kajian tentang quadratic expression and equation in math kbsm

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ReSeaRCH oF Quadratic Expression and Equation in Mathematics KBSM Syllabus

| MUHAMAD ILI NURHAYAT BIN HASSAN 1


ASSIGNMENTS1. Choose one topic from the secondary school mathematics textbook.

2. Carry out the following activities:

a) Discuss the basic structure of the topic.

b) Suggest activities that used to be added to the topic (chapter) so that it

satisfy (10 – 20 pages) the constructivist approach to teaching and

learning.

c) Suggest activities that need to be added to the topic (chapter) so that is

satisfy the integrated approach to teaching and learning the topic.

d) Suggest ways of improving the usage of teaching in the topic.

3. Write your reflection concerning the activities you have done.



INTRODUCTION

First and foremost I would like to express my deepest appreciation to my

SXEP1101 Lecturer, Prof. Nik Aziz Nik Pa for giving us such a wonderful assignment.

Besides, I also would like to thank my course mates, my friends and my family for

helping me and supporting me in doing and completing the assignment.

This assignment actually to me is quite heavy at first sight since we were told to

submit it in 3kg. To me it is quite big task and taking lot of my precious time and since

I’m also not quite understanding about what the needs of this task. However after

referring to the lectures, lecture notes, friends and internet; I finally understand what is

the proposed of this assignment.

After doing some researched, I make a conclusion to the second topic in form of

of our mathematic syllabus which is: Quadratic Expression and Equation since this topic

is one of the important topics that had been learn and taught from the secondary school

and still continued until now. Finding that this topic is one of the important of Algebra

and the Algebra had been included into our mathematic education syllabus for

secondary school since form one until form five states that this Algebra topic is

something big that has to be known for us as a student.

Apart for that, I hope my research here can provide some information that could

help us an educator to apply it in our teaching and learning process (P&P) to make our

education process happened more effective thus uttering our future generation to

become more reliable to our nation and country as proposed in Falsafah Pendidikan

Kebangsaan(FPK).



THE HISTORY BEHIND QUADRATIC EXPRESSION AND EQUATION

This is the quadratic expression, as it has been taught to most of us since in school:

ax2 + bx + c = 0

which gives the formula solution to a generic quadratic equation in the form:

x1,2=(-b/2a) ± (1/2a)(b2-4ac)1/2

The development, or derivation, of a mathematical idea is usually as logical, deducible

and rectilinear as possible. This brings about the common notion that its historical

development is similarly as continuous, logical and rectilinear: one mathematician

picking up an idea where another mathematician left it.

Using the quadratic expression as an example, it will be shown that the historical

development of mathematics is not at all rectilinear. Instead, parallel developments,

interconnections and confluences can be found, which - to complicate this stuff even

further - are also interrelated with social, cultural, political and religious matters.

The quadratic expression has been derived in the course of a few millennia to its current

form, which had been taught in all school in Malaysia and all around the world. So, let

flash back a little story about this quadratic expression and equation’s history.



The Original Problem 2000(or so)BC

Egyptian, Chinese and Babylonian engineers were really smart people - they knew how

the area of a square scales with the length of its side. They knew that it's possible to

store nine times more bales of hay if the side of the square loft is tripled. They also

found out how to calculate the area of more complex designs like rectangles and T-

shapes and so on. However, they didn't know how to calculate the sides of the shapes -

the length of the sides, starting from a given area - which was often what their clients

really needed. And so, this is the original problem: a certain shape1 must be scaled with

a total area, and in the end what's needed is lengths of the sides, or walls to make a

working floor plan.

1500BC The Beginnings - Egypt

The first aspect that finally led to the quadratic equation was the recognition that it is

connected to a very pragmatic problem, which in its turn demanded a 'quick and dirty'

solution. We have to note, in this context, that Egyptian mathematics did not know

equations and numbers like we do nowadays; it is instead descriptive, rhetorical and

sometimes very hard to follow. It is known that the Egyptian Wiseman (engineers,

scribes and priests) were aware of this shortcoming - but they came up with a way to

circumvent this problem: instead of learning an operation, or a formula that could

calculate the sides from the area, they calculated the area for all possible sides and

shapes of squares and rectangles and made a look-up table. This method works much

like we learn the multiplication tables by heart in school instead of doing the operation

proper.

1 For example: the floor-plan of a T-shaped temple with a square patio on a L-shaped lot



So, if someone wanted a loft with a certain shape and a certain capacity to store bales

of papyrus, the engineer would go to his table and find the most fitting design. The

engineers did not have time to calculate all shapes and sides to make their own table.

Instead, the table they used was a reproduction of a master look-up table. The copyists

did not know if the stuff they were copying made sense or not as they didn't know

anything about maths. So, obviously, sometimes errors crept in, and copies of the

copies were known to be less trustworthy2. These tables still exist, and it is possible to

see where errors crept in during the copying of the documents.

400 BCE The Next Step - Babylon and China

The Egyptian method worked fine, but a more general solution - without the need for

tables - seemed desirable. That's where the Babylonian geeks come into play.

Babylonian maths had a big advantage over the one used in Egypt, namely they used a

number-system that is pretty much like the one we use today, albeit on a hexagesimal

basis, or base-60. Addition and multiplication were a lot easier to perform with this

system, so the engineers around 1000 BC could always double-check the values in

their tables. By 400 BC they found a more general method called 'completing the

square' to solve generic problems involving areas. There are no indications that these

people used a specific mathematical procedure to find out the solutions, so probably

some educated guessing was involved. Around the same time, or a bit later, this

method also appears in Chinese documents. The Chinese, like the Egyptians, also did

not use a numeric system, but a double checking of simple mathematical operations

was made astonishingly easy by the widespread use of the abacus

2 Imagine a multiplication table with a typo (8 x 7 = 57), and you learn that by heart!



300BC Geometry - Hellenistic Mediterranean Area

The first attempts to find a more general formula to solve quadratic equations can be

tracked back to geometry (and trigonometry) top-bananas Pythagoras (500 BC in

Croton, Italy) and Euclid (300 BC in Alexandria, Egypt), who used a strictly geometric

approach, and found a general procedure to solve the quadratic equation. Pythagoras

noted that the ratios between the area of a square and the respective length of the side

- the square root - were not always integer, but he refused to allow for proportions other

than rational. Euclid went even further and found out that this proportion might also not

be rational. He concluded that irrational numbers exist.

Euclid's opus Elements covered more or less all the mathematics needed for technical

applications from a theoretical point of view. However, it didn't use the same notation

with formulas and numbers like we use nowadays. For that reason it was not possible

to calculate the square root of any number by hand, in order to obtain a good

approximation for the exact value of the root, which is what the architects and engineers

were after. Because all (theoretically relevant at least) maths seemed to be

complete3 but otherwise useless, the many wars occurring in Europe, and also the early

Middle Ages turned the mathematical world in Europe silent until the 13th Century. In

this period mathematics also suffered a big shift, going from a pragmatic science to a

more mystical, philosophical discipline.

3 Euclidean geometry, for example, was only expanded recently in the late 19th Century!



700AD All Numbers – India

Hindu mathematics has used the decimal system (the one we use) at least since

600AD. One of the most important influences on Hindu mathematics was that it was

widely used in commerce. The average Hindu merchant was pretty fast in simple maths.

If someone had a debt the numbers would be negative, if someone had a credit the

numbers would be positive. Also, if someone had neither credit, nor debt, the numbers

would add up to zero. Zero is an important number in the history of mathematics, and its

relatively late appearance is due to the fact that many cultures had difficulty of

conceiving 'nothing'. The concept of 'nothing', like in 'shunya', the void, or the concept of

'equilibrium', was already anchored in Hindu culture.

Around 700AD the general solution for the quadratic equation, this time using numbers,

was devised by a Hindu mathematician called Brahmagupta, who, among other things,

used irrational numbers; he also recognised two roots in the solution. The final,

complete solution as we know it today came around 1100AD, by another Hindu

mathematician called Baskhara4. Baskhara was the first to recognise that any positive

number has two square roots.

4 In fact the quadratic formula is known in some countries, like Brazil, by the name of 'Baskhara's Formula'.



820AD Powerful Islamic Science - Persia

Around 820AD, near Baghdad, Mohammad bin Musa Al-Khwarismi, a famous Islamic

mathematician5 who knew Hindu mathematics, also derived the quadratic equation. The

algebra used by him was entirely rhetorical, and he rejected negative solutions. This

particular derivation of the quadratic formula was brought to Europe by Jewish

mathematician/astronomer Abraham bar Hiyya (whose Latinised name is Savasorda)

who lived in Barcelona around 1100.

1500AD Renaissance - Europe

With the Renaissance in Europe, academic attention came back to original

mathematical problems. By 1545 Girolamo Cardano, who was a typical Renaissance

scientist (ie, interested in alchemy, occultism and suchlike), and one of the best

algebraists of his time, compiled the works related to the quadratic equations - that is,

he blended Al-Khwarismi's solution with the Euclidean geometry. He was possibly not

the first or only one, but the most famous. In his (mainly rhetorical) works he allows for

the existence of complex, or imaginary numbers - that is, roots of negative numbers. At

the end of the 16th Century the mathematical notation and symbolism was introduced

by amateur-mathematician François Viète, in France. In 1637, when René Descartes

published La Géométrie, modern Mathematics was born, and the quadratic formula has

adopted the form we know today.

5 His name lives on in the English word 'algorithm' ('Khwa' mutated to 'Go' and the 's' mutated to 'th'.).



RAW MATERIAL OF QUADRATIC EXPRESSION AND EQUATION

Quadratic Expression and Equation is one element in algebra and the algebra topic

has been taught and introduced in the early level of primary school especially in Form 1, 2

and 3 but as a basic information and foundation for secondary students to go further after

this. In Form 1, this topic was introduced in Chapter 7 which is “Algebraic Expression” and

this topic continues in Form 2 which is “Algebraic Expression II”. Other than that, there are

other topics that are expands form of algebra which quite related to Quadratic Expression

and Equation and has been taught in Form 2 which is a continuity about this kind of topic

which is “Linear Equation”. Meanwhile in form 3, this topic is touched more deeply in several

topics which are “Algebraic Expression III”, “Algebraic Formulae”, “Linear Equation II”, and

also “Linear Inequalities”.

Since this topic also part and an expand from Algebra, which also quite related to

each other, let we go through about their previous topic that had been taught in Form 1, 2

and 3 first before we elaborate more about this Quadratic Expression and Equation topic as

an introduction to this topic.


Figure 1


ALGEBRAIC EXPRESSION

From the Algebraic Expression I in Form 1, we were introduced about unknown, what is

unknown and its concept (see figure 1). After that we were introduced it in term, its principle

and concept. For example, we were informed that an algebraic term is written as 3x not

x3; a number, example 8 is also a term; x2

is

a term and in 7p: The coefficient of p is 7.

Then, the algebraic expression’s concept

was introduced such as “4s + 8s = 12s”; and

“3k + 4 + 6k –3 = 9 k + 1 ” . Then, in

Algebraic Expression ll in Form 2, we were

introduced about the computation involving

addition, substraction, multiplication and division of

more than 2 terms.

