A-Maths Tuition
A-Math: How Do You Sketch A Modulus Graph?
Last Sunday while coaching my A-Math students on a question on Modulus Functions, we did solving of Modulus equation which is of no big problem as long as you get the basic concept correct.
|x| = x when x >= (more than and equal to) 0 or |x| = -x when x < 0
When we came to the next part of the question which involves Sketching of Modulus Graph, that's where the interesting happens.
Read about the Differences between Drawing and Sketching in this post.
When question involves sketching of graph, we usually do not need
- a table of values.
- axis which are evenly marked out.
Sketching of Graph should however includes
- critical points (i.e x - intercept(s), y-intercept, turning point (if you are sketching a quadratic graph))
Let's take a look at the working of 2 different students:
Student A:
- Sketch the modulus graph using table of values
- Join up the points in a straight line manner
Student B:
- Sketch the modulus graph using a series of 2 other graphs
Note the difference in the shape of the 2 graphs.
I certainly hope that my student A is convinced that using a table of values is not recommended for drawing modulus graphs. Moreover, many questions involving modulus could be that of Trigonometry graphs! So be like student B, draw modulus graph using a transformation of a series of graphs
O Level A-Math and E-Math June Holidays Intensive Revision Programme
We thank you for your patience for our O Level A-Math and E-Math June Holidays Intensive Revision Programme : Be Prepared For Your O Levels This June.
It is less than 24 weeks from the GCE O Level Examinations.
Are your concepts strong or weak?
Are you mentally prepared or not?
Are you afraid of certain A-Math topics like
Trigonometry (Proving Identity, Solving Equations, R-Formula, Sketching of Trigonometry Graphs...), Differentiation & Integration (Very Very Important topics),
Logarithms and Linear Graphs,
Quadratic Equations & Its Applications?
Or are you confused on these E-Math topics like
Vectors
Statistics (Box & Whisker Diagram, Stem & Leaf, Mean, Standard Deviation, Cumulative Frequency, Median, Mode) & Probability
Graphs Mastery (Inclusive of Quadratic Graphs, Distance, Speed Time Graphs, Graphs of Power Functions, Gradient of Curves)
Trigonometry (Angles of Elevation & Depression; Very Very important topic)?
In which is the case, you want to make use of this 'last' holidays to catch up, revise on the topics which you are not strong yet.
Winners Education Group is organizing a June Holidays Intensive Revision Programme for all Secondary 3 & 4s students so that you get prepared with the concepts, strategies to score your distinctions for your GCE O Levels.
Here's the schedule for the O Level A-Math and E-Math June Holidays Intensive Revision Programme:
For more registration details, click here. We will get back to you within 2 days.
“A-Math used to be such an abstract subject for me. I couldn’t make sense of what I was taught and I did not pass any of my A-Math exams or tests previously. Joining this A Math Intensive programme was turning point for me; I wish I was taught the strategies earlier! They were simple and easy to use as I learnt how to “see” A-Math better. Thanks for the belief in me! “ Vanessa Lim, E8 to A2 (3 months), MGS
Click here to hear what others say about us
Leave us your questions or comments at the end of the post.
Announcement: GCE 'O' Level Secondary 3 Additional Math Programme Starts 19th April 2009
For current GCE 'O' Level Secondary 3 Additional Math students:
If you are facing challenges in understanding and making sense of Additional Math, you must seek help NOW! And not wait till end of year, thinking that you still have time. The truth is consistency is the key to good grades!
Our Company, Winners Education Group is launching the Additional Mathematics Ultimate Leap Programme for Secondary 3 this Sunday 19th April 2009.
We would like to invite you to
- understand the abstracts in an easier manner
- "see" Additional Mathematics using everyday life analogies
- expose themselves to real application questions
- build up their confidence for more challenging topics
- instill more interest in Additional Mathematics
Based on our many years of experience, the challenges of Additional Mathematics surface very early in their Secondary 3 and very often left unattended. Misconceptions get accumulated and create very unhappy students who are unmotivated due to the repeated failures.
So start early and build strong foundation, one concept at a time.
To join the programme, please contact us at 
Click here for more programme details.
O level A-Math: 2 Different Approaches To Solve Identity Question In Factor - Remainder Theorem
In O level Additional Mathematics, there is a small section on Identity inside the topic of Factor & Remainder Theorem. Today I am going to share with you the 2 different approaches to solve this kind of questions.
- Substitution Method (My preferred method)
I am going to use the question below to show you the step by step solutions of both methods.
Given that [pmath]3x^2+x-2=A(x-1)(x+2)+B(x-1)+C[/pmath] for all values of x, find the value of A, of B and of C.
Let x = 1,
[pmath]3+1-2 = C[/pmath]
[pmath]C=2[/pmath]
Let x = -2,
[pmath]3(4)-2-2 = B(-3) + 2[/pmath]
[pmath]3(4)-2-2 = B(-3) + 2[/pmath]
[pmath]B= -2[/pmath]
Let x = 0,
[pmath]-2 = -2A+ 2 + 2[/pmath]
[pmath]A = 3[/pmath]
Thus A = 3, B = -2 and C = 2
Concept behind the Subsitution method: The value of x choosen will cause one or more of the unknowns to be "cancel off", leaving just 1 unknown left. For example, when I choose x = 1 in the first subsituition, A & B are eliminated, allowing me to find 'C'.
- Comparing Coefficients Method
Given that [pmath]3x^2+x-2=A(x-1)(x+2)+B(x-1)+C[/pmath] for all values of x, find the value of A, of B and of C.
By comparing coefficient of [pmath]x^2[/pmath]:
LHS: 3 = A => A = 3
By comparing coefficient of [pmath]x[/pmath]:
LHS: 1 = 2A - A + B => B = -2
By comparing coefficient of [pmath]x^0[/pmath]:
LHS: -2 = -2A - B + C => C = 2
Thus A = 3, B = -2 and C = 2
Concept behind the Comparing Coefficient method: Expansion is usually required on one side of the equation. It takes up time. The reason for the insignificant working shown is due to the fact that the expansion is done mentally instead of written. This method is highly recommended if there is more than 1 unknown other than x on the left hand side of the equation. For example, there's an unknown 'D' on the left hand side of the equation.
Which method do you usually use? And which method does your school teach you? Leave me your answer in the comment section below.
Schedule Changes to Sunday O level A-Math Programme
Dear students and parents,
Please note that there are slight changes to Sunday A-Math Ultimate Leap Programme schedule.
These are the new timings:
Sec 4: 1pm - 3pm
Sec 3: 9am - 11am
Find out more about this programme here.
Contact us at 9685 7675 or 9828 7357 for any other enquiries.