Coherence Map: how maths connects from Year 7 to Year 13

Year 7

Number and Algebra

Place value and decimals Year 7, every student Each place is worth ten of the place to its right, which is why whole numbers and decimals follow the same rules. Builds on: nothing earlier on the map Leads to: Powers, squares and roots, Rounding and estimating, Fractions, decimals and percentages, Integers and the number line, Scientific notation Lessons: Year 7, Unit 1, lesson 1: Place value and decimals
Powers, squares and roots Year 7, every student A power is repeated multiplication, and a square root undoes squaring: 7 squared is 49, so the square root of 49 is 7. Builds on: Place value and decimals Leads to: Prime factorization, Order of operations, Powers of negative numbers, Rational and irrational numbers, Pythagoras' theorem Lessons: Year 7, Unit 1, lesson 2: Powers, squares and roots
Factors, multiples and primes Year 7, every student A prime has exactly two factors, which is why 1 is not prime; divisibility tests save a long division. Builds on: nothing earlier on the map Leads to: Prime factorization Lessons: Year 7, Unit 1, lesson 3: Factors, multiples and primes
Prime factorization Year 7, every student Every whole number above 1 splits into primes in exactly one way, written with powers, such as 60 = 2² × 3 × 5. Builds on: Factors, multiples and primes, Powers, squares and roots Leads to: Common factors and multiples Lessons: Year 7, Unit 1, lesson 4: Prime factorization
Common factors and multiples Year 7, every student The highest common factor and the lowest common multiple, read straight from two prime factorizations. Builds on: Prime factorization Leads to: Equivalent fractions, Ratio Lessons: Year 7, Unit 1, lesson 5: Common factors and multiples
Order of operations Year 7, every student Brackets, then powers, then multiplication and division, then addition and subtraction, each worked left to right. Builds on: Powers, squares and roots Leads to: Variables and expressions Lessons: Year 7, Unit 1, lesson 6: Order of operations
Rounding and estimating Year 7, every student Rounding to a place value, and estimating before calculating so that a wrong answer looks wrong. Builds on: Place value and decimals Leads to: Circumference and area of a circle, Significant figures and estimation Lessons: Year 7, Unit 1, lesson 7: Rounding and estimating
Equivalent fractions Year 7, every student A fraction is a point on the number line, and multiplying the top and bottom by the same number names the same point, so 18/24 = 3/4. Builds on: Common factors and multiples Leads to: Adding and subtracting fractions, Multiplying and dividing fractions, Fractions, decimals and percentages, Ratio Lessons: Year 7, Unit 5, lesson 1: Equivalent fractions; Year 7, Unit 5, lesson 2: Improper fractions and mixed numbers; Year 7, Unit 5, lesson 3: Comparing and ordering fractions
Adding and subtracting fractions Year 7, every student Only parts of the same size can be added, so both fractions are first rewritten over a common denominator. Builds on: Equivalent fractions Leads to: Multi-step equations, Algebraic fractions Lessons: Year 7, Unit 5, lesson 4: Adding and subtracting fractions
Multiplying and dividing fractions Year 7, every student A fraction of a fraction multiplies; dividing asks how many of one fit into the other, which is multiplying by the reciprocal. Builds on: Equivalent fractions Leads to: Ratio, Tree diagrams Lessons: Year 7, Unit 5, lesson 5: Multiplying fractions; Year 7, Unit 5, lesson 6: Dividing fractions
Fractions, decimals and percentages Year 7, every student Three ways of writing the same quantity, such as 3/8 = 0.375 = 37.5%, and choosing the one that suits the job. Builds on: Equivalent fractions, Place value and decimals Leads to: Terminating and recurring decimals, Percentages of amounts, Bar charts, pie charts and line graphs, The probability scale Lessons: Year 7, Unit 2, lesson 1: What a percentage is; Year 7, Unit 2, lesson 2: Fractions, decimals and percentages
Terminating and recurring decimals Year 7, every student A fraction's decimal either stops or repeats a block of digits for ever, and a repeating decimal can be turned back into a fraction. Builds on: Fractions, decimals and percentages Leads to: Rational and irrational numbers Lessons: Year 7, Unit 5, lesson 7: Terminating and recurring decimals
Percentages of amounts Year 7, every student Finding a percentage of an amount, and working back to the whole from a part, using benchmarks such as 10% and 1%. Builds on: Fractions, decimals and percentages Leads to: Percentage increase and decrease Lessons: Year 7, Unit 2, lesson 3: Percentage of an amount; Year 7, Unit 2, lesson 4: Finding the whole
Percentage increase and decrease Year 7, every student A change as a percentage of the original amount, and a multiplier such as 1.07 for a rise of 7%. Builds on: Percentages of amounts Leads to: Geometric sequences and growth, Compound interest and depreciation Lessons: Year 7, Unit 2, lesson 5: Increasing and decreasing by a percentage; Year 7, Unit 2, lesson 6: Percentage change
Patterns and sequences Year 7, every student A pattern shown as a diagram, a table, words and a rule: what stays the same and what grows. Builds on: nothing earlier on the map Leads to: Term-to-term and position rules Lessons: Year 7, Unit 3, lesson 1: Growing patterns
Variables and expressions Year 7, every student A letter stands for a number that can vary; 7h means 7 × h, and substituting a value gives the expression a value. Builds on: Order of operations Leads to: Term-to-term and position rules, Collecting like terms, Function machines Lessons: Year 7, Unit 3, lesson 2: Letters and expressions
Term-to-term and position rules Year 7, every student A term-to-term rule says how to get the next term; a position rule such as 4n − 3 gives any term directly. Builds on: Patterns and sequences, Variables and expressions Leads to: Arithmetic sequences Lessons: Year 7, Unit 3, lesson 3: Term-to-term and position rules
Collecting like terms Year 7, every student Only like terms combine, so 3x + 5y + 2x − y simplifies to 5x + 4y. Builds on: Variables and expressions Leads to: Solving simple equations, Expanding and factorizing Lessons: Year 7, Unit 3, lesson 4: Collecting like terms
Solving simple equations Year 7, every student Keeping both sides balanced by applying the same inverse operation to each until the unknown stands alone. Builds on: Function machines, Collecting like terms Leads to: Balancing linear equations Lessons: Year 7, Unit 3, lesson 6: Solving simple equations

