Interactive GCSE lessons. Each includes explanations, worked examples, a live calculator, and practice questions.
Polygons, angles, symmetry, circles, and 3D shapes with Euler's formula.
Draw, read, and interpret pie charts. Interactive builder included.
y = kx with food examples and a live recipe scaler. Drag a slider to scale any recipe.
Convert between F, D and P with an interactive converter and bar model.
Multipliers, % increase/decrease, reverse percentages — all with a live number-line calculator.
Compare value for money using unit rates. Which deal gives the most for your money?
Find and use nth term formulas for arithmetic and quadratic sequences.
Map scales, unit conversions, metric and imperial — with an interactive converter.
The SDT triangle, average speed, and time unit conversions — with a live calculator.
Simplify ratios, share in a given ratio, link ratios to fractions — interactive splitter included.
Long multiplication, division, rounding, significant figures and estimation techniques.
Currency conversions combined with ratio and percentage problems.
Squares, cubes, index notation and the six index laws — with an interactive calculator.
Expressions, substitution, expanding brackets and factorising — step by step.
Rectangles, triangles, parallelograms, trapeziums and composite shapes — with a live calculator.
Inputs, outputs, inverse functions and composite function machines — interactive builder.
Arc length, sector area, cylinder, cone and sphere — with a full 3D calculator.
Column vectors, adding and subtracting, scalar multiplication and magnitude.
Bar charts, line graphs, pictograms, frequency polygons and scatter graphs with correlation.
Mean, median, mode and range from lists, tables and grouped data — with a live calculator.
Tally charts, ungrouped and grouped tables, mean from fx, two-way tables and probability.
Experiments, theoretical probability, expected outcomes, mutually exclusive and independent events.
Sampling methods, organising and representing data, averages, spread and comparing distributions.
Reflection, rotation, translation and enlargement — with scale factors and centres.
Estimation, approximation, calculator methods, bounds and error intervals.
Linear equations, quadratics, simultaneous equations and inequalities on number lines.
Circle theorems, geometric constructions with compass and ruler, and loci problems.
y=mx+c, parallel and perpendicular lines, distance-time and velocity-time graphs.
Arithmetic, geometric, Fibonacci, triangular and square sequences with nth term.
Pythagoras, SOHCAHTOA, exact values, sine and cosine rules, and 3D trig.
Standard form, exact calculations, surds and rationalising the denominator.
Quadratic graphs, completing the square, graph families and real-life interpretation.
Sample spaces, possibility diagrams, tree diagrams and conditional probability.
Substitution, rearranging formulae, identities and composite functions.
Cumulative frequency, box plots, histograms, time series and scatter graphs.
Compound units, direct and inverse proportion, growth and decay.
Sets, union, intersection, complement — with an interactive two and three-circle builder.
Learning Objectives
Tick each off as you go.
Geometry and Measures — both Foundation and Higher papers. Properties must be recalled without prompting and applied to unseen diagrams. They are rarely on the formula sheet.
Polygons
A polygon is a closed 2D shape with straight sides.
All quadrilaterals: interior angles sum to 360°. A square is a special rectangle. A rhombus is a special parallelogram.
Angles in Polygons
Interior and exterior angles follow fixed rules for all polygons.
Hexagon: (6−2)×180÷6 = 120°
Interior + Exterior = 180°
Symmetry
Two types appear in GCSE. Know both for every common shape.
Regular polygon with n sides: lines of symmetry = n, rotational order = n. These values always match for regular polygons.
Circles
Know every term and how it links to the others.
A tangent is always perpendicular to the radius at the point of contact. Mark this 90° on any diagram immediately.
Both are on the formula sheet. Know how to apply them and work backwards.
3D Shapes
Every polyhedron satisfies Euler's formula.
Use to find a missing value when two of the three are known.
A net folds to make a 3D shape. A cube has 11 valid nets. Check all faces are present and correctly positioned before confirming a net is valid.
Practice
Select the correct answer.
A regular polygon has an exterior angle of 45°. How many sides?
Which quadrilateral has exactly one line of symmetry?
Sum of interior angles of a pentagon?
A triangular prism has how many edges?
Exam Tips
The most common errors in this topic.
Do not try to memorise each polygon's sum. One formula works for all cases.
360 ÷ exterior angle = number of sides for any regular polygon.
Mark this right angle immediately on any circle diagram. It unlocks most circle problems.
Write the rule name alongside your working. Missing reasons lose marks on reasoning questions.
If given values do not satisfy Euler's formula, something is wrong.
Congruence
Two shapes are congruent if they are exactly the same shape and size. One may be reflected, rotated or translated — but not scaled.
| Condition | What it means | Sufficient? |
|---|---|---|
| SSS | Three sides equal | Yes ✓ |
| SAS | Two sides and the included angle equal | Yes ✓ |
| ASA / AAS | Two angles and a corresponding side equal | Yes ✓ |
| RHS | Right angle, hypotenuse and one side equal | Yes ✓ |
| SSA | Two sides and a non-included angle | No ✗ (ambiguous) |
| AAA | Three angles equal | No ✗ (only similar) |
Look at what information is given — sides, angles, right angles.
Write each one with a reason (e.g. "AB = DE — given", "angle B = angle E — vertically opposite").
Conclude: "Therefore triangles ABC and DEF are congruent (SAS)" or whichever condition applies.
Similarity
Two shapes are similar if they have the same angles and their corresponding sides are in the same ratio (scale factor).
Two similar triangles. Smaller has sides 3, 4, 5 cm. Larger has longest side 10 cm.
Learning Objectives
Tick each off as you go.
Statistics strand, Foundation and Higher. Questions range from angle calculations to comparing two charts with different sample sizes.
What is a Pie Chart?
A circle divided into sectors. Each sector's size is proportional to its frequency.
Drawing Pie Charts
Follow these six steps every time.
Add all frequencies. Write it down before calculating anything else.
(frequency ÷ total) × 360. Round to nearest degree.
If 359° or 361°, adjust the largest sector by 1°.
Draw a radius from centre to 12 o'clock. This is your baseline.
Place protractor at centre along the current radius. Mark the angle. Draw a new radius. That new radius is the baseline for the next sector.
Category name + angle or percentage. Missing labels cost marks.
Interactive Builder
Add or remove data points — angles calculate automatically and the chart updates live.
Edit labels and frequencies — chart updates instantly
Reading Pie Charts
Always use the formula — never read the angle directly as the frequency.
Example: 48 students, sector 90°. → (90÷360)×48 = 12 students
Example: 72° → (72÷360)×100 = 20%
Favourite Sports — 60 students
Worked Examples
Try each calculation before reading the answer.
30 students asked about travel to school.
| Transport | Frequency | Calculation | Angle |
|---|---|---|---|
| Car | 12 | (12÷30)×360 | 144° |
| Bus | 8 | (8÷30)×360 | 96° |
| Walk | 6 | (6÷30)×360 | 72° |
| Cycle | 4 | (4÷30)×360 | 48° |
| Total | 30 | 360° |
72 people surveyed. Coffee sector = 135°. How many chose coffee?
Four sectors. Known angles: 85°, 110°, 95°. Find the fourth.
Practice
Select the correct answer.
40 people surveyed. 10 chose bananas. What is the sector angle?
120 students surveyed. Science sector = 60°. How many chose Science?
Three known sectors: 95°, 115°, 80°. Find the fourth.
Chart A: 80 students. Chart B: 120 students. Drama sector = 90° in both. Which has more Drama students, by how many?
Exam Tips
The most common mistakes in this topic.
Add all angles first. Fix errors before touching a compass.
Each new sector starts where the previous ended, not from 12 o'clock.
Write (freq÷total)×360 in full. Method marks apply even when wrong.
Name + angle or percentage. Missing labels = free marks lost.
Frequency = (angle÷360)×total. The angle shows proportion, not count.
Same angle in two charts does not mean same count. Calculate separately.
Learning Objectives
Tick each off as you go.
Ratio, Proportion and Rates of Change — Foundation and Higher. Foundation: find k and a missing value. Higher: form equations, use y ∝ x², y ∝ √x, inverse proportion.
What is Direct Proportion?
When one quantity increases, the other increases at the same rate. When one doubles, the other doubles.
A cookie recipe uses 200g of butter to make 20 cookies.
The ratio butter ÷ cookies = 10g per cookie every time. That ratio is the constant k.
Divide y by x for every pair. If the answer is always the same, the relationship is directly proportional. That number is k.
| Cookies (x) | Butter in g (y) | y ÷ x | Conclusion |
|---|---|---|---|
| 10 | 100 | 10 | k = 10 |
| 20 | 200 | 10 | k = 10 |
| 35 | 350 | 10 | k = 10 |
| Any | Any | 10 always | Directly proportional ✓ |
The Formula
Every direct proportion relationship can be written as y = kx.
Substitute one known (x, y) into y = kx and solve for k.
Write y = kx with your value of k. This is the complete equation.
Put the new x or y in and calculate the unknown.
40 doughnuts require 625g of flour. How much flour for 64 doughnuts?
Recipe Scaler
This tool uses direct proportion. Change the number of portions and every ingredient scales automatically.
Direct Proportion Calculator
Step 1: enter a pair of values you already know. Step 2: enter the new value you want to find. The calculator does the rest.
Direct proportion always follows y = kx. To use this tool:
Graphs of Direct Proportion
The graph of y = kx is always a straight line through the origin. The gradient equals k.
Pick two clear points. Divide change in y by change in x. That is k. Write y = kx.
If y ÷ x is constant in every row, the relationship is directly proportional.
| x | y | y ÷ x |
|---|---|---|
| 2 | 6 | 3 |
| 5 | 15 | 3 |
| 10 | 30 | 3 — directly proportional, k=3 |
Practice
Select the correct answer.
y ∝ x. When x = 5, y = 30. What is k?
A recipe uses 200g of flour for 8 muffins. How much flour for 14 muffins?
y = kx. When x = 3, y = 21. Find x when y = 49.
Which table shows direct proportion?
Exam Tips
The most common mistakes.
Do not scale informally. k = y ÷ x, then write y = kx.
The ∝ step earns a method mark in many mark schemes.
If it does not, it is not direct proportion. State this clearly.
x = y ÷ k. Multiplying gives the wrong answer.
Show this check in working. One calculation per row.
The line must cross the y-axis at zero.
Learning Objectives
Tick each off as you go.
Number strand, non-calculator paper. Common equivalences must be memorised. Converting between forms is tested directly and also appears inside other topics like probability, ratio, and percentage questions.
What is FDP?
Three different ways to write the same value. They are equivalent — they describe the same portion of a whole.
Imagine a pizza cut into 10 equal slices. You eat 3 slices.
3 parts out of 10
3 tenths
30 parts out of 100
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/10 | 0.1 | 10% |
| 1/5 | 0.2 | 20% |
| 1/3 | 0.333... | 33.3...% |
| 2/3 | 0.666... | 66.6...% |
| 1/8 | 0.125 | 12.5% |
Converting Between F, D and P
Three conversion routes — each is a simple calculation.
Divide the numerator by the denominator. 3/4 → 3 ÷ 4 = 0.75
Multiply by 100. 0.75 × 100 = 75%
Divide by 100. 75% ÷ 100 = 0.75
Example: 7/20 → 7÷20 = 0.35 → 0.35×100 = 35%
Example: 3/5 → multiply by 20/20 → 60/100 = 60%
Interactive FDP Converter
Enter any fraction, decimal or percentage — the other two are calculated automatically with full working shown.
