## Thursday, 28 February 2013

### 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"
• 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?

### 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...

### 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)

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

## 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.

### What are his contribution?

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

## 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.

### 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.

### 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.

## Sunday, 24 February 2013

### Solution Real Numbers Revision

Hi everyone,

You must be working very hard in preparation for your Level Test. I have great faith this this awesome class will perform very well.

## Saturday, 23 February 2013

### Math Question

Ok this might be old but it remains unsolved
A real life situation

## Thursday, 21 February 2013

### Algebra... Crystal Ball Gazing...

by Mr Johari

Could you UNCOVER THE MYSTERY behind the
crystal ball?

Post an explanation on the Math behind this.

## Tuesday, 19 February 2013

### ESTIMATION_APPROXIMATION 2

By Mr Johari

Key QusWhat's the difference between these 2 words - Approximation Estimation?

On your own, go through the learning activity in the following website
http://www.s-cool.co.uk/gcse/maths/approximations

At the end of the 'visit', you should be able to tell the difference between approximation and estimation, and when each is used in real world...

Write a summary to illustrate the KEY DIFFERENCES between Estimation and Approximation. Submit  your write-up as a Comment under this posting.

## Monday, 18 February 2013

### Math Error Analysis Kai Cheng 19

Diagnostic Test 2 Q4 (c)

QD
Line 1 (Carelessness): 2(11/9) =2x2.
Line 2 (Conceptual) :(-) x (-) = (+), not (-), and it is multiplication, not addition.

QE
Line 1 Conceptual): 2 (11-9) = 2(2), not 22-18.You have to do the bracketed formulas first.

QF
Line 2 (Conceptual):2 (11-9) =2(2), not 22-18, because the bracketed formulas must be done first, same goes for the 16 (-1/2)^2, you have to get (-1/2)^2 first.

Diagnostic Test 2 Q4 (cd)

QA
Line 1( Carelessness): 2/3 x 1/4 should be = to 1/6.
Line 2 (Carelessness): -1/2 / 1/6= -3

QB
Line 2 (Conceptual): -1/2 / 1/6 = 3

QC
Line 1( Conceptual):1/2 - 1 = (-1)
2/3 x 1/4 = 1/6

QD
Line 4(Conceptual) :-1/2 / 1/6 = 1/2 x 6/1, you must flip the second number.

QE
Line 2 (Carelessness) :-0.5 / 0.5/3= -3

### SIGNIFICANCE FIGURES

Courtesy of Ms Loh

## Sunday, 17 February 2013

### Error analysais (Eunice)

Diagnostic test 2Q4 (c)
D)
Nature of errors: careless
Line 1
11-9=2,
Then it is
=2(2)
=4.

Nature of errors: conceptual
Line 2
The student adds the two fractions together instead of multiplying it, where nessasry.
So,
-½ x -½
=-¼

E)
Nature of errors: conceptual
Line 1
When solving an equation, solve the problem in the bracket first.
So, if 11-9=2,
Then it is
2(2)
=4

F)
Nature of error: conceptual
Line 1
2(11-9) is not (22-28), the problem in the bracket must be solved first.
Thus,
11-9=2,
2(2)=4
Line 1
conceptual
Only ½ in the equation should be squared and 16 should not.

Diagnostic test 2 Q4(d)
A)
Line 2
Nature of error: conceptual
2/12 is not 6.
The student have used 12 divided by two instead of using two divided by 12.

B)
Line 2
Nature of error:careless
When
⅔ x ¼ ,it is
2/12
1/12 =⅙
C)
Nature of error:careless
Line 1
The student have added ½ and -1 instead of subtracting 1 rom ½.
Is should be ½-1= -1 ½
The student multiplied ⅔ and ¼ wrongly. It should be ⅙ instead of 11/12

nature of error: conceptual
Line 1
⅔ x ¼ is not ⅓ x ½ , but the student wrote ⅓ x ½ as ⅔ x ¼.

E)
nature of error: conceptual
Line 1
The change of form from fraction form to decimal form is redundant.

### Error Analysis -(Bryan Goh)

Diagnostic Test 2:

Question 4(c):

QD:
Line 1: (Careless)

2(11-9) = (2x2)
not: (2x3)

Line 2: ) (Conceptual)

(-1/2 x -1/2) = +1/4
not: -1

QE:
Line 1: (Conceptual)

(1/2)^2 = +1/4
not: (-0.25)
Note: (-) x (-) = (+)

QF:
Line 2: (Conceptual)

16(-1/2)^2 = 16(1/4)
=4
not:(16 x -1/2)^2 = 8

Question 4 (d):

QA:
Line 2: (Conceptual)

Should be: -1/2
--------
1/6

QB:
Line 3: (Conceptual)

-1/2
------ = -1/2 x 6
1/6
= 3

QC:
Line 1: (Careless)