Functions

Function machines Year 7, every student An input goes in, a rule acts on it and an output comes out; running the machine backwards undoes the rule. Builds on: Variables and expressions Leads to: Solving simple equations, Functions, domain and range Lessons: Year 7, Unit 3, lesson 5: Function machines

Geometry and Trigonometry

Points, lines and angles Year 7, every student Naming points, lines, rays and segments, and measuring and classifying angles as acute, right, obtuse or reflex. Builds on: nothing earlier on the map Leads to: Angles at a point and on a line, Ruler and compass constructions Lessons: Year 7, Unit 4, lesson 1: Points, lines and angles
Angles at a point and on a line Year 7, every student Angles on a straight line sum to 180°, angles at a point to 360°, and vertically opposite angles are equal. Builds on: Points, lines and angles Leads to: Angles with parallel lines, Reflections, rotations and translations Lessons: Year 7, Unit 4, lesson 2: Angles on a line and at a point
Angles with parallel lines Year 7, every student A line crossing two parallel lines makes equal corresponding angles and equal alternate angles. Builds on: Angles at a point and on a line Leads to: Triangles and their angles Lessons: Year 7, Unit 4, lesson 3: Parallel lines and transversals
Triangles and their angles Year 7, every student Triangles sorted by sides and by angles, and alternate angles proving that the three angles sum to 180°. Builds on: Angles with parallel lines Leads to: Conjecture and proof, Similar triangles, Tessellations and polygon angles, Circle theorems Lessons: Year 7, Unit 4, lesson 4: Angles in a triangle
Ruler and compass constructions Year 7, every student Bisecting lines and angles and copying triangles with only a straight edge and a pair of compasses. Builds on: Points, lines and angles Leads to: Perpendicular bisectors Lessons: Year 7, Unit 4, lesson 5: Bisecting lines and angles; Year 7, Unit 4, lesson 6: Constructing triangles and angles
Perimeter and area Year 7, every student Perimeter is the distance around a shape and area counts the square units inside; for a fixed perimeter, the square encloses most. Builds on: nothing earlier on the map Leads to: Area of triangles and parallelograms Lessons: Year 7, Unit 7, lesson 1: Perimeter and area of rectangles; Year 7, Unit 7, lesson 2: Same fence, different area
Area of triangles and parallelograms Year 7, every student Cutting and rearranging shows why a parallelogram has area bh and a triangle half of that. Builds on: Perimeter and area Leads to: Area of compound shapes, Nets and surface area, Volume of cuboids and prisms, Circumference and area of a circle, Pythagoras' theorem Lessons: Year 7, Unit 7, lesson 3: Area of triangles and parallelograms
Area of compound shapes Year 7, every student Splitting a shape into rectangles and triangles to add their areas, and tracing only the outside edges for the perimeter. Builds on: Area of triangles and parallelograms Leads to: Trapeziums and regular polygons Lessons: Year 7, Unit 7, lesson 4: Area of compound shapes
Nets and surface area Year 7, every student Unfolding a solid into a net and adding the areas of its faces. Builds on: Area of triangles and parallelograms Leads to: Surface area and volume of prisms Lessons: Year 7, Unit 7, lesson 6: Nets and surface area
Volume of cuboids and prisms Year 7, every student Volume counts unit cubes; for any prism it is the area of the cross-section times the length. Builds on: Area of triangles and parallelograms Leads to: Surface area and volume of prisms Lessons: Year 7, Unit 7, lesson 5: Volume of cuboids and prisms