Bar Model
A bar model shows the whole split into parts — a visual way to see F, D and P simultaneously.
Bar models make it easy to see which fraction, decimal or percentage is larger without converting. The wider segment is the larger value.
Practice
Select the correct answer.
What is 3/5 as a percentage?
What is 0.35 as a fraction in its simplest form?
Put these in order, smallest first: 0.6, 58%, 3/5
Which of these is NOT equivalent to 1/4?
Exam Tips
Non-calculator — these methods must be fluent.
½, ¼, ¾, 1/5, 1/10, 1/8 — know all three forms for each without calculating.
When ordering a mix of F, D and P, convert everything to decimals first.
35/100 is not fully simplified. Divide by the HCF. Mark schemes often require simplest form.
Move the decimal point two places left. 75% → 0.75. Never divide by 10.
1/3 = 0.333... and 2/3 = 0.666... Use dots or write 0.3̄ to show recurrence.
In a multi-mark question, write the intermediate decimal. Missing this step loses method marks.
Calculations with Fractions
Adding, subtracting, multiplying and dividing fractions — all without a calculator.
Example: 3/4 + 2/5. LCM of 4 and 5 = 20. → 15/20 + 8/20 = 23/20 = 1 3/20
Example: 5/6 − 1/4. LCM = 12. → 10/12 − 3/12 = 7/12
Cancel common factors before multiplying to keep numbers small:
3/8 × 4/9 → cancel 3 and 9 (÷3), cancel 4 and 8 (÷4) → 1/2 × 1/3 = 1/6
Example: 3/4 ÷ 2/5 → 3/4 × 5/2 = 15/8 = 1 7/8
Example: 5/6 ÷ 5 = 5/6 ÷ 5/1 → 5/6 × 1/5 = 5/30 = 1/6
Convert mixed numbers to improper fractions before calculating.
Always divide by the denominator first to find one part, then multiply by the numerator.
Learning Objectives
Tick each off as you go.
Number strand — calculator paper. Multipliers and reverse percentages appear in almost every GCSE paper. The double number line is the key structural tool for every type of percentage question.
Multipliers
A multiplier converts a percentage change into a single multiplication. This is the most efficient method on a calculator.
| Situation | Percentage | Multiplier |
|---|---|---|
| No change | 100% | × 1.00 |
| Increase by 15% | 115% | × 1.15 |
| Increase by 34% | 134% | × 1.34 |
| Decrease by 20% | 80% | × 0.80 |
| Decrease by 12% | 88% | × 0.88 |
| Decrease by 15% | 85% | × 0.85 |
Standard room: £422 per week
| Room type | Percentage | Multiplier | Cost |
|---|---|---|---|
| Economy (20% off) | 80% | × 0.80 | £337.60 |
| Standard | 100% | × 1.00 | £422.00 |
| Deluxe (+15%) | 115% | × 1.15 | £485.30 |
| Super Deluxe (+34%) | 134% | × 1.34 | £565.48 |
Percentage Calculator
Select the type of question, enter your values, and see the multiplier method with full working and a number line.
Types of Percentage Question
There are five types. The multiplier method handles all of them.
What is 30% of £360?
Increase £422 by 15%.
Decrease £422 by 20%.
£12 as a percentage of £60?
After a 20% reduction, a price is £336. What was the original?
Reverse Percentages
You are given the amount after the percentage change. You need to find the original (100%).
The amount you are given is NOT 100%. It is the result after the change. Identify what percentage it represents, then divide.
If a price increased by 30% and is now £360: multiplier = 1.30. Original = 360 ÷ 1.30 = £276.92
If a price decreased by 15% and is now £13,600: multiplier = 0.85. Original = 13600 ÷ 0.85 = £16,000
The value of Michelle's car has decreased by 15%. It is now worth £13,600. Find the original value.
Worked Examples
Holiday shopping context — try each calculation before reading the answer.
Sunglasses cost £32. Jamie has a 14% discount voucher. How much does he pay?
Jamie bought shorts in a 14% off sale. He paid £32. What was the original price?
Laptop weighs 3.2kg. Baggage allowance is 14kg. What percentage does the laptop take?
Suntan cream normally weighs 320g. Special offer has 14% extra. New weight?
July flight = £360. Book 4 months early for 30% off. Early booking cost?
Practice
Select the correct answer.
What is the multiplier for a 23% increase?
A coat costs £85. It is reduced by 30% in a sale. What is the sale price?
A bag weighs 4.5kg. The baggage limit is 20kg. What percentage of the limit is the bag?
After a 25% increase, a price is £250. What was the original price?
Michelle's car decreased by 15% in value. It is now worth £13,600. What was the original value?
Exam Tips
The most common mistakes in this topic.
Finding 10% then 5% then adding is slower and more prone to error. Write the multiplier first.
+23% → 1.23. −23% → 0.77. Check: multiplier for decrease is always less than 1.
Do not take the percentage off the given amount. Divide by the multiplier. 336 ÷ 0.80, not 336 − 67.2.
In reverse questions, the amount given is NOT 100%. State explicitly what percentage it represents before calculating.
Once you find the original, apply the multiplier and check you get back to the given amount.
Make sure you divide the part by the whole (not the other way round).
Learning Objectives
Tick each off as you go.
Ratio, Proportion and Rates of Change — Foundation and Higher. Best buy questions appear in both calculator and non-calculator papers. They are straightforward but require a clear, structured method to avoid errors.
What are Best Buys?
A best buy question asks: which product gives you the most for your money?
Two products cost different amounts and contain different quantities. To compare them fairly, you need to find the cost per unit (per gram, per ml, per item).
Both methods work. Method 1 is slightly more intuitive. Always show which method you are using.
The Methods
Method 1 finds the price per unit. Method 2 finds how much you get per £1. Use either — they always agree.
Compare the results: the smallest price per unit is the best value.
Compare the results: the largest quantity per £1 is the best value.
Note the price and quantity for each. Check units are the same.
Price per unit (÷ quantity) or quantity per £1 (÷ price). Must use same method for both.
Write a clear conclusion naming the best buy. Just giving a number without a conclusion loses the mark.
Best Buy Calculator
Enter the price and quantity for up to three products. The calculator finds the unit rate for each and identifies the best buy.
Worked Examples
Two methods shown side by side. You only need one in an exam.
Pack A: 6 pots for £2.70. Pack B: 4 pots for £1.60. Which is better value?
| Product | Price | Qty | Method 1 (p per pot) | Method 2 (pots per £1) |
|---|---|---|---|---|
| Pack A | £2.70 | 6 pots | 270 ÷ 6 = 45p each | 6 ÷ 2.70 = 2.22 per £1 |
| Pack B | £1.60 | 4 pots | 160 ÷ 4 = 40p each | 4 ÷ 1.60 = 2.50 per £1 |
Small: 1.5 kg for £4.50. Large: 2.5 kg for £7.00. Which is better value?
| Product | Price | Qty | Price per kg |
|---|---|---|---|
| Small | £4.50 | 1.5 kg | £4.50 ÷ 1.5 = £3.00 per kg |
| Large | £7.00 | 2.5 kg | £7.00 ÷ 2.5 = £2.80 per kg |
Shampoo: 200ml for £1.80, 350ml for £2.94, 500ml for £4.25. Best buy?
| Product | Price | Quantity | Pence per ml |
|---|---|---|---|
| 200ml | £1.80 | 200ml | 180 ÷ 200 = 0.90p per ml |
| 350ml | £2.94 | 350ml | 294 ÷ 350 = 0.84p per ml |
| 500ml | £4.25 | 500ml | 425 ÷ 500 = 0.85p per ml |
Practice
Select the correct answer.
Cereal A: 750g for £2.10. Cereal B: 1000g for £2.60. Which is better value?
Which calculation correctly finds the price per gram for a 400g pack costing £2.80?
Orange juice: Small 330ml for 99p. Medium 500ml for £1.45. Large 750ml for £2.10. Which is the best buy?
Pasta: 500g for £0.85 vs 1.2kg for £1.90. Which is better value?
Exam Tips
Best buy questions are reliable marks — do not give them away.
Even if the answer seems obvious, you need both unit rates written down. Both carry marks.
Name the product. "Product B is better value because it has a lower price per gram." One word answers lose the final mark.
If one product is in grams and another in kg, convert to the same unit first.
Multiplying by 100 first (£ to p) avoids small decimals and reduces rounding errors.
Do not assume the larger pack is always cheaper per unit. Always calculate — Example 3 shows this.
Use Method 1 for all, or Method 2 for all. Mixing methods within a question risks comparison errors.
Learning Objectives
Tick each off as you go.
Algebra strand — Foundation and Higher. Linear (arithmetic) nth term is Foundation. Quadratic nth term is Higher only. Sequences questions appear on both calculator and non-calculator papers.
Sequences
A sequence is a list of numbers that follow a pattern. Each number in the sequence is called a term.
| Type | Pattern | Example | Nth term |
|---|---|---|---|
| Arithmetic (linear) | Add or subtract same amount each time | 3, 7, 11, 15, 19... | 4n − 1 |
| Geometric | Multiply by same ratio each time | 2, 6, 18, 54... | 2 × 3ⁿ⁻¹ |
| Quadratic | Second difference is constant | 1, 4, 9, 16, 25... | n² |
| Fibonacci-style | Each term = sum of two previous terms | 1, 1, 2, 3, 5, 8... | No simple formula |
Write the sequence, then write the differences between consecutive terms below them.
Arithmetic (Linear) Sequences
The most common type at GCSE. The nth term formula is always of the form dn + c.
Where d is the common difference and c is found by substituting n = 1 and solving.
Subtract any term from the next: T₂ − T₁ = d. Check it is the same throughout.
Write the coefficient of n. For example, if d = 4, write 4n + c.
Substitute n = 1 and T₁ (the first term). Solve for c. Example: 4(1) + c = 5 → c = 1.
Write the complete formula, then substitute n = 1, 2, 3 to verify it gives back the original sequence.
Example: is 85 in the sequence 4n + 1? → 4n + 1 = 85 → 4n = 84 → n = 21. Yes — it is the 21st term.
Example: is 50 in the sequence 4n + 1? → 4n + 1 = 50 → 4n = 49 → n = 12.25. No — not a whole number.
Nth Term Calculator
Enter the first few terms of your sequence. The calculator finds the common difference, works out the nth term formula, and lists any term you choose.
Quadratic Sequences (Higher)
The second difference is constant. The nth term contains an n² term.
Write the first differences (T₂−T₁, T₃−T₂, etc.), then the second differences. If the second differences are constant and non-zero, the sequence is quadratic.
a = second difference ÷ 2. For second difference = 2: a = 1. The nth term starts with n².
This gives a new sequence (the remainder). Find the nth term of that remainder sequence — it will be linear (dn + c).
Write the full quadratic formula and verify against the original sequence.
Second difference = 2 → a = 1 → start with n²
| n | Term | n² | Term − n² |
|---|---|---|---|
| 1 | 2 | 1 | 1 |
| 2 | 5 | 4 | 1 |
| 3 | 10 | 9 | 1 |
| 4 | 17 | 16 | 1 |
Remainder = 1, 1, 1, 1 → constant → the linear part is just +1.
Worked Examples
Try each before reading the answer.
Sequence: 7, 11, 15, 19, 23...
Sequence: 20, 17, 14, 11, 8...
The nth term is 5n − 3. Find the 12th term.
Practice
Select the correct answer.