Other than that, the students also were teach to perform computations involving algebraic

expressions.

e.g:



Then, in Form 3, the Algebraic Expression III is stressing about the concept of

brackets; expanding the brackets “e.g: (a± b)(a± b) = ( a± b)²”; factorization of algebraic

expressions to solve problems “e.g: ab– ac = a(b– c)” and also to perform computation

on algebraic fraction.

ALGEBRAIC FORMULAE

Meanwhile the “Algebraic Formula” is the continuing of algebraic expression which had

been teaches in Form 1, 2 and 3. This topic more stress on the basic understanding

about the concept of variables and constant which include determination of variables or

constant from a given situation and vice versa. It also stress on understanding the

concept of formulae to solve the problems based on a given statement and situation.

Here, the specified variables is express as the subject of formula involving basic

operations of +,-,x,÷; powers or roots; and also the combination of basic operations and

power or roots.

A. Normal Algebraic formulae

a2 - b2 = ( a + b )( a - b)

( a + b )2 = a2 + 2ab + b2

(a - b)2 = a2 - 2ab + b2



B. Change the subject



C. Finding the variables.

Example:

Given thatW = 3a –12

b, find (a) W when a = 2 and b = 4,

(b) b when a = 3 andW = 5.

Solution:

W= 3 a–12

b

(a)Substitute a = 2 and b = 4 into the formula W = 3(2) –12¿4) = 6 – 2 = 4

(b) 5 = 3(3) –12

b

12b= 9 – 5 = 4 × 2 = 8

D. Solving problems

Example:

The entrance fees to a museum are RM4 for an adult and RM3 for a child. If, on a

certain day, p adults and q children visited the museum, write a formula for the total

collection of that day.

Solution:

Let A = Total collection (in RM)

A = (Number of adults × 4) +(Number of children × 3)



A = 4p + 3q

LINEAR EQUATION

Meanwhile linear equation is a further study of algebraic which involves unknown and

equation. Here, the concept of equality was introduced which are ‘=’ and ‘≠’. And the

discussion included:

1. Solve linear equation = Finding the value of the unknown which satisfies the

equation.

2. The solution of the equation is also known as the root of the equation.

3. A linear equation in one unknown has only one root.

4. To determine whether a given value is a solution of an equation, substitute the

value into the equation. If the sum of the left hand side (LHS) = sum of right hand

side (RHS), then the given value is a solution.

There are 4 different forms of linear equation as follow:



Example

a) If a =b then b =a.

e.g: 2+3 = 4+1 then 4+1 = 2+3

b) If a =b and b =c, then a =c.

e.g: 4+5 = 2+7, then 2+7 = 3+6, then 4+5= 3+6

It’s also stress on the use and the concept of linear equations in one unknown, discuss

why given algebraic terms and expressions are linear, identifying the linear terms, and

also recognizing the linear algebraic terms and linear algebraic expression.

e.g: 3x,xy,x² are the linear term.

x + 3 = 5, x - 2y = 7 are linear equations.

x + 3 = 5is linear equation in one unknown.

2x + 3, x - 2y, xy + 2, x² - 1, 2x + 3, x - 2y are linear expressions.

Other than that is about solving Linear Equations in One Unknown involving combined

operations of +, -, x,÷

Steps:

1. Work on the bracket first, if there is any.



2. Group the terms with the unknown on the left hand side of the equation while the

numbers on the right side.

3. Solve the equation using combined operations.

4. Check your solution by substituting the value into the original equation.

Examples:

Given that 2y + 11 = -5, calculate the value of y.

Solutions:

7 - ( x + 1) = - 4x

7 – x – 1 = - 4x

-x + 4x = -7 + 1

3x = - 6

x = -2

This topic continue in Form 3 where students are also exposed to solve a situation by

forming a linear equation from a given statements or problem and vice versa.

LINEAR INEQUALITIES

Linear inequalities is part of Algebra which almost same as linear equality except that it

emphasis on the use of everyday situation that had been illustrate in the symbol of “>”

which read as “greater than”, “<” read as “less than”, “≥” read as “greater than or equal

to”, and “≤” read as “less than or equal to”.

The process of learning included understand the use and the concept of inequalities

which is identify the relationship of “>”, “<”, “≥” and “≤” between two given numbers and

based on situation, understand and use the concept of linear inequalities in one unknown.

For example x > 0 and x < 2. So, the value of x is 1.

Example:



Linear inequality on number line

Simultaneous inequalities on number line



Other than that, it’s also stress on how to perform the computations, understanding the

concept of simultaneous linear inequalities involving addition, subtraction, multiplication

and division on linear inequalities.

Example: Solve the inequality

(x + 2) / 3 - 2 / 5 < (-x - 1) / 3 - 1 / 6

Solution:

Given

(x + 2) / 3 - 2 / 5 < (-x - 1) / 3 - 1 / 6



Multiply all terms by 30, the LCD

30*(x + 2) /3 - 30*2 / 5 < 30(-x - 1) / 3 - 30*1 / 6

simplify

10(x + 2) - 6*2 < 10(-x - 1) - 5

Multiply factors and group like terms

10x + 20 - 12 < -10x - 10 - 5

10x + 8 < -10x -15

Subtract 8 to both sides and simplify

10x < -10x - 23

Add 10x to both sides and simplify

20x < -23

Divide both sides by 20

x < -23 / 20

Conclusion

The solution set consists of all real numbers in the interval

(- infinity, -23/20).

QUADRATIC EXPRESSION AND EQUATION

A quadratic expression is a mathematical expression that involves at least one

variable raised to the power of two and no variables raised to any higher powers -

quadratic expressions are second degree polynomials. When only one variable has

been raised to the second power, the graphical representation of the quadratic equation

is always a parabola1. Despite the single graphical representation, there are three


http://www.bbc.co.uk/dna/h2g2/A734951

http://www.bbc.co.uk/dna/h2g2/A144280


different common ways to represent parabolas symbolically in mathematical notation.

The form:

y=a(x-h)2+k

Quadratic Expression and Expression is generally preferred above the others, as

the point (h,k) is the vertex of the graph and the value 'a' tells how the graph changes

with respect to the x- and y-axes (that is, how wide the parabola gets, how quickly).

There are two other parabolic forms, however. The form:

y=ax2+bx+c

which is called the 'general form' of a parabola, and from an equation with this form, all

the other forms are usually derived. The third form is known as 'factorised form', and is

symbolically represented by:

y=(ax+b)(cx+d)

This topic has been introduced in chapter 2 in form 4 and is the chosen topic for my

task. Basically all of the main elements in this topic is almost same and related to the

previous topics of Algebra. In this topic, there are a few things that had been introduced:

1. Understanding the concept of Quadratic Expression



a. Realizing what is quadratic and reforming it.

Example:

ax²+bx+c

Above expression is quadratic expression since:

- the higher power is 2

- It has only one unknown

b. Forming quadratic expression from situation.

Example6:

A school hall measuring (3 + x) m long and (4 + x ) m wide is going to be carpeted.

Calculate the area of carpet needed.

Solution:

Area = length × width = (3 + x) × (4 + x) = (12 + 7x + x²) m²

x² + 7x + 12 is a quadratic expression.

2. Factorization of Quadratic Expression.

i. Factorize quadratic expression of the form ax² + c and ax² + bx

Example: 3x²+15 = 3(x² + 5)

ii. Factorize quadratic expressions of the form px² - q, where p and q are perfect

squares.

6 Example from text book form 4 for in the topic 2.1 quadratic expression and equations example 4



Example: px² - q = a²x² - b² = (ax + b) (ax – b)

4p² - 25 = (2p + 5) (2p – 5)

iii. Factorize quadratic expressions of the forms ax² + bx + c where a = 1, and b and

c are not equal to zero.

iv. Example: 3x² + 7x – 6 = (3x – 2) ( x + 3)

3. Understanding the concept of Quadratic equation.

a. Recognizing quadratic equation and forming its general equation.

e.g: ax²+bx+c=0

b. Forming quadratic equation from situation.

Example7

A car is travelling at a constant speed of (2x + 3) km/h in (x+1) hours. If it has travelled a

distance 0f 45 km, form a quadratic speed equation in terms of x.

Solution: Distance = speed × time = (2x+3) (x+1) = 2x² + 5x + 3

The expression for the distance is 2x² + 5x + 3

The quadratic equation is 2x² + 5x + 3 = 45

2x² + 5x – 42 = 0

4. Understanding the concept of roots to solve problems.

Example8:

Encik Ramli’s age is 8 times his son’s age. Three years ago, the product of their age was 74

years. How old is Encik Ramli and his son now?

7 Example 12 topic 2.3 from text book form 4 8 Example 16 chapter 2.4 Quadratic Expression and Equation in text book Form 4



Solution:

i. Read and understand the problems.

Encik Ramli’s age = 8 times his son’s age.

3 years ago: Encik Ramli’s age × His son’s age = 74

ii. Device a plan

Assuming his son’s age is x and Encik Ramli’s age is 8x,

3 Years ago: His son’s age = x-3

Encik Ramli’s age = 8x – 3

(x – 3)(8x – 3) = 74

iii. Carry out the plan

Factorize the equation to get the value of x.

(x – 3) ( 8x – 3) = 74

8x² - 3x -24x + 9 = 74

8x² - 27x – 65 = 0

(8x – 5) ( 8x + 13) = 0

8 = 5 or - 138

Encik Ramli’s son’s age = 5 years old

Encik Ramli’s age = 8× 5= 40 years old

iv. Check the solution

(5 – 3)((8)(5) – 3 ) = (2)(37) = 74

BASIC OPERATION OF QUADRATIC EXPRESSION AND EQUATION

Quadratic Expression and Equation are any algebra unit which has only one

variables and their maximum power of variables is 2 which exist in its general m is



ax2+bx+c for quadratic expression and for quadratic equation where a,

b, c are coefficients and a ≠ 0.

Note that if a=0, then the equation would simply be a linear equation, not quadratic.

Examples

x² + 2x = 4 is a quadratic since it may be rewritten in the form ax² + bx + c = 0 by

applying the Addition Property of Equality and subtracting 4 from both sides of =.

(2 + x)(3 – x) = 0 is a quadratic since it may be rewritten in the form ax2 + bx + c = 0 by

applying the Distributive Property to multiply out all terms and then combining like

terms.

x² - 3 = 0 is a quadratic since it has the form ax² + bx + c = 0 with b=0 in this case.

3x² – 2/x + 4 = 0 is not a quadratic since it has the term 2/x. The term 2/x is the same as

2x-1, and quadratics do not have x raised to any power other than 1 or 2.

Just remember:

Quadratics always have an x² term, possibly an x-term, and possibly a constant term! If

your equation has an x² term or will have an x² term. After multiplying out, it may be a

quadratic, provided the other terms fit the form.

There are number of operations that can be applied to modify quadratic expression and

expression which is addition, subtraction, multiplication, and division.



ADDITION AND SUBTRACTION OF QUADRATIC EXPRESSION AND EQUATION9

Recall that algebraic expression that is a number, a variable, or a product or

quotient of numbers and variables is called terms. Examples of terms are:

7 a -2b -47

y² 0.7ab -5/w

Two or more terms that contain the same variable or variables with

corresponding variables having the same exponents are called like terms or similar

terms. For example, the following pairs are like terms.

6k and k 5x² and -7x² 9ab and 0.4ab92

x²y and -113

x²y

Two terms are unlike terms when they contain different variables, or the same

variables with different exponents. For example, the following pairs are unlike terms.

