IB Math 9
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IB Mathematics · Year 9
Core Mastery
Quiz
20 essential problems across all major units. Type your answer and hit Check. Short form is fine — just the number or expression.
20
Problems
8
Units
★★☆
Difficulty
Unit 1 — Number & Algebra
Indices, surds, standard form, sequences
01 Indices — Laws of Exponents
Number & Algebra
Simplify using index laws.
x3 × x5 ÷ x4
💡
Quick Memory Point
SAME BASE → ADD when ×, SUBTRACT when ÷
xa × xb = xa+b  |  xa ÷ xb = xa-b
Write as x^n (e.g. x^4)
💬 Explanation
Apply: multiply → add exponents, divide → subtract exponents.
x³ × x⁵ = x⁸, then x⁸ ÷ x⁴ = x⁴.
Simply: 3 + 5 − 4 = 4. Answer: x⁴.
02 Surds — Rationalising the Denominator
Number & Algebra
Rationalise the denominator and simplify fully.
6√3 = ?
💡
Quick Memory Point
RATIONALISE → Multiply top AND bottom by √denominator
√a × √a = a (surd × itself = whole number!)
Write as 2√3 (no fraction in denominator)
💬 Explanation
Multiply numerator & denominator by √3:
6/√3 × √3/√3 = 6√3/3 = 2√3
The denominator becomes a rational number. Answer: 2√3.
03 Standard Form — Tricky Multiplication
Number & Algebra
⚠️ Common mistake: forgetting to adjust the power!
(4 × 103) × (7 × 104) = ? × 10?
💡
Quick Memory Point
STANDARD FORM: first number must be 1 ≤ a < 10
If product ≥ 10, move decimal → ADD 1 to power
Answer format: a × 10^n (e.g. 2.8 × 10^8)
💬 Explanation
Multiply numbers: 4 × 7 = 28.
Add powers: 10³ × 10⁴ = 10⁷.
So 28 × 10⁷, but 28 ≥ 10 → write as 2.8 × 10⁸.
(Move decimal 1 place left → add 1 to exponent.) Answer: 2.8 × 10⁸.
04 Arithmetic Sequences — nth Term
Number & Algebra
Find the 15th term of the sequence.
3, 7, 11, 15, 19, …
The sequence increases by the same amount each time. Identify the first term and the common difference, then use the formula.
💡
Quick Memory Point
ARITHMETIC: u_n = a + (n − 1)d
a = first term, d = common difference, n = term number
Just write the number
💬 Explanation
a = 3, d = 4. Use formula: u₁₅ = 3 + (15−1)×4 = 3 + 56 = 59.
Trap: students often use n=15 instead of (n−1)=14. Answer: 59.
Unit 2 — Equations & Inequalities
Linear, simultaneous, inequalities on number line
05 Simultaneous Equations — Substitution
Equations
Solve the simultaneous equations. Find the value of x.
2x + y = 10
y = 3x − 5
💡
Quick Memory Point
SUBSTITUTION: plug one equation INTO the other
y is already isolated → swap y directly
Enter x value only
💬 Explanation
Sub y = 3x−5 into first equation:
2x + (3x−5) = 10 → 5x − 5 = 10 → 5x = 15 → x = 3.
Answer: x = 3.
06 Inequalities — Sign Flip!
Equations
⚠️ Most-missed: flipping the sign when dividing by a negative!
−3x + 6 > 15     Solve for x.
💡
Quick Memory Point
FLIP THE SIGN whenever you × or ÷ by NEGATIVE
−3x > 9 → ÷ by −3 → x < −3 (flip!)
Write as x < or x > value
💬 Explanation
−3x + 6 > 15 → −3x > 9 → x < −3.
CRITICAL: dividing by −3 flips > to <. This is the #1 error! Answer: x < −3.
07 Linear Equations — Variables Both Sides
Equations
Solve for x.
