The Complete Homework Book
Week 1 · Module 1.1
01 / Numbers
The Fundamental Theorem of Arithmetic, canonical factorisation, HCF and LCM
Every integer greater than 1 is either prime or can be written as a product of primes in exactly one way. This is the Fundamental Theorem of Arithmetic, and it is the reason prime factorisation is a reliable structural fingerprint of a number.
A prime factor tree splits a number repeatedly until every branch terminates in a prime. Collecting the terminal primes and writing repeated primes as powers gives the canonical form, e.g. 72 = 2³ × 3².
The Highest Common Factor (HCF) is built from the primes the numbers share, each taken to its lowest power. The Lowest Common Multiple (LCM) takes every prime that appears in either number, each to its highest power.
Because HCF collects the shared core and LCM collects the full union, HCF(a, b) × LCM(a, b) = a × b for any two positive integers.
Key Relationships
3 Bronze · 4 Silver · 3 Gold
List all factors of 60.
BronzeWrite 126 as a product of primes in index form.
BronzeState whether 87 is prime, with justification.
BronzeFind HCF(140, 196) using prime factorisation.
SilverFind LCM(12, 20, 45).
SilverTwo buses leave a station every 18 minutes and every 30 minutes. They left together at 07:00, when will they next leave together?
SilverThe HCF of two numbers is 6 and their LCM is 180. One number is 36. Find the other.
SilverA rectangular floor measures 360 cm by 600 cm. It is to be covered completely with identical square tiles, laid in rows and columns with no tiles cut. Find the side length of the largest possible square tile, and the number of tiles needed.
GoldN = 2² × 3³ × 5. How many factors does N have, and how many are even?
GoldFind the smallest positive integer k such that 1 350 × k is a perfect cube.
GoldSelf-Audit Checkpoint
Week 2 · Module 1.2
01 / Numbers
Part-to-part, part-to-whole, the unitary method, direct and inverse proportion, scale maps
A ratio a : b compares parts. Converting to part-to-whole fractions (a/(a+b)) is the single most useful move when a total is given.
The unitary method finds the value of one part or one unit first, then scales. It converts every ratio problem into a multiplication.
Direct proportion: y = kx, so y/x is constant. Inverse proportion: y = k/x, so xy is constant. Identifying which invariant holds is the whole decision.
Scale on a map, 1 : n, means one unit on the map represents n identical units in reality. Convert units before scaling, not after.
Key Relationships
3 Bronze · 4 Silver · 3 Gold
Simplify the ratio 48 : 84.
BronzeShare 120 stickers in the ratio 5 : 3.
Bronze6 identical pens cost $27. Find the cost of 10 pens.
BronzeIn a ratio 3 : 8, the larger share is 96. Find the total.
SilverA map has scale 1 : 50 000. Two villages are 6.2 cm apart on the map. Find the real distance in km.
Silvery is directly proportional to x, and y = 36 when x = 8. Find y when x = 14.
SilverA recipe for 4 people uses 600 g of rice. How much for 10 people?
SilverThe ratio of red to blue counters is 4 : 3. After adding 15 blue counters the ratio becomes 1 : 2. Find the original number of red counters.
GoldThe time t for a journey varies inversely with speed v. At 60 km/h it takes 50 min. Find the speed needed for 40 min.
GoldC and D share profit in the ratio 5 : 3. D receives $1 800 less than C. Find the total profit.
GoldSelf-Audit Checkpoint
Week 3 · Module 1.3
01 / Numbers
Four-quadrant operations, double negatives, absolute value and the coordinate plane
Subtracting a negative reverses direction twice, so a − (−b) = a + b. Multiplying or dividing two numbers of the same sign gives a positive result; opposite signs give a negative result.
Absolute value |x| is the undirected distance of x from zero, so |−7| = 7. Distance on a number line is |a − b|, which never depends on the order.
The coordinate plane is divided into four quadrants by the axes. Signs are (+,+), (−,+), (−,−), (+,−) reading anticlockwise from the top right.
Horizontal distance between points is |x₂ − x₁| and vertical distance is |y₂ − y₁|; the midpoint averages coordinates.
Key Relationships
3 Bronze · 4 Silver · 3 Gold
Evaluate −17 + 9.
BronzeEvaluate −9 − (−4).
BronzeEvaluate |−15| + |3 − 10|.
BronzeEvaluate (−36) ÷ (−4) × (−3).
SilverState the quadrant of each point: (5, −3), (−2, −8), (−4, 6).
SilverThe temperature falls from 9 °C to −7 °C. Find the drop.
SilverFind the midpoint of P(−5, 6) and Q(9, −2).
SilverSolve |x + 3| = 7.
GoldA(−2, 4), B(5, 4) and C(5, −2) form a right triangle. Find its area.
GoldA submarine at −250 m rises 3 times in steps of 40 m, then dives 45 m. Find its final depth.
GoldSelf-Audit Checkpoint
Week 4 · Module 1.4
01 / Numbers
Index laws, fractional and negative indices, roots and standard form A × 10^k
Indices count repeated multiplication, so multiplying powers of the same base adds indices and dividing subtracts them: aᵐ × aⁿ = aᵐ⁺ⁿ and aᵐ ÷ aⁿ = aᵐ⁻ⁿ.
A power of a power multiplies indices: (aᵐ)ⁿ = aᵐⁿ. Any non-zero base to the power 0 equals 1, and a negative index is a reciprocal: a⁻ⁿ = 1/aⁿ.
Fractional indices are roots: a^(1/n) = ⁿ√a and a^(m/n) = (ⁿ√a)ᵐ. Taking the root first usually keeps the numbers small.