What is the common difference of the sequence: 6, 10, 14, 18, 22...?
What is the nth term of the sequence: 3, 8, 13, 18, 23...?
The nth term is 6n − 1. What is the 15th term?
nth term = 4n + 3. Is 47 in this sequence?
Exam Tips
The most common errors in this topic.
Substitute n = 1, 2, 3 back in. If you do not get the original sequence, recheck c.
A decreasing sequence has a negative common difference. Write −dn + c — keep the sign throughout.
Set nth term = X, solve, check n is a positive integer. Do not just list terms — you may stop too early.
The nth term is a formula, not the next number in the list. Substituting n = 5 gives the 5th term, not the term after the 4th.
Always find the second differences first. If first differences are not constant, move on to second differences.
Write the sequence, then write the differences beneath it. Showing this earns method marks even if the formula is wrong.
Learning Objectives
Tick each off as you go.
Ratio, Proportion and Rates of Change — Foundation and Higher. Unit conversions are non-calculator friendly. Map scale questions appear regularly and combine measurement with ratio.
Metric Units
The metric system uses powers of 10. Every conversion is a multiplication or division by 10, 100 or 1000.
To convert to a smaller unit: multiply. To convert to a larger unit: divide.
Example: 3.4 km = 3.4 × 1000 = 3400 m Example: 750 cm = 750 ÷ 100 = 7.5 m
Imperial & Conversions
Imperial units are not based on powers of 10. Conversion factors are always given in the exam — you just need to know how to use them.
| Imperial | Metric (approx) | Use |
|---|---|---|
| 1 inch | 2.54 cm | length |
| 1 foot | 30 cm | length |
| 1 mile | 1.6 km | distance |
| 1 kg | 2.2 pounds | mass |
| 1 gallon | 4.5 litres | capacity |
| 1 pint | 568 ml | capacity |
Example: Convert 8 miles to km. 1 mile = 1.6 km → 8 × 1.6 = 12.8 km
Example: Convert 5 kg to pounds. 1 kg = 2.2 lb → 5 × 2.2 = 11 pounds
Example: Convert 9 gallons to litres. 1 gallon = 4.5 l → 9 × 4.5 = 40.5 litres
Unit Converter
Select a conversion type, enter a value, and see the result with full working shown.
Map Scales
A map scale tells you how many units in real life are represented by one unit on the map. Written as a ratio 1 : n.
The second number in the ratio tells you how many times bigger reality is than the map.
Example: Scale 1 : 50 000. Map distance = 4 cm. Real = 4 × 50 000 = 200 000 cm = 2000 m = 2 km
Example: Scale 1 : 25 000. Real distance = 3.5 km = 350 000 cm. Map = 350 000 ÷ 25 000 = 14 cm
Practice
Select the correct answer.
How many metres are in 3.7 km?
A car journey is 45 miles. Using 1 mile ≈ 1.6 km, how far is this in km?
A map has scale 1 : 50 000. A road measures 6 cm on the map. What is the real length in km?
Scale 1 : 25 000. A field is 1.5 km long. How long is it on the map in cm?
Exam Tips
The most common errors in this topic.
Never compare cm with m or km with m directly. Convert first — this is the most common error.
Going from km to m (smaller): multiply by 1000. Going from m to km (larger): divide by 1000.
Map → Real: × scale. Real → Map: ÷ scale. Always work in cm.
km × 100 000 = cm. Do this conversion before dividing by the scale factor.
You do not need to memorise exact imperial conversions — but you must know roughly which way to convert (multiply or divide).
5 miles should be about 8 km (not 3 km, not 80 km). Always check your answer is in the right ballpark.
Learning Objectives
Tick each off as you go.
The SDT Triangle
Cover the quantity you want to find — the triangle shows the formula.
Cover D → Distance = S × T Cover S → Speed = D ÷ T Cover T → Time = D ÷ S
SDT Calculator
Select your units, enter two values — the calculator finds the third with full working shown.
Time Conversions
The trickiest part of SDT problems. Always convert time to a decimal before using formulas.
| Time | In hours (decimal) | Calculation |
|---|---|---|
| 30 minutes | 0.5 hours | 30 ÷ 60 |
| 45 minutes | 0.75 hours | 45 ÷ 60 |
| 1 hour 30 min | 1.5 hours | 1 + 30/60 |
| 2 hours 20 min | 2.333... hours | 2 + 20/60 |
| 1 hour 15 min | 1.25 hours | 1 + 15/60 |
Example: 2.75 hours = 2 hours and 0.75 × 60 = 45 minutes = 2 hours 45 minutes
Example: 1.4 hours = 1 hour and 0.4 × 60 = 24 minutes = 1 hour 24 minutes
Worked Examples
Try each before reading the answer.
A car travels 240 km in 3 hours. Find the average speed.
A train travels at 120 km/h. How far does it travel in 45 minutes?
A cyclist travels 36 km at 12 km/h. How long does the journey take?
Stage 1: 60 km at 30 km/h. Stage 2: 90 km at 60 km/h. Find average speed for whole journey.
Practice
Select the correct answer.
A bus travels 180 km in 2.5 hours. What is its average speed?
A car travels at 60 km/h for 1 hour 20 minutes. How far does it travel?
A cyclist rides 20 km at 10 km/h, then 30 km at 15 km/h. What is the average speed?
Exam Tips
Speed, distance and time questions are reliable marks.
Divide minutes by 60 before substituting. 1 hr 30 min = 1.5 h, not 1.3 h.
Always use total distance ÷ total time. Averaging the speeds gives the wrong answer.
Speed in km/h needs distance in km and time in hours. Mixing units is the most common error.
Write S = D ÷ T (or whichever applies) before substituting. Earns a method mark.
If the question asks for time in hours and minutes, multiply the decimal part by 60.
A steeper slope means higher speed. A horizontal line means stationary.
Learning Objectives
Tick each off as you go.
Ratio Basics
A ratio compares two or more quantities. It is written using a colon: 3 : 5 means 3 parts to 5 parts.
Example: Simplify 15 : 25. Factors of 15: 1,3,5,15. Factors of 25: 1,5,25. HCF = 5. → 15÷5 : 25÷5 = 3 : 5
Example: Simplify 24 : 16 : 8. HCF of all three = 8. → 24÷8 : 16÷8 : 8÷8 = 3 : 2 : 1
Like equivalent fractions — multiply or divide all parts by the same number.
Divide both sides by the first number to get a ratio in the form 1 : n. Useful for comparing ratios.
Example: 3 : 7 → divide both by 3 → 1 : 2.33...
Example: 4 : 5 → divide both by 4 → 1 : 1.25
Ratio and Fractions
Every ratio can be written as fractions. This connection is very commonly tested.
Example: Ratio 3 : 5. Total parts = 8.
First share = 3/8 of the whole. Second share = 5/8 of the whole.
Write both fractions over the same denominator, then the numerators form the ratio.
Example: ⅜ and ⅝ → ratio = 3 : 5
Example: ¼ and ¾ → ratio = 1 : 3
Example: ⅓ and ½ → common denominator 6 → 2/6 and 3/6 → ratio = 2 : 3
Example: ⅖ of a bag of sweets are red. The rest are blue and green in ratio 2 : 3. What fraction are green?
Red = 2/5. Remaining = 3/5. Blue : Green = 2 : 3 → green = 3/5 of remaining = 3/5 × 3/5 = 9/25
Interactive Ratio Splitter
Enter a total and a ratio — the splitter divides it and shows each share as a fraction and amount.
Practice
Select the correct answer.
Simplify the ratio 18 : 24.
Share £84 in ratio 3 : 4. How much does the larger share receive?
In a class, boys and girls are in ratio 2 : 3. What fraction of the class are girls?
A and B share money in ratio 3 : 7. B receives £105. How much does A receive?
Exam Tips
Common ratio errors to avoid.
Divide total by total parts. Then scale up. Do not try to do it in one step.
Shares must add back to the original total. If they do not, recheck your working.
Divide by the highest common factor to reach the simplest form in one step.
a : b → fractions are a/(a+b) and b/(a+b). Not a/b.
Divide the given amount by its ratio number to get 1 part. Then multiply for all others.
a : b : c → total parts = a + b + c. Same method, just three shares instead of two.
Percentage Change
Percentage change measures how much a value has increased or decreased relative to the original.
A jumper costs £45 in January and £54 in February. Find the percentage increase.
A car was bought for £12,000 and sold for £9,000. Find the percentage loss.
Example: 15% increase on £240 → 240 × 1.15 = £276
Example: 8% decrease on £500 → 500 × 0.92 = £460
Example: £1000 invested at 3% per year for 4 years.
Example: Car worth £20,000 depreciates 12% per year for 3 years.
Proportion
Direct proportion — as one quantity increases, the other increases at the same rate. Inverse proportion — as one increases, the other decreases.
Example: y ∝ x. When x=4, y=12. Find k: k=12÷4=3. Equation: y=3x. Find y when x=7: y=3×7=21
Example: y ∝ 1/x. When x=3, y=8. Find k: k=3×8=24. Equation: y=24/x. Find y when x=6: y=24/6=4
Inverse proportion graph is a reciprocal curve — as x increases, y decreases but never reaches zero.
| If you double x and y... | Type | Equation |
|---|---|---|
| ...also doubles | Direct (y ∝ x) | y = kx |
| ...quadruples (×4) | Direct squared (y ∝ x²) | y = kx² |
| ...halves (÷2) | Inverse (y ∝ 1/x) | y = k/x |
| ...quarters (÷4) | Inverse squared (y ∝ 1/x²) | y = k/x² |
Learning Objectives
Tick each off as you go.
Long Multiplication
The grid method and column method both work — use whichever you find more reliable.
| × | 300 | 40 | 7 |
|---|---|---|---|
| 20 | 6000 | 800 | 140 |
| 6 | 1800 | 240 | 42 |
Multiply by each digit separately, then add the results.
Ignore the decimal point, multiply as integers, then count total decimal places in the original numbers and insert the point in the answer.
Example: 3.4 × 1.2 → 34 × 12 = 408 → 2 decimal places total → 4.08
Long Division
Work through the dividend digit by digit, left to right.
Once (1 × 8 = 8). Write 1 above. Remainder: 9 − 8 = 1. Bring down the 5 → 15.
Once (1 × 8 = 8). Write 1 above. Remainder: 15 − 8 = 7. Bring down 2 → 72.
9 times (9 × 8 = 72). Write 9 above. Remainder: 0.
Example: 4.5 ÷ 0.3 → multiply both by 10 → 45 ÷ 3 = 15
Example: 2.4 ÷ 0.08 → multiply both by 100 → 240 ÷ 8 = 30
Rounding & Significant Figures
Two different rounding systems — both essential for the non-calculator paper.
Count digits after the decimal point. Look at the next digit: if ≥ 5, round up; if < 5, leave unchanged.
Start counting from the first non-zero digit. Round using the same ≥5 rule.
Estimating
Round every number to 1 significant figure, then calculate. This gives an approximate answer to check your working.
38.4 → 40, 6.7 → 7, 0.48 → 0.5, 312 → 300
Use the simpler numbers to do a quick mental calculation.
Use ≈ (approximately equal to). If your calculator answer is very different, recheck.
Practice
Non-calculator — work these out by hand.
Calculate 36 × 24 without a calculator.
Round 0.005847 to 2 significant figures.
Estimate the value of (48.7 × 3.2) ÷ 4.9
Exam Tips
Non-calculator technique matters.