5(x − 2) = 3x + 4
💡
Quick Memory Point
EXPAND first → COLLECT x-terms on LEFT → numbers on RIGHT
Expand → Collect → Divide
Just the number
💬 Explanation
Expand: 5x − 10 = 3x + 4.
Collect: 2x = 14 → x = 7.
Answer: 7.
Unit 3 — Geometry & Trigonometry
Pythagoras, SOHCAHTOA, area, angle rules
08 Pythagoras — Find the Hypotenuse
Geometry
A right triangle has legs 5 cm and 12 cm. Find the hypotenuse.
a² + b² = c²   →   5² + 12² = c²
💡
Quick Memory Point
PYTHAGORAS: a² + b² = c² (c = hypotenuse, ALWAYS longest)
Memorise: 3-4-5 and 5-12-13 are Pythagorean triples!
Answer in cm (whole number)
💬 Explanation
25 + 144 = 169. Then c = √169 = 13.
This is the 5-12-13 Pythagorean triple. Answer: 13 cm.
09 Trigonometry — SOHCAHTOA
Geometry
⚠️ Tricky: picking the wrong ratio!
In a right triangle, the hypotenuse = 10 cm, and one angle = 30°. Find the side opposite that angle.
sin(30°) = oppositehypotenuse  =  opp10
💡
Quick Memory Point
SOH: Sin = Opposite/Hypotenuse
CAH: Cos = Adjacent/Hypotenuse
TOA: Tan = Opposite/Adjacent
sin 30° = 0.5 (memorise!)
Answer in cm
💬 Explanation
Need: Opposite, have: Hypotenuse → use SIN.
sin(30°) = opp/10 → opp = 10 × 0.5 = 5.
Answer: 5 cm.
10 Circle — Arc Length
Geometry
Find the arc length of a sector with radius 6 cm and angle 60°.
Arc length = θ360 × 2πr
💡
Quick Memory Point
SECTOR = fraction of full circle
Arc = (angle/360) × circumference
Area = (angle/360) × πr²
Leave in terms of π (e.g. 2π)
💬 Explanation
Arc = (60/360) × 2π(6) = (1/6) × 12π = 2π.
Answer: 2π cm.
Unit 4 — Quadratics
Factorising, completing the square, quadratic formula
11 Factorising Quadratics
Quadratics
Factorise completely.
x² − 5x + 6
💡
Quick Memory Point
Find TWO numbers that: MULTIPLY to c (+6), ADD to b (−5)
Answer form: (x + p)(x + q)
Write as (x−?)(x−?)
💬 Explanation
Need two numbers: × gives +6, + gives −5 → −2 and −3.
Check: (−2)×(−3) = 6 ✓, (−2)+(−3) = −5 ✓.
Answer: (x−2)(x−3).
12 Quadratic Formula — Discriminant
Quadratics
⚠️ Most missed: sign errors under the square root!
For x² − 4x + 1 = 0, find the discriminant (b² − 4ac).
Δ = b² − 4ac     a=1, b=−4, c=1
💡
Quick Memory Point
Δ > 0 → 2 real roots | Δ = 0 → 1 root | Δ < 0 → no real roots
Watch: b = −4, so b² = (−4)² = +16, NOT −16!
Just the number
💬 Explanation
Δ = (−4)² − 4(1)(1) = 16 − 4 = 12.
Since Δ > 0, there are 2 distinct real roots. Common error: writing (−4)² = −16.
Answer: 12.
13 Difference of Two Squares
Quadratics
Factorise using the difference of two squares.
9x² − 25
💡
Quick Memory Point
DOTS: a² − b² = (a+b)(a−b)
Spot it: two perfect squares, MINUS sign between them
Write as (?+?)?-?)
💬 Explanation
Recognise: 9x² = (3x)² and 25 = 5².
Apply DOTS: (3x)² − 5² = (3x+5)(3x−5).
Answer: (3x+5)(3x−5).
