Factorisation using Cross Method - Collaboration

Instructions

As a group, you will discuss and fatcorise the expressions that are assigned your group (for each collection).

Present your answers neatly on paper, fill up at least half a page.
On the sheet of paper, write down "Collection #1 Q1(a) by Group 1"

Put it up on the notice board at the back of the classroom.

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Collection #1: Factorise the following

Group 1: (a)
Group 2: (b)
Group 3: (c)
Group 4: (d)

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Collection #2: Factorise the following

Group 1: (g)
Group 2: (h)
Group 3: (i)
Group 4: (j)

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Collection #3: Factorise the following


Group 1: (a)(g)
Group 2: (b)(h)
Group 3: (c)(i)
Group 4: (d)(j)

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Collection #4: Factorise the following
Group 1: (a)(e)
Group 2: (b)(f)
Group 3: (c)(g)
Group 4: (d)(h)

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Source: New Syllabus Mathematics 2 (6th Edition)

Algebra - Cross Method (Aka Shoe Lace Method)

Watch the video clips in the playlist:

• Factorisation (using Cross Method)

The series of clips shows examples of how we can factorise expressions 

Based on observation, are you able to describe the "procedure" clearly to someone who is new to this method?

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Factorisation - Grouping Method

Which is bigger? (Eunice and Nehal)

Let’s say n = -3
5n = 5 x n
    = 5 x -3
    = -15
n^2 = n x n
      = -3 x -3
      = 9
In this case, 5n < n^2

Let’s say n = 2
5n = 5 x n
    = 5 x 2
    = 10
n^2 = n x n
      = 2 x 2
      = 4
In this case, 5n > n^2

Let’s say n = 4
5n = 5 x n
    = 5 x 4
    = 20
n^2 = n x n
     = 4 x 4
     = 16
In this case, 5n > n^2


Let’s say n = 5
5n = 5 x n
    = 5 x 5
    = 25
n^2 = n x n
      = 5 x 5
      = 25

Let’s say n = 6
5n = 5 x n
    = 5 x 6
    = 30
n^2 = nxn
      = 6 x 6
      = 36
In this case, 5n < n^2

Let’s say n = 10
5n = 5 x n
    = 5 x 10
    =50
n^2 = n x n
      = 10 x 10
      = 100
In this case, 5n < n^2

Therefore, in conclusion, n does not have a definite value. We cannot determine which is bigger because we do not know the value of n. In certain cases, 5n is bigger, in other cases, 5n is smaller, or both of them are equal. Thus, we cannot tell which is bigger.

Which is bigger? (Kimberly Tan, 08)

5n is less than, more than and equals to n^2
However, it depends on the number that n represents.
If n=more than 0, 5n will be more than n^2.
If n=a negative number, 5n will be less than n^2.
If n=0, 5n will be equals to  n^2
 

Which is bigger? (Bryan Goh, Ryan Ng, Chelsea, Xue Qin)

If n is smaller than 5, 5n > n^2

However if n is negative, 5n < n^2.

If n is equals to 5, 5n = n^2

If n is larger than 5, 5n < n^2

Which is bigger? (Lynette)

The answer will always depend on the value of n as 5n can be equal, more or less than n^2

Basically, 5n = 5*n wheres n^2 = n*n.

So, if n=3, 5n will be greater than n^2 as 5n = 5*3 which is 15. However, n^2 is 3*3 which is equal to 9. Therefore, 5n will be > n^2.

However, if n=5: 5n=n^2. This is because 5n= 5*5 which is 25. n^2=5*5 which is also 25.