3x and 4y 5x² and 5x 9ab and 0.4a83

x²y and 47

xy

And for Quadratic expression and equation, they only included one term of

variable with the maximum power of two only. To add or subtract these terms, we use

the distributive property of multiplication over addition and subtraction.

9x - 2x = (9 + 2)x = 11x

18y² - 5y² - (18 – 5)y² = 13y²

Since the distributive properties is true for any numbers of terms, we can express

the sum or difference of any number of like terms as a single terms.

Recall that when like terms are added:

9 Taken from http://www.babylon.k12.ny.us/PDF/integrated_algebra/Chapter05.pdf



1. The sum or difference has the same variable or variables as the original terms.

2. The numerical coefficient of the sum or difference is the sum or difference of the

numerical coefficients of the terms that were added.

The sum of unlike terms cannot be expressed as a single term. For example, the sum of

2x and 3 cannot be written as a single term but is written 2x±3y.

EXAMPLE

Express the difference (4x² + 2x- 3) - (2x² - 5x - 3) in simplest form.

Solution

How to Proceed

i. Write the subtraction problem: (4x² + 2x - 3) - (2x² - 5x – 3)

ii. To subtract, add the opposite (4x² + 2x - 3) + (2x² + 5x + 3)

of the polynomial to be subtracted:

iii. Use the commutative and (4x² - 2x²) + (2x + 5x) + (-3 + 3)

associative properties to group like

terms:

iv. Add like terms: 2x² + 7x + 0

2x² + 7x

Answer: 2x² + 7x



MULTIPLICATION OF QUADRATIC EXPRESSION AND EQUATION

Multiplying a Monomial by a Monomial

We know that the commutative property of multiplication makes it possible to

arrange the factors of a product in any order and that the associative property of

multiplication makes it possible to group the factors in any combination. For example:

(3x)(7x) = (3)(7)(x)(x) = (3 . 7)(x . x) = 2x²

(-2x)(+5x) = (2)(x)(+5)(x ) = [(2)(+5)] [(x)(x)] = 10x²

In the preceding examples, the factors may be rearranged and grouped mentally.

Procedure

To multiply a monomial by a monomial:

i. Use the commutative and associative properties to rearrange and group the

factors.This may be done mentally.

ii. Multiply the numerical coefficients.

iii. Multiply powers with the same base by adding exponents.

iv. Multiply the products obtained in Steps 2 and 3 and any other variable factors

by writing them with no sign between them.



EXAMPLE

Represent the area of a rectangle whose length is 3x and whose width is 2x.

Solution

How to Proceed

(1) Write the area formula: A = lw

(2) Substitute the values of l and w: = (3x)(2x)

(3) Perform the multiplication: = (3.2)(x. x)

= 6x²

Multiplying a Polynomial by a Monomial

The distributive property of multiplication over addition is used to multiply a

polynomial by a monomial. Therefore,

a(b + c) = ab + ac

x(4x + 3) = x(4x) + x(3)

= 4x² + 3x

This result can be illustrated geometrically. Let us separate a rectangle, whose length is

4x + 3 and whose width is x, into two smaller rectangles such that the length of one

rectangle is 4x and the length of the other is 3.

Since the area of the largest rectangle is equal to the sum of the areas of the two smaller

rectangles:

x(4x + 3) = x(4x) + x(3) = 4x² + 3x



Procedure

To multiply a polynomial by monomial, use the distributive property:

Multiply each term of the polynomial by the monomial and write the result as the sum of

these products.

Multiplication and Grouping Symbols

When an algebraic expression involves grouping symbols such as parentheses,

we follow the general order of operations and perform operations with algebraic terms.

In the example at the right, first simplify 8y - 2(7y - 4y) + 5

the expression within parentheses: 8y - 2(3y) + 5

Next, multiply: 8y - 6y + 5

Finally, combine like terms by addition 2y + 5

or subtraction:

In many expressions, however, the terms within parentheses cannot be combined

because they are unlike terms. When this happens, we use the distributive property to

clear parentheses and then follow the order of multiplying before adding.

Here, clear the parentheses by using the 3 + 7(2x + 3)

distributive property: 3 + 7(2x) + 7(3)

Next, multiply: 3 + 14x + 21

Finally, combine like terms by addition: 24 + 14x



The multiplicative identity property states that a = 1.a. By using this property, we

can say that 5 + (2x - 3) = 5 + 1(2x - 3) and then follow the procedures shown above.

5 + (2x - 3) = 5 + 1(2x - 3) = 5 + 1(2x) - 1(3) = 5 + 2x - 3 = 2 + 2x

Also, since -a = -1.a, we can use this property to simplify expressions in which a

parentheses is preceded by a negative sign:

6y - (9 - 7y) = 6y - 1(9 - 7y) = 6y - 1(9) - 1(7y) = 6y - 9 - 7y = 13y - 9

Multiplying polynomial

As discussed in previous, to find the product (x + 4)(a), we use the distributive

property of multiplication over addition:

(x + 4)(a) = x(a) + 4(a)

Now, let us use this property to find the product of two binomials, for example,

(x + 4)(x + 3).

(x + 4)(a) = x(a) + 4(a)

(x + 4)(x + 3) = x(x + 3) + 4(x + 3)

= x² + 3x + 4x + 12

= x² + 7x + 12



This result can also be illustrated geometrically

In general, for all a, b, c, and d :

(a + b) (c + d) = a(c + d) + b(c + d)

= ac + ad + bc + bd

Notice that each term of the first polynomial multiplies each term of the second.

At the right is a convenient vertical arrangement of proceeding multiplication,

similar to the arrangement used in arithmetic

multiplication. Note that multiplication is done

from left to right.

The world FOIL serves as a convenient way

to remember the steps necessary to multiply two binomials.



EXAMPLE

Dividing by a Monomial

We know that

We can rewrite this equality interchanging the left and right members.

Using this relationship, we can write:



Procedure

To divide a monomial by a monomial:

i. Divide the numerical coefficients.

ii. When variable factors are powers of the same base, divide by subtracting

exponents.

iii. Multiply the quotients from steps 1 and 2.

If the area of a rectangle is 42 and its length is 6, we can find its width by dividing the

area, 42, by the length, 6. Thus, 42 ÷ 6 = 7, which is the width.

Similarly, if the area of a rectangle is represented by 42x and its length by 6x, we can

find its width by dividing the area, 42x, by the length, 6x:

42x² ÷ 6x = 7x

Therefore, the width can be represented by 7x.

Dividing by a Binomial

When we divide 736 by 32, we use repeated subtraction of multiples of 32 to



determine how many times 32 is contained in 736. To divide a polynomial by a

binomial, we will use a similar procedure to divide x²+ 6x + 8 by x + 2.

How to Proceed

i. Write the usual division form:

ii. Divide the first term of the dividend by the

first term of the divisor to obtain the first

term of the quotient:

iii. Multiply the whole divisor by the first term of

the quotient. Write each term of the product

under the like term of the dividend:

iv. Subtract and bring down the next term of

the dividend to obtain a new dividend:

v. Divide the first term of the new dividend by

the first term of the divisor to obtain the next

term of the quotient:

vi. Repeat steps (3) and (4), multiplying the

whole divisor by the new term of the quotient. Subtract this product from

the new dividend. Here the remainder is zero and the division is complete:



The division can be checked by multiplying the quotient by the divisor to obtain the

dividend:

(x + 4)(x + 2) = x(x + 2) + 4(x + 2)

= x² + 2x + 4x + 8

= x² + 6x + 8

EXAMPLE

Divide 5s + 6s² - 6 by 2s + 3 and check.



FACTORING THE QUADRATIC EXPRESSION AND EQUATION

Other than that, quadratic expression and equation with real coefficients can

have none, one or two distinct real roots. To find them, we have a few methods to solve

which are:

Factoring

The easiest way to solve a quadratic equation is to solve by factoring, if possible.

Here are the steps to solve a quadratic by factoring:

1. Put the equation in the standard form (ax2+bx+c=0)

2. Factor the equation (find two numbers that will not only multiply to equal the

constant term "c", but also add up to equal "b", the coefficient on the x-term).

3. Set each of the two binomial expressions equal to cero.

4. Solve each of the equations.

5. Check you answer.



Solve by factoring x2+2x = 15

1. Put the equation in the standard form: x2+2x-15=0

2. We require to numbers that multiply together to give -15 and add together to give -2.

x2+2x-15=(x-3)(x+5).

3. Set each of the two binomial expressions equal to cero.

x2+2x-15 = 0

(x-3)(x+5) =0

4. Solve each equation:

x-3=0, x=3

x+5=0, x=-5

When the leading coefficient (the number on the x2 term) is not 1, the first step in

factoring will be to multiply "a" and "c"; then we'll need to find factors of the product "ac"

that add up to "b".

Solve by factoring 2x2+4x-6 = 0

We need to find factors of 12 (ac=2·(-6)=-12) that add up to +4.

We will use the pair "-2 and 6".

Draw a two-by-two grid, putting the first term in the upper left-hand corner and the last

term in the lower right-hand corner:

2x2

-6

Take the factors –2 and 6 and put them, complete with their signs and variables, in the

diagonal corners:

2x2 -2x

6x -6



Factor the rows and columns:

2x -2

2x 2x2 -2x

6 6x -6

Then, 2x2+4x-6 = (2x-2)(2x+6)

We will find the solutions of the equations by solving each equation:

2x-2=0, x=1

2x+6=0, x=-3

Sometimes you cannot find integer factors that work, then this quadratic is said to be

"unfactorable over the integers" or "prime". On these cases, you must try to solve the

equation using another method.

Extracting Square Roots

Extracting square roots is a very easy way to solve quadratics, provided the equation is

in the correct form.

Basically, Extracting Square Roots allows you to rewrite x² = k as x = ±√k, where k is

some real number. Algebraically, we are taking square roots of both sides of the

equation as shown below and inserting the ± to account for both a positive and negative

case. Note that the squared quantity must be isolated on one side of = before you can

extract the square roots.

Example: Solve x² = 9 by extracting square roots



Example: Solve (2x – 5) ² + 5 = 3

(2x – 5) ² + 5 = 3 Given

(2x – 5) ² = -2 Addition Property of Equality used to add –5 to both

sides

√ (2x – 5) ² = ±√(-2) Extract Square Roots

2x – 5 = ± i√2 Simplify Radicals and Apply Definition of “i”

2x = 5 ± i√2 Addition Property of Equality

x = (5 ± i√2) / 2 Division Property of Equality

Completing The Square

This method of solving quadratic equations is straightforward, but requires a specific

sequence of steps. Here is the procedure:

Example: Solve 3x² + 4x – 7 = 0 by Completing The Square

1. Isolate the x² and x-terms on one side of = by applying the Addition Property of

Equality.

3x² + 4x = 7

2. Apply the Division Property of Equality to divide all terms on both sides by the

coefficient on x².

(3x2)/3 + (4x)/3 = 7/3

x² + (4/3)x = 7/3 Note: Steps 1 and 2 may be done in either

order.

