 8x – 5

QR = 8 – 5

QR = 3

MATHEMATICS IN OUR LIFE AND HOW TO DEVELOPE IT

Name : Nuraida Lutfi Hastuti
NIM : 11301241031
Class : Mathematics Education 2011

            Mathematics is around us. We always meet mathematics in our activities. Even though we live together with mathematics, but mathematics is not a living creature. Then what happens with mathematics? Mathematics can also experiencing growth. To facilitate the mathematics of human life was made in such a way that it can develope. Of course there are people who was instrumental in developing the mathematical sciences.

            To develop the science of mathematics is not as easy as turning the palm of the hand. Because mathematics is one of the fields of science then there are the steps in its development. It consists of:
            ANALYSIS STEPS
            In this process of problem solving research following the steps that perk do:
1. Problem Definition
At this step there are three main elements to be identified:
(A) Function Tujuan: state to help the destination for directing efforts to fulfill the objectives to be achieved.
(B) Functions Limits / constraints: constraints that affect the issue of the objectives to be achieved.
(C) the decision variables: variables that influence the decision-making problems.
2. Model Development
Collect data to assess the magnitude of the parameters of the problems faced affected. These estimates are used to construct and evaluate a mathematical model of the problem dart.
3. Solving Model
In formulating this problem typically use analytical models, the mathematical model that produces the equation, so that the optimum solution is achieved.
4. Validity Testing Model
Determine whether the model has been constructed to describe real situation accurately. If not, fix or create a new model.
5. Implementation of the final basil
Study or calculation basil translate into everyday language that is easily understood.

              MODELS IN RESEARCH
              In research recognized some form of a model that describes the characteristics and form a system problem. Various kinds of models in between:
Iconic Model
Is a physical replica as the original form with a much smaller scale. Example: The maker of the building, automotive models, and model airplanes.
Analog Model
A physical model but do not have a shape similar to that modeled. Example: thermometer gauge that indicates the level of the model temperatur.
Symbolic Model
Is a model that uses symbols (letters, numbers, shapes, images, etc.) which presents the characteristics and properties of a system darts. Comb: network (network diagrams), flow charts, flow charts, and others.
Mathematical Model
Includes models that represent the real situation of the system mathematical functions. Example: P = a'. Po state population models of living things.

              From the above, it is no less important is its application in life being so close to the mathematical problems in life. Starting from the initial steps that the role of college students is also important, especially students majoring in mathematics. Students can do simple research on the development of mathematics, or by raising issues that arise in the life of mathematics.

              This is one example of someone who has done research in the field of mathematics that is Wittgenstein. Wittgenstein is a unique philosopher. He criticizes his own opinion. Philosophers divide his views in two periods that are early and later. The early of Wittgentein’s view set by Tractatus Logico Philosophicus and the later set by Philosophical Investigations. Wittgenstein's conceptions of mathematics fall in three periods, that are early, midle and later. The early period Wittgenstein's conception of mathematics set by Tractatus Logico Philosophicus, the midle period set by Philosophical Grammar dan Philosophical Remarks, and the later period set by Remarks on the Foundations of Mathematics. The Wittgenstein’s view on mathematics is not belonging to logicism, formalism, or intuitionism. The later Wittgenstein on mathematics is that “Mathematics as a human invention”. He maintains his rejection for infinitely in mathematics.

              There are many other figures in mathematics research. In its development efforts as well as through a variety of ways. We may follow them, but with a different object and discover something new. With the existence of mathematical research is expected to solve problems in everyday life. In addition to the development of mathematics and produce researchers, especially those from Indonesia.
              Math is easy, math is beautiful, and mathematics is the breath of life.

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")