Cambridge IGCSE Mathematics (0580) – Number, Algebra, Geometry & Real‑World Applications
Learning Objectives
- Apply numerical techniques to everyday situations involving money, time, travel and finance.
- Develop fluency with core Number concepts: fractions, decimals, percentages, ratios, rates, sets, powers, roots, indices, standard form, estimation, HCF/LCM, prime, square & cube numbers, irrational numbers and reciprocals.
- Use algebraic methods to simplify expressions, solve linear & quadratic equations, inequalities, simultaneous equations, sequences and interpret straight‑line graphs.
- Work confidently with coordinate geometry, basic geometry terminology, constructions, mensuration and introductory trigonometry.
- Extend learning to surds, exponential growth/decay, compound interest and other Extended‑tier topics.
1. Number (Core C1.1 – C1.17 & Extended)
1.1 Whole Numbers, Prime, Square & Cube Numbers
- Identify prime numbers (only 1 and itself as factors).
- Recognise square numbers (1, 4, 9, 16, …) and cube numbers (1, 8, 27, 64, …).
- Find the highest common factor (HCF) and lowest common multiple (LCM) of two or more integers.
Example:
Numbers 12 and 18 → HCF = 6, LCM = 36.
1.2 Fractions, Decimals & Reciprocals
- Place value, rounding, upper & lower bounds.
- Operations with proper, improper and mixed fractions.
- Convert between fractions, decimals and percentages.
- Reciprocal of a non‑zero number = 1 ÷ number.
Example:
\(\displaystyle \frac{3}{4}=0.75\) and \(\displaystyle \frac{1}{0.75}= \frac{4}{3}=1.\overline{3}\).
1.3 Percentages
- Percentage = \(\dfrac{\text{Part}}{\text{Whole}}\times100\%\).
- Finding a percentage of a quantity, percentage increase/decrease, reverse‑percentage problems.
- Successive percentage changes: multiply the factors.
Discount: \(\displaystyle \text{Discounted price}=P\Bigl(1-\frac{d}{100}\Bigr)\).
Profit / Loss: \(\displaystyle \text{Profit}=S-C,\;\text{Loss}=C-S\).
Reverse‑percentage: Original price = \(\displaystyle \frac{\text{Sale price}}{1-\frac{d}{100}}\).
1.4 Ratios & Rates (C1.9 – C1.10)
- Write, simplify and divide quantities in a given ratio.
- Rates are ratios with a unit (e.g., km h⁻¹, $ kg⁻¹).
- Direct proportion: \(y=kx\).
Inverse proportion: \(y=\dfrac{k}{x}\).
1.5 Sets (Core C1.2)
- Notation: \(\{a,b,c\}\) (set), \(\in\) (member of), \(\subset\) (subset).
- Two‑set Venn diagrams – union \(\cup\), intersection \(\cap\), complement \('\).
- Cardinality \(|A|\) = number of elements in set \(A\).
Example:
\(A=\{1,2,3,4\},\;B=\{3,4,5\}\).
\(A\cup B=\{1,2,3,4,5\},\;A\cap B=\{3,4\},\;|A\cup B|=5.\)
1.6 Powers, Indices & Roots (C1.11 – C1.13)
- Positive, zero and negative indices: \(a^0=1\), \(a^{-n}=1/a^{\,n}\).
- Product, quotient and power‑of‑a‑power rules.
- Square \(\sqrt{a}\) and cube \(\sqrt[3]{a}\) roots; higher roots as needed.
Example: \((2^{3})^{2}=2^{6}=64.\)
1.7 Surds & Irrational Numbers (Extended E1.18)
- Surd = irrational root that cannot be simplified to a terminating decimal (e.g., \(\sqrt{2},\;\sqrt[3]{5}\)).
- Simplify surds: factor out squares/cubes.
\(\sqrt{72}= \sqrt{36\times2}=6\sqrt{2}\).
- Rationalise denominators: \(\dfrac{1}{\sqrt{3}}=\dfrac{\sqrt{3}}{3}\).
1.8 Standard Form & Estimation (C1.14 – C1.15)
- Standard form: \(a\times10^{n}\) with \(1\le a<10\).
- Upper & lower bounds, significant figures, rounding.
Example: \(4\,560\,000 = 4.56\times10^{6}\).
Measurement 3.27 m to 2 sf → bounds \(3.25\le x<3.35\) m.
1.9 Simple & Compound Percentage Problems (Extended)
- Successive changes: multiply each factor \((1\pm p/100)\).
- Compound growth/decay: \(A=P\left(1\pm\frac{r}{100}\right)^{t}\).