3. Take ½ of the coefficient on x. Square this product. Add this square to both sides

using the Addition Property of Equality. In this case, we take ½ of 4/3 which is



(1/2)•(4/3) = 4/6. Square 4/6 to get (4/6) •(4/6) = 16/36 = 4/9 when reduced. Add

4/9 to both sides to get

x² + (4/3)x + 4/9 = 7/3 + 4/9

x² + (4/3)x + 4/9 = 21/9 + 4/9 multiply 7/3 by 3/3 to get common denominator

x² + (4/3)x + 4/9 = 25/9 add fractions

4. Factor the left side.

Note: It will always factor as (x ± the square root of what you added) ²

(x + 2/3) ² = 25/9

5. Solve by extracting square roots.

√ (x + 2/3) ² = ±√(25/9) Extract Square Roots

x + 2/3 = ±5/3 Simplify Radicals

x = -2/3 ± 5/3 Addition Property of Equality

This results in two answers: x = -2/3 + 5/3 = 3/3 = 1 and x = -2/3 – 5/3 = -7/3

You may have noticed that we solved this same problem earlier in a much easier

fashion by factoring! So why learn this method of extracting square roots? Answer: This

method is used in higher levels of math (like calculus) to perform similar or identical

equation rearrangements. Also, we need this method to justify and derive the Quadratic

Formula.



Using The Quadratic Formula

Solving a quadratic equation that is in the form ax² + bx + c = 0 only involves plugging

a, b, and c into the formula.

Example: Solve (x + 3)2 = x – 2

(x + 3)2 = x – 2 Given

(x + 3)(x + 3) = x – 2 Rewrite

x² + 6x + 9 = x – 2 Multiply out with Distributive Property, Combine Like

Terms



x² + 5x + 11 = 0 Addition Property of Equality - add 2, add –x to both

sides

Plug a=1, b=5, c =11 from 1x2 + 5x + 11 = 0 into the Quadratic Formula to get

which simplifies to

after we simplify the radical and rewrite √(-19) as (√19) • i by applying the

definition of i.

CONSTRUCTIVIST APPROACHES IN TEACHING AND LEARNING QUADRATIC

EXPRESSION AND EQUATION

INTRODUCTION

Constructivism is one of the modern approaches in teaching and learning which

applied both to learning theory and to epistemology which both of these stress on how

people learn, and to the nature of knowledge10. The main principle of constructivist

learning is that people construct their own understanding of the world, and in turn their

own knowledge.

10 Constructivism asserts two main principles whose applications have far-reaching consequences for the study of cognitive development and learning as well as for the practice of teaching, psychotherapy, and interpersonal management in general. The two principles are (1) knowledge is mot passively received but actively built up by the experiential world, not the discovery of ontological reality." International Encyclopedia of Education. "Constructivism In Education," 1987.



We can see in our country, Malaysia, a typical teaching and learning process in

mathematics involves presenting the concepts followed by reinforcement and

enrichment exercises. Here rote learning style discourages the mind to think

inquisitively and eventually mathematics will be perceived as a boring subject.

To minimize this, the teaching and learning process must involve engaging

activities which include active participation from the students and relate to real-life

situations which then will construct their thinking skills in mathematics subject especially

Quadratic Expression and Equation

All in all, the aim of constructivist learning is that the teacher-defined

goal to achieve an output, where the task practice environment is designed

to reveal the knowledge and concepts to be learned; feedback is given by

the environment in relation to the extent to which learners’ actions achieved

the intended goal. According to Audrey Gray, the characteristics of a constructivist

classroom are as follows:

the learners are actively involved

the environment is democratic

the activities are interactive and student-centered

the teacher facilitates a process of learning in which students are

encouraged to be responsible and autonomous

ACTIVITIES TO BE ADDED



From the model of Constructivist Learning Environments, there are about six

models of teaching and learning that can be bring forward to satisfy the teaching and

learning process which are:

1. Question/Case/Problem/Project

2. Related Cases

3. Information Resources

4. Cognitive (Knowledge Construction) Tools

5. Conversation and Collaboration Tools

6. Social/ Contextual Support.

1. Question/Case/Problem/Project

The focus is the question or issue, the case, the problem, or the project that

learners attempt to solve or resolve. It constitutes the learning goal. The quadratic

problem as examples or applications of the concepts and principles previously taught.



Students learn domain content in order to solve the problem, rather than solving the

problem to apply the learning.

Constructive Learning Environments at this part stress on question/issue-based,

case-based, project-based, or problem-based learning. Question- or issue-based

learning begins with a question with uncertain or controversial. For example, what is

quadratic? In case-based learning, students acquire knowledge and requisite thinking

skills by studying cases and preparing case summaries or diagnoses. For example,

quadratic is a part of algebra and had been learn at form 1, 2 and 3 which can be a

guide to know the quadratic, how about refer it back to know about it more?

Project-based learning focuses on relatively long-term, integrated units of

instruction where learners focus on complex projects consisting of multiple cases. For

example, what are quadratic and where it’s come from? They debate ideas, plan and

conduct experiments, and communicate their findings11. Meanwhile, problem-based

learning integrates courses at a curricular level, requiring learners to self-direct their

learning while solving numerous cases across a curriculum. For example, x²+2 is part of

quadratic. Why?

Basically, case-, project-, and problem-based learning represent a set of

complexity, but all share the same assumptions about active, constructive, and

authentic learning. By doing this way, it can support constructivist approaches in

learning where the students learn to question the reason of every single thing that

happened and then construct their own knowledge and answer by finding it.

It is important to provide interesting, relevant, and engaging problems to solve.

The problem should not be overly prescribed. Rather, it should be ill-defined or ill-

structured, so that some aspects of the problem are emergent and definable by the

learners. Ill-structured problem have unstated goals and constraints and have multiple

11 Proposed by Krajcik, Blumenfeld, Marx, & Soloway, 1994



solutions, solution paths, or no solutions at all. We need to decide and guided them if

the students possess prerequisite knowledge or capabilities for working on the problem

that we identify.

Other than that, the problems constructivist model generated three integrated

components which are;

1. The problem context

2. The problem representation or simulation, and

3. The problem manipulation space.

2. Related Cases

The students basically lack most in experiences. This lack is especially critical

when trying to solve problems. So, it is important to provide access to a set of related

experiences that novice students can refer to. The primary purpose of describing related

cases is to assist learners in understanding the issues implicit in the problem

representation. Besides, understanding any problem requires experiencing it and

constructing mental models of it. Related cases in constructivist learning environments

support learning in at least two ways: by scaffolding memory and by representing

complexity.

Usually, the lessons that we understand the best are those in which we have

been most involved and invested the greatest amount of effort. Related cases can

scaffold memory by providing representations of experiences that learners have not

had. They cannot replace learners’ involvement, but they can provide referents for

comparison. When humans first encounter a situation or problem, they naturally first

check their memory for similar cases that they may have solved previously12. If they can

12 Proposed by Polya, 1957



recall a similar case, they try to map the previous experiences and its lessons onto the

current problem. If the goals or conditions match, they apply their previous lesson. By

resenting related cases in learning environments, you are providing the learners with a

set of experiences to compare to the current problem or issue. Learners retrieve from

related cases advice on how to succeed, pitfalls that may cause failure, what worked or

didn’t work, and why it didn’t. They adapt the explanation to fit the current problem and

that way of learning is called constructivism learning.

Related cases also help to represent complexity in constructivist learning process

by providing multiple perspectives, themes, or interpretations to the problems or issues

being examined by the learners. Usually, this model stress on the conceptual

interrelatedness of ideas and their interconnectedness by providing multiple

interpretations of content and by using multiple, related cases to convey the multiple

perspectives on most problems. In order to enhance cognitive flexibility, it is important

that related cases provided a variety of viewpoints and perspectives on the case or

project being solved.

For example, while introducing the quadratic, how about related it to the previous

topic that the students had learned. Linear equation for example, is quite similar to

quadratic especially is its general form and their properties. How about co-relate this to

the quadratic to make the student understand more easily.

EXAMPLE

In general, the graph of a quadratic equation



y = ax2 + bx + c is a parabola.

If a > 0, then the parabola has a minimum point and it opens upwards (U-shaped) eg.

y = x2 + 2x − 3

If a < 0, then the parabola has a maximum point and it opens downwards (n-shaped)

eg.

y = -2x2 + 5x + 3



And all of this quadratic equation can be related to the previous topic which had been

learning in form to which is linear equation.

Intercept Form of a Straight Line: ax + by = c

In general, linear equation is often a straight line and is written in the form ax + by = c.

One way we can sketch this is by finding the x- and y-intercepts and then joining those

intercepts.

ax + by = c

by = -ax +c

y = -ab

x + cb

let ab

= m and cb

= C

hence, y = mx + c

Slope-Intercept Form of a Straight Line: y = mx + c

If the slope (also known as gradient) of a line is m, and the -intercept is c, then the

equation of the line is written: y = mx + c

Example:

The line y = 2x + 6 has slope m = 2 and y-intercept c = 6.



By showing the previous topic that quite

relate and almost same to the quadratic,

the students can easily understanding

their new knowledge by constructing the

idea from the previous knowledge.

3. Information Resources

In order to investigate problems, learners need information about the problem in

order to construct their mental models and formulate hypotheses that drive the

manipulation of the problem space. So, when using constructivist approaches in

learning, we should determine what kinds of information the learner will need in order to

understand the problem. Rich sources of information are an essential part of

constructivist learning and since nowadays, we were provide by information technology

to get information and reference, it should provide learner-selectable information just-in-

time with the guidance of us, teacher.

So, what we need to do to make sure that information makes sense not only in

the context of learning quadratic only, but only in solving problems and application. So,

determine what information learners need to interpret the problem. Maybe we naturally

used the text book and syllabus in our lesson. How about using other relevant

information banks and repositories that can be linked to the environment? These may

include text documents, graphics, sound resources, video, and animations that are

appropriate for helping learners comprehend the problem and its principles. And since

now, technology is one of the easiest paths to gain knowledge. The World Wide Web

(WWW) for example, is the default storage medium, as powerful new plug-ins enables

users to access multimedia resources from the net. However make sure that we guide

the students in using it since WWW resources may provide unlimited information that



unnecessary for our students. Actually, our textbook also included the information of

website that can be used as a reference and addition information resource.

EXAMPLE

Below are example of the website information of Quadratic Expression and Equation

that had been included in the Mathematics form 4 KBSM text book.



4. Cognitive (Knowledge Construction) Tools


http://www.analyzemath.com/Algebra2/Tutorials.html


Cognitive tools are generalizable computer tools that are intended to engage and

facilitate cognitive processing13. They are intellectual devices that are used to visualize

(represent), organize, automate or supplant information processing. Some cognitive

tools replace thinking while others engage learners in generative processing of

information that would not occur without the tool. Cognitive tools support learners in a

variety of cognitive processing tasks. For example, visualization tools help learners to

construct those mental images and visualize activities. Numerous visualization tools

provide reasoning-congruent representations that enable learners to reason about

objects that behave and interact14. Examples include the graphical proof tree

representation in the Geometry Tutor15; the Weather Visualizer (colorizes climatological

patterns); the Climate Watcher (colorizes climatological variable).

As students study phenomena, it is important that they articulate their

understanding of the phenomena. Modeling tools provide knowledge representation

formalisms that constrain the ways that learners think about, analyze, and organize

phenomena and provide an environment for encoding their understanding of those

phenomena. For example, creating a knowledge database or a semantic network

requires learners to articulate the range of semantic relationships among the concepts

that comprise the knowledge that they had learn in class. Modeling tools help learners

to answer “what do I know” and “what does it mean” questions. As a teacher who apply

constructivist learning, we must decide when learners need to articulate what they know

and which formalism will best support their understanding.

