## Key Topics for IP Additional Mathematics

The following are some of the most important topics for Integrated Programme (IP) Additional Mathematics. Also applicable for the usual ‘O’ level Additional Mathematics.

Notice that Secondary 3 topics are very important as well, for the final Promo or ‘O’ levels. This can be a major problem for students who only start to study seriously in Secondary 4 — it can be a tough job to catch up with the important Secondary 3 topics.

## Secondary 3 topics

• Binomial Theorem
• Indices and Logarithms
• Coordinate Geometry of Circles
• Linear Law

## Secondary 4 topics

• Trigonometry: R formula and Graphs
• Differentiation and its Applications
• Integration and its applications (including area under the curve)

## IP Math Syllabus (Integrated Programme Mathematics)

Students and parents new to IP (Integrated Programme) may be confused on what is the Mathematics syllabus of IP Math. Indeed, it is very confusing as every school has its own syllabus. In general, the syllabus as a whole is not that different from ‘O’ level Mathematics, but the order in which the school teaches is unique to each school.

In general, the topics can be divided as follows, following the famous assessment book “Mathematics (Integrated Programme)” by Wong-Ng Siew Hiong who is a teacher at RI. This is one of the very few IP Math books available in local bookstores.

## Secondary 3 IP Math Syllabus

1. Geometrical Properties of Circles
3. Matrices & Simultaneous Equations
4. Quadratic Functions, Inequalities & Roots of Equations
5. Sets
6. Relations & Functions
7. Indices & Surds
8. Exponential, Logarithmic & Modulus Functions
9. Polynomials & Partial Fractions
10. Graphical Solutions & Transformations
11. Circular Measure
12. Plane Geometry
13. Coordinate Geometry & Equations of Circles
14. Linear Law
15. Trigonometry
16. Further Trigonometry

## Secondary 4 IP Math Syllabus

1. Binomial Theorem
2. Probability
3. Statistics
4. Vectors
5. Differentiation Techniques
6. Differentiation and its Applications
7. Integrated Techniques
8. Applications of Integration
9. Integration Applications — Area and Kinematics

## Hwa Chong IP dropout shares his story: Student hacked and sabotaged Project Work of rival group

This is the dark side of the Integrated Programme, parents and students should read this (and definitely watch the video).

Something in the video is quite shocking: According to the interviewee Edwin, someone in his HCI batch “hacked into the files of another Project Work (PW) group to sabotage them so that they wouldn’t do well”. This is certainly “absurd”, as Edwin put it. (Note that PW is widely considered the least important of all subjects.) Such is the level of competitiveness in the top tier IP schools? Such unethical students are really the black sheep of IP program and society in general, it reflects the “black heart” of the student.

Good luck to Edwin, he seems like a nice guy. Perhaps this may be a blessing in disguise for him, his path less travelled may lead him to a better life in the Land Down Under (Australia).

Source (Text and Video): https://www.straitstimes.com/singapore/education/i-flunked-the-ip

Excerpt:
“With his top honours in finance and economics, Singaporean Edwin Chaw, who recently graduated from the University of Melbourne, should enjoy good job prospects when he returns home.

But the 26-year-old is applying for jobs in Melbourne, Australia, instead, as he believes that his past as a “dropout” from the elite Integrated Programme at Hwa Chong Institution still haunts him.”

## Promotion Criteria for IP Schools

Entering IP is not all smooth sailing, there is still much work to be done. Most IP schools will set a minimum criteria to proceed in the Integrated Program (i.e. skip ‘O’ levels). Those students who don’t make the criteria may be streamed into the ‘O’ level track.

A recent post from Kiasuparents summarized nicely the situation for the top 4 IP schools:

At the end of the day, it’s each sch’s perogative to set the IP promotion bar at a level it deems fit. For HCI/NYGH case, that bar was set at 65%, while RI set it at 60% and RGS at 50%. The R sch students also have the flexibility to choose the best subjects for GPA calculation (and not have to include all subjects).

Source: Kiasuparents

Hence, basically for HCI/NYGH an average grade of B3 and above is needed to proceed in the IP track, while for RI it is B4, and RGS is just pass (50%). Note that IP exam papers tend to be tougher than usual, hence it is not as easy as it sounds to get B3 in an IP school, or even to pass.

While the ‘O’ level track is not bad in the sense that it provides an ‘O’ level certificate as a backup, the problem is that IP schools are so entrenched in the IP system, and so used to teaching IP syllabus, their teachers may or may not be proficient in teaching ‘O’ level material. See this news: RI’s O Level Scores: Only 1 student out of 10 made it to JC. Hence, there is always an inherent risk in being streamed to the ‘O’ level track.

## A Maths Tuition: Trigonometry Formulas

Many students find Trigonometry in A Maths challenging.

