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🇬🇧 General Certificate of Secondary Education (GCSE) · subject

General Certificate of Secondary Education (GCSE) Mathematics Syllabus

The Mathematics syllabus for General Certificate of Secondary Education (GCSE) in full: 5 chapters, 21 topics and 61 sub-topics, every one listed below, plus 50 Mathematics flashcards with answers.

5Chapters
21Topics
61Sub-topics
50Flashcards

Mathematics syllabus: all 5 chapters

Expand any chapter to see its topics and sub-topics.

  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. 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. 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. 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. 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

General Certificate of Secondary Education (GCSE) Mathematics flashcards — 50 questions with 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, $1, 1, 2, 3, 5, 8, 13,\dots$.

Show the other 30 Mathematics flashcards

  1. 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).

  2. How do you find the gradient of a line through two points $(x_1,y_1)$ and $(x_2,y_2)$?

    $$m=\frac{y_2-y_1}{x_2-x_1}$$ the change in $y$ divided by the change in $x$.

  3. What is the gradient relationship for parallel and perpendicular lines?

    Parallel lines have equal gradients ($m_1=m_2$). Perpendicular lines have gradients whose product is $-1$, i.e. $m_2=-\frac{1}{m_1}$.

  4. Give the general equation of a circle centred at the origin with radius $r$.

    $x^{2}+y^{2}=r^{2}$.

  5. What is function notation $f(x)$, and what is the inverse function $f^{-1}(x)$?

    $f(x)$ gives the output when $x$ is the input. The inverse $f^{-1}(x)$ reverses the function, so $f^{-1}(f(x))=x$. To find it, set $y=f(x)$, swap $x$ and $y$, and solve for $y$.

  6. What does the composite function $fg(x)$ mean?

    Apply $g$ first, then apply $f$ to the result: $fg(x)=f\big(g(x)\big)$. Order matters, so generally $fg(x)\neq gf(x)$.

  7. How do you share an amount in a given ratio, e.g. £$60$ in the ratio $2:3$?

    Add the parts: $2+3=5$. One part $=\frac{60}{5}=£12$. So the shares are $2\times12=£24$ and $3\times12=£36$.

  8. What is the difference between direct and inverse proportion?

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

  9. State the formulas for speed, density and pressure (compound measures).

    $\text{speed}=\frac{\text{distance}}{\text{time}}$, $\quad \text{density}=\frac{\text{mass}}{\text{volume}}$, $\quad \text{pressure}=\frac{\text{force}}{\text{area}}$.

  10. What is the compound interest formula?

    $$A=P\left(1+\frac{r}{100}\right)^{n}$$ where $P$ is the principal, $r$ the percentage rate per period, and $n$ the number of periods. For decay, use $\left(1-\frac{r}{100}\right)^{n}$.

  11. State the sum of interior angles of a polygon and the size of each exterior angle of a regular polygon.

    Sum of interior angles $=(n-2)\times180^{\circ}$. Each exterior angle of a regular $n$-gon $=\frac{360^{\circ}}{n}$, and exterior angles always sum to $360^{\circ}$.

  12. State the angle facts: angles on a straight line, around a point, and in a triangle.

    Angles on a straight line sum to $180^{\circ}$; angles around a point sum to $360^{\circ}$; angles in a triangle sum to $180^{\circ}$.

  13. Give the area formulas for a triangle, parallelogram, trapezium and circle.

    Triangle: $\frac{1}{2}bh$. Parallelogram: $bh$. Trapezium: $\frac{1}{2}(a+b)h$. Circle: $\pi r^{2}$.

  14. Give the circumference of a circle and the arc length / sector area of a sector with angle $\theta$.

    Circumference $=2\pi r=\pi d$. Arc length $=\frac{\theta}{360}\times2\pi r$. Sector area $=\frac{\theta}{360}\times\pi r^{2}$.

  15. State the volume and surface area of a sphere, and the volume of a cone.

    Sphere volume $=\frac{4}{3}\pi r^{3}$, surface area $=4\pi r^{2}$. Cone volume $=\frac{1}{3}\pi r^{2}h$, curved surface area $=\pi r l$.

  16. State Pythagoras' theorem.

    In a right-angled triangle, $a^{2}+b^{2}=c^{2}$, where $c$ is the hypotenuse (the side opposite the right angle).

  17. State the three basic trigonometric ratios (SOHCAHTOA).

    $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$, $\quad\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$, $\quad\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$.