Scientific notation writes a number as A × 10^k where 1 ≤ A < 10. Multiplying adds the exponents; dividing subtracts them; always renormalise A back into range afterwards.
Key Relationships
3 Bronze · 4 Silver · 3 Gold
Simplify 3⁴ × 3².
BronzeEvaluate 5⁰ + 2⁻³.
BronzeWrite 0.000 78 in standard form.
BronzeSimplify (3a²b³)³.
SilverEvaluate 27^(2/3).
SilverSimplify (x⁸y⁻³) ÷ (x²y⁻⁷).
SilverCompute (4 × 10⁶) × (5 × 10⁻³) in standard form.
SilverSolve 3^(x − 1) = 81.
GoldSimplify (√72 + √32) ÷ √2.
GoldA grain of pollen has mass 4 × 10⁻¹² g. Find the mass of 5 × 10⁹ grains in standard form.
GoldSelf-Audit Checkpoint
Week 5 · Module 2.1
02 / Algebra
Double bracket expansion, common monomial factors, difference of two squares and trinomials
Expanding (x + a)(x + b) means every term in the first bracket multiplies every term in the second — the FOIL order (First, Outer, Inner, Last) is just a bookkeeping device for the four products.
The area model shows why: a rectangle of side (x + a) by (x + b) splits into four tiles of areas x², bx, ax and ab.
Factoring reverses expansion. Always look for a common monomial factor first, then check for a difference of two squares a² − b² = (a − b)(a + b), then attempt a trinomial split.
For x² + bx + c, look for two numbers with product c and sum b. For ax² + bx + c, split the middle term using two numbers with product ac and sum b, then factor by grouping.
Key Relationships
3 Bronze · 4 Silver · 3 Gold
Expand 5(2x − 9).
BronzeFactorise 15x² + 20x.
BronzeExpand (x + 7)(x + 2).
BronzeExpand and simplify (2x − 5)².
SilverFactorise 64x² − 25.
SilverFactorise x² − 10x + 21.
SilverExpand and simplify (x + 4)(x − 3) − (x − 2)².
SilverFactorise 6x² − 19x + 15.
GoldFactorise fully 2x³ − 50x.
GoldA rectangle has area 3x² + 10x + 8. Find expressions for its sides.
GoldSelf-Audit Checkpoint
Week 6 · Module 2.2
02 / Algebra
Brackets, variables on both sides, fractional terms and inequalities on number lines
An equation is a balance: whatever you do to one side you must do to the other. Expand brackets first, gather variables on one side, constants on the other, then divide.
Fractional equations are cleared by multiplying every term by the LCM of the denominators — every term, including the ones without a denominator.
Inequalities follow the same rules with one exception: multiplying or dividing by a negative number reverses the inequality sign.
On a number line, a solid dot means the endpoint is included (≤, ≥) and an open circle means it is excluded (<, >).
Key Relationships
3 Bronze · 4 Silver · 3 Gold
Solve 4x − 7 = 29.
BronzeSolve 2(x + 5) = 24.
BronzeSolve x/6 = 8.
BronzeSolve 7x − 3 = 3x + 17.
SilverSolve 3(3x − 2) = 2(x + 11).
SilverSolve (x + 3)/4 − (x − 2)/3 = 1.
SilverSolve 3x + 11 < 5 and show on a number line.
SilverSolve −5 ≤ 2x + 1 < 9.
GoldSolve 8 − 3(x − 2) > 5x − 2.
GoldThe perimeter of a rectangle is 66 cm and the length is 3 cm more than twice the width. Find the width.
GoldSelf-Audit Checkpoint
Week 7 · Module 2.3
02 / Algebra
Completing the square, completed square form, deriving the quadratic formula and the discriminant
Completing the square rewrites x² + bx + c as (x + b/2)² + (c − b²/4). The bracket captures the symmetry of the parabola, so the vertex can be read off directly.
In completed square form y = (x + p)² + q, the turning point is (−p, q) and the axis of symmetry is x = −p.
Completing the square on ax² + bx + c = 0 produces the quadratic formula x = (−b ± √(b² − 4ac)) / 2a — the formula is not a separate rule, it is the general result of the same procedure.
The discriminant Δ = b² − 4ac decides the root structure: Δ > 0 gives two distinct real roots, Δ = 0 gives one repeated root, Δ < 0 gives no real roots.
Key Relationships
3 Bronze · 4 Silver · 3 Gold
Solve x² − 7x + 12 = 0 by factorising.
BronzeWrite x² + 8x + 3 in the form (x + p)² + q.
BronzeFind Δ for x² − 6x + 9 = 0 and state the number of real roots.
BronzeSolve x² + 3x − 18 = 0.
SilverSolve 2x² + 5x − 12 = 0 by factorising.
SilverSolve x² − 4x − 1 = 0, leaving answers in surd form.
SilverState the vertex of y = 3(x + 2)² + 7.
SilverFind k so that x² + kx + 16 = 0 has exactly one root.
GoldSolve 3x² − 4x − 5 = 0 to 2 dp using the formula.
GoldA rectangle has perimeter 30 cm and area 54 cm². Find its dimensions.
GoldSelf-Audit Checkpoint
Week 8 · Module 2.4
02 / Algebra
Slope-intercept form, gradients, intercepts, parallel and perpendicular lines
Gradient measures steepness: m = Δy/Δx = (y₂ − y₁)/(x₂ − x₁). A positive gradient rises left to right, a negative one falls.
In y = mx + c, m is the gradient and c is the y-intercept. The x-intercept is found by setting y = 0.