Lastly, if n=10: 5n < n^2. This is because 5n will be 5*10 which is 50 whereas n^2=10*10 which is 100.

which is bigger? Taufiq

i went through google and searched "what is the value of an awesome n?" the first value i came across was ten. see ten dollar discount? in this case since n equals ten 5n equals 50 whilst n^2 equals 100. therefore in this case 5n<n^2. but that is only when n is qualified awesome. but in typical math if n>5 then 5n<n^2. but if n=5 then 5n=n^2. but if n<5 but n>0 then 5n>n^2. if n is negative then 5n<n^2

Question: Is 5n > or = n^2 [ Sean, Bryan Lee , Kai Cheng , Kenric]

It depends. 
In the situation which both equations are equal , the value of n has to be 5 or 0.
But in the situation that 5n is larger , the value of n has to be smaller than 5 but more than 0.
Finally ,on the other hand , if n^2 is larger , the value of n has to be larger than 5. 
So in conclusion, because of the fact that we do not know what is the value of n, we cannot conclude anything as it just depends. 

Algebra ... How "X" finds its value

Algebra ... Representation

Attempt the Algebra Puzzle at: http://www.mathplayground.com/algebra_puzzle.html

• There are 2 levels.
• Attempt the 3x3 grid until you are able to find the solution of the puzzle.
• Present your solution (together with the screen capture that shows the answers are correct) in your personal blog.
• Label the Blog post as "Chapter 4: Algebra Puzzle"
• Post the permlink of your post to comments in this post.
• The 3x4 grid is a bonus level... It is not compulsory, do challenge yourself to see if you could solve it using algebra (see example below)

The objectives of the puzzle are to...

• Find the value of each of the three objects presented in the puzzle.
• The numbers given represent the sum of the objects in each row or column.
• Sometimes, only one object will appear in a row or column.
• That makes the puzzle easier to solve. Other times, you will have to look for relationships among the objects.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Here is an example of how you should articulate your solutions in your personal blog
The 3 x 3 Grid puzzle

My solution (Method 1, which most of you would use this method to reason out/deduce your answer):

• 3 Apples = 6
• Therefore, 1 Apple represents 2
• 1 Apple + 2 Cars = 4
• Since 1 Apple represents 2,
• 2 + 2 Cars = 4
• 2 Cars represent 2
• Therefore, 1 Car represents 1
• 2 Apples + 1 Pen = 20
• Since 1 Apple represents 2, 2 Apples = 4
• 4 + 1 pen = 20
• Therefore, 1 Pen represents 16

My solution (Method 2, using the algebraic method which you will learn in Chapter 6, in Term 2):

• Let a represents apple
• Let c represents car
• Let p represents pen
• 3a = 6
• a = 6/3
• Therefore, a = 2
• Since 2a + p = 20
• 2(2) + p = 20
• 4 + p = 20
• p = 20 - 4
• Therefore, p = 16
• Since c + 2a = 5
• c + 2(2) = 5
• c + 4 = 5
• c = 5 - 4
• Therefore, c = 1

In conclusion, each apple represents 2, each pear represents 16and each car represents 1.

Bonus Level (We will only cover this method in secondary 2.

• Let m represents ice-cream
• Let p represents pear
• Let f represents flower
• 2m + p = 24 {equation #1}
• m + 2p = 45 {equation #2}
• From equation #2, we can say m = 45 - 2p
• Substitute it into equation 1, we get 2(45 - 2p) + p = 24
• We get 90 - 4p+ p = 24
• 90 - 3p = 24
• - 3p = 24 - 90
• - 3p = - 66
• Therefore, p = 22
• m = 45 - 2(22)
• Therefore, m = 1
• 2m + f = 26
• 2(1) + f = 26
• 2 + f = 26
• f = 26 - 2
• Therefore, f = 24

In conclusion, each ice cream represents 1, each pear represents 22 and each flower represents 24.

Algebra ... Nomenclature

(I) Algebraic Expressions vs Equations
The following are expressions

• 3x
• 5 - 2y
• p + 2q - 45
• mn

The following are equations:

• 2x = 4
• x + 8 = 20
• x² = 36
• 2x + y = 4

Can you tell the difference between an expression and an equation?