13 Proposed by Kommers, Jonassen, & Mayes, 199214 Proposed by Merrill, Reiser, Bekkalaar, & Hamid, 199215 Proposed by Anderson, Boyle, & Yost, 1986



Complex systems contain interactive and interdependent components. In order to

represent the dynamic relationships in a system, learners can use dynamic modeling

tools for building simulations of those systems and processes and testing them.

Programs like Stella use a simple set of building blocks to construct a map of a process.

Learners supply equations that represent causal, contingent, and variable relationships

between the variables identified on the map. Having modeled the system, Stella

enables learners to test the model and observe the output of the system in graphs,

tables, or animations. At the run level, students can change the variable values to test

the effects of parts of a system on the other. Modeling phenomena may also call on

dynamic modeling tools, such as Model IT16 which scaffolds the use of mathematics by

providing a range of qualitative relationships that describe the quantitative relationships

between the factors or allowing them to enter a table of values that they have collected.

Picture of Stella program

16 Proposed by Spitulnik, Studer, Finkel, Gustafson, & Soloway, 1995



In many environments, performing repetitive, algorithmic tasks can rob cognitive

resources from more intensive, higher-order cognitive tasks that need to be performed.

Therefore, constructivist learning should automate algorithmic tasks in order to offload

the cognitive responsibility for their performance. For example, using simple software

like Microsoft office; Microsoft excels especially, we already provided with the

component-component to solve mathematics problem including quadratic problem. So,

it is depends to us whether to use it or not. Most forms of testing in constructivist

learning should be automated so that learners can simply call for test results. And even

generic tools such as calculators or database shells may be embedded to help learners

organize the information that they collect.

Most of this way of learning provide note taking facilities to offload memorization

tasks. Identify in the activity structures those tasks with which learners are facile and

may distract reasoning processes and try to find a tools which supports that

performance.



Another cognitive tool support interactive learning is GeoGebra. This software is

quite famous in all over the educational space since it’s provide the most simple

interactive learning ways to help students understanding the process of learning. It’s

constructions can be made with points, vectors, segments, lines, polygons, conic

sections, and functions. All of them can be changed dynamically afterwards. Elements

can be entered and modified directly on screen, or through the Input Bar. GeoGebra

has the ability to use variables for numbers, vectors and points, find derivatives and

integrals of functions and has a full complement of commands like Root or Extremum.

Teachers can use GeoGebra to make conjectures and prove geometric theorems.

Below are the example of GeoGebra for quadratic.

EXAMPLE OF GEOGEBRA



Using Geogebra is one of the most simple way which help students to understand the

quadratic effectively. For example, as an introduction of quadratic, we just need to

explain a little bit briefing about the quadratic. Then, we can introduce this software to

help student’s visual in understanding Quadratic. The software of GeoGebra will explain

the rest. What the meaning of a, b and c in the general form of quadratic which is

ax²+bx+c and what happened if each of the value in a, b and c changing.

5. Conversation and Collaboration Tools

Learning most naturally occurs not in isolation but by teams of people working

together to solve problems. As teacher, we should provide access to shared information

and shared knowledge building tools to help learners to collaboratively construct socially

shared knowledge.

Problems are solved when groups work toward developing a common conception

of the problem, so their energies can be focused on solving it. Conversations may be

supported by discourse communities, knowledge-building communities, and

communities of learners.

Scardamalia and Bereiter (1996) argue that schools inhibit, rather than support,

knowledge building by focusing on individual student abilities and learning. In

knowledge building communities, the goal is to support students to "actively and

strategically pursue learning as a goal (Scardamalia, Bereiter, & Lamon, 1994, 201). To

enable students to focus on knowledge construction as a primary goal, Computer

Supported Intentional Learning



By doing this approaches, it enable students to produce knowledge databases so

that their knowledge can "be objectified, represented in an overt form so that it could be

evaluated, examined for gaps and inadequacies, added to, revised, and reformulated".

Since this method provide a medium for storing, organizing, and reformulating the ideas

that are contributed by each of the members of the community. The knowledge base

represents the synthesis of their thinking, something they own and for which they can

be proud.

We sometimes think that our students maybe know each other but actually, not

all students are close to each other. So, by doing this activity, we may can collaboration

among groups of students. As teacher, we should encourage conversations about the

problems between the students to make them participates actively. Students write notes

to the teacher and to each other about questions, topics, or problems that arise.

Textualizing discourse among students makes their ideas appear to be as important as

each other’s and the instructor’s comments (Slatin, 1992). When students and teacher

collaborate, they share the same goal which is to make the process of learning

happened effectively.



6. Social/Contextual Support

Most of the process of learning failed to achieve the target is because of poor

implementation. Frequently, our education system tried to implement their innovation

without considering important physical, organizational, and cultural aspects to the

students or learners and environment. Actually, all the innovation, and implement that

need to do need support not just only from the teachers, but everyone along it. As

teacher, it is necessary to provide the learning aids which show our support in teaching

something. The environments of studies also need to reconstruct to make sure it fulfills

the optimal need of process of learning. For the school administrator, it is compulsory to

make sure that the utilities of school and class is in perfect condition. And for the family,

their support is really necessary to make sure that the educators feel that their effort is

worthy.

EXAMPLE ARRANGEMENT OF CLASS

Figure 1: The dance-floor seating chart in all its finery



Figure 2: The runway-model seating chart — effective but underrated.

Figure 3: The independent-nation-state seating chart.



Figure 4: The Battleship seating chart.



INTEGRATED APPROACHES IN TEACHING AND LEARNING QUADRATIC


INTRODUCTION

To satisfy the integrated approaches in teaching and learning process, there are

several things that need to be emphasis upon student’s thinking, active learning,

discovery learning, and interest in Mathematics. Some think that’s, the student-centered

mathematical classrooms are now considered to be more effective in learning

mathematics than the teacher-centered traditional classrooms. These ways was

supported since it requires following characteristics:

1. Emphasize on processes as well as results.

2. Encourage and support various levels of oral and written mathematical

communication.

3. Encourage and empower leadership and authority shared with students, and

4. Encourage reflective mathematical practice with thinking mathematically.

Some suggested problem solving as an approach to learning mathematics. In

this method, the teaching of a mathematical topic begins with a problem situation that

embodies key aspects of the topic, and mathematical concepts and techniques are

developed as reasonable responses to reasonable problems.

And nowadays, since technology had also been introduce in this path, our

education system also prefer the use of computer as a tool for teaching mathematics

because of the availability of appropriate numerical, graphic, and symbolic capability of

software. In using technology, some of them have revealed that by using the computer



as a tool for performing the mathematical procedure, the students can be provided with

an opportunity and time to work on real-life problems in mathematics. However, some of

them agree that there should be less emphasis on algebraic manipulation skills and

more emphasis on underlying concepts and mathematical thinking.

Actually, there are some problems that had been faced by our students in the

beginning lesson of secondary mathematics which is the students are more interested

in the use of mathematics and their basic operation than its justification or proofs of

theorems. Because of this problem, our mathematics had been reorganized into two

types which is basic mathematic and additional mathematics.

This action was done since there are some other problems that had been

discovered from most of our students which are:

1. Are mathematically unprepared — they have gaps in mathematical knowledge

and understanding, poor recall and retention of mathematical knowledge.

2. Have perception of having rusty math skills.

3. Have a short attention span, poor attitude and lack of motivation.

4. Have a lack of meta-cognitive math skill.

5. Are passive listeners and have mental blocks.

6. Have not enough time and show poor attendance patterns.

7. Have a keen desire to have a passing grade with very little efforts.

Such diverse students with variety of their abilities and backgrounds are

constrained by learned or acquired behavior patterns that inhibit advanced learning and

the traditional teaching method of chalk-talk-homework- exam does not work for such

students.



IMPORTANT FEATURES OF INTEGRATED APPROACH

An integrated approach to teaching and learning process in Malaysia nowadays

had been offers a revolution approach that is different from the traditional approach of

chalk-talk-homework-exam. In this approach, the students were given more stress on

the following ten points:

i. Conceptual understanding rather than only computations.

ii. Relational understanding rather than just instrumental understanding.

iii. Exploring patterns and relationships rather than just memorizing formulas.

iv. Variety of pedagogical strategies rather than just chalk and talk.

v. Variety of non-traditional assessments rather than just traditional tests/exams.

vi. Effective and meaningful learning rather than just learning for test/exams.

vii. Listening (hearing, interpreting) to students’ thinking rather than only telling

(speaking, explaining).

viii. Cooperative learning rather than just individualistic learning.

ix. Making sense of mathematics using real life applications rather than just

explaining abstract concepts.

x. Helping students to develop an appreciation of the power of mathematics rather

than a negative view of math.

For interactive and discussion-based teaching, new material is introduced either

with a class discussion or via teacher-made worksheets. The teacher poses problems

and questions for discussions or investigations. During group work, the teacher

observes group interactions and their individual working on the computer or using paper



and pencil. After completion of group work, there is whole-class discussion and the

teacher serves as a facilitator.

To create mathematical culture in the classroom we emphasize four key features:

i. Various levels of math communication.

ii. Processes as well as results.

iii. Leadership and authority shared with the students, and

iv. Reflective mathematical practice with thinking mathematically.

These four elements are chosen because it is user-friendly, easy to learn and it

supports most of the concepts needed for the freshman and sophomore levels of

college mathematics. All computers in our computer labs are equipped with the latest

features and there are enough computers available to provide each student with at least

a computer.

For group work, the class is divided into small groups of two or three members

each. The students are allowed to choose their group members. During the group work,

the teacher observes the group interactions and individual contributions. While

observing the groups, the teacher checks their work, makes corrections, answers

questions and provides motivation. In the class discussion that follows group work, the

teacher generally serves as a facilitator.

Use the small class which is generally kept between 16 and 20 of students and

use a combination of qualitative and quantitative instruments in gathering data. The

students’ performance is assessed by observations of student’s presentations and

discussions, students’ worksheets, out-of-class assignments, pop-quizzes, class tests,

and midterm and final examinations.



The evaluation of the effectiveness of the teaching style is done through formal

assessments such as quizzes and tests, students’ and teachers’ weekly logs, and

informal surveys and peer evaluations.

EXAMPLES HIGHLIGHTING INTEGRATED APPROACH

In this section, we present a variety of our classroom activities and examples that

highlight our integrated approach to teaching and learning mathematics.

Conceptual understanding rather than only computations

Conceptual understanding is essential in teaching and learning mathematics; it

influences the mathematics that is taught and enhances a student’s learning.

Mathematics is not just about computation, but understanding and defining the concept

and problem and solving it by computations. Here is an example that we often use in

our Quadratic Expression and Equation:

Example

What do we mean by a root of a quadratic? Find the roots of this quadratic and explain

briefly about what the meaning of it.

x² + 2x − 8

Answer

1.Root of quadratic is a solution to the quadratic equation.

2.By factorize the equation, the root of the quadratic are −4 and 2. For, we can factor that

quadratic as

(x + 4)(x − 2).


http://www.themathpage.com/alg/factoring-trinomials.htm#one


3. Now, if x = −4, then the first factor will be 0. While if x = 2, the second factor will be

0. But if any factor is 0, then the entire product will be 0. That is, if x = −4 or 2, then

x² + 2x − 8 = 0.