Parallel lines have equal gradients (m₁ = m₂). Perpendicular lines have gradients whose product is −1 (m₁m₂ = −1), so each is the negative reciprocal of the other.
Given a point and a gradient, y − y₁ = m(x − x₁) constructs the equation without solving for c separately.
Key Relationships
3 Bronze · 4 Silver · 3 Gold
State the gradient and y-intercept of y = −2x + 7.
BronzeFind the gradient through (2, 3) and (6, 23).
BronzeFind the x-intercept of y = 4x − 12.
BronzeFind the equation of the line with gradient 4 through (1, −2).
SilverAre y = 3x − 2 and 6x − 2y = 5 parallel? Justify.
SilverFind the gradient of a line perpendicular to y = ¾x − 1.
SilverRewrite 2x + 5y = 20 in slope-intercept form and state both intercepts.
SilverFind the equation of the line through (2, −3) and (2, 6), and describe it.
GoldLine L passes through (0, 3) and is perpendicular to x + 3y = 6. Find its equation.
GoldFind the intersection point of y = 2x + 1 and y = −x + 10.
GoldSelf-Audit Checkpoint
Week 9 · Module 2.5
02 / Algebra
Common difference, the general term and the sum of an arithmetic series
An arithmetic sequence adds a fixed common difference d each step. Because the step is constant, the nth term is linear in n: Tₙ = a + (n − 1)d.
The common difference is found by subtracting consecutive terms: d = T₂ − T₁ = T₃ − T₂.
Gauss pairing gives the sum: pairing the first and last terms repeatedly shows Sₙ = n/2 (a + l) where l is the last term, equivalently Sₙ = n/2 [2a + (n − 1)d].
Sequences model any situation with a constant increment — savings with a fixed deposit, seat rows increasing by a fixed number, staircase patterns.
Key Relationships
3 Bronze · 4 Silver · 3 Gold
Find d and the next term of 5, 9, 13, 17, …
BronzeFind T₁₂ for a = 3, d = 5.
BronzeFind the sum 1 + 2 + 3 + … + 40.
BronzeFind a formula for Tₙ of 30, 25, 20, 15, …
SilverWhich term of the sequence 4, 11, 18, … equals 186?
SilverFind S₁₅ for the sequence 8, 11, 14, …
SilverT₄ = 17 and T₁₀ = 47. Find a and d.
SilverHow many terms of 3, 7, 11, … are needed for a sum of 820?
GoldA stadium section has 20 rows of seats. The first row has 15 seats, and each row after that has 2 more seats than the row in front of it. Find the total number of seats in the section.
GoldThe sum of the first n terms of an arithmetic sequence is Sₙ = 3n² + n. Find a and d.
GoldSelf-Audit Checkpoint
Week 10 · Module 2.6
02 / Algebra
Function notation, domain and range, rational expressions and average rate of change
f(x) names a rule; f(3) means substitute 3 for every x. The domain is the set of permitted inputs, and the range is the set of resulting outputs.
Rational expressions are undefined where the denominator is zero, so those values are excluded from the domain. Simplify by factorising numerator and denominator and cancelling common factors.
The average rate of change of f between x = a and x = b is (f(b) − f(a))/(b − a) — the gradient of the secant line joining the two points on the curve.
For a straight line the average rate of change is constant; for a curve it depends on the interval chosen, which is the first step towards calculus.
Key Relationships
3 Bronze · 4 Silver · 3 Gold
For f(x) = 4x − 3, find f(5).
BronzeFor g(x) = x² − 2, find g(−4).
BronzeState the domain of f(x) = 1/(x + 5).
BronzeFor f(x) = 3x² − 2x, find the average rate of change from x = 0 to x = 2.
SilverSimplify (x² − 9)/(x² + 7x + 12).
SilverFor f(x) = 7 − 3x, solve f(x) = 28.
SilverState the range of f(x) = x² + 3 for real x.
SilverFor f(x) = 3x², find the average rate of change from x = a to x = a + h, where h ≠ 0.
GoldA ball's height is h(t) = 30t − 5t² metres. Find the average velocity from t = 1 to t = 4 s.
GoldFor f(x) = (3x − 2)/(x − 1), find f(3) and state the domain and the range.
GoldSelf-Audit Checkpoint
Week 11 · Module 3.1
03 / Space
Parallel line angle pairs, polygon interior sums and exterior angles
When a transversal cuts parallel lines, corresponding angles are equal (F-shape), alternate angles are equal (Z-shape) and co-interior angles sum to 180° (C-shape).
Angles on a straight line sum to 180°, angles at a point sum to 360°, and vertically opposite angles are equal.
The interior angles of an n-sided polygon sum to (n − 2) × 180°, because the polygon splits into n − 2 triangles.
The exterior angles of any convex polygon always sum to 360°, so a regular n-gon has exterior angle 360°/n.
Key Relationships
3 Bronze · 4 Silver · 3 Gold
Find x if angles on a straight line are x and 62°.
BronzeFind the interior angle sum of an octagon.
BronzeFind the exterior angle of a regular dodecagon (12 sides).
BronzeParallel lines cut by a transversal give co-interior angles 4x and 5x. Find x.
SilverA regular polygon has exterior angle 20°. Find n and the interior angle.
SilverIn a triangle the angles are x, 2x and 3x. Find each angle.
SilverFind the interior angle of a regular hexagon.
SilverA polygon's interior angles sum to 1 440°. Find n.
GoldTwo parallel lines are cut by a transversal; one angle is (3x + 20)° and its corresponding angle is (5x − 10)°. Find x and the angle.
GoldAn irregular hexagon has five angles 110°, 125°, 130°, 95° and 140°. Find the sixth.
