KS3 Maths topics

Place value & ordering

Read, write and order integers and decimals of any size.

Negative numbers

Order, add and subtract with numbers below zero. Read the full explainer →

Builds on: Place value & ordering — “Ordering below zero extends the line you already walk above it.”

Order of operations

Apply the agreed order: brackets, indices, × ÷, + −.

Builds on: Negative numbers · Multiplication & division fluency — “You can't order operations you can't yet perform.”

Fractions of amounts

Find fractions of quantities and simplify fractions.

Builds on: Multiplication & division fluency — “Finding three-fifths of £40 is division and multiplication in costume.” · Equivalent fractions

Reading algebraic notation

Read 3x as “three lots of x” — letters standing for numbers.

Simplifying expressions

Tidy an expression so its structure shows.

Builds on: Reading algebraic notation — “Simplifying starts from reading the notation correctly.”

Ratio notation

Write and simplify ratios; share in a given ratio.

Builds on: Multiplication & division fluency — “Sharing £24 in 3:5 is division with a plan.” · Equivalent fractions

Perimeter & area

Perimeter and area of rectangles, triangles, compound shapes.

Builds on: Multiplication & division fluency — “Area counts squares by multiplying.”

Averages & range

Mean, median, mode and range from raw data.

Builds on: Tally charts & frequency tables · The mean as fair shares — “Compute the mean only after it means something.”

Inverse operations

Every operation has an undo — subtraction undoes addition.

Builds on: Order of operations — “To undo operations in reverse order, you must know the order they happen in.” · Function machines

Multiplication & division fluency

Multiply and divide whole numbers confidently with a written method — long multiplication and short division.

Rounding to powers of ten

Round whole numbers to the nearest 10, 100 or 1,000 — and judge when a rounded answer is good enough.

Builds on: Place value & ordering — “Rounding to the nearest ten means knowing which digit holds the tens.”

Adding & subtracting decimals

Add and subtract decimals with the points lined up, so tenths meet tenths. Read the full explainer →

Builds on: Place value & ordering — “Lining up the decimal point is place value in action.”

Multiplying & dividing decimals

Multiply and divide decimals by shifting place value — 0.3 × 0.2 without a calculator. Read the full explainer →

Builds on: Multiplication & division fluency — “Decimal multiplication is whole-number multiplication plus a place-value shift.” · Adding & subtracting decimals

Factors, multiples & primes

Know what factors, multiples and primes are, and find them for numbers up to 100.

Builds on: Multiplication & division fluency — “Factors are found by dividing — fluently.”

Prime factorisation

Break any number into a product of primes with a factor tree.

Builds on: Factors, multiples & primes — “You can't break a number into primes until you can spot one.”

HCF & LCM

Find the highest common factor and lowest common multiple of two numbers.

Builds on: Factors, multiples & primes — “Common factors start with finding factors at all.” · Prime factorisation

Squares, cubes & roots

Recognise square and cube numbers and undo them with square and cube roots.

Builds on: Multiplication & division fluency — “Squaring is multiplying — just by yourself.”

Equivalent fractions

See that 2/3, 4/6 and 8/12 are the same number in different outfits.

Builds on: Multiplication & division fluency — “Equivalents are made by multiplying top and bottom — tables first.”

Comparing & ordering fractions

Put fractions in order by rewriting them over a common denominator. Read the full explainer →

Builds on: Equivalent fractions — “You compare fractions by rewriting them as equivalents.”

Inequality symbols & number lines

Read and write <, >, ≤ and ≥, and show the numbers they describe on a line.

Builds on: Place value & ordering — “You can't mark numbers on a line you can't yet order.” · Negative numbers

Estimating before calculating

Get a rough answer first — round the numbers, estimate, and use it to catch slips.

Builds on: Multiplication & division fluency · Rounding to powers of ten — “Estimates are built from rounded numbers.”

Substituting into expressions

Swap the letter for a number and evaluate — find 3x + 2 when x = 5.

Builds on: Order of operations — “Substituting drops numbers into an order-of-operations machine.” · Reading algebraic notation — “Letters must stand for numbers before numbers can stand in.”

Writing expressions from words

Turn '5 more than a number, then doubled' into 2(n + 5).

Builds on: Reading algebraic notation — “You write notation you can already read.”