Therefore, −4 and 2 are the solutions to the quadratic equation. They are the roots of

that quadratic.

Relational understanding rather than just instrumental understanding .

New material may be introduced either with a class discussion or via teacher-made

worksheets. Rational understanding can be very handy in motivating students for class

discussion. For example, before we give more detail about types of factorization of

quadratic, make a group work to research about it and followed by class discussion is

found very useful in students’ learning new material.

Example

Giving the equation is

x2 + 6x – 7 = 0

Discuss and show how to solve this equation by completing the square.

Solution


http://www.themathpage.com/alg/quadratic-equations.htm#root

http://www.themathpage.com/alg/reciprocals.htm#zero


Learning via problem solving in developing concepts

It is a well-known fact that students learn math more meaningfully when they can make

sense of what they are talking about and they can connect ideas or skills they learn in

new situations, in real-life and in other subjects. For example, the following worksheet

followed by classroom discussion helped our students in understanding exponential

functions.

Example

A ball is thrown straight up. Its height, h(in metres), after t seconds is give by

h= -5t² + 10t + 2

To the nearest tenth of a second, when is the ball 6m above the ground? Explain why

there are two answers.

Solution

Formula Given:

h= -5t² + 10t + 2



They tell you h=6, find t

So,

-5t² + 10t + 2= h

-5t²+10t+2 = 6

-5t²+10t-4=0

Use the Quadratic Equation on the above formula.

Therefore x, or t = 0.552786 or t=1.44721

Rounding to the nearest tenth of a second, t = 0.6 and 1.4

Why are there 2 values for t?

Because the ball hits the 6m mark on the way up (at t=0.6) and then hits the 6m mark

on the way down (at t=1.4)

Learning for discovering facts

Sometimes, mathematics is not just only about logic. There are also some parts that we

can relate it to the facts. It is because, in quadratic, we are using it in studying the effect

and course of something that happened around us. For the above example, we are

finding the time taken for the ball to reach at the height 6cm. This thing actually not

about logic only, but it is also about fact that we have found by using logic or calculation.

And for the quadratic, it is the fact that the roots it related to the value of x when the

other axis is zero.



Example

Use Derive and sketch the graphs of the following functions. Observe changes in

graphs as you draw. How do these graphs differ? In what ways are they similar?

a) y = (x − 2)(x − 3) and y = x – 3

b) y = (x − 2)(x − 3) and y = (x − 2)2(x − 3)

c) y = x(x − 2)(x − 3) and y = x²(x − 2)2 (x − 3)2

Using problem solving approach for sound pedagogical reasons

Problem solving is designed as a process by which an individual uses previously

acquired knowledge, skills and understanding to satisfy the demands of an unfamiliar

situation. The situation must synthesize what she or he has learned and apply it to new

and different situations17. In our integrated approach, we generally select those

problems that can;

a) engage students in mathematical discussion,

b) promote mathematical thinking,

c) focus on the development of both cognitive and meta-cognitive strategies, and

d) wherever possible, help students to learn math through problem solving.

Our main strategy of problem solving is small group work followed by class discussion.

Also, we encourage students to use Polya’s 4-step approach to problem solving as

outlined in the below figure.

17 Proposed by Krulik & Rudnick in 1989



The following problem is from our Developmental Mathematics, a remedial course,

which is not a credit course, but is a prerequisite for the Introductory Algebra course

especially quadratic.

Example

If one of the roots of the quadratic equation

x2 + mx + 24 = 0

is 1.5, then what is the value of m?

Solution

We know that the product of the roots of a quadratic equation ax2 + bx + c = 0 is

In the given equation, x2 + mx + 24 = 0, the product of the roots = = 24.

The question states that one of the roots of this equation = 1.5

If x1 and x2 are the roots of the given quadratic equation and let x1 = 1.5



Therefore, x2 = = 16.

In the given equation, m is the co-efficient of the x term.

We know that the sum of the roots of the quadratic equation

ax2 + bx + c = 0 is = -m

Sum of the roots = 16 + 1.5 = 17 = -17.5.

Therefore, the value of m = -17.5

Using JAT Approach by Mathematical Connections

Mathematics makes sense and is easier to remember and apply when students can

connect new knowledge to existing knowledge in meaningful ways. We have found that

by using “Just At Time” (JAT) approach by mathematical connections helps students to

meaningfully recall pre-requisite concepts and skills and connect them with new

knowledge. For instance, we use the following example with several follow-up activities

as ‘Warm-Up Problem’ (adapted from Choike (2000)) for our Introductory Algebra

students.

Example

Use the factorization method for solving x2 + 19x + 18=0

Solution:

x2 + 19x + 18



x2 +x + 18x + 18 = 0

(x2 + x) + (18x + 18) = 0

x(x + 1) + 18(x + 1) = 0

(x + 1)(x + 18) = 0

x + 1=0 and x + 18 = 0

x + 1 – 1 = 0 - 1 and x + 18 – 18 = 0 -18

x = -1 and x = -18

Therefore, x1,2 = -1,-18

Using Open Approach

In this approach we select problems that exemplify a diversity of approaches to solving

a problem or multiple correct answers. There are three aspects of the approach: open

process, open-end product, and open problem formulation18. The main purpose of an

open approach is to make mathematics alive and relevant and help students to develop

divergent thinking.

Example

1. Graph the following functions:

i. y = x

ii. y = x²

iii. y = x² - x - 2

18 Refer to Becker & Shimada on 1997



iv. y = −x

v. y = −x²

vi. y = −x² + 2x + 3

2. Write as many properties and difference as you can see from the functions that

you had see above.

Solution

i. y = x



ii. y = x²

iii. y = x² - x - 2

iv. y = -x



v. y = −x² + 2x + 3

Example

Describe how to solve a quadratic equation and give a story problem to illustrate. Solve

your story problem by using as many methods of solving a quadratic equation as

possible.

Writing about Mathematics

The simple exercise of writing an explanation of how a problem was solved not only

helps to clarify a student’s thinking but also may provide other students with fresh

insights gained from viewing the problem from a new perspective.

Example 1

Make up a word problem with a variable having a quadratic relationship.



Find a quadratic formula for this variable. Try your formula for two hypothetical

situations.

Example 2

One of your friends in College Algebra sends you an email and asks you to explain how

to graph a quadratic formula and equation. Starting with quadratic equation, write a

clear instruction to help her with her problem. Can you generalize this to same root

quadratic, different root quadratic, and no root quadratic? Consider cases when a or b

or c in quadratic equation of ax² + bx + c is exchange.

Example 3

Explain the difference between a quadratic equation and a quadratic expression.

Using puzzles and interesting examples for motivation

There are many websites and books that give variety of mathematical puzzles and

games and some of them may be used as motivation prior to teaching a new concept or

skills. The example below gives a well-known puzzle followed by motivation for some

quadratic concepts.

Example



Forward Tracking:

Make a puzzle similar to Example above and explain the mathematical logic you used.

Using worksheets for understanding problems and improving writing

The following example and accompanying worksheet show how a worksheet can serve

as a medium that can permit students to follow sequence of steps in learning math.

Example

Below are some example of quadratic equation worksheet.



SUGGESTED APPROACHES IN TEACHING AND LEARNING QUADRATIC


INTRODUCTION

Algebra is central to proficiency in mathematics and Quadratic Expression and

Equation is part of Algebra. The development and understanding of algebraic concepts

and skills, especially in the learning of quadratic expressions and equations are of

fundamental importance in mathematics, thus can raise students’ mathematical

understanding’s skills. Yet students generally have difficulties in algebraic manipulation,

and in formulating quadratic equations to solve problems. These difficulties are

encountered by students around the world formerly and in Malaysia specially.

We can see the proof by looking our secondary school mathematics’ syllabus

where the students are only introduced by the very basic information about the

Quadratic Expression and Equation. At first, they were introduced by the basic concept

of Quadratic Expression and Equation, their basic operation, and then the very basic

problem about the quadratic related to real life. By doing this way, students basically

only got the very basic elements and the very surface about the quadratic. How can the

students master the quadratic if they only exposed by only this basic thing? Isn’t it?

To me, there are something that needs to be added to satisfy the teaching and

learning process of Quadratic Expression and Equation especially in putting the higher

level of understanding in the learning and teaching process. For me, I will use the latest

learning and teaching approaches in my teaching which is cognitive approaches since

this approach is one of the latest approaches that had been introduced and this

approaches basically overlapped and almost same in certain part of ideas and principle

such as behaviorist and constructivist.



This cognitive approach can be applied according to the situation and condition

regarding the part of the subtopic in the quadratic especially by integrating the variety

types of learning theory which included discovery learning, active learning, meaningful

verbal learning and others can make the teaching and learning process happened

effectively.

THE COGNITIVE APPROACHES IN TEACHING AND LEARNING

QUADRATIC EXPRESSION AND EQUATION

INTRODUCTION

As its name implies, the cognitive approach deals with mental processes like

memory and problem solving. By emphasizing mental processes, it places itself in

opposition to behaviorism, which largely ignores mental processes. Yet, in many ways

the development of the cognitive approach, in the early decades of the 20th century, is

intertwined with the behaviorist approach.

Behaviorist, cognitivist, and constructivist ideas and principles overlap in many

areas. Therefore, classifying some theories fit in more than one classification and

different sources classify the theories in different ways. For example, in some sources

Jerome Bruner‘s Discovery Learning Theory is classified as cognitive and not

developmental. In other sources, Bruner is deemed developmental (Driscoll,

2005/2007). In still other sources, Bruner is considered constructivist (Learning Theories

Knowledgebase, 2009). In addition, Albert Bandura is often classified as a behaviorist;

however, Bandura, himself, claimed that he was never a behaviorist.


http://en.wikipedia.org/wiki/Discovery_learning

http://sites.wiki.ubc.ca/etec510/Cognitive-Construction


All in all, basically, cognitive approaches are devided into two parts which is

cognitive cognitive approaches and cognitive developments approaches. Cognitive

cognitive approaches including:

i. Social Cognitive Theory (Social Learning Theory) by Bandura which focused on

observational learning and self-efficacy19.

ii. Information Processing Theories by various theorists where the computer was

seen as a metaphor for the mind.

iii. Assimilation Theory (Meaningful Learning) by Ausubel which focused on

reception learning; he noted that the learner was active and thus he differentiated

between rote and meaningful learning. Ausubel also stressed the importance of

the advance organizer.

Meanwhile, the cognitive development is divided into three part which are:

i. Genetic Epistemology by Piaget.

- Piaget believed that experience with the environment affected knowledge

acquisition.

- His four stages of development detail how humans develop cognitively.

ii. Sociocultural Theory by Vygotsky

- Vygotsky’s Zone of Proximal Development (ZPD) details the difference

between what a learner can do independently and what the leaner can do

with help; independent learning may not take place, but scaffolded learning

can.

iii. Discovery Learning by Bruner which describes representational stages, and

emphasizes exploring the environment.

19 Proposed by Zeldin, Britner, & Pajares in 2008



Below are the diagram about the Cognitive Cognitive Approaches and Development

Cognitive Approaches.