Split into partial products. Write each one separately. Easy to check and earns method marks.
If 347 × 26 = 9022, check by estimating: 350 × 25 = 8750 ≈ 9022. Reasonable.
0.0047 has 2 significant figures: 4 and 7. The zeros are placeholders, not significant.
Write each number rounded to 1 s.f. before calculating. Missing this step loses marks even if the estimate is correct.
2.4 ÷ 0.08 → × 100 → 240 ÷ 8 = 30. Much easier than dividing by a decimal directly.
Use an estimate to sense-check your calculator answer before writing it down.
Learning Objectives
Tick each off as you go.
Exchange Rates
An exchange rate tells you how many units of one currency equal one unit of another.
Example: Convert £250 to euros at £1 = €1.17. → 250 × 1.17 = €292.50
Example: Convert €180 to pounds at £1 = €1.17. → 180 ÷ 1.17 = £153.85
Convert all prices to the same currency, then compare.
Example: A laptop costs £680 in the UK and $850 in the US. Rate: £1 = $1.28. Is the UK or US cheaper?
Convert $850 to pounds: 850 ÷ 1.28 = £664.06. The US is cheaper by £680 − £664.06 = £15.94.
Some currency exchanges charge commission — a percentage deducted from the amount exchanged.
Example: Exchange £500 at €1.15 per £, 2% commission. Amount after commission = 500 × 0.98 = £490. Then 490 × 1.15 = €563.50.
Currency Calculator
Enter an amount and exchange rate — converts in both directions with working shown.
Ratio and Percentages Combined
Multi-step problems linking ratio, fractions and percentages — very common at Higher tier.
If profit is shared in ratio 3 : 7, the first person gets 3/(3+7) = 3/10 = 30%
Example: Two friends invest in ratio 2 : 3. The investment grows by 18%. How much does each receive if they started with £5000?
Example: In a school, students study French, Spanish, German in ratio 5 : 3 : 2. What percentage study German?
Total parts = 10. German = 2 parts. Fraction = 2/10. Percentage = 20%
Worked Examples
Full multi-step solutions.
Sarah exchanges £350 to euros at £1 = €1.14. The bank charges 1.5% commission. How many euros does she receive?
A watch costs £145 in the UK and $180 in the US. Exchange rate: £1 = $1.27. Which is cheaper and by how much (in £)?
Three siblings share an inheritance of £24 000 in ratio 1 : 2 : 3. The youngest invests her share and earns 8% interest. How much does she have after interest?
Practice
Select the correct answer.
£1 = $1.32. Convert £220 to dollars.
£1 = €1.15. A meal costs €46. What is this in pounds?
Profits are shared in ratio 2 : 3 : 5. What percentage does the largest share receive?
£8000 is shared in ratio 3 : 5. The larger share is invested at 10% interest. What is the final value of that share?
Exam Tips
Exchange rate and combined questions — work methodically.
£ → euros: multiply by the rate. Euros → £: divide by the rate.
Deduct commission from the original amount first, then apply the exchange rate.
Never compare prices in different currencies directly. Convert both to £ or both to the foreign currency.
Split the total into shares using ratio. Then apply the percentage to the relevant share.
Show this step explicitly. The ratio numbers are not the percentages.
After applying % to each ratio share, check that the total matches the expected result (e.g. whole × multiplier).
Learning Objectives
Tick each off as you go.
Algebra and Number — both papers. Non-calculator questions test knowledge of squares and cubes to 15². Calculator questions test index laws and fractional indices.
Powers & Roots
A power tells you how many times a number is multiplied by itself.
The number being raised is the base. The power is the index or exponent.
| n | n² | n³ | n | n² | n³ |
|---|---|---|---|---|---|
| 1 | 1 | 1 | 6 | 36 | 216 |
| 2 | 4 | 8 | 7 | 49 | 343 |
| 3 | 9 | 27 | 8 | 64 | 512 |
| 4 | 16 | 64 | 9 | 81 | 729 |
| 5 | 25 | 125 | 10 | 100 | 1000 |
| 12 | 144 | — | 15 | 225 | — |
Index Laws
Six rules for simplifying expressions with powers. Bases must match for laws 1–3.
Powers Calculator
Enter a base and power, or find a square/cube root.
Practice
Simplify x⁴ × x⁶
What is 4⁻²?
What is 27^(2/3)?
Exam Tips
x³ × y² cannot be simplified — the bases differ.
Any base raised to the power zero equals 1.
a⁻ⁿ = 1/aⁿ. If a is a fraction, it flips.
a^(1/3) = cube root. The denominator tells you which root.
For a^(m/n): find the root first — keeps numbers smaller.
These appear on almost every non-calculator paper.
Factors & Multiples
Factors divide exactly into a number. Multiples are in a number's times table.
Find HCF(24, 36): Factors of 24: 1,2,3,4,6,8,12,24. Factors of 36: 1,2,3,4,6,9,12,18,36. Common: 1,2,3,4,6,12. HCF = 12
Use HCF to simplify fractions: 24/36 → ÷12 → 2/3
Find LCM(8, 12): Multiples of 8: 8,16,24,32… Multiples of 12: 12,24,36… LCM = 24
Use LCM to add fractions with different denominators: 1/8 + 1/12 → denominator = 24 → 3/24 + 2/24 = 5/24
Prime Factor Decomposition
Every integer greater than 1 can be written as a product of prime factors. This is unique (Fundamental Theorem of Arithmetic).
Divide the number by the smallest prime that goes into it.
Circle each prime number as you reach it.
Collect all circled primes, smallest first. Use index notation.
Find HCF and LCM of 360 and 252.
360 = 2³ × 3² × 5 252 = 2² × 3² × 7
Learning Objectives
Tick each off as you go.
Expressions & Like Terms
Like terms share the same variable and power. Only like terms can be added or subtracted.
Example: 5x + 3y − 2x + 4y = 3x + 7y
Example: 4a² + 3a − a² + 2 − 5a = 3a² − 2a + 2
Substitution
Replace letters with given values — then apply BIDMAS carefully.
Copy it out before substituting.
If x = −3, write (−3) not just −3.
Brackets → Indices → ÷× → +−
Find 3x² − 2x + 5 when x = 4.
Find 2a − b when a = 3, b = −5.
Expanding Brackets
Multiply every term inside the bracket by the term outside.
(2x − 1)(x + 4) = 2x² + 8x − x − 4 = 2x² + 7x − 4
(x − 3)² = x² − 6x + 9
Factorising
The reverse of expanding — take out the HCF and write the remainder in a bracket.
12x² − 8x → HCF = 4x → 4x(3x − 2)
x² − 5x + 6 → pairs: −2 and −3 → (x − 2)(x − 3)
x² + x − 12 → pairs: 4 and −3 → (x + 4)(x − 3)
Substitution Calculator
Evaluates ax² + bx + c for any value of x.
Practice
Simplify 4x² + 3x − x² − 7x
Find 2x² − 3x when x = −2
Factorise x² + 2x − 15
Exam Tips
x=−3: write (−3)²=9, not −3²=−9. The bracket is essential.
Always expand your factorised answer to check it matches the original.
12x²−8x: take out 4x, not just 2 or 4. Get the highest factor of all terms.
Do not add them together. They must be kept separate.
First, Outer, Inner, Last — then collect the middle two terms.
a²−b² = (a+b)(a−b). Look for this pattern before trying other methods.
Learning Objectives
Tick each off as you go.
Area & Perimeter Formulas
Every formula — the height must always be the perpendicular height, not a slant side.
Converting m² to cm²: multiply by 10 000 (not 100).
Area & Perimeter Calculator
Select a shape, enter dimensions, and see area and perimeter with full working.
Composite Shapes
Split into simpler shapes, calculate each part, then add or subtract.
Draw lines on the diagram to divide into rectangles, triangles, etc.
Use the total dimensions to calculate unlabelled sides.
Apply the correct formula to each simpler shape.
Add parts that make up the shape. Subtract any removed sections.
Outer rectangle: 8m × 6m. Corner removed: 3m × 4m.
Practice
A trapezium has parallel sides 8 cm and 12 cm, height 5 cm. Find the area.
Triangle: base 10 cm, perpendicular height 7 cm. Find the area.
Exam Tips
h in every formula = height at 90° to base. Never use the slant.
Perimeter in cm/m. Area in cm²/m². Wrong units cost marks.
m²→cm²: ×10 000. km²→m²: ×1 000 000.
Show each sub-area separately. Examiners award marks for method.
Work out unlabelled sides using total dimensions before adding up.
½(a+b)h — add a and b before multiplying. Very common error.
Learning Objectives
Tick each off as you go.
Mean, Median, Mode & Range
Ordered: 2, 3, 4, 7, 7, 7, 9 (n=7)
| Average | Best used when | Affected by outliers? |
|---|---|---|
| Mean | Data is fairly symmetrical | Yes — most affected |
| Median | Data has outliers or is skewed | No |
| Mode | Categorical or discrete data | No |
Averages from Tables
Multiply each value by its frequency, sum the products, divide by total frequency.
| Score (x) | Frequency (f) | f × x |
|---|---|---|
| 1 | 3 | 3 |
| 2 | 5 | 10 |
| 3 | 2 | 6 |
| Totals | 10 | 19 |
Use the midpoint of each class. This gives an estimate, not an exact answer.
| Class | Midpoint (m) | Freq (f) | f × m |
|---|---|---|---|
| 0–10 | 5 | 4 | 20 |
| 10–20 | 15 | 6 | 90 |
| 20–30 | 25 | 2 | 50 |
| Totals | 12 | 160 |
Averages Calculator
Enter a list of numbers separated by commas.
Practice
Find the median of: 5, 2, 8, 1, 9, 4, 3
Value 2 occurs 4 times, value 3 occurs 6 times, value 4 occurs 2 times. Find the mean.
Exam Tips
The most common error. Write the ordered list before identifying the middle.
8 values → median = average of 4th and 5th. Add and divide by 2.
Divide by total frequency, not number of rows. Very common error.
Write "estimated mean" — the answer cannot be exact from grouped data.
Range measures spread. Never call it an average.
Always comment on the average and the range when comparing two distributions.
Learning Objectives
Tick each off as you go.
Algebra strand, Foundation and Higher. Function machines appear on both calculator and non-calculator papers. Inverse and composite functions are mainly Higher. Function notation f(x) is expected at both tiers.
What are Function Machines?
A function machine takes an input, applies one or more operations in sequence, and produces an output.
Think of each operation as a station on a conveyor belt. Whatever goes in one end comes out the other end changed by each station in turn.
This machine applies ×3 then +5. In function notation: f(x) = 3x + 5. When x = 4: f(4) = 17.
Different letters are used — f(x), g(x), h(x) — but they all work the same way. f(3) means "apply the function to 3", not f multiplied by 3.
Always apply operations left to right in the order given. Do not rearrange or skip steps.
Inverse Functions
The inverse function undoes what the original did — it maps outputs back to inputs. Run the machine backwards with opposite operations.
| Operation | Inverse |
|---|---|
| + a | − a |
| × a | ÷ a |
| − a | + a |
| ÷ a | × a |
| x² | √x |
| x³ | ∛x |
Replace f(x) with y.
Apply inverse operations — reverse the order of the original steps.
Swap x and y in your rearranged equation.
Interactive Function Machine
Build a two-step machine. Enter an input to find the output, or enter an output to work backwards and find the input.