This is a list of Trigonometry Formulas that I compiled for A Maths. Students in my A Maths tuition class will get a copy of this, neatly formatted into one A4 size page for easy viewing.

A Maths: Trigonometry Formulas

$\mathit{cosec}x=\frac{1}{\sin x}$$\mathit{sec}x=\frac{1}{\cos x}$

$\cot x=\frac{1}{\tan x}$$\tan x=\frac{\sin x}{\cos x}$

(All Science Teachers Crazy)

$y=\sin x$

$y=\cos x$

$y=\tan x$

$\frac{d}{\mathit{dx}}(\sin x)=\cos x$

$\frac{d}{\mathit{dx}}(\cos x)=-\sin x$

$\frac{d}{\mathit{dx}}(\tan x)=\mathit{sec}^{2}x$

$\int {\sin x\mathit{dx}}=-\cos x+c$

$\int \cos x\mathit{dx}=\sin x+c$

$\int \mathit{sec}^{2}x\mathit{dx}=\tan x+c$

Special Angles:

$\cos 45^\circ=\frac{1}{\sqrt{2}}$

$\cos 60^\circ=\frac{1}{2}$

$\cos 30^\circ=\frac{\sqrt{3}}{2}$

$\sin 45^\circ=\frac{1}{\sqrt{2}}$

$\sin 60^\circ=\frac{\sqrt{3}}{2}$

$\sin 30^\circ=\frac{1}{2}$

$\tan 45^\circ=1$

$\tan 60^\circ=\sqrt{3}$

$\tan 30^\circ=\frac{1}{\sqrt{3}}$

$y=a\sin (\mathit{bx})+c$ Amplitude: $a$; Period: $\frac{2\pi }{b}$

$y=a\cos (\mathit{bx})+c$ Amplitude: $a$; Period: $\frac{2\pi }{b}$

$y=a\tan (\mathit{bx})+c$ Period: $\frac{\pi }{b}$

$\pi \mathit{rad}=180^\circ$

Area of  $\triangle \mathit{ABC}=\frac{1}{2}\mathit{ab}\sin C$

Sine Rule:  $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$

Cosine Rule:  $c^{2}=a^{2}+b^{2}-2\mathit{ab}\cos C$

## O Level Logarithm Question (Challenging)

Question:

Given $\displaystyle\log_9{a} = \log_{12}{b} =\log_{16}{(a+b)}$, find the value of  $\displaystyle\frac{a}{b}$.

Solution:

Working with logarithm is tricky, we try to transform the question to an exponential question.

Let $\displaystyle y=\log_9{a} = \log_{12}{b} =\log_{16}{(a+b)}$

Then, we have $a=9^y=3^{2y}$

$b=12^y=3^y\cdot 2^{2y}$

$a+b=16^y=2^{4y}$.

Here comes the critical observation:

Observe that $\boxed{a(a+b)=b^2}$.

Divide throughout by $b^2$, we get $\displaystyle (\frac{a}{b})^2+\frac{a}{b}=1$.

Hence, $\displaystyle (\frac{a}{b})^2+\frac{a}{b}-1=0$.

Solving using quadratic formula (and reject the negative value since $a$ and $b$ has to be positive for their logarithm to exist),

We get $\displaystyle\frac{a}{b}=\frac{-1+\sqrt{5}}{2}$.

If you have any questions, please feel free to ask me by posting a comment, or emailing me.

(I will usually explain in much more detail if I teach in person, than when I type the solution)

## Hwa Chong IP Sec 2 Maths Question – Equation of Parabola

Question:

Given that a parabola intersects the x-axis at x=-4 and x=2, and intersects the y-axis at y=-16, find the equation of the parabola.

Solution:

Sketch of graph:

Now, there is a fast and slow method to this question. The slower method is to let $y=ax^2+bx+c$, and solve 3 simultaneous equations.

The faster method is to let $y=k(x+4)(x-2)$.

Why? We know that x=4 is a root of the polynomial, so it has a factor of (x-4). Similarly, the polynomial has a factor of (x-2). The constant k (to be determined) is added to scale the graph, so that the graph will satisfy y=-16 when x=0.

So, we just substitute in y=-16, x=0 into our new equation.

$-16=k(4)(-2)$.

$-16=-8k$.

So $k=2$.

In conclusion, the equation of the parabola is $y=2(x+4)(x-2)$.

## Sec 2 IP (HCI) Revision 1: Expansion and Factorisation

This is a nice worksheet on Expansion and Factorisation by Hwa Chong Institution (HCI).

There are no solutions, but if you have any questions you are welcome to ask me, by leaving a comment, or by email.

Hope you enjoy practising Expansion and Factorisation.