Figure 1: The Diagram of Cognitive Cognitive Approaches



Figure 2: The Diagram of Cognitive Cognitive Approaches



SOCIAL LEARNING THEORY

The social learning theory of Bandura emphasizes the importance of observing

and modeling the behaviors, attitudes, and emotional reactions of others. Bandura

states that "Learning would be exceedingly laborious, not to mention hazardous, if

people had to rely solely on the effects of their own actions to inform them what to do.

Fortunately, most human behavior is learned observationally through modeling:

from observing others one forms an idea of how new behaviors are performed, and on

later occasions this coded information serves as a guide for action. Social learning

theory explains human behavior in terms of continuous reciprocal interaction between

cognitive, behavioral, an environmental influences. The component processes

underlying observational learning are:

1. Attention, including modeled events (distinctiveness, affective valence,

complexity, prevalence, functional value) and observer characteristics (sensory

capacities, arousal level, perceptual set, past reinforcement).

2. Retention, including symbolic coding, cognitive organization, symbolic rehearsal,

motor rehearsal)

3. Motor Reproduction, including physical capabilities, self-observation of

reproduction, accuracy of feedback, and

4. Motivation, including external, vicarious and self reinforcement.

Because it encompasses attention, memory and motivation, social learning theory

spans both cognitive and behavioral frameworks. Bandura's theory improves upon the

strictly behavioral interpretation of modeling provided by Miller & Dollard

1941). Bandura’s work is related to the theories of Vygotsky and Lave which also

emphasize the central role of social learning.


http://tip.psychology.org/lave.html

http://tip.psychology.org/vygotsky.html


Scope and application

In the learning process of Quadratic Expression and Equation, social learning

theory can be apply extensively by converting the topic of learning into labels or images

results which is better retention than simply explaining by words. By using technology

for example GeoGebra, it can attract student’s attention to focus on the technology

tools, and by using this software, it can help the student’s understanding by visualizing

what we want to explain about quadratic. At the same time, polish their understanding

by including some example about the application of quadratic in the real life since it can

adopt a modeled behavior if it results in outcomes they value and if the model is similar

to the observer and has admired status and the behavior has functional value.

Example

Below are the example of how to use GeoGebra in introducing the quadratic equation

and relate it to the real life.

Water Fountain and the Parabola

Embed a water fountain picture under the x-y system. Give some random values to a, b,

c, and futher graph the quadratic function water(x)=ax2+bx+c. Manipulate a, b, c so that

the graph of water(x) fits the water stream. If necessary, change the increment step for

a, b, c for accurate fitting. Explain what is the actually the relation of a, b and c to the

graph and correlate it.



After fitting the curve, place a point on the graph and draw a line tangent to the curve at

P. Drag P along the curve and observe the changes in the slope of the tangent

line. What does the slope of that line mean in a physical sense? Also, please try to

explain why the water stream behaves like a parabola? Does it look like that same on

the Moon? How do you interpret the horizontal displacement of the curve?



INFORMATION PROCESSING

Information processing includes theories that focus on the structure and function

of mental processing. They focus on these structures and functions within specific

contexts and environments. In other words, information processing theories focus on

how people pay attention to events occurring within the environment, how they encode

that information by relating it to knowledge currently stored in memory, how the new

information is stored and finally, how that information is later retrieved when needed.

Human Information Processing can metaphorically be compared to computer

processing. In a computer, information is entered using an input device such as a

keyboard or a scanner. In humans, these input devices could be compared to the ears,

eyes and all other senses. The computer takes the inputed information, organizes it in

specific locations and saves it until further use. The human mind does this as well when

it makes connections and organizes information so that it can be recalled later when

needed.

The working memory or short term memory can be compared with a computers

Central Processing Unit. In a computer, the CPU acts as the brain of the computer.

Working memory is the area where we think about the information presented to us and

process it in specific ways. Once the information has been processed and rehearsed, it

then moves on to long term memory. In a computer, information is stored on hard

drives, memory sticks and CDs. Computers can demonstrate the information they have

through displays on the screen or on printed paper. Humans demonstrate their

knowledge by acting in everyday life; walking, talking and doing



Scope and application

Principal Example

1. Gain the students' attention

Use cues to signal when you are ready to

begin.

Move around the room and use voice

inflections.

2. Bring to mind relevant prior knowledge

Review previous day's lesson.

Have a discussion about previously covered

content.

3. Point out important information Provide handouts.

Write on the board or use transparencies.

4. Present information in an organized manner

Show a logical sequence to concepts and

skills.

Go from simple to complex when presenting

new material.

5. Show students how to organize (chunk) related information

Present information in categories.

Teach inductive reasoning



6. Provide opportunities for students to elaborate on new information

Connect new information to something already

known.

Look for similarities and differences among

concepts.

7. Show students how to use coding when memorizing lists

Make up silly sentence with first letter of each

word in the list.

Use mental imagery techniques such as the

keyword method.

8. Provide for repetition of learning

State important principles several times in

different ways during presentation of

information (STM).

Have items on each day's lesson from previous

lesson (LTM).

Schedule presiodic reviews of previously

learned concepts and skills (LTM).

Give exercise

9. Provide opportunities for overlearning of fundamental concepts and skills

Use daily drills for arithmetic facts.

Play form of trivial pursuit with content related

to class.



MEANINGFUL LEARNING

Ausubel's theory is concerned with how individuals learn large amounts of

meaningful material from verbal/textual presentations in a school setting. According

to Ausubel, learning is based upon the kinds of superordinate, representational, and

combinatorial processes that occur during the reception of information. A primary

process in learning is subsumption in which new material is related to relevant ideas in

the existing cognitive structure on a substantive, non-verbatim basis. Cognitive

structures represent the residue of all learning experiences; forgetting occurs because

certain details get integrated and lose their individual identity.

A major instructional mechanism proposed by Ausubel is the use of advance

organizers. Ausubel emphasizes that advance organizers are different from overviews

and summaries which simply emphasize key ideas and are presented at the same level

of abstraction and generality as the rest of the material. Organizers act as a subsuming

bridge between new learning material and existing related ideas.

Scope and Application

Basically this theory stress in creating a meaningful verbal learning and prevent

rote learning among students. From what we can see in our syllabus, basically our

syllabus are tendency to support students in rote learning since it just only introduce by

just a basic and simple exercise and example with almost the same way of problem and

solution. So, how about revolution a little bit approaches in our teaching by:



1. Derivative Subsumption

- Is a situation in which the new information that an individual learns is an

instance or example of a concept that has already been learned.

- Introduce the previous topic that had been learn in the previous that related to

the topic such as “Algebraic Expression”, “Algebraic Formulae”, “Linear

Equation”, and also “Linear Inequalities” which had been taught in form 1, 2 and 3

as a briefing.

2. Correlative Subsumption

- Is a situation in which the new material that the individual learns is an

elaboration or extension or alteration of what has already been learned.

- Correlate the previous topic that had been learn into the quadratic such as

quadratic expression is ax² + bx + c meanwhile linear expression is ax + b.

3. Superordinate Learning

- Is a situation in which the new information to be learned is a concept that

relates known examples of a concept. In this case an individual is able to

provide examples of a concept but does not know the concept itself.

- Now, the students may be able to correlate what is the meaning of a, b and c

in the quadratic expression ax² + bx + c after refer to the function of a and b in

ax + b of linear expression.

4. Combinatorial Learning

- It is describes as a process by which the new idea is derived from another

idea that is neither higher nor lower in the hierarch which is similar to learning

by analogy.



- Students now understanding deeply about quadratic expression and all the

basic concept that related to the quadratic and can correlate it to the real life.

GENETIC EPISTEMOLOGY

Genetic epistemology is a study of the origins of knowledge (epistemology),

which was established by Jean Piaget.

The goal of genetic epistemology is to link the validity of knowledge to the model

of its construction. In other words, it shows that the method in which the knowledge was

obtained/created affects the validity of that knowledge. For example, our direct

experience with gravity makes our knowledge of it more valid than our indirect

experience with black holes or in Quadratic is how can an unknown be connected to an

equation.

Genetic epistemology also explains the process of how the process of

intelligence growth:-

1. Assimilation

o which occurs when the perception of a new event or object occurs to

the learner in an existing schema and is usually used in the context of

self motivation.

2. Accommodation

o one accommodates the experiences according to the outcome of the

tasks.

3. Equilibration

o encompasses both assimilation and accommodation as the learner

changes their way of thinking in order to arrive at a correct or different

answer. This is the upper level of development.


http://en.wikipedia.org/wiki/Accommodation_(psychology)

http://en.wiktionary.org/wiki/Assimilation

http://en.wikipedia.org/wiki/Black_hole

http://en.wikipedia.org/wiki/Jean_Piaget

http://en.wikipedia.org/wiki/Epistemology


Scope and Application

By using this approach, teachers can guide the students by following the process of

intelligence growth which is assimilation, accommodation and equilibration.

1. Assimilation

- Using the environment which relate to the use of quadratic such as the Paris

Eiffel Tower or just a Penang Bridge or just as simple U bridge as a main

focus of study.

2. Accommodation

- Correlate the bridge to the quadratic such as in the picture below and explain

what the meaning of quadratic and it’s concept after that to polish student’s

thinking.



3. Equilibration

- Give the students chance to think and regenerate their own knowledge and

ask them what they had understands from the learning. Guide them to make

sure they got the right information

SOCIOCULTURAL THEORY

Current conceptualizations of sociocultural theory draw heavily on the work of

Vygotsky (1986). This sociocultural perspective has profound implications for teaching,

schooling, and education. A key feature of this emergent view of human development is

that higher order functions develop out of social interaction.



Vygotsky’s theory lies the understanding of human cognition and learning as

social and cultural rather than individual phenomena. The sociocultural approach are

items in the culture such as computers, books, and traditions that teach children about

the expectations of the group. By participating in the cultural events and using the tools

of the society, human learns what is important in his culture.

This approach rely on three aspects since understanding of human cognition and

learning as social and cultural rather than individual phenomena is what it’s stress on:

1. Zone of proximal development (ZPD)

- Which explain about the distance between what a person can do with and

without help

2. Symbolic tools

- Intellectual tools such as language which is the basic elements of

socialization.

3. Scaffolding

- Metaphor to describe and explain the role of teachers or more knowledgeable

peers in guiding student’s learning and development.

Scope and Application:

This approach basically stress on the sociocultural theory of learning process

which means the learning process happened through communication whether between

the students, between the students and the teachers or between the environments.



In teaching and learning process of quadratic expression and equation, the study

can continue by:

1. Using the scaffolding.

- Arrange the lesson to become more systematic and effective by doing a

lesson plan that enhances understanding in the important concept of

quadratic

2. Activate the Zone of Proximal Development(ZPD)

- Arrange some group work containing a various level of intelligent students in

each group where the students can depend on each other in doing the

learning process.

3. Naturalized the students using the symbolic tools

- Courage the students to participate actively in group work by giving the

variety of Quadratic problems and question to make them discuss between

each other.

- Give them chance to share what they had understand to the others by giving

them group presentation to make them become more reliable and confident in

learning something new and challenging.

Example

Example of the simple group work task:



Give them time to answer the question and ask them to prepare an explanation of the

answer to share with all the class.

DISCOVERY LEARNING

Discovery learning is an inquiry-based, constructivist learning theory that takes

place in problem solving situations where the learner draws on his or her own past

experience and existing knowledge to discover facts and relationships and new truths to

be learned.