2. Algebra (Core C2.1 – C2.10 & Extended)
2.1 Substituting & Simplifying (C2.1)
Replace variables with numbers and use the order of operations.
Example: For \(3x^{2}-2x+5\) with \(x=2\):
\(3(2)^{2}-2(2)+5=13.\)
2.2 Expanding & Factorising (C2.2 – C2.3)
- Common factor: \(3a+3b=3(a+b)\).
- Bracket expansion: \((x+4)(x-3)=x^{2}+x-12\).
- Quadratic factorising: \(x^{2}+5x+6=(x+2)(x+3)\).
- Completing the square (Extended E2.5): \(x^{2}+6x= (x+3)^{2}-9\).
2.3 Solving Linear Equations (C2.4)
Standard form \(ax+b=c\). Isolate \(x\) by inverse operations.
Example: \(4x-7=9\) → \(x=4\).
2.4 Solving Inequalities (C2.6)
- Treat like equations, but reverse the inequality sign when multiplying/dividing by a negative number.
- Represent solutions on a number line (open/closed circles).
Example: \(3x-5>4\) → \(x>\dfrac{9}{3}=3\). Solution: \((3,\infty)\).
2.5 Sequences (C2.7)
- Arithmetic sequence: \(a_{n}=a_{1}+(n-1)d\).
- Geometric sequence (Extended): \(a_{n}=a_{1}r^{\,n-1}\).
Example: First term 7, common difference 4 → \(a_{5}=7+4(4)=23\).
2.6 Simultaneous Linear Equations (C2.5)
- Substitution or elimination.
Example (Elimination):
\[
\begin{aligned}
2x+3y&=12\\
4x-y&=5
\end{aligned}
\]
Multiply second by 3: \(12x-3y=15\). Add to first: \(14x=27\) → \(x=\dfrac{27}{14}\).
Substitute: \(y=\dfrac{12-2x}{3}= \dfrac{12-2\cdot27/14}{3}= \dfrac{30}{42}= \dfrac{5}{7}\).
2.7 Quadratic Equations (Extended E2.2)
- Factorising (when possible).
- Completing the square.
- Quadratic formula: \(x=\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}\).
Example: \(x^{2}-5x+6=0\) → \((x-2)(x-3)=0\) → \(x=2\) or \(x=3\).
2.8 Linear Graphs (C2.8 – C2.10)
- Equation of a straight line: \(y=mx+c\).
- Gradient \(m=\dfrac{\Delta y}{\Delta x}\); intercept \(c\) is the value of \(y\) when \(x=0\).
- Reading values from a graph and solving equations graphically.
- Parallel lines: same gradient; perpendicular lines: product of gradients \(-1\).
Example: Line through \((0,2)\) and \((4,10)\) → \(m=\dfrac{10-2}{4-0}=2\) → equation \(y=2x+2\).
3. Coordinate Geometry (Core C3.1 – C3.7)
3.1 Plotting Points & Reading Coordinates
- Ordered pairs \((x,y)\) on a Cartesian plane.
- Quadrants, axes and origin.
3.2 Gradient & Equation of a Straight Line
- Gradient formula \(m=\dfrac{y_{2}-y_{1}}{x_{2}-x_{1}}\).
- Point‑slope form: \(y-y_{1}=m(x-x_{1})\).
- Slope‑intercept form: \(y=mx+c\).
3.3 Mid‑point Formula (C3.3)
\[
M\left(\frac{x_{1}+x_{2}}{2},\;\frac{y_{1}+y_{2}}{2}\right)
\]
3.4 Distance Formula (C3.4)
\[
d=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}
\]
3.5 Parallel & Perpendicular Lines (C3.5 – C3.6)
- Parallel: same gradient \(m\).
- Perpendicular: gradients satisfy \(m_{1}m_{2}=-1\).
3.6 Equation of a Circle (Extended E3.7)
\[
(x-h)^{2}+(y-k)^{2}=r^{2}
\] where \((h,k)\) is the centre and \(r\) the radius.
4. Geometry (Core C4.1 – C4.8)
4.1 Geometrical Terminology
| Term | Definition / Example |
|---|
| Point | Exact location, no size. |
| Line | Infinite length, no thickness. |
| Line segment | Part of a line bounded by two points. |
| Ray | Starts at a point and extends infinitely in one direction. |
| Angle | Space between two rays sharing a vertex (measured in degrees). |
| Polygon | Closed figure formed by straight lines (triangle, quadrilateral, …). |
| Circle | Set of points equidistant from a centre. |
4.2 Angles & Bearing (C4.2 – C4.4)
- Types: acute < 90°, right = 90°, obtuse > 90° & < 180°, reflex > 180°.