GoldSelf-Audit Checkpoint
Week 12 · Module 3.2
03 / Space
2D and 3D Pythagoras, SOH-CAH-TOA and angles of elevation and depression
In a right triangle, a² + b² = c² where c is the hypotenuse (always opposite the right angle). Rearranged, a shorter side is c² − b².
In three dimensions, the space diagonal of a cuboid follows d² = l² + w² + h², which is Pythagoras applied twice.
SOH-CAH-TOA: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. Label the triangle relative to the angle you are using before choosing a ratio.
An angle of elevation is measured upward from the horizontal; an angle of depression is measured downward. They are equal when measured between the same two points.
Key Relationships
3 Bronze · 4 Silver · 3 Gold
Find the hypotenuse of a right triangle with legs 8 cm and 15 cm.
BronzeFind the missing length if c = 29 and a = 20.
BronzeFind x if sin 30° = x/22.
BronzeFind θ if tan θ = 9/4, to 1 dp.
SilverA 5 m ladder rests against a wall with its foot 1.4 m out. Find the height reached.
SilverFind the side adjacent to a 35° angle if the hypotenuse is 20 cm, to 1 dp.
SilverFind the space diagonal of a cube of side 9 cm, to 2 dp.
SilverFrom a cliff 120 m high, the angle of depression to a boat is 28°. Find the horizontal distance, to the nearest metre.
GoldA cuboid is 10 × 6 × 5 cm (height 5). Find the angle the space diagonal makes with the base, to 1 dp.
GoldAn isosceles triangle has equal sides 17 cm and base 16 cm. Find its height and area.
GoldSelf-Audit Checkpoint
Week 13 · Module 3.3a
03 / Space
The Sine Rule, the ambiguous SSA case and the area formula ½ab sin C
In any triangle, a/sin A = b/sin B = c/sin C. Use it whenever you have a matched side-and-opposite-angle pair.
For missing sides put the sides on top; for missing angles invert the ratios so the sines sit on top: sin A/a = sin B/b.
The ambiguous case arises with SSA data: because sin θ = sin(180° − θ), a second obtuse solution may exist if it keeps the angle sum below 180°.
The area of a triangle with two sides and the included angle is ½ab sin C — the included angle is essential.
Key Relationships
3 Bronze · 4 Silver · 3 Gold
A = 35°, B = 70°, a = 12. Find b to 1 dp.
BronzeFind the area of a triangle with sides 9 cm and 12 cm and an angle of 50° in between, to 1 dp.
BronzeA = 30°, a = 8, b = 12. Find sin B.
BronzeIn triangle ABC, C = 52°, c = 18, A = 58°. Find a to 1 dp.
Silvera = 16, A = 40°, b = 12. Find B to 1 dp.
SilverFind the third angle and remaining side: A = 50°, B = 68°, a = 11. Give c to 1 dp.
SilverThe area of a triangle is 24 cm² with sides 12 cm and 8 cm. Find the possible angles in between (it can be acute or obtuse).
Silvera = 8, b = 12, A = 32°. Determine whether two triangles exist and find both B values.
GoldTwo observers 600 m apart see a balloon between them at elevations 35° and 50°. Find the balloon height, to the nearest metre.
GoldIn triangle ABC, a = 14, A = 110°, b = 10. Explain why only one triangle exists and find B.
GoldSelf-Audit Checkpoint
Week 14 · Module 3.3b
03 / Space
The Cosine Rule for SAS and SSS, and three-figure navigation bearings
The Cosine Rule c² = a² + b² − 2ab cos C generalises Pythagoras; when C = 90°, cos C = 0 and it collapses to a² + b² = c².
Use the Cosine Rule when you have two sides and the included angle (SAS) to find the third side, or three sides (SSS) to find any angle.
For SSS, rearrange: cos C = (a² + b² − c²)/(2ab). A negative value means the angle is obtuse.
Bearings are measured clockwise from true north and always written with three figures, e.g. 065°. Sketch the north lines before computing anything.
Key Relationships
3 Bronze · 4 Silver · 3 Gold
Find c if a = 6, b = 8, C = 60°.
BronzeWrite the bearing of due south in three figures.
BronzeFind cos C if a = 5, b = 12, c = 13.
BronzeFind the largest angle of a triangle with sides 6, 8, 11, to 1 dp.
SilverFind b if a = 12, c = 9, B = 110°, to 2 dp.
SilverA boat sails 15 km on 050° then 20 km on 140°. Find the distance from the start.
SilverThe back bearing of 230° is what?
SilverSides 7, 10 and 14 form a triangle. Find its largest angle and its area, to 1 dp.
GoldTwo towns: B is 30 km from A on bearing 040°; C is 50 km from A on bearing 130°. Find BC to 1 dp.
GoldIn triangle ABC, a = 8, b = 15, c = 17. Verify it is right-angled and find its area.
GoldSelf-Audit Checkpoint
Week 15 · Module 3.4
03 / Space
Prisms, cylinders, pyramids, cones and spheres, including composite solids
Any prism or cylinder has volume V = A_base × h, because identical cross-sections stack uniformly.
Pyramids and cones taper to a point and take exactly one third of the matching prism or cylinder: V = ⅓A_base × h.
A sphere has V = 4/3 πr³ and surface area 4πr². A hemisphere has half the volume and a curved surface of 2πr², plus πr² if the flat face is included.
For composite solids, add volumes but be careful with surface area: any face hidden at a join is not part of the exterior surface.
Key Relationships
3 Bronze · 4 Silver · 3 Gold
Find the volume of a cuboid 5 × 7 × 8 cm.