Coordinates in four quadrants

Plot and read points anywhere on the grid, including negative coordinates. Read the full explainer →

Builds on: Negative numbers — “Two of the four quadrants live below zero.”

Function machines

Numbers go in, a rule acts, numbers come out — the machine picture of a rule.

Term-to-term sequences

Continue a sequence by describing how each term makes the next.

Expanding a single bracket

Multiply a term over a bracket: 3(x + 4) = 3x + 12.

Builds on: Substituting into expressions · Simplifying expressions — “You can't multiply over a bracket until simplifying feels routine.”

Term, expression, equation, formula

Learn what the four words mean, so instructions like 'simplify the expression' make sense.

Builds on: Reading algebraic notation — “The four words sort symbols you can now read.”

Substituting into formulae

Use a real formula — put numbers into P = 2(l + w) and get a perimeter.

Builds on: Substituting into expressions — “Formulae are expressions with a job title.”

Sequences from a position rule

Generate a sequence from its position: the 10th term of 3n + 1 without listing nine others.

Builds on: Substituting into expressions — “Position rules are substitution in disguise.” · Term-to-term sequences — “A position rule is a shortcut through a sequence you can already continue.”

Recognising special sequences

Know the famous families by sight: odds, evens, squares and triangle numbers.

Builds on: Squares, cubes & roots · Term-to-term sequences — “You recognise families by continuing their members.”

Ratios with mixed units

Simplify ratios like 50p : £2 by putting both sides in the same unit first.

Builds on: Ratio notation — “Mixed units complicate a notation that must already be solid.” · Metric unit conversions — “Same units first — then the ratio tells the truth.”

Ratio ↔ fraction links

See that 3:5 means 3/8 of the whole for one share — part-to-part versus part-of-whole.

Builds on: Equivalent fractions — “A ratio and a fraction describe the same split through different doors.” · Ratio notation — “The link needs both ends — ratio notation is one of them.”

Proportion tables

Complete tables of values that scale together — 3 pens cost 90p, so 5 pens cost…

Builds on: Multiplication & division fluency — “A proportion table is multiplication holding its shape.”

The unitary method

Find the value of one, then scale to any amount — the workhorse of proportion.

Builds on: Multiplying & dividing decimals · Proportion tables — “Find one first — the table shows why that works.”

Metric unit conversions

Change between mm, cm, m and km — and g, kg; the metric system runs on tens. Read the full explainer →

Builds on: Place value & ordering — “Kilometres to metres is place value ×1000.”

Percentage as a proportion

Per cent means 'per hundred' — a way to compare parts on a common base of 100.

Builds on: Equivalent fractions — “Per cent is an equivalent fraction with denominator 100.”

Simple scale drawings

Let 1 cm stand for 1 m — draw and read plans where the paper is smaller than the world.

Builds on: Ratio notation · Metric unit conversions — “Scale drawings convert units: centimetres standing for metres.”

Measuring & drawing angles

Use a protractor to measure and draw angles to the nearest degree.

Angles at a point & line

Angles on a straight line total 180°; around a point, 360°; vertically opposite angles match.

Builds on: Measuring & drawing angles — “The facts are about turns you can already measure.”

Angles in a triangle

The three angles of any triangle total 180° — and you can show why.

Builds on: Angles at a point & line — “The 180° argument leans on angles on a line.”

Special triangles

Isosceles, equilateral and right-angled triangles, and what their symmetries force their angles to be.

Builds on: Angles in a triangle — “Isosceles facts are the angle sum with symmetry added.” · Reflection symmetry

Quadrilateral properties

Know the square, rectangle, parallelogram, rhombus, trapezium and kite by their sides, angles and symmetries. Read the full explainer →

Builds on: Special triangles · Reflection symmetry — “Most quadrilateral facts are symmetry facts in disguise.”

Constructing triangles

Draw accurate triangles from given sides and angles with ruler, protractor and compasses. Read the full explainer →

Builds on: Measuring & drawing angles — “Accurate triangles need accurately drawn angles.”

Nets of 3D shapes

Unfold a solid flat — and predict which flat patterns fold back up. Read the full explainer →

Builds on: Naming 3D solids — “You can't unfold a solid you can't name.”

Naming 3D solids

Know prisms, pyramids, cylinders, cones and spheres — and count faces, edges and vertices.