Statistics and Probability

Types of data Year 7, every student Categorical or numerical, discrete or continuous, primary or secondary: the type decides which graph is fair. Builds on: nothing earlier on the map Leads to: Frequency tables, Mean, median and mode, Scatter graphs and correlation Lessons: Year 7, Unit 6, lesson 1: Types of data; Year 7, Unit 6, lesson 2: Asking and collecting
Frequency tables Year 7, every student Tallies and frequency tables turn a list of raw values into something a reader can take in. Builds on: Types of data Leads to: Bar charts, pie charts and line graphs, Mean, median and mode Lessons: Year 7, Unit 6, lesson 3: Tally and frequency tables
Bar charts, pie charts and line graphs Year 7, every student Choosing the chart that fits the data, with a title, labelled axes and an even scale, and spotting a chart that misleads. Builds on: Frequency tables, Fractions, decimals and percentages Leads to: Comparing distributions Lessons: Year 7, Unit 6, lesson 4: Bar charts and histograms; Year 7, Unit 6, lesson 5: Pie charts; Year 7, Unit 6, lesson 6: Line graphs; Year 7, Unit 6, lesson 7: Choosing and honest charts

Year 8

Number and Algebra

Ratio Year 8, every student Comparing quantities in the same units, simplifying the way fractions are (12 : 16 = 3 : 4), and sharing an amount in a given ratio. Builds on: Common factors and multiples, Equivalent fractions, Multiplying and dividing fractions Leads to: Proportion, Enlargement Lessons: lessons coming
Proportion Year 8, every student Two equal ratios, solved by scaling up or down or by cross-multiplying. Builds on: Ratio Leads to: Rates and unit rates, Proportional relationships, Similar triangles Lessons: lessons coming
Rates and unit rates Year 8, every student Comparing quantities in different units, such as kilometres per hour, and using the rate for one unit to compare prices. Builds on: Proportion Leads to: Converting units and currencies, Constant rate of change Lessons: lessons coming
Converting units and currencies Year 8, every student A conversion factor is a rate: centimetres to inches, litres to gallons, one currency to another. Builds on: Rates and unit rates Leads to: Units, prefixes and conversions Lessons: lessons coming
Integers and the number line Year 8, every student Positive and negative whole numbers, opposites, ordering, and absolute value as the distance from zero. Builds on: Place value and decimals Leads to: Adding and subtracting integers, Linear inequalities, Coordinates in four quadrants, Absolute value Lessons: Year 8, Unit 3, lesson 1: Numbers on both sides of zero; Year 8, Unit 3, lesson 2: Absolute value and distance
Adding and subtracting integers Year 8, every student Zero pairs show why subtracting a negative number is the same as adding its opposite. Builds on: Integers and the number line Leads to: Multiplying and dividing integers Lessons: Year 8, Unit 3, lesson 3: Adding and subtracting integers
Multiplying and dividing integers Year 8, every student The sign of a product depends on how many negative numbers are in it, and patterns explain why. Builds on: Adding and subtracting integers Leads to: Powers of negative numbers, Expanding and factorizing Lessons: Year 8, Unit 3, lesson 4: Multiplying and dividing integers
Powers of negative numbers Year 8, every student (−3)² = 9 but −3² = −9: the brackets decide what is being squared. Builds on: Multiplying and dividing integers, Powers, squares and roots Leads to: Laws of indices Lessons: Year 8, Unit 3, lesson 5: Powers of negative numbers
Expanding and factorizing Year 8, every student The distributive law in both directions: 9(n + 8) = 9n + 72, and back again by taking out a common factor. Builds on: Collecting like terms, Multiplying and dividing integers Leads to: Balancing linear equations, Expanding double brackets Lessons: lessons coming