BronzeFind the volume of a cylinder r = 4, h = 9, to 1 dp.
BronzeFind the volume of a sphere r = 6, to 1 dp.
BronzeFind the surface area of a cylinder r = 5, h = 8, to 1 dp.
SilverFind the volume of a square pyramid with base 12 cm and height 10 cm.
SilverA cone has r = 6 and h = 8. Find its volume, to 1 dp.
SilverA triangular prism has base area 18 cm² and length 12 cm. Find its volume.
SilverA hemisphere of radius 5 cm sits on a cylinder of the same radius and height 12 cm. Find the total exterior surface area (including the base), to 1 dp.
GoldA sphere of radius 3 cm is melted and recast into a cylinder of radius 2 cm. Find the cylinder height.
GoldAn inverted cone-shaped container has a radius of 8 cm and a vertical height of 12 cm. Water fills the cone from its vertex up to half its vertical height. Find the volume of water, to 1 decimal place.
GoldSelf-Audit Checkpoint
Week 16 · Module 3.5
03 / Space
Congruence tests and the scale factor ladder k, k² and k³
Congruent figures are identical in size and shape. The valid tests are SSS, SAS, ASA (and AAS) and RHS for right triangles. AAA is not a congruence test — it only proves similarity.
Similar figures have equal corresponding angles and proportional corresponding sides. The constant of proportionality is the linear scale factor k.
Areas scale by k² and volumes (and masses of the same material) scale by k³. This is the single most tested idea in the topic.
To find k from areas take the square root; from volumes take the cube root, then use k for lengths.
Key Relationships
3 Bronze · 4 Silver · 3 Gold
Two similar rectangles have corresponding sides 4 cm and 14 cm. Find the scale factor k from the smaller rectangle to the larger one.
BronzeIf k = 4, state the area and volume ratios.
BronzeName the congruence test for two triangles with two equal sides and the equal angle between them.
BronzeSimilar triangles: sides 3, 4, 5 and the largest side of the second is 20. Find its other sides.
SilverTwo similar cones have radii 3 and 6. The smaller has volume 40π. Find the larger.
SilverTwo similar posters have areas 150 cm² and 600 cm². Find the length scale factor k from the smaller poster to the larger one.
SilverTwo triangles both have angles 50°, 60° and 70°. Explain why they must be similar but need not be congruent.
SilverTwo similar solids have surface areas 50 cm² and 162 cm². The smaller has volume 125 cm³. Find the larger volume.
GoldA model boat at scale 1 : 20 has a 8 cm² sail. Find the real sail area in cm².
GoldTriangle ABC has DE parallel to BC with AD = 3, DB = 9 and DE = 4. Find BC.
GoldSelf-Audit Checkpoint
Week 17 · Module 4.1
04 / Data
Sample spaces, tree diagrams, independent events and mutually exclusive addition
Theoretical probability is favourable outcomes divided by equally likely total outcomes. Experimental (relative frequency) probability is observed successes divided by trials, and it approaches the theoretical value as trials increase.
A sample space lists every possible outcome. Two-way tables and tree diagrams are structured ways of enumerating it without omission.
Independent events do not influence each other, so P(A and B) = P(A) × P(B). On a tree, multiply along branches.
Mutually exclusive events cannot both happen, so P(A or B) = P(A) + P(B). On a tree, add across completed paths. In general P(A or B) = P(A) + P(B) − P(A and B).
Key Relationships
3 Bronze · 4 Silver · 3 Gold
A fair die is rolled. Find P(a number greater than 4).
BronzeP(late) = 0.42. Find P(not late).
BronzeTwo coins are tossed. List the sample space and find P(exactly one head).
BronzeA spinner has 10 equal sectors numbered 1–10. Find P(prime).
SilverIndependent events with P(A) = 0.6, P(B) = 0.3. Find P(A and B) and P(A or B).
SilverA coin is tossed 200 times and shows heads 116 times. Find the experimental probability and compare with theory.
SilverA bag has 5 red and 3 blue balls; two are drawn without replacement. Find P(both red).
SilverTwo dice are rolled. Find P(sum = 7) and P(sum > 10).
GoldP(A) = 0.4, P(B) = 0.5, P(A or B) = 0.7. Are A and B independent?
GoldA test is 95% accurate. Two independent tests are run. Find P(at least one error).
GoldSelf-Audit Checkpoint
Week 18 · Module 4.2
04 / Data
The counting principle, factorials, permutations and combinations
The fundamental counting principle: if a first choice has m options and a second has n, the pair has m × n outcomes. This extends to any number of independent stages.
n! = n × (n − 1) × … × 1 counts the arrangements of n distinct objects, with 0! defined as 1.
Permutations count ordered selections: nPr = n!/(n − r)!. Combinations count unordered selections: nCr = n!/(r!(n − r)!).
The decisive question is always: does order matter? Seating, ranking, passwords and podium places are permutations; committees, teams and hands of cards are combinations.
Key Relationships
3 Bronze · 4 Silver · 3 Gold
Evaluate 5!.
BronzeA menu has 3 starters and 6 mains. How many different two-course meals can you have?
BronzeEvaluate 8C3.
BronzeHow many 3-letter arrangements from the 7 distinct letters of JUMPING (using each once)?
SilverHow many ways can 6 people be seated in a row?
SilverFrom 9 books, choose 3 to take on holiday. How many different selections can you have?
SilverA lock code uses 3 digits 0–9 with repetition allowed. How many different possible codes are there?
SilverHow many different arrangements of the letters of LETTER are there?
GoldFrom 6 boys and 4 girls, choose a team of 4 with at least 3 girls.