Composite Functions — Higher
A composite function applies one function to the result of another. fg(x) means apply g first, then f.
f(x) = x² + 1, g(x) = 3x. Find fg(4).
f(x) = 2x − 1, g(x) = x + 4. Find fg(x) as an expression.
Practice
A function machine applies ×4 then −3. What is the output when the input is 7?
f(x) = 5x + 2. Find f⁻¹(x).
A machine does ×2 then +6. An output is 20. What was the input?
f(x) = x + 3 and g(x) = 2x. Find fg(5).
Exam Tips
Function machines work in sequence. Do not change the order of operations.
f(x) = ×3 then +5. Inverse = −5 first, then ÷3. Both the operation and the sequence flip.
Apply f then f⁻¹ — you should get back to the original input. If not, recheck.
The function closest to x is applied first. fg(x) = f(g(x)), not g(f(x)).
Function notation means "evaluate f at x=3". It is not multiplication.
For multi-step machines, write the value after each step. Examiners award method marks for this.
Learning Objectives
Tick each off as you go.
At GCSE you are given: Volume of cone = ⅓πr²h, Volume of sphere = ⁴⁄₃πr³, Surface area of sphere = 4πr². You are NOT given cylinder, cuboid or prism formulas — these must be memorised.
Arc Length & Sector Area
An arc is a fraction of the circumference. A sector is a pizza-slice fraction of the full circle.
Both formulas work the same way: take the fraction of the full circle (angle ÷ 360), then multiply by the full circle formula.
θ = angle of the sector in degrees. Both are just a fraction of the full circle formula.
Using the example above: P = 4π + 2(6) = 4π + 12 ≈ 24.57 cm
Volume
Volume measures how much 3D space a shape occupies. Units are always cubed (cm³, m³).
A prism is any 3D shape with a constant cross-section running along its length. The key step is identifying the 2D cross-section shape and finding its area first.
Surface Area
Surface area is the total area of all faces of a 3D shape. Add the area of every face — do not miss any.
Surface area of cone uses slant height l. Volume of cone uses vertical height h. These are different values.
Check whether the question asks for a closed shape (all faces) or an open shape (e.g. a cup with no base, or a half-sphere on a cylinder). Only include the faces that are part of the surface.
3D Shape Calculator
Select a shape, enter its dimensions, and see volume and surface area with full working shown.
Practice
Use π where needed — give exact answers in terms of π unless told otherwise.
A cylinder has radius 4 cm and height 9 cm. Find the volume in terms of π.
Find the surface area of a sphere with radius 5 cm. Give answer in terms of π.
A sector has radius 9 cm and angle 80°. Find the area. Give answer to 3 significant figures.
A cone has radius 3 cm and vertical height 4 cm. Find the total surface area in terms of π.
Exam Tips
You must memorise cylinder (πr²h) and cuboid (lwh). Cone and sphere are given.
SA uses slant height l. Volume uses vertical height h. Find l = √(r²+h²) when needed.
24π is exact. 75.4 is not. Many mark schemes require the exact form.
V = cross-section area × length. Identify the 2D shape, find its area, then multiply.
Do not give just the arc length when asked for the perimeter of a sector.
Volume in cm³. Surface area in cm². Wrong units cost marks every time.
Learning Objectives
Tick each off as you go.
Geometry and Measures — Higher tier mainly, though basic column vectors appear at Foundation. Vector proof questions are exclusively Higher and are often worth 4–5 marks. They test whether you can navigate a diagram systematically.
Introduction to Vectors
A vector has both magnitude (size) and direction. A scalar has magnitude only.
| Scalar (magnitude only) | Vector (magnitude + direction) |
|---|---|
| Speed: 30 mph | Velocity: 30 mph due north |
| Distance: 5 km | Displacement: 5 km east |
| Mass: 70 kg | Force: 70 N downwards |
| Temperature: 20°C | Acceleration: 9.8 m/s² downwards |
Positive x → right Negative x → left Positive y → up Negative y → down
Vector Operations
All operations work component by component — x with x, y with y.
Geometrically: place the second vector at the tip of the first. The sum goes from the start of the first to the end of the second.
A positive scalar scales the magnitude but keeps the direction. A negative scalar also reverses the direction. A scalar of −1 gives the negative vector.
Two vectors are parallel if one is a scalar multiple of the other.
If two vectors share a common point and are parallel, then the three points are collinear (lie on a straight line).
Magnitude of a Vector
The magnitude is the length of the vector — calculated using Pythagoras' theorem.
The vertical bars | | mean "magnitude of". This is exactly Pythagoras applied to the horizontal and vertical components.
Notice: (3, 4) and (−3, 4) have the same magnitude even though they point in different directions. Magnitude is always positive.
A unit vector has magnitude 1. To find the unit vector in the direction of v:
Example: v = (3, 4), |v| = 5. Unit vector = (3/5, 4/5) = (0.6, 0.8).
Vector Paths — Higher
Vector path problems ask you to find the vector for a route by combining known vectors. The key rule: going against an arrow negates the vector.
Build up any path step by step. Each step either uses a vector directly (going with the arrow) or negates it (going against the arrow).
OA⃗ = a, OB⃗ = b. M is the midpoint of AB. Find OM⃗.
To prove three points P, Q, R are collinear:
Express both in terms of the given vectors.
e.g. PR⃗ = 2PQ⃗ means they are parallel.
Since PQ and PR both pass through P and are parallel, P, Q and R are collinear.
Vector Calculator
Enter two vectors — the calculator shows all combinations with full working.
Practice
a = (4, −2) and b = (−1, 5). Find a + b.
Find the magnitude of vector (5, 12).
a = (2, −5). Find 3a.
OA⃗ = a and OB⃗ = b. M is the midpoint of OB. Find AM⃗.
Exam Tips
Top + top, bottom + bottom. Never mix x and y components.
|v| = √(x²+y²). Square each component separately before adding — never add then square.
If OA⃗ = a, then AO⃗ = −a. Travelling against the arrow direction means negate all components.
Write a not just a. Examiners distinguish between vectors and scalars.
If b = 3a, the vectors are parallel. Use this to prove collinearity.
Proving two vectors are parallel is not enough. You must also state they share a common point.
Write each step of the route separately. AC⃗ = AB⃗ + BC⃗. Systematic working earns all marks even if the algebra gets complex.
3-4-5, 5-12-13, 8-15-17 appear constantly in magnitude questions on non-calculator papers.
Learning Objectives
Tick each off as you go.
Statistics strand — Foundation and Higher. Chart reading and drawing appears on both papers. Scatter graphs with correlation descriptions are very common. Frequency polygons and back-to-back stem and leaf diagrams appear more often on Higher.
Chart Types
Each chart has a specific purpose. Using the wrong type — or misreading one — is a common error.
| Chart | Best used for | Key reading rule |
|---|---|---|
| Bar chart | Comparing discrete categories | Height of bar = frequency |
| Dual bar chart | Comparing two groups across categories | Read each bar separately, compare side by side |
| Line graph | Change over time (continuous) | Read from the line — gradient shows rate of change |
| Pictogram | Simple frequency — visual appeal | One symbol = fixed number — ALWAYS check the key |
| Pie chart | Proportions of a whole | Angle ÷ 360 × total = frequency |
| Frequency polygon | Grouped continuous data | Plot at midpoints, join with straight lines |
| Scatter graph | Relationship between two variables | Line of best fit — read from line not from points |
| Stem and leaf | Raw data — shows distribution shape | Leaves increase away from stem |
Scatter Graphs
Scatter graphs show whether two variables are related. Each point represents one item with two measurements.
| Type | What the graph looks like | Real example |
|---|---|---|
| Strong positive | Points close to an upward line | Height and shoe size |
| Weak positive | Roughly upward trend, spread out | Revision hours and exam score |
| Strong negative | Points close to a downward line | Speed and journey time |
| Weak negative | Roughly downward, spread out | Temperature and hot drink sales |
| No correlation | No pattern — points scattered randomly | Shoe size and exam score |
Only interpolate — do not extrapolate far beyond the range of the data. The relationship may not continue outside the range you measured.
Correlation means two variables tend to change together. Causation means one variable directly causes the other to change. Correlation does not prove causation.
Classic example: Ice cream sales and drowning incidents both increase in summer. They are correlated — but eating ice cream does not cause drowning. Both are caused by hot weather (a lurking variable).
An outlier is a point that does not fit the general trend. In an exam question, identify it by its approximate coordinates and explain that it does not follow the pattern shown by the other data points.
Frequency Polygons
Used for grouped continuous data. Plot the midpoint of each class against its frequency, then join with straight lines.
Midpoint = (lower boundary + upper boundary) ÷ 2. For 10 ≤ x < 20: midpoint = (10+20)÷2 = 15.
Midpoint on the x-axis, frequency on the y-axis.
Use a ruler. Do not use a curve. Do not extend the line back to zero at either end (unless the question asks you to close the polygon).
Missing labels cost marks.
| Class (age) | Frequency | Midpoint | Plot point |
|---|---|---|---|
| 0 ≤ x < 10 | 4 | 5 | (5, 4) |
| 10 ≤ x < 20 | 9 | 15 | (15, 9) |
| 20 ≤ x < 30 | 12 | 25 | (25, 12) |
| 30 ≤ x < 40 | 7 | 35 | (35, 7) |
| 40 ≤ x < 50 | 3 | 45 | (45, 3) |
Plot the five points and join with four straight line segments.
When two polygons are drawn on the same axes, you can compare their distributions directly. In exam answers, comment on:
Stem and Leaf Diagrams
Show the actual data values while also displaying the distribution shape. Every value is preserved.
The stem is the tens digit. Each leaf is a units digit. Leaves are written in order, smallest closest to the stem.
From this diagram: min = 12, max = 46, range = 34, median = 28 (8th of 15 values), mode = 23 and 35.
Compares two groups using a shared stem in the middle. Leaves for Group A go left, leaves for Group B go right.
Interactive Scatter Plotter
Enter data points to plot a scatter graph, see the correlation described, and draw an automatic line of best fit.
Practice
A scatter graph shows that as temperature increases, the number of coats sold decreases. Which best describes this?
For the class 20 ≤ x < 30 with frequency 8, where is the point plotted on a frequency polygon?
A stem and leaf diagram has: 1 | 3 5 8, 2 | 0 4 7, 3 | 2 6. Key: 1|3 = 13. What is the median?
A scatter graph shows a strong positive correlation between shoe size and reading ability in children. What can we conclude?
Exam Tips
Never say one variable causes another based on a scatter graph. Always use "correlation" language.
Equal numbers of points above and below. Use a ruler. It does not need to pass through any specific point.
Never plot at the class boundary. Always calculate (lower + upper) ÷ 2 first.
"Strong negative correlation" earns more marks than just "negative." Both words count.
One symbol might equal 5, 10 or any number. Reading the key is the first step, not the last.
Leaves must be written in order (smallest nearest the stem). Disordered leaves lose marks.
The left side of a back-to-back diagram increases away from the stem, not towards it.
Using a line of best fit beyond the measured range gives unreliable estimates. State this limitation when asked.
Learning Objectives
Tick each off as you go.
Probability and Statistics — Foundation and Higher. Two-circle diagrams appear on both tiers. Three-circle diagrams and the addition rule are Higher. Venn diagrams often link to frequency trees, two-way tables and conditional probability.