- Sum of interior angles of an \(n\)-gon: \((n-2)\times180^\circ\).
- Bearings measured clockwise from north (0°–360°).
4.3 Construction Tasks (C4.5)
- Construct a triangle given three sides (SSS) or two sides and the included angle (SAS).
- Construct a perpendicular bisector of a given segment.
4.4 Similar & Congruent Figures (C4.6 – C4.7)
- Similar: same shape, proportional sides, equal corresponding angles.
- Scale factor \(k\): side lengths multiplied by \(k\); areas multiplied by \(k^{2}\).
- Congruent: identical size and shape (all corresponding sides and angles equal).
4.5 Pythagoras’ Theorem (C6.1 – Core)
In a right‑angled triangle, \(a^{2}+b^{2}=c^{2}\) where \(c\) is the hypotenuse.
5. Mensuration (Core C5.1 – C5.5)
5.1 Perimeter & Circumference
- Rectangle: \(P=2(l+w)\).
- Circle: \(C=2\pi r =\pi d\).
5.2 Area of Plane Figures
- Triangle: \(\displaystyle A=\frac{1}{2}bh\) or \(\displaystyle A=\frac{1}{2}ab\sin C\).
- Parallelogram: \(A=bh\).
- Trapezium: \(\displaystyle A=\frac{1}{2}(b_{1}+b_{2})h\).
- Circle: \(\displaystyle A=\pi r^{2}\).
5.3 Surface Area & Volume of Solids (Core)
- Cuboid: \(SA=2(lw+lh+wh),\; V=lwh\).
- Cylinder: \(SA=2\pi r h+2\pi r^{2},\; V=\pi r^{2}h\).
- Cone: \(SA=\pi r(r+\sqrt{r^{2}+h^{2}}),\; V=\dfrac{1}{3}\pi r^{2}h\).
- Sphere: \(SA=4\pi r^{2},\; V=\dfrac{4}{3}\pi r^{3}\).
5.4 Compound Shapes
Break a figure into known shapes, calculate each area/volume, then add or subtract as required.
6. Trigonometry (Core C6.1 – C6.2 & Extended)
6.1 Right‑Angle Trigonometric Ratios
- For a right‑angled triangle with angle \(\theta\):
\(\displaystyle \sin\theta=\frac{\text{opposite}}{\text{hypotenuse}},\;
\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}},\;
\tan\theta=\frac{\text{opposite}}{\text{adjacent}}.\)
- Use calculator in degree mode unless otherwise specified.
6.2 Solving Practical Problems
- Finding heights or distances using bearings and trigonometric ratios.
- Angle of elevation/depression problems.
Example: A lighthouse is 30 m tall. From a boat the angle of elevation to the top is \(15^\circ\). Distance from boat to base \(= \dfrac{30}{\tan15^\circ}\approx 109.5\) m.
7. Money – Real‑World Applications
7.1 Currency Conversion & Budgeting
| From | To | Rate (1 unit = ?) |
|---|
| GBP (£) | USD ($) | 1.28 |
| GBP (£) | EUR (€) | 1.18 |
| USD ($) | EUR (€) | 0.92 |
Convert £120 to dollars: \(120\times1.28= \$153.60\).
7.2 Budgeting Example
Weekly allowance £25. Expenditure: transport £8, food £6.
Remaining: \(25-8-6 = £11\).
7.3 Profit, Loss & Mark‑up
- Mark‑up % = \(\dfrac{S-C}{C}\times100\%\).
- Discount % = \(\dfrac{d}{100}\) where \(d\) is the percent reduction.
Example: Cost = £45, selling price = £55 → Mark‑up = \(\dfrac{10}{45}\times100\%≈22.2\%\).
8. Time – Conversions & Clock Skills
8.1 Unit Conversions
\[
1\text{ hour}=60\text{ min}=3600\text{ s},\qquad
1\text{ day}=24\text{ h}=1440\text{ min}=86\,400\text{ s}.
\]
| Unit | Equivalent in Seconds |
|---|
| 1 minute | 60 |
| 1 hour | 3 600 |
| 1 day | 86 400 |
| 1 week | 604 800 |
8.2 12‑hour ↔ 24‑hour Clock
- pm → add 12 (except 12 pm stays 12:00).
- am → keep hour (except 12 am = 00:00).
Examples: 3:45 pm = 15:45; 22:10 = 10:10 pm; 12:00 am = 00:00.