1/2 - 1 is not 1/2 + 1
(-1/2)            (1.5)

2/3 x 1/4 = 1/6

QD:
Line 4: (Conceptual)

(-1/2)
------- = (-1/2) x 6/1
1/6
not: (-2/1) x 1/6

QE:
Line 1: (Conceptual)

2/3 x 1/4 is not 2/3
------
4/1

## Saturday, 16 February 2013

### Getting to know MacRitchie Reservior Park: Tree Top Walk, presented by Heng Yee Ying Beverly

Approximated figures:
Total length of walkway, Time, Distances to the entrance

Exact Figures:
Date, Highest point from forest floor, The connecting of Bukit Peirce and Bukit Kallang

## Friday, 15 February 2013

### Error analysis Luke

Diagnostic Test 2 Q 4 (c)
QD
Nature of errors:Careless
Error:
Line 1:
2 x 2 - 16 x -1/2 x -1/2=0
Line 2:
2 x 2 - 16 x 4= -60

Diagnostic Test 2 Q 4 (c)
QE
Nature of errors:Conceptual
Error:
Line 1:
22 - 18 - 16 (0.25)=-3

Diagnostic Test 2 Q 4 (c)
QF
Nature of errors:Conceptual
Error:
Line 1:
22 - 18 - 16 x -1/2 x -1/2 =4

Diagnostic Test 2 Q 4 (d)
QA
Nature of errors:Careless
Error:
Line 2:
-1/2 / 1/6 = -3

Diagnostic Test 2 Q 4 (d)
QB
Nature of errors:Carless
Error:
Line 3:
-1/2 x 6/1 = -3

Diagnostic Test 2 Q 4 (d)
QC
Nature of errors:Careless
Error:
Line 1:
-1/2 / 1/6 = -3
Line 3:
-18/12 /11/12 = -1 7/11

Diagnostic Test 2 Q 4 (d)
QD
Nature of errors:Conceptual
Error:
Line 1:
-1/2 / 1/3 x 1/2
Line 4:
-1/2 x 6/1

Diagnostic Test 2 Q 4 (d)
QE
Nature of errors:Careless
Error:
Line 3:
-0.5/0.5/3 = -3

### Getting to know Singapore Discovery Centre

Source:http://www.sdc.com.sg

### Getting to know Chinatown presented by Nehal Janakraj

Exact figures:
- 10 non-chinese restaurants
- 5 places to visit in Chinatown Charm
- 10 Food markets

Approximated figures:
- 20 000 people that took part in the light-up ceremony
- More than 10 000 people taking part in the mass-lantern walk
- Attracts more than 10 000 a week

Source: http://www.chinatown.sg/

### Getting to Know Marina Barrage (Taufiq)

this is my poster of marina barrage

by Mr Johari
courtesy of Ms Loh Kwai Yin

Source: My Paper Friday August 8, 2008

1. Number of days from Opening to Closing Ceremony = 16 days [exact]
2. Cost of Opening Ceremony Ticket = 5000 yuan [exact]
3. Tickets sold for the game = 7,000,000 [approximated]
4. Number of athletes competing = 10, 500 [approximated]
5. Number of sports = 28 [exact]
6. Number of nations competing = 205 [exact]
7. Number of permanent residents in Beijing (2007) = 16,330,000 [approximated]
8. Number of bicycles in Beijing = 10,000,000 [approximated]
9. Number of accredited journalists = 21,600 [approximated]
10. Cost of ticket for general admission for softball preliminaries = 50 yuan [exact]
11. Number of official Olympic and paralympic volunteers = 100,000 [approximated]
12. Number of city volunteers to provide tourist services and bolster security = 400,000 [approximated]
13. Number of social volunteers at community level = 1,000,000 [approximated]
14. Weight of rubbished produced in the city (in 2006) = 5,850,000 tonnes [approximated]
15. Forecast number of cars in Beijing by the time of games = 3,300,000 [approximated]
16. Expected number of visitors to Beijing during the games = 2, 500,000 [approximated]
17. Investment in environmental improvement from 1998 to 2006 = 120 billion yuan [approximated]
18. Number of old buses and taxis to be taken off the road before the games = 65,000 [approximated]
19. Length of train track in Beijing by 2008 = 198 km [approximated]

### Getting to Know Singapore! (Assignment)

by Mr Johari
Courtesy Ms Loh

In this activity, you are going to assume the role of a Singapore Ambassador, to make recommendations of interests to tourists visiting Singapore. You will introduce a significant landmark by highlighting the features of the landmark using some ‘vital statistics’ in a digital poster.