  18. Give the exact values of $\sin$, $\cos$ and $\tan$ for $30^{\circ}$, $45^{\circ}$ and $60^{\circ}$.

    $\sin30^{\circ}=\frac{1}{2}$, $\cos30^{\circ}=\frac{\sqrt{3}}{2}$, $\tan30^{\circ}=\frac{1}{\sqrt{3}}$; $\sin45^{\circ}=\cos45^{\circ}=\frac{\sqrt{2}}{2}$, $\tan45^{\circ}=1$; $\sin60^{\circ}=\frac{\sqrt{3}}{2}$, $\cos60^{\circ}=\frac{1}{2}$, $\tan60^{\circ}=\sqrt{3}$.

  19. State the sine rule and the cosine rule for any triangle.

    Sine rule: $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$. Cosine rule: $a^{2}=b^{2}+c^{2}-2bc\cos A$. Area $=\frac{1}{2}ab\sin C$.

  20. Name the four types of transformation and what defines each.

    Translation (described by a column vector), reflection (in a mirror line), rotation (about a centre, with angle and direction), and enlargement (by a scale factor from a centre). The first three preserve size and shape; enlargement changes size.

  21. What does a negative or fractional enlargement scale factor do?

    A fractional scale factor (between $0$ and $1$) makes the shape smaller. A negative scale factor enlarges through the centre of enlargement, producing an image on the opposite side that is inverted.

  22. How do you add column vectors and find the magnitude of a vector?

    Add component-wise: $\begin{pmatrix}a\\b\end{pmatrix}+\begin{pmatrix}c\\d\end{pmatrix}=\begin{pmatrix}a+c\\b+d\end{pmatrix}$. Magnitude $=\sqrt{a^{2}+b^{2}}$.

  23. How do you construct the perpendicular bisector of a line segment $AB$?

    Place the compass at more than half of $AB$. Draw arcs centred on $A$ and on $B$ above and below the line; they intersect at two points. Draw the straight line through those intersections — it bisects $AB$ at right angles, and every point on it is equidistant from $A$ and $B$.

  24. What is the locus of points a fixed distance from a point, and from a line?

    The locus a fixed distance from a point is a circle of that radius centred on the point. The locus a fixed distance from a straight line is a pair of parallel lines (and, around a line segment, semicircular ends forming a 'racetrack').

  25. State the probability scale and the formula for the probability of an event.

    Probability runs from $0$ (impossible) to $1$ (certain). For equally likely outcomes, $P(\text{event})=\frac{\text{number of favourable outcomes}}{\text{total number of outcomes}}$.

  26. State the AND and OR rules for independent / mutually exclusive events.

    For independent events (AND): $P(A\text{ and }B)=P(A)\times P(B)$. For mutually exclusive events (OR): $P(A\text{ or }B)=P(A)+P(B)$. Also $P(\text{not }A)=1-P(A)$.

  27. What is the difference between a primary and secondary data source, and between a sample and a population?

    Primary data is collected first-hand by the investigator; secondary data is taken from an existing source. The population is the whole group being studied; a sample is a representative subset used to make inferences about it.

  28. How do you calculate the mean, median, mode and range of a data set?

    Mean $=\frac{\text{sum of values}}{\text{number of values}}$. Median is the middle value when ordered. Mode is the most frequent value. Range $=\text{largest}-\text{smallest}$.

  29. What does a box plot show, and how is the interquartile range found?

    A box plot shows the minimum, lower quartile $Q_1$, median $Q_2$, upper quartile $Q_3$ and maximum. The interquartile range (IQR) $=Q_3-Q_1$ and measures the spread of the middle $50\%$ of the data.

  30. How do you estimate the mean from a grouped frequency table?

    Use the midpoint $x$ of each class. Estimated mean $=\frac{\sum fx}{\sum f}$, the sum of (frequency $\times$ midpoint) divided by the total frequency.

General Certificate of Secondary Education (GCSE) Mathematics FAQ

What is in the General Certificate of Secondary Education (GCSE) Mathematics syllabus?

General Certificate of Secondary Education (GCSE) Mathematics is split into 5 chapters: Number, Algebra, Ratio, Proportion and Rates of Change, Geometry and Measures and Probability and Statistics. Together they contain 21 topics and 61 sub-topics.

How many chapters are in General Certificate of Secondary Education (GCSE) Mathematics?

5 chapters, containing 21 topics and 61 sub-topics. The largest chapter is Algebra, with 5 topics.

Are there flashcards for General Certificate of Secondary Education (GCSE) Mathematics?

Yes — a 50-card Mathematics deck, printed in full in the flashcards section of this page and available in the Examius app with spaced repetition. The free plan covers one exam of your choice, including its flashcards; Examius Pro, a one-time purchase, unlocks every other exam and deck.