(II) Coefficients, Variables and Constants
In the following algebraic expressions:

(a) 6x

• x is a variable
• 6 is the coefficient of x
• There is no constant

(b) 2y + 5

• y is a variable
• 2 is the coefficient of y
• 5 is the constant

(c) 3m + 7n + 9

• m and n are variables
• 3 and 7 are coefficients of m and n respectively
• 9 is the constant

(d) p + 2q - 45

• p and q are variables
• 1 and 2 are coefficients of p and q respectively
• -45 is the constant

(e) 5a - 3b + 1

• a and b are variables
• 5 and -3 are coefficients of a and b respectively
• 1 is the constant

(f) x - y

• x and y are variables
• 1 and -1 are coefficients of x and y respectively
• There is no constant

When given an expression, are you able to pick out the variable(s), coefficient(s) and constant terms?


Click HERE to read more about these...

Algebra - An Introduction... constants and variables

In algebra, we make friends with variables & constants... Who are they? How do they look like? Let's get to know VARIABLES first...



Click here to view consolidated responses.

Algebra - Collaboration research work ( Sean Chiu , Chelsea , Qayyum , Bryan Goh)

Al - Kwarizmi 
Who was he ? 

 Khawarizmi was a mathematician, astronomer and geographer.
 He was perhaps one of the greatest mathematicians who ever lived. 
His contribution to algebra was substantial , almost making him sort of the founder or person who "made" algebra. 
 He not only launched the subject in a systematic form but he also developed it to the extent of giving analytical solutions of linear and quadratic equations, which established him as the founder of Algebra. 

What are his contributions?

 He launched the subject algebra in a systematic form but he also developed it to the extent of giving analytical solutions of linear and quadratic equations, which established him as the founder of Algebra. He introduced the Arabic numerals to Europe and made Algebra famous. He convinced European mathematicians to use these numbers as they are easier to use. He wrote a book called "Al-Jabr Wal' Muqibla", in which he introduced his own number system and introduced Algebra. TheRomans and Greeks named his books "So said Algorizmi". The word algorithm is also named after him.

Origins of algebra

It was not developed or invented by a single person but it evolved over the centuries.The  basics or traces of the beginning of algebra leads back to the Babylonians but it was further developed by Civilisations , any many people , like Al Kwarizmi.

What is algebra and how can it be applied to real life ?

Algebra is the part of mathematics in which letters and other symbols are used to represent numbers and quantities in formulae and equations. In other words , alphabets 

are used to solve math questions and the alphabets are know as "unknowns".

How can it be applied to real life ?

Lets say you want to buy a Xbox. However , your budget is only $800 . You know a new system costs $400 and extra controller $40. Assuming a game costs $60 how many games could you get?

Let x= the number of Games

$800 = $400 + $40 +$60x
800=460+60x
360=60x
x = 6 Games

So that is how algebra is used in real life.

Algebra - Collaboration research work ( Lynette, Xue Qin, Kai Cheng, Yu Hin)

Muḥammad ibn Mūsā al-Khwārizmī
        He was a Persian mathematician,astronomer and geographer during the Abbasid Empire, a scholar in the House of Wisdom in Baghdad.        His contributions; Algebra, Arithmetic, Astronomy, Trigonometry, Geography, Jewish Calendar

Origin Of Algebra: The origins of algebra can thus be traced back to ancient Babylonian mathematicians roughly four thousand years ago. The word "algebra" is derived from the Arabic word Al-Jabr, and this comes from the treatise written in 820 by the medieval Persian mathematician, Muhammad ibn Mūsā al-Khwārizmī


Application:
1)    When filling your car up with gas you can use a form of algebra. Lets say you only have $20.00 to spend on gas today and gas is $3.50 a gallon. How many gallons could you buy?
Let x = # of gallons of gas
3.50x=20.00 x=5.71 gallons
2)     Lets say you need to buy a NEW XBox 360. You have $800 to spend on everything. You know a new system costs $400 and extra controller $40. Assuming a game costs $60 how many games could you get?
Let x= the number of Games
$800 = $400 + $40 +$60x 800=460+60x
360=60x
x = 6 Games
3)     For the last one lets say you are all grown up now and have to move across country for a new job. Lets use Buffalo, NY to Sacramento, CA which is roughly 2500 miles of driving. How much money do you need to save for gas if the national average is $3.23/gallon.