Balancing linear equations Year 8, every student Equations with brackets and with the unknown on both sides, solved by keeping the balance. Builds on: Solving simple equations, Expanding and factorizing Leads to: Linear inequalities, Multi-step equations Lessons: lessons coming
Linear inequalities Year 8, every student A solution that is a range of values on a number line, and why multiplying by a negative number reverses the sign. Builds on: Balancing linear equations, Integers and the number line Leads to: Inequalities and linear programming Lessons: lessons coming

Functions

Coordinates in four quadrants Year 8, every student Ordered pairs locate any point, with negative values to the left of and below the origin. Builds on: Integers and the number line Leads to: Proportional relationships, Gradient and y = mx + c, Distance and midpoint, Reflections, rotations and translations, Scatter graphs and correlation, Trigonometry beyond 90° Lessons: Year 8, Unit 3, lesson 6: Coordinates in four quadrants
Proportional relationships Year 8, every student A table, a straight line through the origin and y = kx are three views of one relationship. Builds on: Proportion, Coordinates in four quadrants Leads to: Constant rate of change, Direct and inverse proportion Lessons: lessons coming
Constant rate of change Year 8, every student A steady rate gives a straight-line graph, which need not pass through the origin. Builds on: Rates and unit rates, Proportional relationships Leads to: Gradient and y = mx + c Lessons: lessons coming

Geometry and Trigonometry

Trapeziums and regular polygons Year 8, every student A trapezium's area averages its parallel sides; a regular polygon is identical triangles around its centre. Builds on: Area of compound shapes Leads to: Surface area and volume of prisms, The trapezoidal rule Lessons: lessons coming
Circumference and area of a circle Year 8, every student π is the ratio of circumference to diameter, and slicing a circle into sectors shows why its area is πr². Builds on: Area of triangles and parallelograms, Rounding and estimating Leads to: Cylinders, Arcs and sectors, Volumes of revolution Lessons: lessons coming
Surface area and volume of prisms Year 8, every student Nets and cross-sections for any prism, keeping the units of length, area and volume apart. Builds on: Volume of cuboids and prisms, Nets and surface area, Trapeziums and regular polygons Leads to: Cylinders Lessons: lessons coming

Statistics and Probability

The probability scale Year 8, every student Chance from 0 (impossible) to 1 (certain), written as a fraction, a decimal or a percentage. Builds on: Fractions, decimals and percentages Leads to: Sample spaces Lessons: lessons coming
Sample spaces Year 8, every student Organized lists, tables and tree diagrams make sure every outcome is counted exactly once. Builds on: The probability scale Leads to: Theoretical probability, Tree diagrams, Counting principles Lessons: lessons coming
Theoretical probability Year 8, every student For equally likely outcomes, favourable outcomes divided by all outcomes, and P(not A) = 1 − P(A). Builds on: Sample spaces Leads to: Experimental probability, Probability from sample spaces Lessons: lessons coming
Experimental probability Year 8, every student Relative frequency settles towards the theoretical value as trials increase, and a simulation can stand in for the real event. Builds on: Theoretical probability Leads to: Probability and expected frequency Lessons: lessons coming
Mean, median and mode Year 8, every student Three ways to describe a typical value, from lists, tables and stem-and-leaf diagrams, and why an outlier drags the mean but barely moves the median. Builds on: Frequency tables, Types of data Leads to: Quartiles and box plots, Line of best fit, Standard deviation Lessons: lessons coming
Quartiles and box plots Year 8, every student Range, quartiles and the interquartile range, and the five-number summary drawn as a box plot. Builds on: Mean, median and mode Leads to: Grouped data and cumulative frequency, Describing grouped data Lessons: lessons coming