Sets & Notation
A set is a collection of objects. In GCSE we deal with numbers or events.
| Symbol | Name | Meaning | Example |
|---|---|---|---|
| ξ | Universal set | Everything being considered | Integers 1–10 |
| A∪B | Union | In A OR B (or both) | Evens or multiples of 3 |
| A∩B | Intersection | In A AND B | Even multiples of 3: {6} |
| A' | Complement | NOT in A | Odd numbers: {1,3,5,7,9} |
| n(A) | Cardinality | Number of elements in A | n(A)=5 means 5 elements |
| ∅ | Empty set | No elements | A∩B=∅ means no overlap |
Two-Circle Venn Diagrams
Four regions: A only, the overlap (A∩B), B only, and outside both. Always start with the intersection.
The intersection goes in the overlap. This is usually given or stated as "both".
Subtract what is already in the overlap from the total in A.
Same idea for B.
Subtract the sum of all three inner regions from the total.
30 students. 18 study French (F), 12 study Spanish (S), 7 study both. Find all regions.
| Region | Calculation | Value |
|---|---|---|
| F∩S (both) | Given directly | 7 |
| F only | 18 − 7 | 11 |
| S only | 12 − 7 | 5 |
| Neither | 30 − (11+7+5) | 7 |
Three-Circle Venn Diagrams — Higher
Seven regions plus the outside. Always work from the centre outwards.
The centre value. Always start here.
e.g. A∩B only = (A∩B) − (A∩B∩C). Subtract the centre from each pair total.
A only = n(A) − (A∩B only) − (A∩C only) − (A∩B∩C).
Subtract the sum of all seven inner regions.
Probability from Venn Diagrams
Once the diagram is complete, read probabilities from the regions.
F only=11, F∩S=7, S only=5, Neither=7. Total n(ξ)=30.
| Probability | Calculation | Answer |
|---|---|---|
| P(F) | (11+7) ÷ 30 | 18/30 = 3/5 |
| P(F∩S) | 7 ÷ 30 | 7/30 |
| P(F∪S) | 18/30+12/30−7/30 | 23/30 |
| P(F') | 1 − 18/30 | 12/30 = 2/5 |
| P(neither) | 7 ÷ 30 | 7/30 |
| P(F|S) Higher | P(F∩S) ÷ P(S) = (7/30)÷(12/30) | 7/12 |
Interactive Venn Builder
Enter counts for each region — the diagram and all probabilities update live.
Practice
What does A∩B represent?
50 people surveyed. 28 like coffee, 19 like tea, 11 like both. How many like neither?
Using the coffee/tea data (50 people, coffee=28, tea=19, both=11). Find P(coffee ∪ tea).
P(A) = 0.45. What is P(A')?
Exam Tips
Fill A∩B first. Then A only=n(A)−intersection. B only=n(B)−intersection.
Add all four regions. If they do not sum to n(ξ), recheck every step.
P(A∪B) = A only + overlap + B only. Do not forget the overlap is shared.
P(A∪B)=P(A)+P(B)−P(A∩B). Forgetting to subtract is the most common error.
Start A∩B∩C, then each pair's exclusive overlap, then each circle's exclusive region, then outside.
Write 0 in empty regions. Blank regions look like mistakes and lose marks.
Confusing these reverses the answer. Memorise both symbols before the exam.
P(A|B)=P(A∩B)÷P(B). The denominator is P(B) — you are restricted to set B.
Learning Objectives
Tick each off as you go.
Statistics — Foundation and Higher, both papers. Frequency tables are a foundation skill that feeds into averages, probability, histograms and cumulative frequency. Two-way tables appear on almost every paper.
Tally Charts
A tally chart records data as it is collected. Each mark represents one item. Every fifth mark crosses the previous four — making counting in fives easy.
The diagonal crossing stroke on every 5th tally makes counting efficient. Always count in groups of five when reading tallies.
Write them in the left column before you start recording.
Mark one tally stroke for each item in the correct row.
Count in fives. Write the total in the Frequency column.
Add all frequencies. They must equal the total number of items.
Ungrouped Frequency Tables
Data recorded as exact values. You can find all four averages directly from the table.
Add a third column (f × x). Sum it. Divide by the total frequency Σf.
| Score (x) | Frequency (f) | f × x |
|---|---|---|
| 1 | 3 | 3 |
| 2 | 7 | 14 |
| 3 | 5 | 15 |
| 4 | 4 | 16 |
| 5 | 1 | 5 |
| Total | 20 | 53 |
Use a cumulative frequency column to find which value is in position 10 and 11.
| Score | Frequency | Cumulative frequency |
|---|---|---|
| 1 | 3 | 3 |
| 2 | 7 | 10 |
| 3 | 5 | 15 |
| 4 | 4 | 19 |
| 5 | 1 | 20 |
Position 10 → score 2. Position 11 → score 3. Median = (2+3)÷2 = 2.5
Mode = 2 (highest frequency). Range = 5 − 1 = 4
Grouped Frequency Tables
Data is placed into class intervals. Exact values are lost — so averages are estimates only.
| Class (height cm) | Freq (f) | Midpoint (m) | f × m |
|---|---|---|---|
| 140 ≤ h < 150 | 4 | 145 | 580 |
| 150 ≤ h < 160 | 11 | 155 | 1705 |
| 160 ≤ h < 170 | 9 | 165 | 1485 |
| 170 ≤ h < 180 | 6 | 175 | 1050 |
| Total | 30 | 4820 |
Example: P(height is 160–170) = 9 ÷ 30 = 3/10 = 0.3
Frequency Table Calculator
Enter your ungrouped frequency table — the calculator finds all averages with full working.
Two-Way Tables
Show two pieces of categorical information simultaneously. Row totals, column totals and the grand total must all be consistent.
| Cats | Dogs | Neither | Total | |
|---|---|---|---|---|
| Boys | 8 | 12 | 5 | 25 |
| Girls | 14 | 6 | 5 | 25 |
| Total | 22 | 18 | 10 | 50 |
Row totals and column totals must both sum to the grand total (50). Use this to find missing values.
Each row and each column must sum to its total. Write these as equations.
Fill in what you know. The missing value is whatever makes the row or column sum correctly.
Using the cats/dogs table above (50 students total):
| Question | Calculation | Answer |
|---|---|---|
| P(likes cats) | 22 ÷ 50 | 11/25 = 0.44 |
| P(girl who likes dogs) | 6 ÷ 50 | 3/25 = 0.12 |
| P(boy | likes cats) | 8 ÷ 22 | 4/11 ≈ 0.364 |
| P(girl | likes dogs) | 6 ÷ 18 | 1/3 ≈ 0.333 |
Practice
A frequency table shows: value 2 (freq 3), value 4 (freq 5), value 6 (freq 2). What is the mean?
A grouped table has class 20≤x<30 with frequency 8 and class 30≤x<40 with frequency 12. What is the midpoint of the second class?
In a two-way table: 40 students total. 22 are girls. 15 boys play sport. 10 girls play sport. How many students play sport in total?
Classes: 0–10 (f=3), 10–20 (f=12), 20–30 (f=8), 30–40 (f=5). What is the modal class?
Exam Tips
Mean = Σ(fx) ÷ Σf. Dividing by the number of rows (e.g. 5) instead of total frequency is the most common error.
Always write out the f×x column. It earns method marks and prevents arithmetic errors.
Midpoints are approximations. The word "estimated" is required in your answer.
For grouped data you can only state the modal class (e.g. 10≤x<20), not a specific mode value.
Check every row and column sums to its marginal total. Use this to find missing values.
P(A|B) — the denominator is the B total, not the grand total.
Add a running total column. Find the row that contains the median position.
Sum the frequency column before doing any calculations. A wrong total cascades through everything.
Learning Objectives
Tick each off as you go.
Probability — Foundation and Higher, both papers. Basic probability and expected outcomes are Foundation. Independent events, relative frequency as an estimate, and combining probabilities are Higher. Links to Venn diagrams and tree diagrams (Combined Events lesson).
Basic Probability
Probability measures how likely an event is — always between 0 (impossible) and 1 (certain).
Example: rolling a fair die. P(even) = 3/6 = 1/2 (three even numbers: 2, 4, 6 out of six outcomes)
Example: picking a red card from a standard deck. P(red) = 26/52 = 1/2
Example: P(rain tomorrow) = 0.35. P(no rain) = 1 − 0.35 = 0.65
For combined events, list all outcomes in a sample space diagram.
Example: two coins. Outcomes: HH, HT, TH, TT. P(two heads) = 1/4. P(at least one head) = 3/4.
Experimental Probability
Also called relative frequency — based on actual results from an experiment or survey rather than theory.
Example: a biased coin is flipped 200 times. Heads occurs 130 times.
This is an estimate for P(heads). The more trials, the more reliable the estimate.
| Theoretical | Experimental | |
|---|---|---|
| Based on | Equally likely outcomes | Actual results |
| When to use | Fair coins, dice, cards | Biased equipment, real data |
| Improves with | More outcomes listed | More trials |
| Symbol | P(A) = exact value | P(A) ≈ relative frequency |
If experimental probability consistently differs from theoretical probability after many trials, the equipment is likely biased.
Example: if a die shows 6 on 1/4 of trials (theoretical = 1/6), it is likely biased towards 6.
Expected Outcomes
Expected frequency predicts how many times an event will occur in a given number of trials.
Example: P(six on a fair die) = 1/6. Roll 300 times. Expected sixes = 1/6 × 300 = 50
Example: P(defective item) = 0.03. Produce 2500 items. Expected defective = 0.03 × 2500 = 75
If you know the expected frequency and the probability, you can find the number of trials.
Mutually Exclusive Events
Two events are mutually exclusive if they cannot both happen at the same time.
Example: rolling a die. P(1 or 2) = P(1) + P(2) = 1/6 + 1/6 = 2/6 = 1/3
Cannot roll a 1 and a 2 on the same throw → mutually exclusive.
A set of events is exhaustive if one of them must happen. The probabilities of an exhaustive set of mutually exclusive events sum to 1.
Use this to find missing probabilities: if three mutually exclusive outcomes have P = 0.3, 0.45, and x, then x = 1 − 0.3 − 0.45 = 0.25.
If events CAN both happen, use the addition rule with the intersection subtracted (from Venn Diagrams):
Independent Events
Two events are independent if the outcome of one does not affect the outcome of the other.
Example: flipping a coin and rolling a die. P(heads and 4) = 1/2 × 1/6 = 1/12
The coin flip does not affect the die roll → independent.
Events A and B are independent if P(A∩B) = P(A) × P(B).
Example: P(A) = 0.4, P(B) = 0.3, P(A∩B) = 0.12. Check: 0.4 × 0.3 = 0.12 ✓ → independent.
Example: P(A) = 0.5, P(B) = 0.4, P(A∩B) = 0.25. Check: 0.5 × 0.4 = 0.20 ≠ 0.25 → NOT independent.
Example: P(exactly 2 heads in 3 flips) = ³C₂ × (1/2)² × (1/2)¹ = 3 × 1/4 × 1/2 = 3/8
Probability Calculator
Calculate theoretical probability, expected frequency, or check independence.
Practice
A bag contains 3 red, 5 blue and 2 green counters. What is P(blue)?
P(scoring a goal) = 0.4. A player takes 35 shots. How many goals are expected?
P(A) = 0.3, P(B) = 0.45. A and B are mutually exclusive. Find P(A or B).
P(A) = 0.6 and P(B) = 0.5. A and B are independent. Find P(A and B).
Exam Tips
If your answer is greater than 1 or negative, you have made an error.