GoldSix people sit in a row but two specific people must sit together. How many arrangements can there be?
GoldSelf-Audit Checkpoint
Week 19 · Module 4.3
04 / Data
Five-number summary, interquartile range, box plots and the 1.5 × IQR outlier rule
The five-number summary is minimum, Q₁, median, Q₃, maximum. It must be computed from ordered data.
The median splits the data in half; Q₁ is the median of the lower half and Q₃ the median of the upper half. IQR = Q₃ − Q₁ measures the spread of the middle 50%.
A box-and-whisker plot draws the box from Q₁ to Q₃ with the median inside, and whiskers to the extreme values that are not outliers.
A value is an outlier if it lies below Q₁ − 1.5 × IQR or above Q₃ + 1.5 × IQR. IQR is preferred over range because it resists outliers.
Key Relationships
3 Bronze · 4 Silver · 3 Gold
Find the median of 4, 7, 10, 13, 18, 21, 25.
BronzeFind the range of 14, 6, 31, 22, 9.
BronzeFind Q₁ and Q₃ of 3, 5, 7, 9, 11, 13, 15, 17.
BronzeFind the IQR of 10, 14, 15, 19, 22, 27, 30.
SilverQ₁ = 30, Q₃ = 50. Find both outlier fences.
SilverSketch the box plot for the summary 4, 10, 15, 22, 30 and state the IQR.
SilverTwo teams have equal median scores but IQRs of 8 and 20. Compare consistency.
SilverFor 5, 9, 10, 12, 14, 16, 18, 19, 21, 40, find the five-number summary and identify outliers.
GoldA box plot has whiskers at 10 and 50, a box from 18 to 30 and a median of 22. Describe the skew.
GoldA data set has IQR 10 and Q₁ 30. A new value 12 is added (assume the quartiles do not change). Is it an outlier?
GoldSelf-Audit Checkpoint
Week 20 · Module 4.4
04 / Data
Scatter plots, line of best fit, correlation coefficient r, causation and sampling design
A scatter plot shows paired data. Its pattern reveals the direction (positive/negative), form (linear/curved) and strength of association.
The line of best fit minimises the vertical residuals. Predicting inside the data range is interpolation (reasonably safe); outside it is extrapolation (unreliable).
Pearson's r lies between −1 and 1. Values near ±1 indicate strong linear association; near 0 indicates weak linear association — which does not rule out a curved relationship.
Correlation is not causation: a lurking third variable or reverse causation may explain the pattern. Sound conclusions require a well-designed sample: random, representative and large enough, with stratified sampling used when the population has distinct subgroups.
Key Relationships
3 Bronze · 4 Silver · 3 Gold
Describe the correlation if r = 0.88.
BronzeFor ŷ = 8 + 4x, predict y when x = 6.
BronzeMore firefighters at a fire are linked with more damage. Do firefighters cause the damage?
BronzeObserved y = 38 while ŷ = 42. Find the residual.
SilverData ranges from x = 5 to x = 20. Classify predictions at x = 12 and x = 2.
SilverA survey about reading habits is conducted only inside a library. Name the bias.
SilverA school has 600 boys and 400 girls. Design a stratified sample of 40.
SilverFor ŷ = 75 − 1.8x with r = −0.82, interpret the gradient and strength.
GoldA scatter plot forms a clear U-shape but r ≈ 0. Explain.
GoldPoints (1, 7) and (6, 22) lie on the line of best fit. Find its equation and predict y at x = 4.
GoldSelf-Audit Checkpoint
Week 21 · Module 5.1
05 / Finance
Discounts, markups, profit margins, VAT/GST and reverse percentages
A percentage change is applied with a multiplier: an increase of r% multiplies by (1 + r/100) and a decrease by (1 − r/100). Chained changes multiply their multipliers.
Markup is measured on cost price; profit margin is measured on selling price. The same dollar profit gives a larger markup percentage than margin percentage.
VAT/GST is added to the pre-tax price, so tax-inclusive price = pre-tax × (1 + rate).
Reverse percentages undo a multiplier: divide, do not subtract. If a price after a 20% discount is $96, the original is 96 ÷ 0.8 = $120.
Key Relationships
3 Bronze · 4 Silver · 3 Gold
Find 15% of $360.
BronzeIncrease $180 by 35%.
BronzeAn $80 jacket is discounted 25%. Find the sale price.
BronzeCost $60, selling price $81. Find the markup percentage.
SilverA price is $221 after a 15% discount. Find the original price.
SilverAdd 12% GST to $425.
SilverCost $90, selling price $120. Find the profit margin.
SilverA price rises 10% then falls 10%. Find the overall percentage change.
GoldA shop must make a 20% margin on a $240 cost item. Find the selling price.
GoldAfter a 12% pay rise, a salary is $56 000. Find the rise in dollars.
GoldSelf-Audit Checkpoint
Week 22 · Module 5.2
05 / Finance
Linear interest I = Prt, compound growth A = P(1 + r/n)^(nt), EAR and the Rule of 72
Simple interest is earned only on the original principal, so the balance grows linearly: I = Prt and A = P(1 + rt).
Compound interest is earned on the accumulated balance, so growth is exponential: A = P(1 + r/n)^(nt), where n is the number of compounds per year.
The effective annual rate converts any compounding frequency into a single yearly figure: EAR = (1 + r/n)ⁿ − 1. More frequent compounding gives a higher EAR for the same nominal rate.
The Rule of 72 estimates doubling time: years ≈ 72 ÷ (annual percentage rate). It is an approximation, useful for sanity-checking an exponential answer.