Unit 5 — Functions & Graphs
Gradient, y-intercept, transformations, domain/range
14 Linear Graphs — Gradient
Functions
Find the gradient of the line through (2, 3) and (6, 11).
m = y₂ − y₁x₂ − x₁
💡
Quick Memory Point
GRADIENT = Rise ÷ Run = (change in y) ÷ (change in x)
Order matters: keep y₂−y₁ and x₂−x₁ in the SAME order
Just the number
💬 Explanation
m = (11−3)/(6−2) = 8/4 = 2.
Answer: 2.
15 Function Notation & Substitution
Functions
⚠️ Students forget: f(3) means SUBSTITUTE x = 3!
Given f(x) = 2x² − 3x + 1, find f(3).
💡
Quick Memory Point
f(x) is just a name for a formula
f(3) → replace EVERY x with 3, then calculate
Just the number
💬 Explanation
f(3) = 2(3)² − 3(3) + 1 = 2(9) − 9 + 1 = 18 − 9 + 1 = 10.
Answer: 10.
Unit 6 — Statistics & Probability
Mean, median, mode, probability rules
16 Probability — Combined Events
Statistics & Probability
⚠️ Trap: "at least one" vs "exactly one"
A bag has 3 red and 7 blue balls. Two are drawn without replacement. Find the probability both are red.
💡
Quick Memory Point
WITHOUT replacement → denominator DECREASES each draw
P(A and B) = P(A) × P(B|A) — second pick depends on first!
Write as a fraction (e.g. 1/15)
💬 Explanation
P(1st red) = 3/10. After removing one red, P(2nd red) = 2/9.
P(both red) = 3/10 × 2/9 = 6/90 = 1/15.
Answer: 1/15.
17 Mean from Frequency Table
Statistics & Probability
Calculate the mean from this frequency table.
Score: 2, 3, 4, 5
Freq:  1, 3, 4, 2
──────────────────
Total frequency = 10
💡
Quick Memory Point
MEAN from table = Σ(x × f) ÷ Σf
Multiply each value by its frequency, sum up, divide by total
Decimal answer
💬 Explanation
Σ(xf) = 2×1 + 3×3 + 4×4 + 5×2 = 2 + 9 + 16 + 10 = 37.
Mean = 37/10 = 3.7.
Answer: 3.7.
Unit 7 — Ratio, Proportion & Percentages
Percentage change, reverse percentage, direct/inverse proportion
18 Reverse Percentage — Original Value
Ratio & Proportion
⚠️ Top mistake: dividing by the percentage instead of finding 1%!
After a 20% increase, a price is £84. What was the original price?
💡
Quick Memory Point
REVERSE %: New value = Original × multiplier
+20% → multiplier = 1.2 → Original = New ÷ 1.2
NEVER subtract 20% from the new price!
Answer in £
💬 Explanation
After +20%, the price is 120% of original.
Original = 84 ÷ 1.2 = 70.
Common wrong answer: 84 × 0.8 = 67.2 — this is WRONG because 20% of 84 ≠ 20% of original.
Answer: £70.
19 Direct Proportion — Formula
Ratio & Proportion
y is directly proportional to x. When x = 4, y = 20. Find y when x = 7.
y ∝ x  →  y = kx
💡
Quick Memory Point
DIRECT: y = kx → find k first, then use it
k = y/x = constant (never changes!)
Just the number
💬 Explanation
Find k: k = 20/4 = 5. So y = 5x.
When x = 7: y = 5 × 7 = 35.
Answer: 35.
20 Geometric Sequences — nth Term
Number & Algebra
🏆 Final Challenge — Geometric Sequence
A geometric sequence starts: 2, 6, 18, 54, … Find the 6th term.
u_n = a × r^(n−1)     r = common ratio
💡
Quick Memory Point
GEOMETRIC: MULTIPLY by r each time (not add!)
r = any term ÷ previous term | u_n = a × r^(n−1)
Just the number
💬 Explanation
a = 2, r = 6/2 = 3.
u₆ = 2 × 3^(6−1) = 2 × 3⁵ = 2 × 243 = 486.
Answer: 486.
🎓
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