Cambridge IGCSE Mathematics (0580) Syllabus

Mathematics (0580) syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Mathematics (0580) in Cambridge IGCSE, not a summary of it.

1. 1

   Number

   6 topics

   • Types of number and operations
     • Natural numbers, integers, primes, squares, cubes
     • HCF and LCM by prime factorisation
     • Order of operations and use of brackets
   • Fractions, decimals and percentages
     • Converting between forms
     • Percentage increase, decrease and reverse percentages
     • Recurring decimals to fractions
   • Ratio, proportion and rate
     • Sharing in a given ratio
     • Direct and inverse proportion
     • Unit conversions and compound measures
   • Standard form and surds
     • Writing numbers in standard form
     • Calculations in standard form
     • Simplifying and rationalising surds
   • Indices and limits of accuracy
     • Laws of indices including negative and fractional
     • Upper and lower bounds
   • Money and finance
     • Simple and compound interest
     • Exchange rates and profit/loss
2. 2

   Algebra and Graphs

   6 topics

   • Manipulating expressions
     • Expanding brackets and factorising
     • Algebraic fractions
   • Equations and inequalities
     • Linear and simultaneous equations
     • Quadratic equations by factorising, formula and completing the square
     • Linear inequalities and number lines
   • Sequences
     • nth term of linear and quadratic sequences
     • Geometric and special sequences
   • Functions
     • Function notation and composite functions
     • Inverse functions
   • Graphs of functions
     • Linear, quadratic, cubic, reciprocal and exponential graphs
     • Gradient and equation of a straight line
     • Estimating gradients of curves and area under graphs
   • Differentiation
     • Differentiating powers of x
     • Turning points and rates of change
3. 3

   Geometry and Mensuration

   5 topics

   • Angles and constructions
     • Angles on lines, at a point and in parallel lines
     • Angles in polygons
     • Ruler-and-compass constructions and loci
   • Circle geometry
     • Circle theorems
     • Tangent and chord properties
   • Similarity and congruence
     • Congruent triangle conditions
     • Areas and volumes of similar shapes
   • Mensuration
     • Perimeter and area of plane shapes
     • Surface area and volume of solids
     • Arc length and sector area
   • Transformations and vectors
     • Reflection, rotation, translation and enlargement
     • Vector notation and operations
4. 4

   Trigonometry

   4 topics

   • Right-angled trigonometry
     • Pythagoras' theorem
     • Sine, cosine and tangent ratios
     • Angles of elevation and depression
   • Non-right-angled triangles
     • Sine rule
     • Cosine rule
     • Area of a triangle using ½ab sin C
   • Trigonometric graphs and equations
     • Graphs of sine, cosine and tangent
     • Solving simple trigonometric equations
   • Bearings and 3D problems
     • Three-figure bearings
     • Trigonometry in three dimensions
5. 5

   Statistics and Probability

   4 topics

   • Data collection and display
     • Bar charts, pie charts and pictograms
     • Histograms with unequal intervals
     • Scatter diagrams and correlation
   • Averages and spread
     • Mean, median and mode
     • Range and interquartile range
     • Cumulative frequency and box plots
   • Probability fundamentals
     • Theoretical and experimental probability
     • Probability of combined events
   • Probability diagrams
     • Tree diagrams
     • Venn diagrams and set notation

Mathematics (0580) flashcards for Cambridge IGCSE

25 of 55 cards from the Mathematics (0580) deck — real questions with worked answers.

1. What is the difference between a rational and an irrational number?

   A rational number can be written as a fraction $\frac{p}{q}$ where $p,q$ are integers and $q \neq 0$ (its decimal terminates or recurs). An irrational number cannot be written as such a fraction; its decimal is non-terminating and non-recurring, e.g. $\sqrt{2}$ and $\pi$.

2. Define the HCF and LCM of two numbers, and how do you find them from prime factorisations?

   The HCF (Highest Common Factor) is the largest number dividing both; take the product of common prime factors to the lowest power. The LCM (Lowest Common Multiple) is the smallest number both divide into; take every prime factor to the highest power. Also $\text{HCF}\times\text{LCM}=a\times b$.

3. List the BIDMAS/BODMAS order of operations.

   Brackets, Indices (powers/roots), Division and Multiplication (left to right), then Addition and Subtraction (left to right).

4. How do you convert a recurring decimal such as $0.\overline{27}$ to a fraction?

   Let $x=0.272727\ldots$ Multiply by $100$ (two recurring digits): $100x=27.2727\ldots$ Subtract: $99x=27$, so $x=\frac{27}{99}=\frac{3}{11}$.

5. How do you find a percentage increase or decrease between two values?

   $$\text{Percentage change}=\frac{\text{new value}-\text{old value}}{\text{old value}}\times 100\%$$ A positive result is an increase, a negative result a decrease.

6. How do you find the original (pre-change) value after a known percentage change — reverse percentages?

   Divide the final amount by the multiplier. If a price rose by $20\%$ to $\$72$, the original is $\frac{72}{1.20}=\$60$. For a $20\%$ decrease, divide by $0.80$.

7. To compare or order fractions, decimals and percentages, what is the most reliable method?

   Convert them all to the same form — usually decimals (or to fractions with a common denominator) — then compare. E.g. $\frac{3}{8}=0.375$, $37\%=0.37$, so $\frac{3}{8}>37\%$.