Year 9

Number and Algebra

Rational and irrational numbers Year 9, every student Rational numbers are fractions of integers; π and √2 are not, and their decimals never settle into a repeating block. Builds on: Terminating and recurring decimals, Powers, squares and roots Leads to: Sets and number systems, Surds Lessons: lessons coming
Laws of indices Year 9, every student Multiplying powers of the same base adds the indices, dividing subtracts them, and a power of a power multiplies them. Builds on: Powers of negative numbers Leads to: Zero and negative indices, Surds, Geometric sequences and growth, Logarithms and their laws, Exponents and logarithms Lessons: lessons coming
Zero and negative indices Year 9, every student Continuing the pattern downwards forces a⁰ = 1 and a⁻ⁿ = 1/aⁿ. Builds on: Laws of indices Leads to: Scientific notation, Fractional indices, Differentiating powers of x Lessons: lessons coming
Scientific notation Year 9, every student Very large and very small numbers written as a × 10ⁿ with 1 ≤ a < 10. Builds on: Zero and negative indices, Place value and decimals Leads to: Units, prefixes and conversions, Standard form and order of magnitude Lessons: lessons coming
Multi-step equations Year 9, every student Linear equations with brackets, fractions and the unknown on both sides. Builds on: Balancing linear equations, Adding and subtracting fractions Leads to: Systems of linear equations, Rearranging formulae Lessons: lessons coming
Systems of linear equations Year 9, every student Two equations with one shared solution: where two lines cross, found by graph, by substitution or by elimination. Builds on: Multi-step equations, Gradient and y = mx + c Leads to: Linear models and break-even, Inequalities and linear programming, Solving with technology, Systems of three equations, Matrices Lessons: lessons coming
Expanding double brackets Year 9, every student Every term in one bracket multiplies every term in the other, which an area grid makes visible. Builds on: Expanding and factorizing Leads to: Generalizing with algebra, Factorizing quadratics, The binomial theorem Lessons: lessons coming

Functions

Gradient and y = mx + c Year 9, every student Gradient as a rate of change and the intercept as a starting value, read from a graph and from an equation. Builds on: Constant rate of change, Coordinates in four quadrants Leads to: Systems of linear equations, Forms of a straight line, Parallel and perpendicular lines, Line of best fit, Functions, domain and range, Arithmetic sequences, Limits and the derivative Lessons: lessons coming
Forms of a straight line Year 9, every student Gradient-intercept, point-gradient and standard form, and the equations of horizontal and vertical lines. Builds on: Gradient and y = mx + c Leads to: Linear models and break-even, Equations of straight lines Lessons: lessons coming
Parallel and perpendicular lines Year 9, every student Parallel lines share a gradient; the gradients of perpendicular lines multiply to −1. Builds on: Gradient and y = mx + c Leads to: Equations of straight lines, Perpendicular bisectors Lessons: lessons coming
Linear models and break-even Year 9, every student Cost and income as straight lines, and the break-even point where they meet. Builds on: Systems of linear equations, Forms of a straight line Leads to: Modelling with technology Lessons: lessons coming

