1. Number Foundation & Higher
The building blocks: working with numbers, fractions, percentages,
standard form and surds. Get this section solid — it underpins
everything else.
Make sure you can add, subtract, multiply and divide whole numbers
and decimals without a calculator. Always line up the decimal
points when adding or subtracting.
Example: Work out \(4.6 + 0.85\).
Line up the decimal points: \(4.60 + 0.85 = 5.45\)
Tip: When multiplying decimals, ignore the decimal
points, multiply as whole numbers, then count the total number of
decimal places in the question and put the point back in.
- A factor divides exactly into a number.
- A multiple is in the times table of a number.
- A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11, 13…). Note: 1 is not prime.
Example — Prime factorisation: Write 72 as a product of its prime factors.
\(72 = 8 \times 9 = 2 \times 2 \times 2 \times 3 \times 3 = 2^3 \times 3^2\)
Tip: Use a factor tree. Keep splitting numbers
until every branch ends in a prime.
To find the HCF (highest common factor), list the
common prime factors using the lowest power. For the
LCM (lowest common multiple), use every prime
factor that appears, with the highest power.
- Adding/subtracting: find a common denominator first, then add or subtract the numerators.
- Multiplying: multiply the numerators and multiply the denominators. Simplify at the end (or cancel first).
- Dividing: flip the second fraction (find its reciprocal) and multiply — "flip and times".
Example: Work out \(2\tfrac{1}{3} \div 1\tfrac{1}{4}\).
Convert to improper fractions: \(\dfrac{7}{3} \div \dfrac{5}{4}\).
Flip and times: \(\dfrac{7}{3} \times \dfrac{4}{5} = \dfrac{28}{15} = 1\tfrac{13}{15}\)
- Percentage of an amount: divide the percentage by 100, then multiply by the amount.
- Percentage change: \(\dfrac{\text{change}}{\text{original}} \times 100\).
- Percentage increase/decrease (multiplier method): multiply by \((1 \pm \text{percentage as a decimal})\).
- Reverse percentages: if you're given the amount after a change, divide by the multiplier to find the original.
Example — Multiplier: Increase £80 by 15%.
Multiplier \(= 1 + 0.15 = 1.15\)
\(£80 \times 1.15 = £92\)
Example — Reverse percentage: A jacket costs £51 after a 15% discount. Find the original price.
Multiplier \(= 1 - 0.15 = 0.85\)
Original price \(= £51 \div 0.85 = £60\)
Tip: For compound interest over n years at rate r%, use
\[ \text{amount} = \text{original} \times \left(1 + \frac{r}{100}\right)^n \]
Standard form is \(A \times 10^n\), where \(1 \le A < 10\) and \(n\) is
an integer.
- For large numbers, \(n\) is positive (e.g. \(3\,400\,000 = 3.4 \times 10^6\)).
- For small numbers (less than 1), \(n\) is negative (e.g. \(0.00045 = 4.5 \times 10^{-4}\)).
Example: Calculate \((3 \times 10^5) \times (4 \times 10^3)\), giving your answer in standard form.
\(3 \times 4 = 12\), and \(10^5 \times 10^3 = 10^8\), so the result is \(12 \times 10^8\).
Adjust so \(A\) is between 1 and 10: \(12 \times 10^8 = 1.2 \times 10^9\)
- \(a^u \times a^v = a^{u+v}\)
- \(a^u \div a^v = a^{u-v}\)
- \((a^u)^v = a^{uv}\)
- \(a^0 = 1\)
- \(a^{-n} = \dfrac{1}{a^n}\)
- \(a^{1/n} = \sqrt[n]{a}\) and \(a^{m/n} = \left(\sqrt[n]{a}\right)^m\)
Example: Work out \(8^{2/3}\).
Cube root of 8 is 2, then square it: \(2^2 = 4\)
A surd is a root that doesn't simplify to a whole number, e.g. \(\sqrt{2}\), \(\sqrt{5}\).
- \(\sqrt{a} \times \sqrt{b} = \sqrt{ab}\)
- \(\sqrt{a} \div \sqrt{b} = \sqrt{a/b}\)
- To rationalise a denominator like \(\dfrac{1}{\sqrt{3}}\), multiply top and bottom by \(\sqrt{3}\) to get \(\dfrac{\sqrt{3}}{3}\).
Example: Simplify \(\sqrt{50}\).
\(\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2}\)
2. Algebra Foundation & Higher
Letters instead of numbers — but the same rules of arithmetic
apply. This section covers expanding, factorising, solving
equations, sequences, graphs and functions.
Collect like terms by adding/subtracting their coefficients.
Only terms with exactly the same letters and powers can be
combined.
Example: Simplify \(5a + 3b - 2a + 7b\).
