IP Maths Notes (Lower Sec, Year 1-2): 02) Algebraic Expressions & Equations

Study guide17 Nov 2025, 00:00 ZUpdated 30 Nov 2025

Expand and factor expressions, solve linear equations and inequalities, and translate word problems into algebraic models.

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Q: What does IP Maths Notes (Lower Sec, Year 1-2): 02) Algebraic Expressions & Equations cover?
A: Expand and factor expressions, solve linear equations and inequalities, and translate word problems into algebraic models.

The core idea is simple: Algebra is pattern language: expand, factor, solve, and explain each line.

Use it as a working check: Keep like terms together, factor common structures first, and solve equations by doing the same operation to both sides.

Then go one layer deeper: Use the worked examples to practise four moves in order: expand cleanly, collect terms, factor when useful, and translate a word problem into variables and equations.

Lower-sec algebra is a language: you model patterns, manipulate unknowns, and justify each line. This guide covers expansion, factorisation, equation solving, inequalities, and proportional reasoning.

These notes align with MOE Lower Secondary Mathematics syllabus used in IP pathways (aligned to O-Level Mathematics 4052 foundations).

Status: MOE Lower Secondary Mathematics syllabus (latest release) checked 2025-11-30 - scope unchanged; remains the reference for these lower-sec notes.

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Learning targets

Expand binomials and collect like terms accurately.
Factor expressions using common factors, grouping, and difference of squares.
Solve single-variable linear equations and inequalities, showing balanced steps.
Set up algebraic models from contextual problems (rates, mixtures, and ages).

1 Expansion and simplification

1.1 Distributive law refresher

For constants $a, b, c$ : $a(b + c) = ab + ac$

Reviewed by

Marcus Pang·Managing Director (Maths)

Sources

SEAB - Mathematics (4052) GCE O-Level 2026 syllabus (PDF)

Expression part	First expansion line	Check before collecting
$-3(2x - 4)$	$-6x + 12$	The negative multiplier changes both terms inside the bracket.
$-(x + 5)$	$-x - 5$	The invisible multiplier is $-1$ , not just a minus on the first term.
$4 - (2x - 7)$	$4 - 2x + 7$	Subtracting a bracket means subtracting every term inside it.
$(x - 4)(2x + 5)$	$2x^2 + 5x - 8x - 20$

What you notice	First move	Why it helps	Common trap
Every term shares a number or letter factor	Factor out the highest common factor.	It makes the remaining expression smaller and easier to recognise.	Trying difference of squares before removing the common factor.
Two terms with a minus sign between square terms	Check for difference of squares: $a^2-b^2=(a+b)(a-b)$ .	Two square terms can split into conjugate brackets.	Using this pattern on a sum such as $a^2+b^2$ .
Four terms with repeated groups	Group pairs of terms, then factor each pair.	A common bracket may appear after grouping.	Grouping terms randomly without checking for a repeated bracket.
Three terms where the first and last are square terms	Check whether it is a perfect square trinomial.	It may become $(a+b)^2$ or $(a-b)^2$ .	Matching the first and last terms but ignoring the middle term.

Equation type	First move	Why it helps
$3(x - 2) = 18$	Divide by 3 or expand cleanly.	Removes the multiplier before isolating $x$ .
$\frac{x + 5}{4} = 7$	Multiply both sides by 4.	Clears the denominator in one balanced step.
$5x - 3 = 2x + 12$	Collect $x$ -terms on one side and constants on the other.	Keeps the unknown in one place before dividing.
$2(3x - 1) = x + 9$	Expand the bracket, then collect like terms.	Makes every term visible before balancing.

Move	Does the sign flip?	Why
Add or subtract the same number	No	The order of the two sides is unchanged.
Multiply by a positive number	No	Both sides scale in the same direction.
Divide by a positive number	No	Both sides scale in the same direction.
Multiply or divide by a negative number	Yes	Negative scaling reverses which side is larger.

Sentence clue	Define the variable as...	Algebraic move	Check before solving
"A number increased by 7 is 19."	Let $x$ be the number.	$x + 7 = 19$	Does $x$ stand for the original number, not the answer after adding 7?
"Two numbers differ by 8."	Let $x$ be the smaller number.	Larger number is $x + 8$ .	Have you named both numbers before using their sum?
"A shirt after 20\% discount costs \$32."	Let $x$ be the original price in dollars.	$0.8x = 32$	Is the 20\% discount subtracted from 100\%, not used as the final price?
"A tap fills a tank in 30 minutes."	Let one full tank be $1$ .	Rate is $\frac{1}{30}$ tank per minute.	Are all rates measured in the same unit per minute?

IP Maths Notes (Lower Sec, Year 1-2): 02) Algebraic Expressions & Equations

Learning targets

1 Expansion and simplification

1.1 Distributive law refresher

Sources

1.2 Binomial products

Bracket-sign checkpoint

1.3 Collecting like terms

2 Factorisation toolkit

Factorisation pattern checkpoint

3 Solving equations and inequalities

3.1 Linear equations

3.1A Balance step checkpoint

3.2 Inequalities

3.2A Inequality sign checkpoint

3.3 Compound inequalities

4 Word problem modelling

4.1 Translation checkpoint

Worked example - Rate mixture

Checklist for algebra word problems

Practice Quiz

Try it yourself

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