Geometry and Trigonometry

Conjecture and proof Year 9, every student The difference between noticing a pattern and proving it: axiom, conjecture and theorem. Builds on: Triangles and their angles Leads to: Pythagoras' theorem, Generalizing with algebra, Circle theorems Lessons: lessons coming
Pythagoras' theorem Year 9, every student In a right-angled triangle a² + b² = c², and the converse tests whether an angle is a right angle. Builds on: Area of triangles and parallelograms, Powers, squares and roots, Conjecture and proof Leads to: Distance and midpoint, Sine, cosine and tangent, Pyramids, cones and spheres, Right-angled triangles in 3D, Sine and cosine rules, Vectors Lessons: lessons coming
Distance and midpoint Year 9, every student Pythagoras on the coordinate grid gives the distance between two points, and averaging gives the midpoint. Builds on: Pythagoras' theorem, Coordinates in four quadrants Leads to: Perpendicular bisectors Lessons: lessons coming
Similar triangles Year 9, every student Equal angles force proportional sides, so one scale factor connects every pair of corresponding sides. Builds on: Proportion, Triangles and their angles Leads to: Sine, cosine and tangent Lessons: lessons coming
Sine, cosine and tangent Year 9, every student In similar right-angled triangles the side ratios depend only on the angle, and sin, cos and tan record those ratios. Builds on: Similar triangles, Pythagoras' theorem Leads to: Elevation, depression and bearings, Exact trigonometric values, Sine and cosine rules Lessons: lessons coming
Cylinders Year 9, every student A prism with a circular cross-section: volume πr²h, and a curved surface that unrolls into a rectangle. Builds on: Surface area and volume of prisms, Circumference and area of a circle Leads to: Pyramids, cones and spheres Lessons: lessons coming
Pyramids, cones and spheres Year 9, every student A pyramid or cone holds a third of the matching prism; a sphere has surface area 4πr² and volume 4/3 πr³. Builds on: Cylinders, Pythagoras' theorem Leads to: Similar shapes and solids, Solids and angles in 3D Lessons: lessons coming
Reflections, rotations and translations Year 9, every student Moving a shape without changing its size or shape, described exactly with coordinates. Builds on: Coordinates in four quadrants, Angles at a point and on a line Leads to: Enlargement, Tessellations and polygon angles, Transforming graphs, Combining transformations, Vectors Lessons: lessons coming
Enlargement Year 9, every student A scale factor from a centre changes the size but keeps the shape, so the image is similar rather than congruent. Builds on: Reflections, rotations and translations, Ratio Leads to: Similar shapes and solids Lessons: lessons coming
Tessellations and polygon angles Year 9, every student Interior angles of (n − 2) × 180° ÷ n explain why only triangles, squares and hexagons tile on their own. Builds on: Reflections, rotations and translations, Triangles and their angles Leads to: Combining transformations Lessons: lessons coming

Statistics and Probability

Scatter graphs and correlation Year 9, every student Plotting pairs of values and describing a relationship by its direction, form and strength. Builds on: Coordinates in four quadrants, Types of data Leads to: Line of best fit, Correlation, causation and bias Lessons: lessons coming
Line of best fit Year 9, every student A line through the mean point that follows the trend, and how far it can be trusted for prediction. Builds on: Scatter graphs and correlation, Gradient and y = mx + c, Mean, median and mode Leads to: The correlation coefficient, Interpolation and extrapolation Lessons: lessons coming
The correlation coefficient Year 9, every student r runs from −1 to 1 and measures how closely points follow a straight line; one outlier can change it a lot. Builds on: Line of best fit Leads to: Pearson's r and regression Lessons: lessons coming
Correlation, causation and bias Year 9, every student A link in data is not proof of a cause: a lurking variable or a biased sample can explain it. Builds on: Scatter graphs and correlation Leads to: Sampling methods Lessons: lessons coming