Volume of cuboids

Volume as layers of unit cubes: length × width × height.

Builds on: Perimeter & area — “Volume stacks areas, layer by layer.”

Reflection symmetry

Spot and draw lines of symmetry — the fold that maps a shape onto itself.

Rotational symmetry

How many times a shape looks the same in one full turn — its order of rotational symmetry.

Reflecting shapes

Reflect a shape in a mirror line on the grid, including the axes and y = x.

Builds on: Reflection symmetry — “Performing a reflection needs the mirror idea already in place.” · Coordinates in four quadrants — “Mirror lines like y = x live on the coordinate grid.”

Reading measurement scales

Read rulers, jugs and weighing scales — including the marks between the numbers.

Parts of a circle

Radius, diameter, chord, arc, sector, segment and tangent — the circle's vocabulary. Read the full explainer →

The language of chance

Impossible, unlikely, evens, likely, certain — words for how much to expect something.

The probability scale

Chance drawn as a line from 0 to 1 — every event has an address on it.

Builds on: The language of chance — “Words go on the line before numbers do.” · Inequality symbols & number lines

Listing outcomes systematically

Write out everything that could happen — in an order that guarantees nothing is missed.

Fair or biased?

Fair means every outcome is equally likely — and biased means the counting shortcut breaks.

Builds on: The language of chance — “Fairness is a claim made in the language of chance.” · Listing outcomes systematically — “Equally likely means counting the outcomes first.”

Tally charts & frequency tables

Record data as you collect it — tallies in fives, then totals in a frequency column.

Bar charts

Draw and read bar charts — equal-width bars, honest gaps, a labelled scale.

Builds on: Tally charts & frequency tables — “Bars are drawn from the frequency table.”

Pictograms & their keys

Pictures standing for counts — where half a symbol means half the key's value.

Builds on: Tally charts & frequency tables — “Reading half a symbol means reading the table behind it.”

Reading tables & timetables

Pull the right number out of a real table — bus timetables, price grids, league tables.

Line graphs over time

Plot measurements against time and read the story — rises, falls and steady spells.

Builds on: Reading tables & timetables · Coordinates in four quadrants — “A time series is coordinates with time along the bottom.”

The data-handling cycle

Pose a question, collect data, present it, interpret it — statistics as a full loop, not a chart exercise.

Builds on: Tally charts & frequency tables — “You can't plan a cycle whose collection step you've never done.” · Bar charts

The mean as fair shares

The mean is what everyone would get if the total were shared out equally.

Builds on: Multiplication & division fluency — “Fair sharing is division with a story.”

Collecting like terms

Gather matching terms so long expressions shrink. Read the full explainer →

Builds on: Simplifying expressions — “Collecting terms is simplifying, made a habit.”

Fractions ↔ decimals ↔ %

Move fluently between the three notations.

Builds on: Fractions of amounts — “Converting starts from knowing what a fraction is.” · Percentage as a proportion — “Per cent must mean 'per hundred' before the third notation joins.”

One-step equations

Solve x + 4 = 11 or 3x = 12 by undoing one operation.

Builds on: Inverse operations — “Solving is undoing — inverse operations are the tool.” · Reading algebraic notation — “You can’t solve a sentence you can’t read.”

Solving two-step linear equations

Undo operations in reverse order to find the unknown in 3x + 4 = 19. Read the full explainer →

Builds on: Collecting like terms · One-step equations — “Two steps are one step done twice — make one step automatic first.”

Ratio & proportion problems

Solve sharing and scaling problems — recipes, maps, best buys.

Builds on: Fractions of amounts · Ratio notation — “Proportion problems need the notation first.”

Angles in polygons

Interior and exterior angles; angle sums.

Builds on: Angles at a point & line · Angles in a triangle — “Every polygon's angle sum is triangles glued together.” · Perimeter & area

Probability of single events

Express chance as a fraction on the 0–1 scale.

Builds on: The probability scale — “Fractions land on a scale you've already drawn.” · Listing outcomes systematically — “The fraction counts outcomes — you must be able to list them.” · Fractions of amounts — “You can’t weigh chances you can’t write down.”

Multiplying & dividing negatives

Use the sign rules: same signs make positive, different signs make negative.

Builds on: Negative numbers — “The sign rules extend adding and subtracting below zero.” · Multiplication & division fluency — “Sign rules sit on top of the times tables.”