\((5a - 2a) + (3b + 7b) = 3a + 10b\)
Example — Substitution: If \(a = 3\) and \(b = -2\), work out \(2a^2 - b\).
\(2(3)^2 - (-2) = 2(9) + 2 = 20\)
Multiply every term inside the bracket by the term outside.
Example — Single bracket: Expand \(4(2x - 3)\).
\(8x - 12\)
Example — Double brackets (FOIL): Expand \((x + 3)(x - 5)\).
\(x^2 - 5x + 3x - 15 = x^2 - 2x - 15\)
Tip: FOIL = First, Outer, Inner, Last. Multiply each pair, then collect like terms.
- Common factor: take out the highest common factor, e.g. \(6x^2 + 9x = 3x(2x + 3)\).
- Quadratics \((x^2 + bx + c)\): find two numbers that multiply to \(c\) and add to \(b\).
- Difference of two squares: \(a^2 - b^2 = (a + b)(a - b)\).
Example: Factorise \(x^2 + 5x + 6\).
Find two numbers that multiply to 6 and add to 5: 2 and 3.
\((x + 2)(x + 3)\)
Example — Difference of two squares: Factorise \(x^2 - 49\).
\((x + 7)(x - 7)\)
Whatever you do to one side, do to the other. Work towards
getting the unknown on its own.
Example: Solve \(3(x - 4) = 2x + 1\).
Expand: \(3x - 12 = 2x + 1\)
Subtract \(2x\): \(x - 12 = 1\)
Add 12: \(x = 13\)
- Factorising: rearrange to "= 0", factorise, then set each bracket to zero.
- Quadratic formula: for \(ax^2 + bx + c = 0\),
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
- Completing the square (Higher): write \(ax^2 + bx + c\) in the form \(a(x + p)^2 + q\).
Example — Factorising: Solve \(x^2 - 2x - 15 = 0\).
Factorise: \((x - 5)(x + 3) = 0\)
So \(x = 5\) or \(x = -3\)
Example — Quadratic formula: Solve \(2x^2 + 3x - 4 = 0\) to 2 d.p.
\(a = 2, b = 3, c = -4\)
\(x = \dfrac{-3 \pm \sqrt{9 + 32}}{4} = \dfrac{-3 \pm \sqrt{41}}{4}\)
\(x \approx 0.85\) or \(x \approx -2.35\)
Treat the formula like an equation — isolate the required letter
by undoing operations in reverse order (reverse BIDMAS).
Example: Make \(r\) the subject of \(A = \pi r^2\).
Divide by \(\pi\): \(\dfrac{A}{\pi} = r^2\)
Square root: \(r = \sqrt{\dfrac{A}{\pi}}\)
For linear pairs, add or subtract the equations to eliminate one
variable, after scaling so the coefficients match.
Example: Solve \(3x + 2y = 16\) and \(5x - 2y = 8\).
Add the equations (the \(y\)-terms cancel): \(8x = 24\), so \(x = 3\).
Substitute: \(3(3) + 2y = 16 \Rightarrow 2y = 7 \Rightarrow y = 3.5\)
Higher: for one linear and one quadratic equation, substitute the linear equation into the quadratic one and solve.
Solve like an equation, but flip the inequality sign if you
multiply or divide by a negative number.
Example: Solve \(4 - 2x > 10\).
Subtract 4: \(-2x > 6\)
Divide by \(-2\) (flip the sign): \(x < -3\)
Number lines: use an open circle (○) for \(<\) or \(>\), and a filled circle (●) for \(\le\) or \(\ge\).
- Linear (arithmetic) sequences: \(n\)th term \(= dn + (a - d)\), where \(d\) is the common difference and \(a\) is the first term.
- Quadratic sequences: look at the second differences — if they're constant, the \(n\)th term includes an \(n^2\) term.
- Geometric sequences: each term is multiplied by a common ratio.
Example: Find the \(n\)th term of \(5, 8, 11, 14, \ldots\)
Common difference \(d = 3\). \(a - d = 5 - 3 = 2\).
\(n\text{th term} = 3n + 2\)
The equation of a straight line is \(y = mx + c\), where
\(m\) is the gradient and \(c\) is the \(y\)-intercept.
- Gradient \(= \dfrac{\text{change in } y}{\text{change in } x}\).
- Parallel lines have the same gradient.
- Perpendicular lines (Higher) have gradients that multiply to give \(-1\).
Example: Find the equation of the line through \((0, 3)\) and \((2, 7)\).
Gradient \(= \dfrac{7 - 3}{2 - 0} = 2\). The line crosses the \(y\)-axis at 3.
\(y = 2x + 3\)
- \(y = x^2\) gives a parabola (U-shape).