Let x = amount of money you need to save
2500 = 3.23x x=$773.99 

Algebra - Collaboration research work (Bryan Lee, Myat Noe, Luke, Nehal)

Al-Khwārizmī is the father of algebra.  He was born in Baghdad, Iraq.  He was a mathematician, geographer and astronomer.  His method of solving linear and quadratic equations worked by first reducing the equation to one of six standard forms (where b and c are positive integers) by dividing out the coefficient of the square and using the two operations restoring or completion and balancing. Al-jabris the process of removing negative units, roots and squares from the equation by adding the same quantity to each side.

Algebra is one of the broad parts of mathematics, together with number theory, geometry and analysis.

For historical reasons, the word "algebra" has several related meanings in mathematics, as a single word or with qualifiers.

Uses Of Algebra

Architects use it to approximation  the measurements to build buildings. 

In recipes, you add an x amount of flour and x-2 amount of sugar. 

Algebra - Collaboration research work (HOng Yi, Timothy and Kenric)

This is our Maths assignment

Algebra - Collaboration research work (Beverly, Sabrina, Khairul, Taufiq)

Al-Khwarizmi

Who is Al-Khwarizmi?

Abū ʿAbdallāh Muḥammad ibn Mūsā al-Khwārizmi (Arabic: عَبْدَالله مُحَمَّد بِن مُوسَى اَلْخْوَارِزْمِي‎), earlier transliterated as Algoritmi or Algaurizin,  was a Persian mathematician, astronomer and geographer during the Abbasid Empire, a scholar in the House of Wisdom in Baghdad. The word al-Khwarizmi is pronounced in classical Arabic as Al-Khwarithmi hence the Latin transliteration.

Source: http://en.wikipedia.org/wiki/Mu%E1%B8%A5ammad_ibn_M%C5%ABs%C4%81_al-Khw%C4%81rizm%C4%AB#Contributions

What are his contribution?

1. Algebra
2. Arithmetic
3. Astronomy
4. Trigonometry
5. Geography
6. Jewish Calendar

Source: http://en.wikipedia.org/wiki/Mu%E1%B8%A5ammad_ibn_M%C5%ABs%C4%81_al-Khw%C4%81rizm%C4%AB#Contributions

Algebra

Origin/History

The roots of algebra can be traced to the ancient babylonians, who developed an advanced arithmetical system with which they were able to do calculations in an  fashion.

The word algebra comes from the Arabic language (الجبر al-jabr "restoration") and much of its methods from Arabic/Islamic mathematics. Earlier traditions discussed above had a direct influence on Muhammad ibn Mūsā al-Khwārizmī (c. 780–850). He later wrote The Compendious Book on Calculation by Completion and Balancing, which established algebra as a mathematical discipline that is independent of geometry and arithmetic.

Source: http://en.wikipedia.org/wiki/Algebra

Application

One of the primary uses of equations in algebra is to model and solve application problems. In

fact, much of the remainder of this book is based on the application of algebra to real-world 

situations. The purpose of this section is to introduce the use of variables in equations as a

method of solving applications, and to familiarize you with some of the common applications in 

algebra.

Source: http://infinity.cos.edu/algebra/Rueger%20Text/Chapter04/sec%204.5%20(some%20applications%20of%20algebraic%20equations).pdf

Algebra - Collaboration research work (Ryan, Chester, Kimberly, Eunice)

Who Is Ai Khwarizmi

He was a mathematician and astronomer who wrote many books on arithmetic and algebra. 

What Are His Contributions

He invented Algebra and derived methods to solve quadratic equations in a simple and easy way. 

Origin of Algebra

 The word algebra is a latin variant of the Arabic word al-jabr.

What is Algebra and its Application 

Algebra is a way to solve quadratic equations in a simple and easy way. 

In architecture algebra is used to put the correct scale of the building onto the blueprint. 

In engineering, algebra is used to solve physical problems such as how to build a bridge or design an airplane. 

Algebra... Crystal Ball Gazing...

by Mr Johari

Could you UNCOVER THE MYSTERY behind the 

crystal ball?

click: http://www.flashlightcreative.net/swf/mindreader/

Post an explanation on the Math behind this.