Year 10

Number and Algebra

Sets and number systems Year 10, Standard and Extended Set notation, and the number sets nested inside each other: ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ. Builds on: Rational and irrational numbers Leads to: Venn diagrams Lessons: lessons coming
Significant figures and estimation Year 10, Standard and Extended Rounding to significant figures, and estimating to one significant figure to check a calculation. Builds on: Rounding and estimating Leads to: Upper and lower bounds, Approximation and error Lessons: lessons coming
Units, prefixes and conversions Year 10, Standard and Extended Metric prefixes as powers of ten, and why area conversions square the factor while volume conversions cube it. Builds on: Scientific notation, Converting units and currencies Leads to: Standard form and order of magnitude Lessons: lessons coming
Surds Year 10, Standard and Extended Exact roots: simplifying √50 to 5√2, multiplying surds and rationalizing a denominator. Builds on: Rational and irrational numbers, Laws of indices Leads to: Fractional indices, Exact trigonometric values Lessons: lessons coming
Absolute value Year 10, Standard and Extended |x| as a distance from zero, and solving equations such as |x − 3| = 5. Builds on: Integers and the number line Leads to: Modulus functions Lessons: lessons coming
Fractional indices Year 10, Extended only x to the power 1/n is the nth root of x, which keeps every law of indices working. Builds on: Surds, Zero and negative indices Leads to: Logarithms and their laws, Rational exponents Lessons: lessons coming
Upper and lower bounds Year 10, Extended only A rounded value hides an interval, and a calculation with rounded values carries that interval with it. Builds on: Significant figures and estimation Leads to: Approximation and error Lessons: lessons coming
Generalizing with algebra Year 10, Standard and Extended Writing even, odd and consecutive numbers as 2n, 2n + 1 and n + 1 to prove that a pattern holds every time. Builds on: Conjecture and proof, Expanding double brackets Leads to: Networks, Deductive proof Lessons: lessons coming
Factorizing quadratics Year 10, Standard and Extended Reversing double brackets, including the difference of two squares and expressions with a leading coefficient. Builds on: Expanding double brackets Leads to: Solving quadratic equations, Quadratic functions, Algebraic fractions Lessons: lessons coming
Solving quadratic equations Year 10, Standard and Extended If ab = 0 then a = 0 or b = 0; when factorizing fails, complete the square or use the formula. Builds on: Factorizing quadratics Leads to: The discriminant, Quadratic and rational inequalities, Solving with technology, Eigenvalues and eigenvectors Lessons: lessons coming
The discriminant Year 10, Standard and Extended b² − 4ac decides whether a quadratic has two, one or no real roots before it is solved. Builds on: Solving quadratic equations Leads to: Complex numbers, The discriminant and inequalities Lessons: lessons coming
Inequalities and linear programming Year 10, Extended only Regions of a graph that satisfy several inequalities at once, and the best value at a corner of the feasible region. Builds on: Linear inequalities, Systems of linear equations Leads to: Quadratic and rational inequalities Lessons: lessons coming

Functions

Functions, domain and range Year 10, Standard and Extended A function gives each input exactly one output: f(x) notation, mapping diagrams, domain and range. Builds on: Function machines, Gradient and y = mx + c Leads to: Quadratic functions, Exponential growth and decay, Domain, range and inverses, Graphs and key features Lessons: lessons coming
Quadratic functions Year 10, Standard and Extended Standard, factorized and vertex forms, each showing a different feature of the same parabola. Builds on: Functions, domain and range, Factorizing quadratics Leads to: Transforming graphs, Quadratic sequences, Modelling with technology, The quadratic function Lessons: lessons coming
Transforming graphs Year 10, Extended only f(x − h) + k, −f(x), f(−x) and af(x): how each change to the rule moves or stretches the graph. Builds on: Quadratic functions, Reflections, rotations and translations Leads to: Reciprocal and rational functions, Graphs of sine, cosine and tangent, Transformations of graphs Lessons: lessons coming

Geometry and Trigonometry

Similar shapes and solids Year 10, Standard and Extended Scaling lengths by k scales areas by k² and volumes by k³. Builds on: Enlargement, Pyramids, cones and spheres Leads to: Solids and angles in 3D Lessons: lessons coming
Combining transformations Year 10, Standard and Extended One transformation after another, and why every isometry is a reflection, a rotation, a translation or a glide reflection. Builds on: Reflections, rotations and translations, Tessellations and polygon angles Leads to: Matrix transformations Lessons: lessons coming