- \(y = \dfrac{1}{x}\) gives a reciprocal graph with two curved branches.
- \(y = a^x\) gives an exponential growth/decay curve.
Tip: When sketching, find where the graph crosses the axes (set \(x = 0\) and \(y = 0\)), and identify any turning points.
3. Ratio, Proportion & Rates of Change Foundation & Higher
Comparing quantities, sharing in a ratio, scaling recipes, speed,
density and pressure, and direct/inverse proportion.
To share an amount in a given ratio, add the parts of the ratio
to find the total number of parts, then work out the value of
one part.
Example: Share £180 in the ratio \(2:3:4\).
Total parts \(= 2 + 3 + 4 = 9\). One part \(= £180 \div 9 = £20\).
Shares: \(£40 : £60 : £80\)
Tip: To simplify a ratio, divide every part by their HCF — just like simplifying a fraction.
- Direct proportion: \(y = kx\) — as \(x\) increases, \(y\) increases at the same rate. Doubling \(x\) doubles \(y\).
- Inverse proportion: \(y = \dfrac{k}{x}\) — as \(x\) increases, \(y\) decreases. Doubling \(x\) halves \(y\).
Example: \(y\) is directly proportional to \(x\). When \(x = 4\), \(y = 20\). Find \(y\) when \(x = 7\).
\(y = kx \Rightarrow 20 = k \times 4 \Rightarrow k = 5\).
\(y = 5 \times 7 = 35\)
- \(\text{Speed} = \dfrac{\text{Distance}}{\text{Time}}\)
- \(\text{Density} = \dfrac{\text{Mass}}{\text{Volume}}\)
- \(\text{Pressure} = \dfrac{\text{Force}}{\text{Area}}\)
Example: A car travels 150 km in 2.5 hours. Find its average speed.
Speed \(= 150 \div 2.5 = 60\) km/h
Tip: Cover up the quantity you want with your finger in a "formula triangle" — the other two tell you whether to multiply or divide.
Repeated percentage change uses a multiplier raised to a power:
\(\text{final} = \text{initial} \times (\text{multiplier})^n\), where \(n\) is the
number of time periods.
Example: A car worth £18,000 depreciates by 12% per year. Find its value after 3 years.
Multiplier \(= 1 - 0.12 = 0.88\)
\(£18{,}000 \times 0.88^3 \approx £12{,}277\)
4. Geometry & Measures Foundation & Higher
Angles, shapes, area, volume, Pythagoras, trigonometry,
transformations and vectors.
- Angles on a straight line add up to \(180°\).
- Angles around a point add up to \(360°\).
- Angles in a triangle add up to \(180°\).
- Angles in a quadrilateral add up to \(360°\).
- Parallel lines: alternate angles are equal (Z-shape), corresponding angles are equal (F-shape), and co-interior angles add to \(180°\) (C-shape).
Tip: Always state the angle rule you're using — "angles on a straight line = 180°" — as you can get marks for the reason, not just the answer.
- Rectangle: \(A = \text{length} \times \text{width}\)
- Triangle: \(A = \tfrac{1}{2} \times \text{base} \times \text{height}\)
- Parallelogram: \(A = \text{base} \times \text{height}\)
- Trapezium: \(A = \tfrac{1}{2}(a + b) \times h\) (\(a\) and \(b\) are the parallel sides)
- Circle: \(A = \pi r^2\), \(\text{Circumference} = 2\pi r\) (or \(\pi d\))
Example: Find the area of a circle with radius 6 cm (to 1 d.p.).
\(A = \pi \times 6^2 = \pi \times 36 \approx 113.1 \text{ cm}^2\)
- Cuboid: \(V = \text{length} \times \text{width} \times \text{height}\)
- Prism: \(V = \text{cross-sectional area} \times \text{length}\)
- Cylinder: \(V = \pi r^2 h\)
- Cone: \(V = \tfrac{1}{3}\pi r^2 h\)
- Sphere: \(V = \tfrac{4}{3}\pi r^3\)
Example: Find the volume of a cylinder with radius 4 cm and height 10 cm (to 1 d.p.).
\(V = \pi \times 4^2 \times 10 = \pi \times 160 \approx 502.7 \text{ cm}^3\)
For a right-angled triangle with hypotenuse \(c\) and shorter sides \(a\)
and \(b\): \(a^2 + b^2 = c^2\)
Example: A ladder of length 5 m leans against a wall, reaching 4 m up. How far is the base from the wall?
\(5^2 = 4^2 + b^2 \Rightarrow 25 = 16 + b^2 \Rightarrow b^2 = 9 \Rightarrow b = 3 \text{ m}\)
Tip: The hypotenuse is always the longest side and is opposite the right angle.
