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.
Misconception check: the minus before (x+3)2 applies after the square is expanded. Do not change only the first term of the bracket.
1.3 Collecting like terms
Group coefficients with identical literal parts (same variables and exponents). Ensure linear terms, constants, and powers stay distinct.
2 Factorisation toolkit
Common factor:6x2−9x=3x(2x−3).
Grouping:ax+ay+bx+by=(a+b)x+(a+b)y=(a+b)(x+y).
Difference of squares:a2−b2=(a+b)(a−b).
Perfect square trinomials:x2+6x+9=(x+3)2.
Worked example: Factorise 12y2−27.
12y2−27=3(4y2−9)=3(2y+3)(2y−3).
Factorisation pattern checkpoint
Before factorising, scan for the simplest structure first. A common factor should be taken out before trying special products.
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: a2−b2=(a+b)(a−b).
Two square terms can split into conjugate brackets.
Using this pattern on a sum such as a2+b2.
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.
Worked check: 8x2−18 has a common factor first:
8x2−18=2(4x2−9)=2(2x+3)(2x−3).
Misconception check: factorisation can need more than one move. Stopping at 2(4x2−9) is only partly factorised because 4x2−9 is still a difference of squares.
3 Solving equations and inequalities
3.1 Linear equations
Maintain balance: whatever operation you apply to one side must be applied to the other.
Example: Solve 3(2x−1)=5(x+3).
6x−3=5x+15⟹6x−5x=15+3⟹x=18.
Verify by substitution: LHS =3(36−1)=105, RHS =5(21)=105.
3.1A Balance step checkpoint
Before moving terms, decide what is making the unknown hard to isolate. The first move should remove that obstacle without changing the solution set.
Equation type
First move
Why it helps
3(x−2)=18
Divide by 3 or expand cleanly.
Removes the multiplier before isolating x.
4x+5=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.
Misconception check: changing sides is not magic. Each "move" is really adding, subtracting, multiplying, or dividing both sides by the same non-zero value.
3.2 Inequalities
Flip the inequality sign only when multiplying or dividing by a negative number.
Example: Solve 4−3x>13.
−3x>9⟹x<−3.
Represent solutions on a number line with open (strict) or closed (inclusive) dots.
3.2A Inequality sign checkpoint
Before writing the next line, name the operation applied to both sides. The sign only flips when that operation reverses the order of numbers.
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.
Worked check:
−2x≤10⟹x≥−5
The sign flips because both sides were divided by −2. Check with a test value: x=0 satisfies 0≥−5, but substituting into the original gives 0≤10, which is true.
Common trap: do not flip the sign just because a negative term appears. Flip only when the step multiplies or divides both sides by a negative number.
3.3 Compound inequalities
For 2<3x+1≤11, isolate x:
2<3x+1≤11⟹1<3x≤10⟹31<x≤310.
4 Word problem modelling
4.1 Translation checkpoint
Before solving a word problem, turn each sentence into one algebraic job. This prevents the common mistake of writing an equation before the variable has a clear meaning.
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 301 tank per minute.
Are all rates measured in the same unit per minute?
Common trap: do not let x mean different things in the same solution. If x starts as the original price, it cannot later become the discounted price.
Worked example - Rate mixture
A tap fills a tank in 30 minutes. A drain empties the same tank in 45 minutes. Both are opened together and the tank already contains 40% of its capacity. How long to fill the tank?
Let the tank capacity be 1 unit of volume.
Tap rate: 301 per minute.
Drain rate: −451 per minute.
Net rate: 301−451=901 per minute.
Remaining volume: 0.6 unit.
Time required: t=9010.6=54 minutes.
Checklist for algebra word problems
Define variables clearly (e.g., let x be the number of weeks, not the value of goods).
Translate each sentence into an equation or inequality.
Keep units consistent and annotate the meaning of intermediate expressions.
Practice Quiz
Run through expansion, factorisation, balancing steps, and inequality checks with the interactive quiz before tackling the written set.
Try it yourself
Expand and simplify (x−4)(2x+5)−(x+3)2.
Factorise 15p2−21p.
Solve 35−2x=x−4.
Solve the inequality 2(3y−5)≤4−y and plot the solution on a number line.
Two numbers differ by 8 and their sum is 30. Form and solve simultaneous equations to find the numbers.