8. How do you divide a quantity in a given ratio, e.g. share $\$60$ in the ratio $2:3$?

   Add the parts: $2+3=5$. Divide the total by the number of parts: $\frac{60}{5}=12$. Multiply: $2\times12=\$24$ and $3\times12=\$36$.

9. What is the difference between direct and inverse proportion, including their equations?

   Direct proportion: $y\propto x$ so $y=kx$ (as one increases the other increases). Inverse proportion: $y\propto\frac{1}{x}$ so $y=\frac{k}{x}$ (as one increases the other decreases). $k$ is the constant of proportionality.

10. Write the formula relating speed, distance and time, and density, mass and volume.

    $$\text{speed}=\frac{\text{distance}}{\text{time}}\qquad \text{density}=\frac{\text{mass}}{\text{volume}}$$

11. What is standard form (scientific notation) and its required form?

    A number written as $a\times 10^{n}$ where $1\leq a<10$ and $n$ is an integer. Large numbers have positive $n$, small numbers negative $n$, e.g. $4500=4.5\times10^{3}$ and $0.0067=6.7\times10^{-3}$.

12. How do you rationalise a denominator of the form $\frac{1}{\sqrt{a}}$ and $\frac{1}{b+\sqrt{a}}$?

    For $\frac{1}{\sqrt{a}}$ multiply top and bottom by $\sqrt{a}$ to get $\frac{\sqrt{a}}{a}$. For $\frac{1}{b+\sqrt{a}}$ multiply by the conjugate $b-\sqrt{a}$: $\frac{b-\sqrt{a}}{b^{2}-a}$.

13. State the laws of surds for multiplication and division.

    $$\sqrt{a}\times\sqrt{b}=\sqrt{ab}\qquad \frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}\qquad (\sqrt{a})^{2}=a$$

14. State the laws of indices for $a^{m}\times a^{n}$, $a^{m}\div a^{n}$, $(a^{m})^{n}$, $a^{0}$, $a^{-n}$ and $a^{1/n}$.

    $$a^{m}\times a^{n}=a^{m+n},\quad a^{m}\div a^{n}=a^{m-n},\quad (a^{m})^{n}=a^{mn}$$ $$a^{0}=1,\quad a^{-n}=\frac{1}{a^{n}},\quad a^{1/n}=\sqrt[n]{a},\quad a^{m/n}=\sqrt[n]{a^{m}}$$

15. What is the upper bound and lower bound of a measurement given to the nearest unit, e.g. $L=12\text{ cm}$ to the nearest cm?

    Add and subtract half the rounding unit. For nearest cm, half a cm is $0.5$: lower bound $=11.5\text{ cm}$, upper bound $=12.5\text{ cm}$ (the upper bound is the value the measurement is just below).

16. When combining bounds, which bounds give the maximum of $\frac{a}{b}$ and the maximum of $a-b$?

    Maximum of $\frac{a}{b}$: use the largest $a$ and smallest $b$, i.e. $\frac{a_{\max}}{b_{\min}}$. Maximum of $a-b$: use the largest $a$ and smallest $b$, i.e. $a_{\max}-b_{\min}$.

17. State the formulas for simple interest and compound interest.

    Simple interest: $I=\frac{P\,R\,T}{100}$. Compound amount: $$A=P\left(1+\frac{r}{100}\right)^{n}$$ where $P$ is the principal, $r$ the percentage rate and $n$ the number of periods.

18. What does it mean to factorise an expression, and what should you always look for first?

    Factorising means writing an expression as a product of factors. Always take out the highest common factor first, e.g. $6x^{2}+9x=3x(2x+3)$.

19. Expand $(a+b)^{2}$, $(a-b)^{2}$ and $(a+b)(a-b)$.

    $$(a+b)^{2}=a^{2}+2ab+b^{2}$$ $$(a-b)^{2}=a^{2}-2ab+b^{2}$$ $$(a+b)(a-b)=a^{2}-b^{2}\ \text{(difference of two squares)}$$

20. State the quadratic formula for solving $ax^{2}+bx+c=0$.

    $$x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$$

21. What does the discriminant $b^{2}-4ac$ tell you about the roots of a quadratic?

    If $b^{2}-4ac>0$ there are two distinct real roots; if $b^{2}-4ac=0$ there is one repeated (equal) real root; if $b^{2}-4ac<0$ there are no real roots.

22. When solving an inequality, what operation reverses the inequality sign?

    Multiplying or dividing both sides by a negative number reverses the inequality sign, e.g. $-2x<6 \Rightarrow x>-3$.

23. Give the formula for the $n$th term of an arithmetic sequence with first term $a$ and common difference $d$.

    $$u_{n}=a+(n-1)d$$

24. How do you recognise a geometric sequence and what is its $n$th term?

    Each term is multiplied by a constant ratio $r$ (consecutive terms have a common ratio). The $n$th term is $u_{n}=a\,r^{\,n-1}$, where $a$ is the first term.

25. What is the $n$th-term rule for the square numbers, cube numbers and triangular numbers?

    Square numbers: $n^{2}$. Cube numbers: $n^{3}$. Triangular numbers: $\frac{n(n+1)}{2}$.

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