Adding fractions, unlike denominators

Add and subtract fractions by first rewriting them over a common denominator.

Builds on: HCF & LCM · Equivalent fractions — “A common denominator is an equivalent fraction, chosen twice.”

Multiplying fractions

Multiply fractions — top by top, bottom by bottom — and see why the answer shrinks.

Builds on: Equivalent fractions · Fractions of amounts — “'Half of a third' only makes sense once 'of' does.”

Dividing by a fraction

Understand dividing by a fraction as asking 'how many fit?' — which is why you flip and multiply.

Builds on: Multiplying fractions — “Flip-and-multiply is meaningless until multiplying fractions isn't.”

Mixed numbers & improper fractions

Switch between 2¾ and 11/4 — the same amount, packaged differently. Read the full explainer →

Builds on: Equivalent fractions — “Repackaging 11/4 as 2¾ is equivalence doing the lifting.”

Percentages of amounts

Find percentages of quantities with and without a calculator — 35% of £80.

Builds on: Fractions of amounts — “35% of £80 is a fraction-of-amount wearing a % sign.” · Percentage as a proportion — “Per cent has to mean 'per hundred' before you take 35% of anything.”

Percentage increase & decrease

Increase or decrease an amount by a percentage — find the change, then apply it. Read the full explainer →

Builds on: Percentages of amounts — “You can't add 20% on until you can find 20% of.”

Index notation

Write repeated multiplication as a power: 2⁵ means five 2s multiplied together.

Builds on: Squares, cubes & roots — “Powers generalise the squares and cubes you've already met.”

Laws of indices

Multiply and divide powers of the same base by adding or subtracting the indices.

Builds on: Index notation — “The laws compress notation you can already unpack.”

Rounding to decimal places

Round decimals to a given number of decimal places.

Builds on: Rounding to powers of ten — “Same rounding decision, smaller columns.” · Adding & subtracting decimals

Significant figures

Round to significant figures — keeping the digits that carry the information. Read the full explainer →

Builds on: Rounding to decimal places — “Decimal places first — significant figures make the same cut smarter.”

Calculator fluency

Use a scientific calculator well — fractions, powers, brackets — and read the display critically.

Builds on: Order of operations — “A calculator obeys the order of operations, so you must know it.” · Estimating before calculating

Order of operations with indices

Run the full priority order when powers and nested brackets join the queue.

Builds on: Order of operations — “Indices join a queue you must already run in order.” · Index notation — “You can't prioritise powers you can't read.”

Factorising into a bracket

Pull the common factor out: 6x + 9 = 3(2x + 3).

Builds on: Factors, multiples & primes — “No factor sense, no common factor to pull out.” · Expanding a single bracket — “Factorising is expanding run in reverse.”

Expanding and simplifying

Expand two brackets and collect the pieces: 3(x + 2) + 2(x − 1).

Builds on: Collecting like terms — “Expanded brackets leave like terms to collect.” · Expanding a single bracket — “Two brackets means the single-bracket move, twice.”

Equations with brackets

Solve 3(x + 2) = 18 — expand first, or divide first, and know both routes. Read the full explainer →

Builds on: Solving two-step linear equations — “After expanding, a two-step equation is what remains.” · Expanding a single bracket — “One route through the equation runs straight over the bracket.”

Unknowns on both sides

Solve 5x + 2 = 3x + 10 by collecting the unknowns on one side first.

Builds on: Collecting like terms · Solving two-step linear equations — “Both-sides moves assume two-step solving is automatic.”

nth term of linear sequences

Find the position-to-term formula of an arithmetic sequence — 5, 8, 11 becomes 3n + 2.

Builds on: Writing expressions from words · Sequences from a position rule — “The nth term formalises the position rule you've been using.”

Rearranging simple formulae

Change the subject: turn v = u + at into a = (v − u)/t.

Builds on: One-step equations — “Changing the subject is solving, with letters left in.” · Substituting into formulae — “Rearrange only formulae you can already use forwards.”

Inequalities on a number line

Show x > 2 or −1 ≤ x < 3 as a shaded stretch of the number line. Read the full explainer →

Builds on: Inequality symbols & number lines — “The symbols come before the pictures.”

Solving linear inequalities

Solve 2x + 1 < 9 the way you solve an equation — and keep the direction honest.

