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IGCSE Maths: Algebra and graphs

Topic 2 of the Cambridge IGCSE Mathematics (0580) syllabus: from substitution to differentiation. Here is what each part asks, a practice question for each, and the mistakes that cost the most marks.

Practise algebra and graphs as a game

The 13 parts of the topic

Each part with one practice question, made by the same engine your child plays. Parts marked Extended are for the Extended paper only.

2.1 Introduction to algebra: letters for numbers, substitution

p = 3 and q = −3.
Find 3p² − 2q.

  1. 21
  2. 87
  3. 24
  4. 33
Show the answer

33. 3 × (3)² = 27. 2 × −3 = −6. 27 − (−6) = 33.

2.2 Algebraic manipulation: collect like terms, expand, factorise, complete the square

Expand and simplify
(x − 6)(x − 5)

  1. x² − 11x + 30
  2. x² − 11x − 30
  3. x² + 30
  4. x² + 30x − 11
Show the answer

x² − 11x + 30. x × x = x². The x terms: −6x and −5x make −11x. The numbers: −6 × −5 = 30.

2.3 Algebraic fractions: simplify, add, subtract, multiply, divide Extended

Simplify (x/5) ÷ (x²/2).

  1. 2/(5x)
  2. x³/10
  3. 5x/2
  4. 2/5
Show the answer

2/(5x). Dividing is multiplying by the flipped SECOND fraction: (x/5) × (2/x²) = 2x/(5x²) = 2/(5x).

2.4 Indices II: the rules of indices with algebra, including negative and fractional indices

Simplify x⁵ × x⁴

  1. x¹
  2. 2x⁹
  3. x²⁰
  4. x⁹
Show the answer

x⁹. Multiplying: add the powers. 5 + 4 = 9.

2.5 Equations: construct; solve linear, fractional, simultaneous and quadratic equations; change the subject

Solve the simultaneous equations:
y = x² − 1
y = 7x − 13

  1. (3, 8) only
  2. (−4, 15) and (−3, 8)
  3. (3, 8) and (4, 15)
  4. (3, 15) and (4, 8)
Show the answer

(3, 8) and (4, 15). Set them equal: x² − 1 = 7x − 13, so x² − 7x + 12 = 0 = (x − 3)(x − 4). x = 3 or 4; put each back in for y. A line can cross a curve twice.

2.6 Inequalities: number lines, solving, and regions on a graph (Extended)

How many integers satisfy
−5 ≤ x < 1 ?

  1. 3
  2. 5
  3. 7
  4. 6
Show the answer

6. From −5 up to 0: that is 6 integers. < leaves the end number out, ≤ keeps it.

2.7 Sequences: continue, term-to-term and nth term for linear, quadratic, cubic and exponential

4, 9, 16, 25, 36, …
Which is the nth term of this sequence?

  1. n² + 2n + 2
  2. n² + 2n + 1
  3. n² + 3n + 1
  4. 5n − 1
Show the answer

n² + 2n + 1. The differences are 5, 7, 9, 11: they go up by 2 each time, so the rule has n² in it. Take n² away from each term and what is left is linear: n² + 2n + 1.

2.8 Direct and inverse proportion in algebra Extended

y is directly proportional to x.
When x = 7, y = 42.
Find y when x = 6.

  1. 6
  2. 36
  3. 49
  4. 41
Show the answer

36. y = kx. From the first pair, k = 42 ÷ 7 = 6. So y = 6 × 6 = 36.

2.9 Graphs in practical situations: travel graphs, conversion graphs, speed-time graphs

The questions for this part come with a picture or are played as a game, so they are best tried in Huh Lab.

2.10 Graphs of functions: tables of values, drawing and reading, solving equations from a graph

The curve y = a × bˣ passes through (0, 5) and (1, 10).
Find b.

  1. 7
  2. 2
  3. 10
  4. 5
Show the answer

2. At x = 0, y = a × b⁰ = a, so a = 5. At x = 1, y = a × b = 10, so b = 10 ÷ 5 = 2.

2.11 Sketching curves: recognise and sketch linear, quadratic, cubic, reciprocal and exponential graphs; turning points

y = (x + 1)² − 2
Where is its turning point?

  1. (1, −2)
  2. (−1, −2)
  3. (−1, 2)
  4. (−2, −1)
Show the answer

(−1, −2). The bracket is zero, and y is smallest, when x = −1. Then y = −2. The sign inside the bracket is the OPPOSITE of the x-coordinate.

2.12 Differentiation: gradients, stationary points and their nature Extended

y = x² + 8x + 4
Find the x-coordinate of the stationary point.

  1. x = −4
  2. x = 8
  3. x = 4
  4. x = −8
Show the answer

x = −4. dy/dx = 2x + 8. At a stationary point the gradient is 0: 2x = −8, so x = −4.

2.13 Functions: notation, domain and range, inverse and composite functions Extended

f(x) = 1 / (x + 6)
Which value of x must be excluded from the domain?

  1. x = 0
  2. x = 1
  3. x = −6
  4. x = 6
Show the answer

x = −6. You cannot divide by zero. The bottom is zero when x = −6.

The mistakes that cost the most marks

Huh Lab checks every wrong answer against known mistakes like these, and goes back to the step behind the one your child keeps making.

Loses the negative sign

Use a number line: below zero the numbers keep counting, in the other direction.

Gets the end number wrong

< and > leave the end number OUT. ≤ and ≥ keep it in. Check each end separately.

Moves a term across without reversing it

Whatever is done to x must be UNDONE on the other side: + becomes −, × becomes ÷.

Picks the wrong side of a boundary line

Take a point in the middle of the shaded region: it must make every inequality true.

Treats a sequence as linear from its first gap

Check the gaps between terms. If the gaps themselves change, the rule is not "add the same each time".

Scales in a straight line when the rule has a square or a root

Find k first from y = kx² (or k√x). Doubling x does not double y here.

Stops one step before the end

After removing the number, there is still the multiplying to undo: divide as well.

Multiplies by the index

A power of 2/3 means "cube root, then square". It is not "times two thirds".

Other IGCSE Maths topics

Number17 parts Coordinate geometry7 parts Geometry8 parts Mensuration5 parts Trigonometry6 parts Transformations and vectors4 parts Probability4 parts Statistics7 parts

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