8.3 Time‑Word Problems
Work in minutes or seconds, then convert back.
Example: Film starts 19:20, lasts 2 h 15 min → finishes 21:35.
9. Travel – Speed, Distance & Time
9.1 Fundamental Relationship
\[
v=\frac{d}{t},\qquad d=v\,t,\qquad t=\frac{d}{v}
\]
\(v\) in km h⁻¹, \(d\) in km, \(t\) in hours.
9.2 Worked Example
Car travels 150 km at 75 km h⁻¹ → \(t=150/75=2\) h.
If journey must be 1.5 h, required speed \(v=150/1.5=100\) km h⁻¹.
9.3 Timetable Calculations
| Service | Departure | Travel Time | Arrival |
|---|
| Train A | 08:15 | 2 h 30 min | 10:45 |
| Bus B | 09:00 | 1 h 45 min | 10:45 |
Interval between services: convert to minutes (08:15 = 495 min, 09:00 = 540 min) → 45 min.
9.4 Gradient Interpretation (Link to Algebra)
On a distance‑time graph, gradient = speed. Line through \((0,0)\) and \((3,150)\) → gradient \(=150/3=50\) km h⁻¹.
10. Finance – Interest, Loans & Growth
10.1 Simple Interest (Core)
\[
I = P\,r\,t
\]
\(P\) = principal, \(r\) = annual rate (decimal), \(t\) = years.
Example: £800 at 5 % for 3 years → \(I=800\times0.05\times3=£120\); total = £920.
10.2 Compound Interest (Core & Extended)
\[
A = P\left(1+\frac{r}{n}\right)^{n t}
\]
\(n\) = number of compounding periods per year.
Example: £1 200 at 4 % compounded quarterly for 2 years → \(A=1200(1+0.04/4)^{8}\approx£1\,299.48.\)
10.3 Exponential Growth / Decay (Extended)
\[
A = P\left(1+\frac{r}{100}\right)^{t}\quad\text{(growth)},
\qquad
A = P\left(1-\frac{r}{100}\right)^{t}\quad\text{(decay)}.
\]
10.4 Loan Repayment – Annuity Formula (Core)
\[
R = P\;\frac{r/12}{1-(1+r/12)^{-m}}
\]
\(R\) = monthly repayment, \(P\) = loan amount, \(r\) = annual rate (decimal), \(m\) = total months.
Example: £5 000 loan, 6 % p.a., 36 months → \(R\approx£152.99\) per month.
10.5 Investment & Depreciation (Extended)
- Future value of regular deposits (ordinary annuity):
\(A = d\;\dfrac{(1+r)^{n}-1}{r}\) where \(d\) is the deposit each period.
- Depreciation using straight‑line method:
\(\displaystyle \text{Annual depreciation}= \frac{\text{Cost}-\text{Residual}}{\text{Life}}\).
11. Quick Reference Tables
11.1 Common Formulas
| Topic | Formula |
|---|
| Percentage change | \( \text{New}=P\left(1\pm\frac{p}{100}\right) \) |
| Speed‑distance‑time | \(v=\dfrac{d}{t},\;d=vt,\;t=\dfrac{d}{v}\) |
| Simple interest | \(I=Prt\) |
| Compound interest | \(A=P\left(1+\dfrac{r}{n}\right)^{nt}\) |
| Gradient | \(m=\dfrac{y_{2}-y_{1}}{x_{2}-x_{1}}\) |
| Mid‑point | \(\left(\dfrac{x_{1}+x_{2}}{2},\dfrac{y_{1}+y_{2}}{2}\right)\) |
| Distance | \(d=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}\) |
| Area of triangle (SAS) | \(A=\tfrac12ab\sin C\) |
| Volume of cylinder | \(V=\pi r^{2}h\) |
| Quadratic formula | \(x=\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}\) |
11.2 Common Conversions
| From | To | Factor |
|---|
| Hours → Minutes | Minutes | \(\times60\) |
| Minutes → Seconds | Seconds | \(\times60\) |
| Days → Hours | Hours | \(\times24\) |
| £ → $ | $ | \(\times1.28\) |
| $ → € | € | \(\times0.92\) |
12. Tips for Exam Success
- Read the question carefully – identify which syllabus code(s) are being tested.
- Show all working; marks are awarded for method as well as final answer.
- Check units and convert early to avoid mistakes.
- For percentage and finance problems, keep intermediate results to 3‑4 sf then round only at the end.
- Use a calculator only where permitted; practice mental estimation for speed.
- When working with graphs, label axes clearly and use the correct scale.