Integrated Programme (IP) Math Year-by-Year Roadmap and 12-Week Grade-Jump Plan

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12 Jun 2025, 00:00 Z

Singapore's Integrated Programme (IP) squeezes four years of mainstream mathematics into three and adds selected A Level ideas one to two years early. This article gives you a clear year-by-year map, explains how the 2025 H2 syllabus changes flow downward, and ends with a 12-week plan to lift grades.

1 Year-by-Year Topic Panorama

Year	New ideas beyond Express stream	Typical assessment timings
1	standard form, proportionality equations, financial math, solving inequalities	WA 1 (Term 1 week 8), WA 2 (Term 2 week 8), end-year exam
2	quadratic sketching, changing the subject, trigonometry with obtuse angles, basic probability	Two WAs, mid-year practical task, end-year exam
3	binomial theorem, surds, logarithms, sigma notation, derivative from first principles	Two WAs, Paper 1: 1.5 h, Paper 2: 2 h
4	recurrence relations, product, quotient, chain rules, kinematics functions, first hypothesis test	WA 1, block test, prelim, school mock

2 Five Core Strands

2.1 Number and Structure

Y1: factors, multiples, indices
Y2: radicals and rational exponents
Y3: full surd manipulation, logarithm laws
Y4: divisibility proof, basic recurrence

Example (Y3): Rationalise the following irrational Number

\[ \frac{5}{(3-\sqrt{5})}. \]

\[ \frac{5}{(3-\sqrt{5})} = \frac{5(3+\sqrt{5})}{(3-\sqrt{5})(3+\sqrt{5})} = \frac{5(3+\sqrt{5})}{3^2-(\sqrt{5})^2} = \frac{15+5\sqrt{5}}{4} \]

2.2 Algebra and Functions

Y1: linear functions
Y2: quadratic sketching, simultaneous linear-quadratic systems
Y3: sigma notation, binomial expansion $ (a+b)^n $ for integer n <= 10
Y4: remainder theorem, exponential and logarithmic models

2.3 Geometry and Trigonometry

Angle work in Y1-Y2 grows into full trigonometric equations and circle proofs by Y3-Y4.

2.4 Calculus

Year	Milestone
3	Derivative from first principles $$ f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h} $$
4	Product, quotient, chain rules; Definite integrals; Area under velocity-time graphs

2.5 Probability and Statistics

Probability starts with Venn diagrams in Y2 and ends with normal-approximation hypothesis tests in Y4.

3 Worked Examples

3.1 Y2 Quadratic Sketch

Sketch $ y = -x^2 + 6x - 5 $ and find intercepts.

Complete the square:

\[ -(x^2 - 6x) - 5 = -(x-3)^2 + 4 \]

Vertex (3, 4). Roots from $ x^2 - 6x + 5 = 0 $ give $x = 1$ or $x = 5$.

3.2 Y3 Binomial Coefficient

Coefficient of $ x^3 $ in $ (2+x)^6 $:
$$ {6 \choose 3} \times 2^3 \times x^3 = 20 \times 8 \times x^3 = 160x^3 $$

3.3 Y4 Chain Rule Kinematics

Displacement $ s(t) = 3 e^{2t} $.
Velocity $ v(t) = 6 e^{2t} $.
Acceleration $ a(t) = 12 e^{2t} $.
So $$ a(0.5) = 12e = 32.6 \space \pu{m.s-2} $$

4 2025 Syllabus Tweaks and IP Impact

H2 change	Effect on IP
Method of differences removed	Shifted to Olympiad CCA only
Parametric-area topic removed	Extra WA time for sigma proofs
Sampling distributions added	Y4 WA now tests normal approximation

5 Traffic-Light Audit

Print the list and mark each topic G, A, or R: indices and surds, quadratic sketch, logarithm laws, binomial theorem, basic differentiation, chain rule, definite integrals, sigma proofs, recurrence relations, normal distribution.

6 12-Week Grade-Jump Plan

Weeks	Focus	Concrete action
1-2	Foundation Patch	Write the full formula list nightly; 15 MCQs on one red topic each day
3-4	Algebra Speed	30-minute binomial and surd drills every other day
5-6	Graph-Calculus Link	Alternate derivative-sketch tasks with coordinate-geometry proofs
7-8	Timed Paper 1 Sprints	25-mark mini paper in 37 min, three per week
9-10	Full Rehearsals	Paper 1 on Saturday, Paper 2 on Sunday; log every error
11-12	Stretch and Teach	Solve one past H2 question nightly; record a 2-minute teach-back clip

7 When to Call a Specialist

Seek help if two red topics persist, if timed papers remain under 70 percent complete, or if the same error repeats three times.