Weekly Question

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: 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]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.

[Video] E-Math Popular Exam Question: Finding Ratio of Areas in Vectors (Includes 3 Strategies & Revision of Similar Triangles)

In this post, we are going to discuss on the applications of vectors - Finding Ratio of Areas. This is a popular section in examinations and based on my many years of experience, students simply don't like it due to many of them disliking and not making sense of the topic on Similar Triangles.

Strategy #1 : Similar Triangles

In Similar Triangles, the ratio of 2 similar triangles can be easily found by squaring the ratio of their corresponding length.

[pmath]({A_1}/{A_2})=({l_1}/{l_2})^2[/pmath]

Strategy #2 : Common Base/Common Height

I am going to use this examination question below to illustrate the application of Strategy #2. This strategy works when the triangles shared either a common base or a common height. And that the triangles are not similar.

Watch the video below to find out if your answers are correct. Included in this video is a trick which will help you to 'see' your answer faster!

Rate the video or leave me a comment or question.

Strategy #3 : Overlap

When strategy 1 or 2 do not work and the question involves repeated triangles, overlap is the strategy you can apply. Overlap involves equalizing of ratio of THE triangle which overlaps.

A-Math Binomial Expansion: Finding Term Independent of x By A Shortcut Method

In the earlier post on free Math Exam Papers, we received very good response. Almost 200 copies were downloaded in less than 7 days. We have a subscriber requesting for step by step solution for the questions though we have provided the answer keys. I am sorry I am unable to provide the step by step solutions due to my busy schedule. However, subscribers can email me their workings I can assist and advice you on the incorrect workings. I hope this would be useful. Moreover, by providing the step by step solution will also not be useful as most students will perhaps take the easy way out to just "read" the solution and think that they understand them. Mastery of Mathematics is not by "reading" but it's the knowing and applying of the strategies.

I have picked up one question on Binomial Expansion (another tricky A-Math topic) for discussion. Specifically on finding Term Independent of x.

Allow me to discuss the common mistake that students make.

Most students will expand the expression term by term

Disadvantages:

Too time consuming
Higher tendency to make careless mistakes!

So the following step by step solution is what I taught my students during my A-Math Ultimate Leap Programme (For Sec 4s who still wish to join, call me @ 9685 7675. For Sec 3s, we are opening up the classes in March 2009! More info will be released in Feb. Keep reading this blog)

Features to take note:

General Term is applied (No memorization is required, just refer to the formula sheet if you aren't sure)
Constant (numbers) & variable (which is x in the question) are separated. (so that we can focus on the important part first)
Power of x is circled (in red) so that you focus all your attention on it. (Reduces careless mistakes too!)
Since this is a 4 marks question, 4 minutes is the working time to complete the solution. (Time management is part of examination techniques)

Skills required:

Understanding of Term Independent of x (i.e it's x to the power of 0 NOT x is zero!)
Usage of Binomial Formula
Basic application of Indice law (Observe that [pmath]{1}/{x^7}[/pmath] is rewritten as [pmath]x^-7[/pmath])

Evaluate the term which is independent of x in the expansion of .

So do you do your working in a similar manner or you have your own style? I would love to hear from you if you know how to do this question initially. If no, which part did you not understand?

A Goodie For You! Monthly Math Exam Papers Questions (With Answer Key Included)

Update (2011): Download of Maths Exam Papers is not available as of now.

Photo Credit:psd

In response to the poll conducted in Dec '08, we have launched our first value add service for 2009 to all your readers.

Free Monthly Math Exam Papers Questions & Answers

For those who want to receive Math Exam Paper Questions sent on a monthly basis, click here.

What you will expect:

4 sets of Math (Either Elementary Math or Additional Math) Exam Papers sent monthly with answer keys. (You can print it out and work on it)
Frequent updates on discussion of Math questions, Math tips & strategies which could be time-saving and reduce careless mistakes.

More consistent practice = more exposure = more marks!

For your information, almost 50 readers have subscribed for the free exam papers in a short 30 minutes.

PS: Did I mention you are getting all these exam papers with hundreds of questions for f*ree? :)

E-Math: Drawing and Understanding of Cumulative Frequency Curve (Step by step working included)

Let's use this question to discuss on drawing and using cumulative frequency curve.

(Click on image to enlarge)

(a) Drawing of cumulative freq curve

(Click on image to enlarge)

Step by step solutions:

Notes:

140 pupils weigh 72.5 kg or less. So if you are interested to find out the number of pupils who weigh more than 72.5 kg. It will be 280 (total number of pupils) - 140 = 140 pupils.

Special attention to (iii) no. of students who weigh more than 65kg. By going UP & ACROSS, we get 20 students. This only means 20 students weigh 65 kg or less. So the rest (280 - 20 = 260) will weigh more than 65 kg

In my next post, I will be discussing the features of box and whisker plot.