General Certificate of Secondary Education (GCSE) Mathematics Syllabus

Mathematics syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Mathematics in General Certificate of Secondary Education (GCSE), not a summary of it.

1. 1

   Number

   4 topics

   • Structure and calculation
     • Order of operations (BIDMAS)
     • Place value and standard form
     • Operations with negative numbers
     • Factors, multiples, primes, HCF and LCM
   • Fractions, decimals and percentages
     • Converting between forms
     • Four operations with fractions
     • Percentage change, increase and decrease
     • Reverse percentages and compound interest
   • Powers and roots
     • Index laws
     • Negative and fractional indices
     • Surds and rationalising denominators
   • Approximation and accuracy
     • Rounding and significant figures
     • Estimation and error intervals
     • Bounds of accuracy
2. 2

   Algebra

   5 topics

   • Notation and manipulation
     • Simplifying and collecting terms
     • Expanding brackets and factorising
     • Algebraic fractions
   • Equations and inequalities
     • Linear equations
     • Quadratic equations (factorising, formula, completing the square)
     • Simultaneous equations
     • Linear and quadratic inequalities
   • Sequences
     • nth term of linear sequences
     • Quadratic, geometric and special sequences
   • Graphs
     • Straight line graphs and gradients
     • Quadratic, cubic and reciprocal graphs
     • Equation of a circle and tangents
     • Graph transformations
   • Functions
     • Function notation
     • Composite and inverse functions
3. 3

   Ratio, Proportion and Rates of Change

   4 topics

   • Ratio
     • Simplifying and dividing in a given ratio
     • Combining and comparing ratios
   • Proportion
     • Direct and inverse proportion
     • Proportion formulae and graphs
   • Real-life rates
     • Speed, distance and time
     • Density, mass and volume
     • Pressure and compound units
   • Growth and decay
     • Compound growth and depreciation
     • Iterative methods
4. 4

   Geometry and Measures

   5 topics

   • Properties of shapes
     • Angles in polygons and parallel lines
     • Congruence and similarity
     • Circle theorems
   • Mensuration
     • Area and perimeter of 2D shapes
     • Surface area and volume of prisms, cylinders, cones and spheres
     • Arc length and sector area
   • Pythagoras and trigonometry
     • Pythagoras' theorem in 2D and 3D
     • Right-angled trigonometry (SOHCAHTOA)
     • Sine and cosine rules, area of a triangle
     • Exact trig values
   • Transformations and vectors
     • Reflection, rotation, translation, enlargement
     • Vector arithmetic and geometric proof
   • Construction and loci
     • Ruler and compass constructions
     • Bearings and loci
5. 5

   Probability and Statistics

   3 topics

   • Probability
     • Theoretical and experimental probability
     • Sample space and Venn diagrams
     • Tree diagrams and conditional probability
   • Data collection and representation
     • Sampling methods
     • Bar charts, pie charts and frequency tables
     • Histograms and frequency polygons
   • Analysing data
     • Averages and range from grouped data
     • Cumulative frequency and box plots
     • Scatter graphs and correlation

Mathematics flashcards for General Certificate of Secondary Education (GCSE)

21 of 50 cards from the Mathematics deck — real questions with worked answers.

1. What is a prime number? Give the first six prime numbers.

   A prime number is a natural number greater than $1$ with exactly two factors: $1$ and itself. The first six are $2, 3, 5, 7, 11, 13$. Note that $2$ is the only even prime.

2. State the BIDMAS/order of operations.

   Brackets, Indices, Division and Multiplication (left to right), Addition and Subtraction (left to right).

3. How do you find the Highest Common Factor (HCF) and Lowest Common Multiple (LCM) using prime factorisation?

   Write each number as a product of prime factors. The HCF is the product of the lowest powers of common primes; the LCM is the product of the highest powers of all primes appearing. Also $\text{HCF}(a,b)\times\text{LCM}(a,b)=a\times b$.