Students interact with the world by exploring and manipulating objects, wrestling

with questions and controversies, or performing experiments. As a result, students may

be more likely to remember concepts and knowledge discovered on their own. Models

that are based upon discovery learning model include: guided discovery, problem-based

learning, simulation-based learning, case-based learning, incidental learning, among

others.

Proponents of this theory believe that discovery learning has many advantages,

including:

encourages active engagement

promotes motivation

promotes autonomy, responsibility, independence

the development of creativity and problem solving skills.

a tailored learning experience

Scope and Application:

GUIDED DISCOVERY LEARNING



Guided discovery learning is a constructivist instructional design model that

combines principles from discovery learning and sometimes radical constructivism with

principles from cognitivist instructional design theory.

In here, teachers just play the role by giving the hint about the topic, giving

something that interesting and attract the students to get interest to learn about the

topic. The teacher also give some guided about the topic by giving reference that can

be use as a guide so that the students did not lost.

Example

The first page of the chapter Quadratics Expression and Equation in Mathematics

KBSM for 4 is one of the best reference to attract students’ interests, provided guided

links about the topic and the focus of the topic which can be seen as guided tool to

support the guided discovery learning.(see picture at the next pages)


http://edutechwiki.unige.ch/en/Instructional_design

http://edutechwiki.unige.ch/en/Radical_constructivism

http://edutechwiki.unige.ch/en/Discovery_learning

http://edutechwiki.unige.ch/en/Instructional_design_model

Information related to real life

Picture to attract interest


PROBLEM BASED LEARNING (PBL)



Problem-based learning (PBL) is a student-centered instructional strategy in

which students collaboratively solve problems and reflect on their experiences.

Characteristics of PBL are:

Learning is driven by challenging, open-ended, ill-defined and ill-structured

problems.

Students generally work in collaborative groups.

Teachers take on the role as "facilitators" of learning.

In Quadratic Expression and Equation topic, this technique can be applied by

giving the students variety of question and problems from the basic concept until the

application and advance part of quadratic. This will cause them to find the answer and

at the same time, learning the topic independently. The teachers just be a reference or

a guidance when it is in need only and all the students all the learning process.

CASE-BASED LEARNING



Case-based learning (CBL) is an instructional design model that is a variant

of project-oriented learning. It is popular in business and law schools. CBL in a narrow

sense is quite similar to to problem-based learning, but it may also be more open ended

as in our definition of project-based learning.

Case-based learning (CBL) features a learner-centered, collaboration and

cooperation between the participants, discussion of specific situations, typically real-

world examples and also questions with no single right answer.

The process occurs by students engaged with the characters and circumstances

of the story, identifies problems as they perceive it, connect the meaning of the story to

their own lives, bring their own background knowledge and principles, raise points and

questions, and defend their positions and formulate strategies to analyze the data and

generate possible solutions.

Meanwhile, the teachers functioning as facilitators which encourages exploration

of the case and consideration of the characters' actions in light of their own decisions.

The learning process in quadratic occurs by giving the complex problems written

to stimulate classroom discussion and collaborative analysis which involves the

interactive, student-centered exploration of realistic and specific situations.


http://edutechwiki.unige.ch/en/Project-based_learning

http://edutechwiki.unige.ch/en/Problem-based_learning

http://edutechwiki.unige.ch/en/Project-oriented_learning

http://edutechwiki.unige.ch/en/Instructional_design_model


INCIDENTAL LEARNING

Incidental learning is unintentional or unplanned learning that results from other

activities. It occurs often in the workplace and when using computers, in the process of

completing tasks. It happens in many ways: through observation, repetition, social

interaction, and problem solving by watching or talking to colleagues or experts about

tasks from mistakes, assumptions, beliefs, and attributions or from being forced to

accept or adapt to situations. This "natural" way of learning has characteristics of what

is considered most effective in formal learning situations: it is situated, contextual, and

social.

In learning the quadratic expression and equation, teachers can also included

pop quiz, interactive exercise or maybe use the cognitive tools such as computer games

since nowadays, we can get so many types of educational games by the tip of our

finger. Below are some example of interactive games that can be used as incidental

learning process in teaching quadratic expression and equation.

Example



1. Quadratics Equation Matching Games



2. Bug Match Games

3. Quadratic Equation Hang Man



4. Study Stack



REFLECTION ABOUT THIS ASSIGNMENT

I look upon this assignment as opportunities. It will give me a chance to talk

about the Quadratic Expression and Equation that I had learn in my secondary school

and still learning till now, which is an excellent way to enhance my knowledge. My

efforts will go more smoothly and be more successful if my approach these assignments

the same way I should approach doing mathematics; methodically, and with a spirit of

play and discovery. The information which follows is intended to make assignments

more manageable for me, and to give me an edge in doing the best papers possible.

When doing this assignment, I concentrate first on the topic itself and save

formatting and proofing till last – but then be thorough. In an academic paper, searching

for what I want to say can take time, and then saying it clearly can take several

attempts. The biggest part of my grade comes from organizing my ideas and presenting

them clearly. The perfect grade is apt to go to that paper which is free of glaring spelling

and grammar errors, is in the proper format, and presents its material clearly, supporting

it well from the sources. Other than that, this assignment also force me to look back all

the theory of education that I had learn in the previous as a reference and basic material

to use in this task.

I can foresee that objective of all of this assignment is to promote grow through

self-awareness and self-research. Besides, this assignment is designed to foster

communication among the students with students and students with lecturer. By

sharing information, we are perceptiveness, and insightfulness as well as the efforts that

have been made to improve, learn, and grow. Many misconceptions are avoided when

the student lecturer communicates this information to each others.



This assignment helps me to understand certain concepts of quadratics and help

me to build certain skills. I also try to understand the process of the specific problem.

Classify problems in the assignment by problem type. I try to figure out why I missed the

ones my friends did instead of just working toward the answer

It helps me to build on a concept or skill I did not posses before. Before doing this

assignment, I look back over the assignment questions many times and try to explain to

myself what the assignment was about, what each kind of problem was asking, how I

got the answers and what the answers tell me. This process will help me understand the

material and will help me discover what I don’t understand.

I noticed three things that I have never had to do for a math assignment before.

These goals may take me a while to get used to and/or get good at. They include

defining quadratics basic element, quadratics raw materials, basic operation and its

manipulation and so on. When I was asking the question, I can understand better about

the questions demand. We were never allowed to stray too far off topic. As a result of

this I am a little apprehensive of this assignment but hope that I will learn quickly. As a

saying goes “Be patient with the process. You are right in noting wholesale changes in

expectations. Give yourself time to adjust.”

In the process of completing this assignment, rarely before have I ever written

narrations of my discoveries and my teaching techniques. I believe that if I am capable

of explaining my actions in writing, then I would definitely be able to understand what I

was doing since writing and explaining could be considered as my weaknesses.



But even more important is the question of what skills will really prepare today’s

students for the future. Surely the next decades will be ones of rapid change where old

answers don’t always work, where employers demand communication and human

relations skills as well as the ability to think incisively and imagine creative solutions to

unforeseen problems. Many of today’s computer applications offer poor preparation for

such abilities.

The future also will favor those who have learned how to learn, who can respond

flexibly and creatively to challenges and master new skills. At the moment, the computer

is a shallow and pedantic companion for such a journey. We should think long and

carefully about whether our purpose is to be trendy or to prepare students to be

intelligent, reasoning human beings whose skills extend far beyond droid-like button

clicking.

As a result, I can say that the approach in this assignment offer possibilities for a

real interaction to take place between the students. This might provide information

which normally would not be available and that can be used to help students develop

our mathematical understanding. I hope that my analysis will contribute to my

understanding of the characteristics of this approach and its implications for

mathematics educations.



I now believe that straight teaching Quadratics should be replaced by what is

appropriate to student’s settings. I can no longer ignore the inefficiency of the traditional

teaching method. I feel that communication between the mathematics and mathematics

education communities is vital to enriching the teaching of Algebra. In fact research

results do not always have an instantaneous impact on the teaching and learning of

Algebra especially in the quadratic topics. Rather, they show ways to understand

student learning that can later be used as a means of improving instruction.



REFERENCE

An Approach to Curriculum Design, (2010, April), Institute of Education University of

London (IOE), http://www.lkl.ac.uk/ltu/files/publications/Laurillard-

An_Approach_to_Curriculum_Design-WIP.pdf

Brooks, J.G., & Brooks, M.G. (1999). "In search of understanding: The case for

constructivist classrooms." Alexandria, VA: Association for Supervision and

Curriculum Development.

Chee, S.N, Hamzah Sahrom, & Thiam C.L (2005) “Mathematic KBSM Form .4” KDEB

ANZAGAIN .

Constructivism and the 5Es, Miami Museum of Science, 2001

http://www.miamisci.org/ph/lpintro5e.html

ERIC Clearinghouse for Science Mathematics and Environmental Education, 2003

(ERIC). http://www.ericdigests.org/2004-3/views.html

Ernest, P. (1996). Varieties of constructivism: A framework for Comparison. In L.P.

Steffe, P. Nesher, P. Cobb, G.A Goldin, and B. Greer (Eds.), "Theories of

mathematical learning." Nahwah, NJ: Lawrence Erlbaum.

Good, R.G., Wandersee, J.H., & St. Julien, J. (1993). Cautionary notes on the appeal of

the new "ism" (constructivism) in science education. In K. Tobin (Ed.), "The

practice of constructivism in science education." Hillsdale, NJ: Lawrence

Erlbaum.


http://www.ericdigests.org/2004-3/views.html

http://www.miamisci.org/ph/lpintro5e.html

http://www.lkl.ac.uk/ltu/files/publications/Laurillard-An_Approach_to_Curriculum_Design-WIP.pdf

http://www.lkl.ac.uk/ltu/files/publications/Laurillard-An_Approach_to_Curriculum_Design-WIP.pdf


Huraian Sukatan Pelajaran Matematik Tingkatan 4 KBSM, 2006, Pusat Perkembangan

Kurikulum Kementerian Pendidikan Malaysia(PPK),

http://www.scribd.com/doc/493831/Matematik-Tingkatan-4

Inquiry Education Information for the Muzeum Education, 1996 (Institutet for Inquiry).

http://www.exploratorium.edu/IFI/resources/constructivistlearning.html

Integrated Approaches to Teaching and Learning in the Senior Secondary School,

2008, Western Australian Certificate of Education(WACE),

http://www.curriculum.wa.edu.au/internet/_Documents/General/Guidelines+for+in

tegrated+approaches+to+teaching+and+learning+in+senior+secondary+school+

V5.pdf

Lakenheath, Can the Quadratic Formula Kiss-off Completing the Square, Pat Ballew

United Kingdom, http://pballew.net/quadform.doc

Mathematics Education: Constructivism in the Classroom,1994-2010. (The Math Forum)

http://mathforum.org/mathed/constructivism.html

Packer, J. & Bain, J. (1978). Cognitive style and student-teachers compatibility. Journal

of Educational Psychology, 70, 864 - 871.

Von Glaserfeld, E. (1993). Questions and answers about radical constructivism. In K.

Tobin (Ed.), "The practice of constructivism in science education." Hillsdale, NJ:

Lawrence Erlbaum.


http://www.exploratorium.edu/IFI/resources/constructivistlearning.html

http://www.scribd.com/doc/493831/Matematik-Tingkatan-4