Builds on: Solving two-step linear equations — “Solve it like an equation — then keep the direction honest.” · Inequalities on a number line — “You should see what the answer looks like before hunting it.”

Plotting lines from tables

Build a table of values for y = 2x + 1, plot the points, join the line.

Builds on: Substituting into expressions — “Each row of the table is one substitution.” · Coordinates in four quadrants — “Every point in the table needs a home on the grid.”

Horizontal & vertical lines

Know x = 3 is a vertical wall and y = 2 a horizontal floor.

Builds on: Coordinates in four quadrants — “x = 3 only makes sense as all the points whose x is 3.”

Index laws in algebra

Simplify a³ × a⁴ and a⁸ ÷ a² — the index laws now wearing letters.

Builds on: Laws of indices — “The laws were learned on numbers; the letters change nothing.” · Simplifying expressions

Map scales

Use scales like 1:25,000 to turn map centimetres into real kilometres.

Builds on: OS maps & grid references (GEOG) · Ratios with mixed units · Simple scale drawings — “Maps are scale drawings that left the classroom.”

Best-buy comparisons

Decide which deal is better by comparing the price of one unit.

Builds on: Multiplying & dividing decimals — “Unit prices are decimal division.” · The unitary method — “Price-per-one is the unitary method out shopping.”

Speed, distance, time

Connect the three: speed is the distance covered in one unit of time.

Builds on: The unitary method — “Speed is a unit rate: the distance for one hour.” · Metric unit conversions · Speed = distance ÷ time (PHYS)

Density as a rate

Density is mass per unit of volume — how much stuff is packed into each cubic centimetre.

Builds on: Particle model & states (CHEM) · Volume of cuboids · The unitary method — “Density is 'how much stuff per one' — a rate.”

Change as a percentage

Express a rise or fall as a percentage of where it started.

Builds on: Percentage increase & decrease — “Apply changes forwards before expressing them backwards.” · The unitary method

Ratio problems with differences

Solve puzzles like 'shared 3:5, and one gets £8 more' by valuing one part.

Builds on: One-step equations · Ratio ↔ fraction links — “You must see what slice of the whole each share is.”

Currency conversion

Use an exchange rate to convert money both ways between two currencies.

Builds on: Multiplying & dividing decimals — “Exchange rates multiply decimals.” · The unitary method — “An exchange rate is the value of one unit — the unitary method again.”

Direct proportion graphs

Quantities in direct proportion plot as a straight line through the origin.

Builds on: Plotting lines from tables — “Plotting from a table is the skill this borrows.” · Proportion tables — “The graph is the proportion table, drawn.”

Translating shapes

Slide a shape across the grid without turning it — same shape, new address.

Builds on: Coordinates in four quadrants — “You can't slide a shape you can't place.”

Angles in parallel lines

Alternate, corresponding and co-interior angles — the F, Z and C patterns, properly named. Read the full explainer →

Builds on: Angles at a point & line — “New angle facts stack on the point-and-line ones.”

Area of a parallelogram

Base × perpendicular height — because a parallelogram is a rectangle in disguise.

Builds on: Perimeter & area — “Shear the rectangle: same base, same height, same area.”

Area of a trapezium

Average the parallel sides, then multiply by the height between them.

Builds on: Area of a parallelogram — “The trapezium formula is two parallelogram stories averaged.”

Circumference of a circle

The distance round a circle is π times its diameter — a touch over three of them.

Builds on: Parts of a circle — “Diameter and radius must be distinct before π multiplies one.” · Multiplying & dividing decimals — “Multiplying by 3.14 is decimal multiplication.”

Area of a circle

A = πr² — and why the radius gets squared while the diameter doesn't.

Builds on: Circumference of a circle — “Meet π in the perimeter before squaring anything.” · Squares, cubes & roots

Rotating shapes

Turn a shape about a centre — by 90° or 180°, clockwise or anti — and land it exactly.

Builds on: Rotational symmetry — “Rotational symmetry gives the feel; now perform the turn.” · Coordinates in four quadrants — “Centres of rotation are coordinates.”

Enlarging shapes

Enlarge a shape from a centre by a whole-number scale factor — every length multiplied, every ray from the centre.

Builds on: Coordinates in four quadrants — “Centres and image points are coordinate work.” · Simple scale drawings — “Enlargement is a scale drawing that stayed on the page.”