For a right-angled triangle, with angle \(\theta\):
- \(\sin\theta = \dfrac{\text{Opposite}}{\text{Hypotenuse}}\)
- \(\cos\theta = \dfrac{\text{Adjacent}}{\text{Hypotenuse}}\)
- \(\tan\theta = \dfrac{\text{Opposite}}{\text{Adjacent}}\)
Example: A right-angled triangle has a hypotenuse of 10 cm and an angle of \(30°\). Find the opposite side.
\(\sin 30° = \dfrac{\text{opp}}{10} \Rightarrow \text{opp} = 10 \times \sin 30° = 5 \text{ cm}\)
Higher: the sine rule \(\left(\dfrac{a}{\sin A} = \dfrac{b}{\sin B}\right)\) and cosine rule \((a^2 = b^2 + c^2 - 2bc\cos A)\) work for any triangle, not just right-angled ones.
- Translation: described by a vector, e.g. \(\binom{3}{-2}\) means 3 right, 2 down.
- Reflection: described by a mirror line, e.g. the line \(y = x\).
- Rotation: described by an angle, direction, and centre of rotation.
- Enlargement: described by a scale factor and centre of enlargement. A scale factor between 0 and 1 makes the shape smaller.
A vector has both magnitude (size) and direction, written as
a column vector \(\binom{x}{y}\) or in bold, e.g.
\(\mathbf{a}\).
- To add vectors, add the corresponding components.
- A negative vector reverses the direction.
- A scalar multiple scales the vector's length but keeps its direction (or reverses it if negative).
5. Probability Foundation & Higher
Working with probabilities, sample spaces, tree diagrams and
combining events.
- Probability is always between 0 (impossible) and 1 (certain).
- \(P(\text{event}) = \dfrac{\text{number of successful outcomes}}{\text{total number of outcomes}}\)
- The probabilities of all possible outcomes add up to 1.
- \(P(\text{not } A) = 1 - P(A)\)
- "AND" — independent events: multiply the probabilities. \(P(A \text{ and } B) = P(A) \times P(B)\)
- "OR" — mutually exclusive events: add the probabilities. \(P(A \text{ or } B) = P(A) + P(B)\)
Example: A fair coin is flipped and a fair six-sided die is rolled. Find \(P(\text{heads and a } 6)\).
\(P(\text{heads}) = \tfrac{1}{2}\), \(P(6) = \tfrac{1}{6}\).
\(P(\text{heads and } 6) = \tfrac{1}{2} \times \tfrac{1}{6} = \tfrac{1}{12}\)
Tree diagrams show the outcomes of two or more events. Multiply
along the branches to find the probability of a combination, and
add the relevant end probabilities for an "or" question.
Tip: Check that the probabilities on each pair of branches add up to 1 — a common mistake is forgetting to recalculate probabilities for "without replacement" questions.
- \(A \cup B\) means \(A\) or \(B\) (union).
- \(A \cap B\) means \(A\) and \(B\) (intersection).
- \(A'\) means "not \(A\)" (everything outside \(A\)).
- \(P(A \mid B)\) means the probability of \(A\) given \(B\) has happened (conditional probability).
6. Statistics Foundation & Higher
Collecting, displaying and interpreting data — averages, spread,
charts and sampling.
- Mean: total of all values \(\div\) number of values.
- Median: the middle value when data is in order (average of the two middle values if there's an even number of values).
- Mode: the most frequent value.
- Range: largest value \(-\) smallest value (a measure of spread, not an average).
Example: Find the mean of 4, 7, 7, 9, 13.
Sum \(= 40\), count \(= 5\). Mean \(= 40 \div 5 = 8\)
Tip: For grouped frequency tables, use the midpoint of each group to estimate the mean:
\[ \text{estimated mean} = \frac{\sum (\text{midpoint} \times \text{frequency})}{\sum \text{frequency}} \]
- Bar charts: compare categories — bars don't touch.
- Pie charts: show proportions of a whole. Each sector's angle \(= \dfrac{\text{frequency}}{\text{total}} \times 360°\).
- Scatter graphs: show correlation between two variables. A line of best fit can be used to make estimates.
- Cumulative frequency graphs: plot running totals — used to estimate the median and interquartile range.
- Box plots: show the minimum, lower quartile, median, upper quartile and maximum.
The interquartile range (IQR) measures the spread of the middle
50% of the data: \(\text{IQR} = Q_3 - Q_1\) (upper quartile minus
lower quartile). It's a more useful measure of spread than the
range when there are outliers.
A sample should be representative of the population it's taken
from. Common methods include random sampling (every member has
an equal chance of being chosen) and stratified sampling
(the sample reflects the proportions of different groups in the
population).
Tip: When asked to comment on a sampling method, think about bias — could the way the sample was chosen make certain results more or less likely?