Key Relationships
3 Bronze · 4 Silver · 3 Gold
Find the simple interest on $3 000 at 5% p.a. for 4 years.
BronzeFind the total amount for $2 500 at 4% simple for 5 years.
BronzeFind the value of $2 000 after 2 years at 5% compounded annually.
Bronze$5 000 at 6% p.a. compounded annually for 4 years, to the cent.
SilverEstimate the doubling time at 6% p.a. using the Rule of 72.
Silver$8 000 at 4% compounded quarterly for 3 years.
SilverFind the EAR for 8% p.a. compounded monthly.
SilverHow long for $2 000 to reach $3 000 at 7% compounded annually? Give whole years.
GoldCompare $5 000 for 10 years at 6%: simple versus annual compounding. Find the difference.
GoldWhat principal grows to $15 000 in 5 years at 8% compounded semi-annually?
GoldSelf-Audit Checkpoint
Week 23 · Module 5.3
05 / Finance
The 50/30/20 framework, fixed and variable costs, emergency funds and debt amortisation
A budget starts from net (after-tax) income. The 50/30/20 framework allocates 50% to needs, 30% to wants and 20% to savings and debt repayment.
Fixed costs (rent, insurance, loan repayments) stay constant each period; variable costs (food, fuel, entertainment) change with behaviour and are where adjustments are made first.
An emergency fund is usually 3–6 months of essential expenses, held liquid. It is sized from needs, not from total income.
In an amortisation schedule each payment is split into interest on the outstanding balance and principal reduction. Early payments are interest-heavy; as the balance falls, more of each payment reduces the principal.
Key Relationships
3 Bronze · 4 Silver · 3 Gold
Net income is $3 200. Find the 50/30/20 allocations.
BronzeClassify a phone contract and entertainment as fixed or variable.
BronzeIncome $2 900, expenses $3 150. Find the net cash flow.
BronzeMonthly needs are $1 600. Find a 3-month emergency fund.
SilverAn $8 000 loan at 12% p.a. has a $300 monthly payment. Each payment first covers that month's interest; the rest reduces the principal (the amount owed). Find how much the principal is reduced by in month 1.
SilverNet income $3 800 with needs $2 280. What percentage goes to needs, and does it pass 50/30/20?
SilverSavings are $450 per month. How long to reach an $8 100 goal?
SilverIncome $4 000; fixed $2 100; variable $1 500. Find savings, the savings rate, and the cut needed to reach 20%.
GoldFor a $10 000 loan at 12% p.a. with $400 monthly payments, find the balance after 2 months.
GoldA student earns $2 000 net, spends $1 300 on needs and saves 20%. Find the wants budget and whether needs fit 50%.
GoldSelf-Audit Checkpoint
Appendix
Complete worked solutions for all 230 homework questions.
1. Answer: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
2. Answer: 2 × 3² × 7
3. Answer: Not prime: 87 = 3 × 29
4. Answer: 28
5. Answer: 180
6. Answer: 08:30 (after 90 min)
7. Answer: 30
8. Answer: 120 cm; 15 tiles
9. Answer: 24 factors; 16 even
10. Answer: k = 20
1. Answer: 4 : 7
2. Answer: 75 and 45
3. Answer: $45
4. Answer: 132
5. Answer: 3.1 km
6. Answer: 63
7. Answer: 1 500 g
8. Answer: 12
9. Answer: 75 km/h
10. Answer: $7 200
1. Answer: −8
2. Answer: −5
3. Answer: 22
4. Answer: −27
5. Answer: Q4, Q3, Q2
6. Answer: 16 °C
7. Answer: (2, 2)
8. Answer: x = 4 or x = −10
9. Answer: 21 square units
10. Answer: −175 m
1. Answer: 3⁶ = 729
2. Answer: 1 1/8
3. Answer: 7.8 × 10⁻⁴
4. Answer: 27a⁶b⁹
5. Answer: 9
6. Answer: x⁶y⁴
7. Answer: 2 × 10⁴
8. Answer: x = 5
9. Answer: 10
10. Answer: 2 × 10⁻² g
1. Answer: 10x − 45
2. Answer: 5x(3x + 4)
3. Answer: x² + 9x + 14
4. Answer: 4x² − 20x + 25
5. Answer: (8x − 5)(8x + 5)
6. Answer: (x − 3)(x − 7)
7. Answer: 5x − 16
8. Answer: (2x − 3)(3x − 5)
9. Answer: 2x(x − 5)(x + 5)
10. Answer: (3x + 4) and (x + 2)
1. Answer: x = 9
2. Answer: x = 7
3. Answer: x = 48
4. Answer: x = 5
5. Answer: x = 4
6. Answer: x = 5
7. Answer: x < −2
8. Answer: −3 ≤ x < 4
9. Answer: x < 2
10. Answer: 10 cm
1. Answer: x = 3 or x = 4
2. Answer: (x + 4)² − 13
3. Answer: Δ = 0, one repeated root
4. Answer: x = 3 or x = −6
5. Answer: x = 3/2 or x = −4