Often easier to calculate the probability of something NOT happening and subtract from 1.
P(A or B) = P(A) + P(B) only when events cannot both happen.
P(A and B) = P(A) × P(B) only when events do not affect each other.
A small experiment gives unreliable estimates. State this when commenting on results.
Use a sample space table or ordered list. Missing outcomes is a common error.
Write "expected" not "will". Probability gives long-run averages, not certainties.
Use this to find missing probabilities in a table or list.
Learning Objectives
Tick each off as you go.
Statistics — Foundation and Higher. Sampling, data collection and representation appear on both papers. Comparing distributions and critiquing statistical methods are common Higher questions worth 3–4 marks.
Sampling
A sample is a subset of the population selected for investigation. The goal is a representative sample that avoids bias.
| Type | Definition | Examples |
|---|---|---|
| Qualitative | Non-numerical categories | Colour, gender, favourite subject |
| Discrete quantitative | Countable whole numbers | Number of siblings, shoe size |
| Continuous quantitative | Measured on a scale — any value in a range | Height, weight, temperature, time |
| Primary data | Collected directly by the researcher | Surveys, experiments, interviews |
| Secondary data | Already collected by someone else | Census data, internet statistics |
| Method | How it works | Pros / Cons |
|---|---|---|
| Random sampling | Each member has equal chance of selection | Fair, unbiased — needs numbered list |
| Systematic sampling | Select every nth member from ordered list | Simple — could miss patterns |
| Stratified sampling | Sample in proportion to subgroup sizes | Representative — more complex |
| Convenience sampling | Use whoever is available | Easy — often biased |
| Quota sampling | Fill set quotas for each subgroup | Practical — interviewer bias possible |
Example: School has 400 Year 10 and 600 Year 11 students. Sample of 50 needed.
Year 10: (400÷1000)×50 = 20 Year 11: (600÷1000)×50 = 30
Organising Data
Raw data is hard to interpret. Organising it into tables makes patterns visible.
Record data as it is collected. Five-bar gate method — every 5th tally crosses the previous four. Count in fives when reading.
Show two categorical variables simultaneously. Row and column totals must both equal the grand total. Use known totals to find missing values.
| Walk | Bus | Car | Total | |
|---|---|---|---|---|
| Boys | 12 | 8 | 5 | 25 |
| Girls | 9 | 11 | 5 | 25 |
| Total | 21 | 19 | 10 | 50 |
Representing Data
Choosing the right chart for the data type is as important as drawing it accurately.
| Data type | Best chart | Key rule |
|---|---|---|
| Discrete categories | Bar chart / pie chart | Gaps between bars; angle = proportion × 360° |
| Continuous grouped | Histogram / frequency polygon | No gaps; plot midpoints for frequency polygon |
| Change over time | Line graph / time series | Join points; gradient = rate of change |
| Two variables | Scatter graph | Line of best fit; describe correlation |
| Raw data distribution | Stem and leaf | Ordered leaves; median and range easy to read |
In a histogram the area of each bar represents the frequency, not the height. Bars have no gaps.
| Class | Freq | Width | Freq density |
|---|---|---|---|
| 0 ≤ x < 10 | 20 | 10 | 2.0 |
| 10 ≤ x < 15 | 25 | 5 | 5.0 |
| 15 ≤ x < 30 | 30 | 15 | 2.0 |
A running total of frequencies. Plot upper class boundary against cumulative frequency, join with a smooth curve. Read off median (50th percentile), LQ (25th) and UQ (75th).
Averages & Spread
Summary statistics reduce a dataset to a few key values describing centre and spread.
Example: 3, 5, 7, 8, 10, 12, 15, 18, 20. n=9. Median=10 (5th). LQ=6 (median of 3,5,7,8). UQ=16.5 (median of 12,15,18,20). IQR=16.5−6=10.5
Show the five-number summary: minimum, LQ, median, UQ, maximum.
The box shows the middle 50% of data (IQR). Whiskers show the full range. Outliers may be plotted as separate points.
Comparing Distributions
Always compare both a measure of average (centre) AND a measure of spread. One sentence for each.
| Compare | Use | What it tells you |
|---|---|---|
| Centre | Mean or median | Which group has higher/lower typical values |
| Spread | Range or IQR | Which group is more consistent/variable |
Always include: the statistic, the values for both groups, and a contextual interpretation.
Practice
A school has 300 Year 10 and 200 Year 11 students. A stratified sample of 50 is needed. How many Year 10 students should be included?
A histogram bar for class 20≤x<30 has frequency density 3.5. What is the frequency?
Group A: median = 72, IQR = 18. Group B: median = 65, IQR = 28. Which statement is correct?
Which of these is a problem with the question: "How many hours of TV do you watch? □ 0–2 □ 2–4 □ 4–6"?
Exam Tips
(Stratum size ÷ total population) × sample size. Show this calculation in full.
Use ≤ x < to ensure every value fits exactly one class. 10–20 and 20–30 is wrong because 20 fits both.
Frequency = frequency density × class width. Bars with different widths have different scales.
One mark for median comparison, one mark for IQR comparison. Never just say "Group A did better."
Say "The median for A (72) is greater than for B (65)..." — values must appear in your answer.
IQR ignores outliers. Prefer IQR when comparing spread, especially if data is skewed.
Neutral language, mutually exclusive classes, exhaustive options. No "sometimes/often" without definition.
Never use a curve. Plot at class midpoints, not boundaries.
Learning Objectives
Tick each off as you go.
Geometry and Measures — Foundation and Higher. Describing transformations precisely is essential — missing one detail loses the mark. Negative and fractional scale factors for enlargement are Higher only.
Reflection
A reflection flips a shape in a mirror line. Every point maps to an equal distance on the other side.
| Mirror line | Effect on coordinates (x, y) |
|---|---|
| x-axis (y = 0) | (x, y) → (x, −y) |
| y-axis (x = 0) | (x, y) → (−x, y) |
| y = x | (x, y) → (y, x) |
| y = −x | (x, y) → (−y, −x) |
| y = a (horizontal) | (x, y) → (x, 2a−y) |
| x = a (vertical) | (x, y) → (2a−x, y) |
Example: "Reflection in the line y = x"
Rotation
A rotation turns a shape about a fixed point (centre of rotation) through a given angle.
| Rotation | Effect on (x, y) |
|---|---|
| 90° clockwise | (x, y) → (y, −x) |
| 90° anticlockwise | (x, y) → (−y, x) |
| 180° (either direction) | (x, y) → (−x, −y) |
| 270° clockwise = 90° anti | (x, y) → (−y, x) |
Example: "Rotation of 90° anticlockwise about (0, 0)"
Draw lines joining each vertex to its image. The perpendicular bisectors of these lines intersect at the centre of rotation.
Translation
A translation slides a shape without rotating or reflecting it. Described using a column vector.
Example: vector (3 above, −2 below) moves each point 3 right and 2 down.
Example: "Translation by vector (5 above, −3 below)"
The inverse translation uses the negative vector: if one way is (3, −2), going back is (−3, 2).
Enlargement
An enlargement changes the size of a shape while keeping angles the same. The shape and its image are similar.
| Scale factor k | Effect |
|---|---|
| k > 1 | Larger image, same side as original relative to centre |
| 0 < k < 1 | Smaller image, same side as original |
| k = 1 | No change (identity) |
| k < 0 (negative) — Higher | Image on opposite side of centre, rotated 180° |
These rays are your guide lines.
New distance = original distance × scale factor.
All new sides are k times the original sides.
Example: "Enlargement with scale factor 3 centred at (1, 2)"
Draw lines from each vertex on the original to the corresponding vertex on the image. Extend them — they all meet at the centre of enlargement.
Combined Transformations
Two or more transformations applied in sequence. Order matters — the result of AB is generally different from BA.
A point that does not move under a transformation is called an invariant point.
Practice
Point A is at (3, 5). It is reflected in the x-axis. What are the new coordinates?
Which information is NOT needed to fully describe a rotation?
A shape has area 12 cm². It is enlarged by scale factor 3. What is the area of the image?
Exam Tips
If you say "reflection" but give a rotation description, you get zero. Always name the transformation.
All three are required. Missing any one loses the mark.
Multiplying area by k (not k²) is the most common enlargement error.
You are allowed tracing paper in the exam. Use it for rotations and reflections in diagonal lines.
Write the vector in column form. "3 right and 2 down" is acceptable but vector notation is cleaner.
Image is on the opposite side of the centre, rotated 180°. Only tested at Higher.
Learning Objectives
Tick each off as you go.
Estimation & Approximation
Round every value to 1 significant figure before calculating. This gives a quick check on whether your exact answer is reasonable.
38.4 → 40, 6.7 → 7, 0.048 → 0.05, 312 → 300
Use simple mental arithmetic.
Show all rounded values in your working — they earn marks.
Rounding
Two systems — decimal places (d.p.) and significant figures (s.f.). Know both.
Count digits after the decimal point. Look at the next digit: ≥5 round up, <5 leave unchanged.
Count from the first non-zero digit. Leading zeros are NOT significant.
Error Bounds — Upper and Lower Bounds
When a measurement is rounded, the true value lies within an interval. Finding the upper and lower bounds defines that interval.
Example: a length is 7.4 cm (rounded to 1 d.p.). Half a unit = 0.05.
Lower bound = 7.4 − 0.05 = 7.35 cm
Upper bound = 7.4 + 0.05 = 7.45 cm
Written as: 7.35 ≤ length < 7.45 (note: upper bound is NOT included)
| Operation | Maximum result uses | Minimum result uses |
|---|---|---|
| a + b | Upper(a) + Upper(b) | Lower(a) + Lower(b) |
| a − b | Upper(a) − Lower(b) | Lower(a) − Upper(b) |
| a × b | Upper(a) × Upper(b) | Lower(a) × Lower(b) |
| a ÷ b | Upper(a) ÷ Lower(b) | Lower(a) ÷ Upper(b) |
A rectangle has length 8.3 cm (1 d.p.) and width 4.7 cm (1 d.p.). Find the upper bound of the area.
Calculator Methods
Efficient calculator use avoids rounding errors and saves time.
Never round intermediate steps in a multi-step calculation. Only round the final answer.
Practice
Round 0.006247 to 2 significant figures.
A mass is given as 5.6 kg, rounded to 1 decimal place. What is the lower bound?
a = 12 (to nearest whole), b = 5 (to nearest whole). Find the maximum value of a − b.
Exam Tips
Write each number rounded to 1 s.f. before calculating. Missing this step loses marks even if the estimate is correct.
0.0062 has 2 sig figs (6 and 2). The zeros are placeholders.
Rounded to nearest 10: half unit = 5. Rounded to 1 d.p.: half unit = 0.05. Always halve the precision unit.
Write 7.35 ≤ x < 7.45 — the upper bound uses strict inequality.
Subtracting a smaller number gives a bigger result. Max(a−b) = UB(a) − LB(b).
Store full precision throughout a multi-step calculation. Round only the final answer.
Learning Objectives
Tick each off as you go.
Linear Equations
A linear equation has no powers higher than 1. Solve by performing the same operation to both sides until x is isolated.
Example: A rectangle has length (2x+3) and width (x−1). Perimeter = 28 cm. Find x.
Quadratic Equations
A quadratic has an x² term. Three methods — factorising (fastest), completing the square, and the quadratic formula (always works).
Rearrange to ax² + bx + c = 0 first. Then factorise and set each bracket to zero.