Volume of prisms

Cross-section area × length — any prism is layers of its end face.

Builds on: Volume of cuboids — “Cuboids first: volume as layers of one face.” · Area of a parallelogram

Surface area of prisms

Total the area of every face — the net makes sure you miss none. Read the full explainer →

Builds on: Nets of 3D shapes — “The net makes sure no face goes uncounted.” · Perimeter & area — “It's area, added carefully.”

Converting area & volume units

A square metre is 10,000 cm² — area and volume units convert by the scale squared or cubed.

Builds on: Perimeter & area — “You convert area units by re-counting squares.” · Metric unit conversions — “Length conversions come first; these are their squares and cubes.”

Bearings

Directions as three digits measured clockwise from north — 070°, not 'roughly north-east'.

Builds on: OS maps & grid references (GEOG) · Measuring & drawing angles — “A bearing is a measured angle with compass manners.” · Map scales

Constructing bisectors

With compasses alone: cut an angle perfectly in half, or find the line midway between two points.

Builds on: Constructing triangles — “Compass control comes from building triangles.”

Sample space diagrams

A grid of every combined outcome — two dice become a 6-by-6 map of possibilities. Read the full explainer →

Builds on: Listing outcomes systematically — “The grid is your list, organised so it can't lie.”

Probabilities sum to 1

All the possibilities together are certain — so P(not A) is 1 minus P(A).

Builds on: Probability of single events — “P(not A) only clicks once probabilities are fractions.” · Adding fractions, unlike denominators

Experimental probability

Estimate a chance by doing the experiment: successes over trials.

Builds on: Tally charts & frequency tables · Probability of single events — “Relative frequency is a fraction with data on top.”

Theory versus experiment

Compare what should happen with what did — and reason about why they differ.

Builds on: Fair or biased? · Experimental probability — “You can't compare theory with an experiment you can't run.”

Expected frequency

If the chance is 1/6 and you roll 60 times, expect about 10 — probability times trials. Read the full explainer →

Builds on: Probability of single events — “Expectation multiplies a probability you must first have.” · Multiplying fractions

Drawing pie charts

Turn category counts into slices — each category's share of 360 degrees.

Builds on: Measuring & drawing angles — “No protractor, no pie.” · Tally charts & frequency tables — “The angles come from frequencies you must first trust.” · Fractions of amounts — “Each slice is a fraction of 360 degrees.”

Interpreting pie charts

Slices show proportions, not counts — two pies can only be compared with their totals. Read the full explainer →

Builds on: Bar charts · Percentage as a proportion — “Slices are proportions, not counts.”

Averages from frequency tables

Find the mean, median and mode when the data arrives already tallied. Read the full explainer →

Builds on: Tally charts & frequency tables — “The table's structure is the method.” · Averages & range — “The shortcuts compress a list you could already average.”

Choosing the right average

Mean, median or mode — which one tells this dataset's truth?

Builds on: Outliers & their effect · Averages & range — “You can't choose between measures you can't compute.”

Grouped frequency tables

Bundle messy continuous data into classes — 0–10, 10–20 — and keep the boundaries honest. Read the full explainer →

Builds on: Tally charts & frequency tables — “Grouping is the frequency table meeting messy data.”

Dual & compound bar charts

Two datasets on one chart — side-by-side bars or stacked ones, for honest comparison.

Builds on: Bar charts — “Two datasets, same axes — the bar chart grows up.”

Outliers & their effect

One extreme value can drag the mean and stretch the range — spot it and say what it does.

Builds on: The mean as fair shares · Averages & range — “An outlier is judged by how it drags the summaries.”

Linear graphs y = mx + c

Plot lines and read gradient and intercept.

Builds on: Solving two-step linear equations — “A graph is an equation drawn — solve before you draw.” · Plotting lines from tables — “Tables of points come before the y = mx + c shortcut.”

Scatter graphs & correlation

Plot paired data; describe correlation honestly.

Builds on: Coordinates in four quadrants — “Paired data plots as coordinates.” · Averages & range — “Summaries first, patterns second.”

Standard form: large numbers

Write big numbers as A × 10ⁿ — 4,500,000 becomes 4.5 × 10⁶.

Builds on: Place value & ordering · Index notation — “Standard form is index notation given a job.”