4. How do you convert a fraction to a percentage and a percentage to a decimal?

   Fraction to percentage: multiply by $100$, e.g. $\frac{3}{4}\times100=75\%$. Percentage to decimal: divide by $100$, e.g. $75\%=0.75$.

5. How do you add or subtract two fractions with different denominators?

   Find a common denominator (e.g. the LCM), rewrite each fraction with it, then add or subtract the numerators: $\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}$.

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

   Let $x=0.\overline{27}$. Since two digits recur, multiply by $100$: $100x=27.\overline{27}$. Subtract: $99x=27$, so $x=\frac{27}{99}=\frac{3}{11}$.

7. State the percentage-change and reverse-percentage methods.

   To increase by $p\%$, multiply by $\left(1+\frac{p}{100}\right)$; to decrease, multiply by $\left(1-\frac{p}{100}\right)$. To reverse a percentage change, divide the final amount by that same multiplier to find the original.

8. State the three index (power) laws for multiplication, division and powers of powers.

   $a^{m}\times a^{n}=a^{m+n}$, $\quad a^{m}\div a^{n}=a^{m-n}$, $\quad (a^{m})^{n}=a^{mn}$.

9. What do a zero index, a negative index and a fractional index mean?

   $a^{0}=1$ (for $a\neq0$), $\quad a^{-n}=\frac{1}{a^{n}}$, $\quad a^{\frac{1}{n}}=\sqrt[n]{a}$, and $a^{\frac{m}{n}}=\sqrt[n]{a^{m}}=\left(\sqrt[n]{a}\right)^{m}$.

10. How do you simplify a surd such as $\sqrt{50}$, and how do you rationalise $\frac{1}{\sqrt{2}}$?

    Simplify by extracting square factors: $\sqrt{50}=\sqrt{25\times2}=5\sqrt{2}$. Rationalise by multiplying top and bottom by the surd: $\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}$.

11. How do you write a number in standard form (scientific notation)?

    As $A\times10^{n}$ where $1\leq A<10$ and $n$ is an integer. Example: $52000=5.2\times10^{4}$ and $0.0034=3.4\times10^{-3}$.

12. How do you round a number to a given number of significant figures?

    Count significant figures from the first non-zero digit. Look at the next digit: round up if it is $5$ or more, otherwise round down, then fill remaining place-value positions with zeros. E.g. $3847$ to $2$ s.f. is $3800$.

13. What are the upper and lower bounds of a measurement rounded to the nearest unit?

    They lie half a unit either side of the value. For a length given as $24\,\text{cm}$ to the nearest cm, the lower bound is $23.5\,\text{cm}$ and the upper bound is $24.5\,\text{cm}$ (i.e. $23.5\leq x<24.5$).

14. State the difference of two squares and the expansion of $(a+b)^{2}$.

    $a^{2}-b^{2}=(a+b)(a-b)$ and $(a+b)^{2}=a^{2}+2ab+b^{2}$.

15. How do you factorise a quadratic of the form $x^{2}+bx+c$?

    Find two numbers that multiply to give $c$ and add to give $b$. For example $x^{2}+5x+6=(x+2)(x+3)$ because $2\times3=6$ and $2+3=5$.

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

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

17. What does the discriminant tell you about the roots of a quadratic?

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

18. What happens to an inequality when you multiply or divide both sides by a negative number?

    The inequality sign reverses. For example, from $-2x<6$ dividing by $-2$ gives $x>-3$.

19. State the formula for the $n$th term of an arithmetic sequence.

    $a_{n}=a+(n-1)d$, where $a$ is the first term and $d$ is the common difference.

20. Give the rule and first terms for the triangular numbers and the Fibonacci sequence.

    Triangular numbers: $a_{n}=\frac{n(n+1)}{2}$, giving $1, 3, 6, 10, 15,\dots$. Fibonacci: each term is the sum of the two before it, $1, 1, 2, 3, 5, 8, 13,\dots$.

21. What is the equation of a straight line, and what do $m$ and $c$ represent?

    $y=mx+c$, where $m$ is the gradient (steepness) and $c$ is the $y$-intercept (where the line crosses the $y$-axis).

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