Example: 2x² + 3x − 5 = 0. a=2, b=3, c=−5.
| b² − 4ac | Number of solutions | Graph |
|---|---|---|
| > 0 | Two distinct real roots | Crosses x-axis twice |
| = 0 | One repeated root | Touches x-axis once |
| < 0 | No real roots | Does not cross x-axis |
Simultaneous Equations
Two equations, two unknowns. Solve by elimination (add/subtract to remove one variable) or substitution.
Make the coefficients of one variable equal, then add or subtract the equations.
Rearrange one equation for one variable, substitute into the other.
Substitution is essential when one equation is quadratic — Higher.
Inequalities
Solve like equations but keep the inequality sign. One key difference: multiplying or dividing by a negative number reverses the sign.
| Symbol | Meaning | Circle on number line |
|---|---|---|
| < or > | Strict inequality (not equal) | Open circle ○ |
| ≤ or ≥ | Includes the boundary value | Closed circle ● |
Closed circle at −2 (included), open circle at 4 (not included)
Solve the equation first to find critical values, then sketch the parabola to determine which region satisfies the inequality.
Equation Solver
Solve linear and quadratic equations with full working shown.
Practice
Solve 4x − 3 = 2x + 9
Solve x² + x − 12 = 0
Solve −3x + 2 > 14
Solve: 2x + y = 7 and x + y = 4
Exam Tips
x² = 5x is NOT rearranged. Subtract 5x: x²−5x=0 → x(x−5)=0 → x=0 or x=5.
−3x > 9 → x < −3. Forgetting to reverse is the most common inequality error.
Substitute both values back into BOTH equations. One equation is not enough.
Calculate b²−4ac before doing anything else. It tells you how many solutions to expect.
Strict inequalities (< >) use open circles. ≤ ≥ use closed circles. Always draw the line.
Method marks apply even when the final answer is wrong. Write each step on a new line.
Learning Objectives
Tick each off as you go.
Circle Theorems
Eight theorems — each must be stated by name when used as a reason in a proof.
| # | Theorem | Key fact |
|---|---|---|
| 1 | Angle at centre | Angle at centre = 2 × angle at circumference (same arc) |
| 2 | Angle in semicircle | Angle in a semicircle = 90° (diameter is the chord) |
| 3 | Angles in same segment | Angles subtended by same arc are equal |
| 4 | Cyclic quadrilateral | Opposite angles sum to 180° |
| 5 | Tangent-radius | Tangent ⊥ radius at point of contact (90°) |
| 6 | Two tangents from external point | Equal length; line to centre bisects angle between tangents |
| 7 | Alternate segment theorem | Angle between tangent and chord = angle in alternate segment |
| 8 | Perpendicular from centre to chord | Bisects the chord (and the arc) |
Write the values you know directly on the figure.
Look for the key features: diameter, tangent, cyclic quadrilateral, same arc.
Write "angle in semicircle = 90°" or "opposite angles of cyclic quad = 180°" as the reason.
Theorem 1
Theorem 2
Geometric Constructions
Use only a compass and ruler — no protractor. Leave all construction arcs visible for marks.
| Construction | Method summary |
|---|---|
| Perpendicular bisector of AB | Open compass to >½AB. Draw arcs above and below from A and B. Join the two intersections. |
| Angle bisector | Arc from vertex crosses both arms. Arcs from those intersections meet inside — join to vertex. |
| Perpendicular from point P to line | Arc from P cuts line at two points. Perpendicular bisect that chord. |
| Perpendicular at point P on line | Equal arcs either side of P. Perpendicular bisect those two points. |
| 60° angle | Draw arc from vertex. Same radius arc from where it crosses the line — join. |
| Triangle given SSS/SAS/ASA | Draw base. Set compass to given length. Intersecting arcs give the third vertex. |
Loci
A locus is the set of all points satisfying a given condition. Always draw with a compass and ruler.
| Condition | Locus | Drawn using |
|---|---|---|
| Fixed distance from a point P | Circle, centre P, radius d | Compass set to d |
| Fixed distance from a line AB | Two parallel lines distance d from AB (with semicircles at ends) | Ruler + compass for ends |
| Equidistant from two points A and B | Perpendicular bisector of AB | Perpendicular bisector construction |
| Equidistant from two lines | Angle bisector of the two lines | Angle bisector construction |
Construct each boundary line or curve using the appropriate method.
Shade the area that satisfies ALL conditions simultaneously.
"Less than" or "within" — boundary not included. "No more than" or "up to" — boundary included.
Treasure is within 5 m of point A, and closer to B than to C.
Practice
A, B and C are points on a circle. Angle ABC = 38°. What is the angle at the centre subtended by arc AC?
What is the locus of points equidistant from two points A and B?
ABCD is a cyclic quadrilateral. Angle DAB = 112°. What is angle BCD?
Exam Tips
"Angle in semicircle = 90°" scores the mark. "It looks right" does not.
Arcs are evidence of method. Leave every arc — even the "messy" ones.
Mark this angle on every circle diagram involving a tangent before doing anything else.
Not equal — they sum to 180°. The most commonly confused circle theorem.
Freehand curves lose marks. A circle must be drawn with a compass.
"Within 3 m" — boundary not included (dashed). "Up to 3 m" — boundary included (solid).
Learning Objectives
Tick each off as you go.
y = mx + c
Every straight line graph can be written in this form. m is the gradient, c is the y-intercept.
| Equation | Gradient (m) | y-intercept (c) |
|---|---|---|
| y = 3x + 2 | 3 (positive — slopes up) | 2 |
| y = −2x + 5 | −2 (negative — slopes down) | 5 |
| y = ½x − 3 | ½ (gentle positive slope) | −3 |
| y = 4 | 0 (horizontal line) | 4 |
| x = 3 | undefined (vertical line) | none |
Example: points (1, 5) and (4, 11). m = (11−5) ÷ (4−1) = 6 ÷ 3 = 2
Use the widest possible points for accuracy — small triangles give rounding errors.
Sometimes equations are given in other forms. Rearrange to identify m and c.
Parallel & Perpendicular Lines
Two lines are parallel if they have the same gradient. Perpendicular lines have gradients that multiply to −1.
A line parallel to y = 3x + 2 passing through (0, 7) has equation y = 3x + 7
Example: line perpendicular to y=2x+3 through point (4,1).
Finding the Equation of a Line
Two methods — both reliable. Choose based on what information you are given.
Example: gradient = 3, passes through (2, 7).
m = (y₂ − y₁) ÷ (x₂ − x₁)
Rearrange to y = mx + c form.
Example: points (1, 3) and (4, 12). m = (12−3)÷(4−1) = 3. Then: y−3 = 3(x−1) → y = 3x
Kinematic Graphs
Distance-time and velocity-time graphs are the two main types. Gradient is the key — it has a physical meaning in both.
| Feature | Meaning |
|---|---|
| Gradient of line | Speed (distance ÷ time) |
| Steep line | Fast speed |
| Gentle line | Slow speed |
| Horizontal line | Stationary (not moving) |
| Line going down | Moving back towards start |
| Curved section | Changing speed (acceleration or deceleration) |
| Feature | Meaning |
|---|---|
| Gradient of line | Acceleration (velocity ÷ time) |
| Positive gradient | Accelerating |
| Negative gradient | Decelerating |
| Horizontal line | Constant velocity |
| Area under graph | Distance travelled |
Line Graph Plotter
Enter m and c to plot y = mx + c and see key features.
Practice
What is the gradient of the line y = −3x + 7?
A line has gradient 4. What is the gradient of a perpendicular line?
A distance-time graph shows a horizontal line for 10 minutes. What does this mean?
Exam Tips
Always rearrange before reading off m and c. 2y=4x+6 → y=2x+3 → m=2, c=3.
Flip the fraction and change the sign. m=⅔ → perp = −3/2.
Bigger rise and run values reduce rounding error. Use the grid carefully.
Steeper = faster. Horizontal = stopped. Downward slope = returning.
Split the area into triangles and rectangles. Do not confuse gradient (acceleration) with area (distance).
Always substitute the given point into your equation to verify it satisfies it.
Learning Objectives
Tick each off as you go.
Arithmetic (Linear) Sequences
Add the same number each time. The nth term is always of the form dn + c.
Example: 5, 9, 13, 17… d = 4. c = 5 − 4 = 1. nth term = 4n + 1
Example: 20, 17, 14, 11… d = −3. c = 20 − (−3) = 23. nth term = −3n + 23
Set nth term = the value. Solve for n. If n is a positive integer, it is in the sequence.
Geometric Sequences
Multiply by the same ratio each time. The common ratio r is found by dividing any term by the previous term.
Example: first term 5, r = 2. nth term = 5 × 2ⁿ⁻¹
T(6) = 5 × 2⁵ = 5 × 32 = 160
If 0 < r < 1, the terms decrease. Example: 80, 40, 20, 10… r = ½. Terms halve each time.
If r is negative, terms alternate between positive and negative: 3, −6, 12, −24… r = −2.
Special Sequences
These sequences appear by name in exam questions — know them all.
| Sequence | First terms | nth term |
|---|---|---|
| Square numbers | 1, 4, 9, 16, 25… | n² |
| Cube numbers | 1, 8, 27, 64, 125… | n³ |
| Triangular numbers | 1, 3, 6, 10, 15, 21… | n(n+1)/2 |
| Powers of 2 | 2, 4, 8, 16, 32… | 2ⁿ |
| Powers of 10 | 10, 100, 1000… | 10ⁿ |
| Prime numbers | 2, 3, 5, 7, 11, 13… | No simple formula |
| Fibonacci sequence | 1, 1, 2, 3, 5, 8, 13… | Each term = sum of previous two |
The nth triangular number = 1 + 2 + 3 + … + n = n(n+1)/2.
Example: 6th triangular number = 6×7÷2 = 21
Useful check: triangular numbers are always of the form n(n+1)/2. Is 45 triangular? n(n+1)/2 = 45 → n²+n−90=0 → n=9. Yes: 9×10÷2=45 ✓
Quadratic Sequences — Higher
The second differences are constant and non-zero. The nth term contains an n² term.
First differences, then differences of those. If constant, the sequence is quadratic.
This is the coefficient of n².
The remainders form a linear sequence. Find its nth term (dn+c).
Verify by substituting n=1,2,3 into your formula.
1st differences: 5, 7, 9, 11 2nd differences: 2, 2, 2 → a = 2÷2 = 1
| n | Term | n² | Term − n² |
|---|---|---|---|
| 1 | 3 | 1 | 2 |
| 2 | 8 | 4 | 4 |
| 3 | 15 | 9 | 6 |
| 4 | 24 | 16 | 8 |
Remainders: 2, 4, 6, 8 → linear nth term = 2n. Full nth term = n² + 2n = n(n+2)
Sequence Calculator
Enter the first four terms — get the nth term formula, check any term, and list the first 10.
Practice
Find the nth term of: 7, 11, 15, 19…
A geometric sequence has first term 5 and common ratio 3. What is the 5th term?
What is the 8th triangular number?
Exam Tips
Always check your formula gives the correct first few terms before moving on.
Set nth term = value, solve for n. Only accept positive whole number answers.
Divide each term by the previous one. If the ratio is not constant, it is not geometric.
If first differences are not constant, find second differences. If those are constant, use the quadratic method.
1,1,2,3,5,8,13,21,34… If asked to continue a Fibonacci-type sequence, add the previous two values.
Know this by heart — it comes up regularly and the formula is rarely given.