Standard form: small numbers

Use negative powers of ten for tiny numbers — 0.00032 is 3.2 × 10⁻⁴.

Builds on: Standard form: large numbers — “Small numbers reuse the big-number machinery.” · Negative & zero indices — “10⁻³ has to mean something first.”

Negative & zero indices

Extend the pattern: anything to the power 0 is 1, and negative powers mean 'divide'.

Builds on: Laws of indices — “Zero and negative powers extend the pattern the laws create.”

Reverse percentages

Find the original amount after a percentage change — the price before the sale.

Builds on: Percentage multipliers · Percentage increase & decrease — “You can only undo a change you can run forwards.”

HCF & LCM by prime factors

Use prime factorisations to find the HCF and LCM of numbers too big to list.

Builds on: Prime factorisation — “No prime factorisation, no shortcut.” · HCF & LCM — “Meet HCF and LCM by listing before you scale the method up.”

Choosing a calculation method

Decide: mental, written or calculator — then sanity-check whichever you chose.

Builds on: Estimating before calculating — “Checking by estimate is half of choosing well.” · Calculator fluency

Gradient as a rate

Read a gradient as 'how much y changes for each step of x' — a rate, not just a slope.

Builds on: Linear graphs y = mx + c — “Gradient as steepness first; gradient as rate second.” · Speed, distance, time

Finding a line's equation

Take a drawn line and write its equation — gradient from the slope, c from the crossing.

Builds on: Linear graphs y = mx + c — “You can't write m and c until you can read them.” · Gradient as a rate

Real-life graphs

Read distance–time and container-filling graphs as stories: steep means fast, flat means stopped.

Builds on: Line graphs over time · Gradient as a rate — “A story graph is read through its rates.” · Speed = distance ÷ time (PHYS)

Geometric sequences

Sequences that multiply — 3, 6, 12, 24 — and how fast they grow.

Builds on: Percentage multipliers · Term-to-term sequences — “Multiplying sequences are met by continuing them.”

Solving equations with graphs

Where two graphs cross, both equations are telling the truth — read solutions off the picture.

Builds on: Linear graphs y = mx + c — “The crossing point only speaks if you can draw the lines.” · Unknowns on both sides

Equations with fractions

Solve x/3 + 2 = 7 — clear the fraction, then it's an equation you know.

Builds on: Multiplying fractions · Unknowns on both sides — “Fraction-clearing lands you in a both-sides equation.”

Building equations from problems

Turn a word problem into an equation — the sentence becomes symbols before it becomes a number.

Builds on: Writing expressions from words — “You can't build an equation from words you can't phrase in symbols.” · Unknowns on both sides

Graphs of quadratics

Plot y = x² and friends — meet the parabola and its symmetry.

Builds on: Plotting lines from tables — “Plot from a table — the table just gained an x² column.” · Substituting negatives & squares — “One wrong (−2)² and the parabola grows a kink.”

Substituting negatives & squares

Evaluate 3x² when x = −2 without the classic sign slip.

Builds on: Multiplying & dividing negatives — “The sign rules do the heavy lifting.” · Substituting into expressions — “Ordinary substitution first; the traps come after.”

Direct proportion y = kx

When one quantity doubles, so does the other — and k is the constant that links them.

Builds on: Direct proportion graphs — “The line you drew becomes the formula you write.” · Substituting into formulae

Inverse proportion

More workers, less time — when one quantity doubles, the other halves.

Builds on: Direct proportion y = kx — “Inverse is direct proportion's mirror — meet direct first.” · Dividing by a fraction

Scale factors as multipliers

Scaling is multiplying — by 2, by ¾, by 1.1 — one number that does the whole job.

Builds on: Multiplying fractions — “A scale factor of ¾ is fraction multiplication.” · Simple scale drawings

Percentage multipliers

Pack a percentage change into one multiplication: +20% is ×1.2, −15% is ×0.85.

Builds on: Multiplying & dividing decimals — “Multiplying by 0.85 is decimal multiplication.” · Percentage increase & decrease — “See the two-step version before packing it into one.”

Inverse proportion graphs

Inverse proportion draws a falling curve that never quite touches the axes.

Builds on: Direct proportion graphs · Inverse proportion — “The curve pictures a relationship you must already believe.”

Proportional or not?

Pause before calculating: does this situation actually scale? Not everything does.

Builds on: Direct proportion y = kx — “Spotting proportion means knowing its fingerprint.” · Inverse proportion

Combining ratios

Merge a:b and b:c into a:b:c by lining up the shared part.

Builds on: HCF & LCM · Ratios with mixed units — “Aligning the shared part is equivalent-ratio work.”

Pythagoras: the hypotenuse

In a right-angled triangle, the squares on the two short sides add up to the square on the longest.

Builds on: Squares, cubes & roots — “The theorem trades in squares and square roots.” · Perimeter & area

Pythagoras: shorter sides

Rearrange the theorem to find a shorter side — subtract before you square-root.

Builds on: Pythagoras: the hypotenuse — “Rearranging the theorem comes after trusting it.” · Rearranging simple formulae

Congruence criteria

When are two triangles guaranteed identical? SSS, SAS, ASA and RHS — and why AAA isn't enough. Read the full explainer →

Builds on: Special triangles · Constructing triangles — “SSS convinces because you've built triangles from three sides.”

Similar shapes

Same shape, different size — matching angles equal, all lengths scaled by one factor.

Builds on: Scale factors as multipliers — “Missing sides fall to a multiplier.” · Enlarging shapes — “Similarity is what enlargement preserves.”

Volume of cylinders

A cylinder is a prism with a circular face: V = πr²h.

Builds on: Area of a circle — “The layer is a circle — you need its area.” · Volume of prisms — “A cylinder is a prism whose face happens to be round.”

Surface area of cylinders

Two circles plus a rolled-up rectangle whose width is the circumference. Read the full explainer →

Builds on: Circumference of a circle — “The curved face unrolls into a rectangle — circumference wide.” · Surface area of prisms — “Surface-area habits transfer; the net just gained curves.”

Compound shapes with circles

Perimeters and areas of shapes built from rectangles and half- or quarter-circles.

Builds on: Circumference of a circle — “Curved edges in the perimeter come from circumference.” · Area of a circle — “Half a circle's area is no use without the whole one.”

Describing transformations

Name a transformation completely: which kind, plus the mirror line, centre, angle or vector it needs.

Builds on: Reflecting shapes — “You can only name precisely what you can do.” · Rotating shapes — “Rotations demand the fullest descriptions — centre, angle, direction.” · Enlarging shapes

Geometric reasoning chains

Find an angle through several steps, citing a named reason for every claim — the start of proof.

Builds on: Angles in parallel lines — “Parallel-line facts are the standard links in the chain.” · Angles in polygons — “Angle-sum facts supply the reasons you'll cite.”

Mutually exclusive events

Events that can't happen together — only then may you add their probabilities.

Builds on: Probabilities sum to 1 — “Adding chances is only legal when events can't overlap — sum-to-1 sets the stage.”

Venn diagrams for probability

Sort outcomes into overlapping circles — the overlap is where 'both' lives.

Builds on: Listing outcomes systematically · Mutually exclusive events — “Venn regions show exactly when adding would double-count.”

Two-way tables

Cross-classify people or outcomes by two questions at once, then read chances from the cells.

Builds on: Sample space diagrams — “A two-way table is a sample space that grew labels.” · Tally charts & frequency tables

Frequency trees

Follow counts through two splits — 200 people branch into groups at every fork.

Builds on: Experimental probability · Two-way tables — “Two-way tables split once; trees split twice.”

Choosing a probability model

Decide how to get a probability: count equally likely outcomes, run an experiment, or use data.

Builds on: Fair or biased? · Theory versus experiment — “Choosing a model means knowing how theory and experiment each fail.”

Estimating a grouped mean

When data is grouped, use each class midpoint to estimate the mean — an estimate, and honestly so.

Builds on: Averages from frequency tables — “Frequency-times-value first; midpoints are that trick under uncertainty.” · Grouped frequency tables — “Midpoints stand in for values the grouping hid.”

Comparing two distributions

Compare two datasets properly: quote an average and a measure of spread, in context.

Builds on: Choosing the right average · Averages & range — “A comparison quotes an average and the spread.”

Misleading graphs

Truncated axes, stretched scales, cherry-picked windows — how honest data gets dishonest clothes.

Builds on: Bar charts — “You spot a rigged axis by knowing the honest one.” · Interpreting pie charts