6. Answer: x = 2 ± √5
7. Answer: (−2, 7)
8. Answer: k = ±8
9. Answer: x ≈ 2.12 or x ≈ −0.79
10. Answer: 6 cm by 9 cm
1. Answer: m = −2, c = 7
2. Answer: 5
3. Answer: (3, 0)
4. Answer: y = 4x − 6
5. Answer: Yes, both m = 3
6. Answer: −4/3
7. Answer: y = −⅖x + 4; (10, 0) and (0, 4)
8. Answer: x = 2, a vertical line
9. Answer: y = 3x + 3
10. Answer: (3, 7)
1. Answer: d = 4, next 21
2. Answer: 58
3. Answer: 820
4. Answer: Tₙ = 35 − 5n
5. Answer: the 27th
6. Answer: 435
7. Answer: a = 2, d = 5
8. Answer: 20 terms
9. Answer: 680
10. Answer: a = 4, d = 6
1. Answer: 17
2. Answer: 14
3. Answer: all real x except −5
4. Answer: 4
5. Answer: (x − 3)/(x + 4), x ≠ −3, −4
6. Answer: x = −7
7. Answer: y ≥ 3
8. Answer: 6a + 3h
9. Answer: 5 m/s
10. Answer: f(3) = 3.5; domain x ≠ 1; range y ≠ 3
1. Answer: 118°
2. Answer: 1 080°
3. Answer: 30°
4. Answer: 20°
5. Answer: n = 18, interior 160°
6. Answer: 30°, 60°, 90°
7. Answer: 120°
8. Answer: 10
9. Answer: x = 15, angle 65°
10. Answer: 120°
1. Answer: 17 cm
2. Answer: 21
3. Answer: 11
4. Answer: 66.0°
5. Answer: 4.8 m
6. Answer: 16.4 cm
7. Answer: 15.59 cm
8. Answer: 226 m
9. Answer: 23.2°
10. Answer: 15 cm; 120 cm²
1. Answer: 19.7
2. Answer: 41.4 cm²
3. Answer: 0.75
4. Answer: 19.4
5. Answer: 28.8°
6. Answer: C = 62°, c ≈ 12.7
7. Answer: 30° or 150°
8. Answer: Two: B ≈ 52.6° or 127.4°
9. Answer: ≈ 265 m
10. Answer: B ≈ 42.2°, unique
1. Answer: √52 ≈ 7.2
2. Answer: 180°
3. Answer: 0
4. Answer: 102.6°
5. Answer: ≈ 17.29
6. Answer: 25 km
7. Answer: 050°
8. Answer: ≈ 109.6°; area ≈ 33.0
9. Answer: 58.3 km
10. Answer: Right angle at C; area 60
1. Answer: 280 cm³
2. Answer: 452.4 cm³
3. Answer: 904.8 cm³
4. Answer: 408.4 cm²
5. Answer: 480 cm³
6. Answer: 301.6 cm³
7. Answer: 216 cm³
8. Answer: ≈ 612.6 cm²
9. Answer: 9 cm
10. Answer: ≈ 100.5 cm³
1. Answer: 3.5
2. Answer: 16 and 64
3. Answer: SAS
4. Answer: 12 and 16
5. Answer: 320π
6. Answer: 2
7. Answer: Equal angles fix shape, not size
8. Answer: 729 cm³
9. Answer: 3 200 cm²
10. Answer: 16
1. Answer: 1/3
2. Answer: 0.58
3. Answer: HH, HT, TH, TT; 1/2
4. Answer: 2/5
5. Answer: 0.18 and 0.72
6. Answer: 0.58 vs 0.5
7. Answer: 5/14
8. Answer: 1/6 and 1/12
9. Answer: Yes
10. Answer: 0.0975
1. Answer: 120
2. Answer: 18
3. Answer: 56
4. Answer: 210
5. Answer: 720
6. Answer: 84
7. Answer: 1 000
8. Answer: 180
9. Answer: 25
10. Answer: 240
1. Answer: 13
2. Answer: 25
3. Answer: Q₁ = 6, Q₃ = 14
4. Answer: 13
5. Answer: 0 and 80
6. Answer: IQR = 12
7. Answer: The IQR 8 team is more consistent
8. Answer: 5, 10, 15, 19, 40; 40 is an outlier
9. Answer: Right (positive) skew
10. Answer: Yes
1. Answer: Strong positive linear
2. Answer: 32
3. Answer: No — a lurking variable (size of the fire)
4. Answer: −4
5. Answer: Interpolation; extrapolation
6. Answer: Selection bias (unrepresentative sample)
7. Answer: 24 boys, 16 girls
8. Answer: Each unit of x lowers y by 1.8; strong negative
9. Answer: r measures only linear association
10. Answer: ŷ = 3x + 4; 16
1. Answer: $54
2. Answer: $243
3. Answer: $60
4. Answer: 35%
5. Answer: $260
6. Answer: $476
7. Answer: 25%
8. Answer: 1% decrease
9. Answer: $300
10. Answer: $6 000
1. Answer: $600
2. Answer: $3 000
3. Answer: $2 205
4. Answer: $6 312.38
5. Answer: ≈ 12 years
6. Answer: $9 014.60
7. Answer: ≈ 8.30%
8. Answer: 6 years
9. Answer: ≈ $954.24
10. Answer: $10 133.46
1. Answer: $1 600 / $960 / $640
2. Answer: Phone contract fixed; entertainment variable
3. Answer: −$250
4. Answer: $4 800
5. Answer: $220
6. Answer: 60% — over target
7. Answer: 18 months
8. Answer: $400, 10%, cut $400
9. Answer: $9 397
10. Answer: $300 wants; needs are 65% — over
Curriculum Architecture
01 / Numbers
Factors, Prime Trees, Ratios, Signed Coordinates & Index Laws
02 / Algebra
Quadratics, Formula Derivations, Linear Graphs, Sequences & Rates of Change
03 / Space
Angle Deductions, 3D Pythagoras, Sine/Cosine Rules & Scale Similarity
04 / Data
Combinatorics, Quartile Spreads, Box Plots & Bivariate Regression
05 / Finance
Commercial Percentages, Compound Interest Trajectories & Budgeting
Vector