The digital poster
1. should have the title that tells the reader brief info about the place
2. should have at least 3 exact figures and 3 approximated figures to provide useful information about the landmark to the tourists; remember to provide context to the figures
3. should have relevant images of the landmark
4. can be created using Glogster or any digital medium that enables you to create a digital poster to be embedded in the blog post
You will embed the digital poster in the Maths blog. In the same blog post,
• indicate which figures are exact and which are approximated that are found in the poster
• cite the sources where the information and images that you use in the poster
Submission:
• Subject Title: Getting to know "location", presented by "your full name". e.g. "Getting to know the Singapore Civilisation Museum, presented by Paula Phua"
• Deadline: 16 February 2013, 2359h

Here is the list
1. Attraction: Singapore Changi Airport
2. Attraction: The Singapore Flyer
3. Attraction: Singapore EXPO
4. Attraction: Singapore Suntec City
5. Attraction: Istana Kampong Glam
6. Attraction: Chinatown
7. Attraction: The Singapore Zoo
8. Attraction: Night Safari Singapore
9. Attraction: Jurong Bird Park
11. Attraction: NEWater Visitor Centre
12. Attraction: National Museum of Singapore
13. Attraction: Asian Civilisations Museum
14. Attraction: The ArtScience Museum
15. Attraction: Singapore Philatelic Museum
16. Attraction: Singapore Discovery Centre
17. Attraction: Marina Barrage
18. Attraction: Singapore Botanic Gardens
19. Attraction: Bukit Timah Nature Reserve
20. Attraction: Sungei Buloh Wetland Reserve
22. Attraction: MacRitchie Reservoir Park: Tree Top Walk
23. Attraction: Pulau Ubin: Chek Jawa
24. Attraction: Sentosa: Underwater World Singapore
25. Attraction: Sentosa: Butterfly Park and Insect Kingdom
26. Attraction: Sentosa: Fort Siloso
27. Attraction: Universal Studio, Singapore
28. Attraction: The Singapore Science Centre
29. Attraction: Vegetable Farm - AeroGreen Technology
30. Attraction: Goat Farm - Hay Dairies

This is an example of a Digital Poster in the blog post:

Key information in the FACT SHEET
• 42% of women and 55% of men are approximated figures
• 11 million is an approximated figure
• 20 sports & recreation centres is an exact figure
• 24 swimming complexes is an exact figure
Source:

### Viva Voce 2 by Timothy Teng (23), Beverly Heng (4), Chester Chua (13)

We did question 1.
This is the video that I made with my group mates. Enjoy!

## Thursday, 14 February 2013

### ERROR ANALYSIS - NUMBERS

By Mr Johari
courtesy of Ms Loh KY

Identify the Errors in each of the following solutions as shown by some of the students.
State the nature of errors (conceptual or careless)  and why part of the solution is wrong.

Example 1
Diagnostic Test 2 Q 4 (c)
QA
Nature of errors:  Conceptual
Error:
Line 1   (-1/2)^2 = 1/4
Line 4   (-) x (-) = +

Example 2
Diagnostic Test 2 Q 4 (c)
QB
Nature of errors:  Conceptual
Error:
Line 2   (-16)(-1/2) = +8
Line 4   (+8) x (-1/2) = -4 incidentally answer right but method wrong therefore 0 mark

Example 3
Diagnostic Test 2 Q 4 (c)
QC
Nature of errors:  Careless
Error:
Line 2   (-1/2)^2 = 0.25

### WS Integers, rational and real numbers (SOLN)

by Mr Johari

QUICK REVISION ON OPERATIONS

========================================================================
SOLUTION TO CLASS WORK

## Wednesday, 6 February 2013

### PRESENTATION - COLLABORATIVE

This is a class collaborative work. You are required to work in groups of not more than 3 students.

Topic: ARITHMETIC, Numbers - Fractions and Decimals

Objective:
To demonstrate your understanding of the concepts through clear articulation on your approach to solve a problem.

Instructions (I):
• This is a collaborative task. You are to complete the viva voce practice with your team between 2 to 3 members.
• It will also help you to seek and articulate your understanding of concepts as you attempt to explain your approach and solution to the problem.
• You should not take more than 20 minutes to complete the entire task.

Instructions (II):
• Choose ONE question
• Two questions are presented below. Read through carefully.
• Identify a question that you think is challenging but manageable.
• Do not go for the easiest question.
• Plan
• Work out the solution on a piece of paper and think through how you would explain the solution.
• Explaining your solution does not mean simply read out the lines of the working, but inform the audience how you identify the key information, the method you use, and how the method works (e.g. begin the repeated division using the smallest prime number because...)
• Recording
• Use any application/ platform (e.g. QuickTime Player, imovie etc) that is suitable to record your solution.
• Remember to keep the file size small.
• Submission of work
• Post your video clip in the submit folder

Source of questions: ACE Learning

Question 1:

Question 2:

Part 1: Problem Solving Skills and Strategies
Part 2: Concept and Mathematical Communication

Sample work: