Sec 1 Maths Chapter 4

Basic Algebra and Algebraic Manipulation

Learn how algebra uses letters, variables and expressions to represent numbers, patterns and relationships.

This page teaches the key ideas and checks for common algebra mistakes. Use the PDF for the full printable classwork and homework practice.

Download PDF Take Quick Self-Check Preview practice
Algebra infographic showing like terms expansion factorisation and algebra tiles for Sec 1 Maths
Learning goals

What this chapter trains

Basic algebra is not just about writing letters. Students must understand what letters represent, how terms combine, and when expressions are equivalent.

1
Use algebraic notationWrite unknowns, variables, generalised numbers and expressions clearly.
2
Combine like termsAdd or minus only terms with the same variable part.
3
Expand and factoriseUse the distributive law and recognise factorisation as the reverse of expansion.
4
Handle fractional coefficientsSimplify expressions with fractions carefully using common denominators.
Concept 1

Algebraic notation and variables

Algebra uses letters to represent numbers. Sometimes the letter represents a specific unknown. Sometimes it represents a general number or a variable quantity.

Specific unknown

A letter can stand for one fixed number that we want to find.

x + 4 = 9

So x = 5.

Generalised number

A letter can represent any number when we describe a rule.

a + b = b + a

This shows addition can be done in either order.

Variable

A letter can represent a quantity that changes.

P = 4l

The perimeter changes when the side length changes.

Exam trap Do not assume every letter must be solved immediately. Sometimes a letter is used to describe a relationship or a general rule.
Concept 2

Algebraic expressions

An algebraic expression is made up of variables, constants and operations. Expressions do not need an equals sign.

Parts of an expression

In 2x − 3xy + 4:

2x, −3xy and 4 are terms.

2 and −3 are coefficients.

4 is a constant term.

Substitution example

Evaluate 3x − 2y when x = 5 and y = −3.

Reveal teaching answer

3x − 2y = 3(5) − 2(−3)

= 15 + 6

= 21

Trap: −2(−3) becomes +6.

Concept 3

Like and unlike terms

Like terms have the same variable part. To add or minus like terms, add or minus their coefficients only.

1
Like terms2x and 5x are like terms because both contain x.
2
Unlike terms2x and 2y are unlike terms because the variables are different.
3
Variable power mattersx and x² are unlike terms because their powers are different.
4
Combine coefficients3x + 5x = 8x, but 3x + 5y cannot be simplified further.
Concept 4

Expansion and factorisation

Expansion uses the distributive law. Factorisation reverses the process by taking out a common factor.

Expansion

3(a + 2) = 3a + 6

−2(x + y) = −2x − 2y

4(2x − 3) = 8x − 12

Factorisation

6x + 12 = 6(x + 2)

3x − 3y = 3(x − y)

−2x − 10 = −2(x + 5)

Exam trap When factorising with a negative common factor, the signs inside the bracket change. For example, −6x − 15 = −3(2x + 5).
Concept 5

Linear expressions with fractional coefficients

Fractional coefficients require careful simplification. The usual rules still apply: group like terms and use equivalent fractions.

Equivalent notation

n ÷ 2 = n2

12 × n = n2

They represent the same value.

Example for projection

Simplify 12x + 34x.

Reveal teaching answer

12x + 34x = 24x + 34x

= 54x

= 5x4

Trap: the variable part is the same, so combine the fractional coefficients.

Trap-focused review

Common algebra traps that cost marks

These are the mistakes students should check before starting the full PDF practice.

Combining unlike terms

2x + 3y cannot become 5xy or 5x. The variables are different.

Missing negative signs

−2(x + 3) = −2x − 6, not −2x + 6.

Wrong expansion

3(a + 2) means 3a + 6. Every term inside the bracket must be multiplied.

Wrong factorisation

6x + 9 factorises as 3(2x + 3), not 6(x + 9).

x and x² are unlike

x + x² cannot be simplified into 2x² or 2x.

Fraction mistakes

12x + 13x = 56x, not 25x.

Readiness diagnostic

Quick Self-Check before PDF practice

This short quiz checks whether students are ready for the full Chapter 4 PDF. It focuses on common algebra traps, not practice volume.

Suggested classroom use: students attempt the quiz first. If they score low, review the trap cards before starting the PDF.

1. Simplify 2x + 3x.

2. Which expression cannot be simplified further?

3. Expand −3(x − 2).

4. Factorise 8x + 12 completely.

5. Simplify 12x + 13x. Type your answer as a fraction, e.g. 5x/6.

Practice preview

Diagnostic practice questions

These questions are for projection and discussion. Answers are hidden so students can attempt first. Use the PDF for the full classwork and homework set.

Write an algebraic expression for the sum of 2x and 3y.

Skill tested: writing expressions.
Reveal diagnostic answer

1 mark2x + 3y.

Trap: do not write 5xy. The terms are not like terms.

Evaluate 3x − 2y when x = 5 and y = −3.

Skill tested: substitution with negative values.
Reveal diagnostic answer

1 mark3(5) − 2(−3).

1 mark15 + 6 = 21.

Trap: minus a negative becomes plus.

Simplify 4x + 7 − 2x + 5.

Skill tested: grouping like terms and constants.
Reveal diagnostic answer

2 marks4x − 2x + 7 + 5 = 2x + 12.

Trap: combine x-terms with x-terms and constants with constants.

Simplify 3x + 2y − x + 5y.

Skill tested: two-variable like terms.
Reveal diagnostic answer

2 marks3x − x + 2y + 5y = 2x + 7y.

Trap: x-terms and y-terms are not like terms.

Expand 5(a − 3).

Skill tested: distributive law.
Reveal diagnostic answer

2 marks5(a − 3) = 5a − 15.

Trap: multiply every term inside the bracket by 5.

Expand −2(x + 4).

Skill tested: expansion with negative multiplier.
Reveal diagnostic answer

2 marks−2(x + 4) = −2x − 8.

Trap: the negative multiplier affects both terms.

Factorise 12x − 18 completely.

Skill tested: highest common factor.
Reveal diagnostic answer

2 marks12x − 18 = 6(2x − 3).

Trap: use the highest common factor, not just any common factor.

Factorise −6x − 15 completely by taking out a negative common factor.

Skill tested: negative common factor.
Reveal diagnostic answer

2 marks−6x − 15 = −3(2x + 5).

Trap: the signs inside the bracket change when −3 is taken out.

Simplify 12x + 13x.

Skill tested: fractional coefficients.
Reveal diagnostic answer

2 marks12x + 13x = 36x + 26x = 56x.

Trap: use a common denominator before combining coefficients.

Expand and simplify 2(x + 3) + 4x.

Skill tested: expansion followed by simplification.
Reveal diagnostic answer

1 mark2(x + 3) + 4x = 2x + 6 + 4x.

1 mark= 6x + 6.

Trap: expand first, then group like terms.

Download full Chapter 4 PDF Take Quick Self-Check
PDF-first practice

Use the PDF for classwork and homework

The webpage is designed for teaching, diagnosis and discussion. The PDF is the main practice asset for students.

Full chapter practice

The PDF covers notation, like terms, expansion, factorisation and fractional coefficients.

Printable answers

Unlike the webpage, the PDF keeps answers and explanations visible for offline use.

Teacher-friendly

Use the page to teach and project. Use the PDF for student attempts and follow-up marking.

Download Chapter 4 PDF
Teacher notes

How to use this page in class

Suggested lesson flow

1
Teach the conceptsUse notation, like-term and expansion sections as a clean projection canvas.
2
Run the Quick Self-CheckUse the score to decide whether students are ready for the PDF exercise.
3
Attempt the PDFUse the full worksheet for classwork, homework or revision.
4
Open toggles only after attemptsKeep answers hidden during teaching so students can think before checking.

What to watch for

Students combine unlike terms incorrectly.

Students forget to multiply every term inside a bracket.

Students lose negative signs during expansion.

Students factorise using a common factor but not the highest common factor.

Students mishandle fractional coefficients because they do not use common denominators.

Ready for Chapter 4 practice?

Download the PDF for the full worksheet, then use the self-check and diagnostic reveals when you need help with common algebra traps.

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