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 GCSE, not a summary of it.

1. 1

   Number

   4 topics

   • Integers
   • Fractions, Decimals, and Percentages
   • Indices and Standard Form
   • Surds
2. 2

   Algebra

   4 topics

   • Expressions and Equations
   • Quadratics
   • Inequalities
   • Sequences
3. 3

   Geometry and Measures

   4 topics

   • Properties of Shapes
   • Perimeter, Area, and Volume
   • Transformations
   • Trigonometry
4. 4

   Statistics and Probability

   3 topics

   • Data Collection
   • Data Representation
   • Probability

Mathematics flashcards for GCSE

21 of 50 cards from the Mathematics 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 $\frac{a}{b}$ where $a$ and $b$ are integers and $b \neq 0$ (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. $\pi$ and $\sqrt{2}$.

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

   Write each number as a product of primes. The HCF is the product of the prime factors common to both (lowest powers); the LCM is the product of all prime factors present (highest powers). For example, for $12=2^{2}\times3$ and $18=2\times3^{2}$: $\text{HCF}=2\times3=6$ and $\text{LCM}=2^{2}\times3^{2}=36$.

3. State the correct order of operations (BIDMAS).

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

4. How do you convert a recurring decimal such as $0.\overline{7}$ to a fraction?

   Let $x=0.\overline{7}$, so $10x=7.\overline{7}$. Subtracting gives $9x=7$, hence $x=\frac{7}{9}$. In general multiply by a power of $10$ that shifts the recurring block, then subtract the original.

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

   Rewrite each fraction over a common denominator (the LCM of the denominators), then add or subtract the numerators. For example $\frac{1}{4}+\frac{2}{3}=\frac{3}{12}+\frac{8}{12}=\frac{11}{12}$.

6. How do you divide one fraction by another?

   Multiply the first fraction by the reciprocal of the second: $\frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\times\frac{d}{c}=\frac{ad}{bc}$.

7. How do you find the multiplier for a percentage increase or decrease?

   For an increase of $p\%$ multiply by $\left(1+\frac{p}{100}\right)$; for a decrease multiply by $\left(1-\frac{p}{100}\right)$. E.g. a $15\%$ increase uses the multiplier $1.15$.

8. What is the formula for compound interest / repeated percentage change?

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

9. How do you express one quantity as a percentage of another, and find a reverse percentage (original amount)?

   Percentage $=\frac{\text{part}}{\text{whole}}\times100\%$. For a reverse percentage, divide the final amount by its multiplier: if $\pounds60$ is the price after a $20\%$ increase, the original is $\frac{60}{1.2}=\pounds50$.

10. State the laws of indices for multiplication, division and powers.

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

11. Evaluate $a^{0}$, $a^{-n}$ and $a^{\frac{1}{n}}$.

    $a^{0}=1$ (for $a\neq0$); $\;a^{-n}=\frac{1}{a^{n}}$; $\;a^{\frac{1}{n}}=\sqrt[n]{a}$. More generally $a^{\frac{m}{n}}=\sqrt[n]{a^{m}}=\left(\sqrt[n]{a}\right)^{m}$.

12. What is standard form, and how is a number written in it?

    Standard form writes a number as $A\times10^{n}$ where $1\leq A<10$ and $n$ is an integer. For example $4\,500=4.5\times10^{3}$ and $0.0072=7.2\times10^{-3}$.

13. State the three main rules for simplifying surds.

    $\sqrt{ab}=\sqrt{a}\times\sqrt{b}$, $\quad\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$, $\quad\sqrt{a}\times\sqrt{a}=a$. E.g. $\sqrt{50}=\sqrt{25}\times\sqrt{2}=5\sqrt{2}$.

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

    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 $b-\sqrt{a}$ to get $\frac{b-\sqrt{a}}{b^{2}-a}$.

15. What is the difference between an expression, an equation, an identity and a formula?

    An expression is a set of terms with no equals sign (e.g. $3x+2$); an equation has an equals sign and can be solved (e.g. $3x+2=8$); an identity is true for all values, written with $\equiv$; a formula relates variables (e.g. $A=\pi r^{2}$).

16. How do you expand double brackets such as $(x+a)(x+b)$?

    Multiply every term in the first bracket by every term in the second (FOIL): $$(x+a)(x+b)=x^{2}+(a+b)x+ab.$$

17. How do you make $x$ the subject of a formula?

    Use inverse operations to isolate $x$, doing the same to both sides. If $x$ appears more than once, collect all $x$ terms on one side, factorise out $x$, then divide. E.g. from $ax=b+cx$: $ax-cx=b$, so $x=\frac{b}{a-c}$.

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

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

19. 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 real root; if $b^{2}-4ac<0$ there are no real roots.

20. 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 $b$, then write $x^{2}+bx+c=(x+p)(x+q)$. For example $x^{2}+5x+6=(x+2)(x+3)$.

21. How do you complete the square for $x^{2}+bx+c$?

    Write $$x^{2}+bx+c=\left(x+\frac{b}{2}\right)^{2}-\left(\frac{b}{2}\right)^{2}+c.$$ The turning point of $y=x^{2}+bx+c$ is then at $\left(-\frac{b}{2},\,c-\left(\frac{b}{2}\right)^{2}\right)$.

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