Statistics and Probability

Venn diagrams Year 10, Standard and Extended Union, intersection and complement for two or three sets, and n(A ∪ B) = n(A) + n(B) − n(A ∩ B). Builds on: Sets and number systems Leads to: Probability from sample spaces, Venn and tree diagrams Lessons: lessons coming
Probability from sample spaces Year 10, Standard and Extended Probability as a share of a sample space, read from lists, tables and Venn diagrams, and adding the probabilities of mutually exclusive events. Builds on: Venn diagrams, Theoretical probability Leads to: Tree diagrams, Independence and conditional probability, Probability and expected frequency Lessons: lessons coming
Tree diagrams Year 10, Standard and Extended Multiplying along the branches, with and without replacement, and adding the routes that count. Builds on: Sample spaces, Multiplying and dividing fractions, Probability from sample spaces Leads to: Independence and conditional probability, Venn and tree diagrams, The binomial distribution Lessons: lessons coming
Independence and conditional probability Year 10, Extended only P(B | A) from trees, tables and Venn diagrams, and the test P(A ∩ B) = P(A) × P(B) for independence. Builds on: Tree diagrams, Probability from sample spaces Leads to: Conditional probability, formally Lessons: lessons coming
Grouped data and cumulative frequency Year 10, Extended only An estimated mean from midpoints, modal and median classes, quartiles from a cumulative frequency curve, and the 1.5 × IQR rule for outliers. Builds on: Quartiles and box plots Leads to: Presenting data Lessons: lessons coming
Standard deviation Year 10, Extended only Spread measured as the typical distance from the mean, worked by hand once and then by technology. Builds on: Mean, median and mode Leads to: The bell curve, Pearson's r and regression, Measures of centre and spread Lessons: lessons coming
The bell curve Year 10, Extended only A symmetric bell shape with about 68%, 95% and 99.7% of values within 1, 2 and 3 standard deviations of the mean. Builds on: Standard deviation Leads to: The normal distribution Lessons: lessons coming

Year 11

Number and Algebra

Arithmetic sequences Year 11, Standard and Extended A constant difference gives a linear rule, so the nth term is a straight-line function of n. Builds on: Term-to-term and position rules, Gradient and y = mx + c Leads to: Quadratic sequences, Geometric sequences and growth, Arithmetic sequences and series Lessons: lessons coming
Quadratic sequences Year 11, Standard and Extended A constant second difference means the nth term has an n² term whose coefficient is half that difference. Builds on: Arithmetic sequences, Quadratic functions Leads to: The quadratic function Lessons: lessons coming
Geometric sequences and growth Year 11, Standard and Extended A constant ratio between terms models compound interest, depreciation and doubling. Builds on: Arithmetic sequences, Percentage increase and decrease, Laws of indices Leads to: Series and sigma notation, Exponential growth and decay, Geometric sequences and series Lessons: lessons coming
Series and sigma notation Year 11, Extended only Adding the terms of a sequence, written compactly with Σ, and when an infinite sum settles to a value. Builds on: Geometric sequences and growth Leads to: Arithmetic sequences and series Lessons: lessons coming
Rearranging formulae Year 11, Standard and Extended Making another letter the subject, including when it appears twice or sits under a power or a root. Builds on: Multi-step equations Leads to: Direct and inverse proportion Lessons: lessons coming
Direct and inverse proportion Year 11, Standard and Extended y = kx, y = k/x and versions with squares and roots, each with its graph. Builds on: Proportional relationships, Rearranging formulae Leads to: Modelling with functions Lessons: lessons coming
Algebraic fractions Year 11, Extended only Simplifying, multiplying, dividing, adding and subtracting fractions with letters in them, factorizing first. Builds on: Factorizing quadratics, Adding and subtracting fractions Leads to: Reciprocal and rational functions, Partial fractions Lessons: lessons coming
Logarithms and their laws Year 11, Extended only A logarithm answers the question 'what power?', so it undoes an exponential, and its laws come straight from the laws of indices. Builds on: Laws of indices, Fractional indices Leads to: Exponents and logarithms Lessons: lessons coming
Quadratic and rational inequalities Year 11, Extended only Sign diagrams and graphs show where a quadratic or rational expression is positive or negative. Builds on: Inequalities and linear programming, Solving quadratic equations Leads to: Polynomial and rational inequalities Lessons: lessons coming

Functions

Exponential growth and decay Year 11, Standard and Extended Models of the form y = a × bˣ, recognized by a constant ratio rather than a constant difference. Builds on: Geometric sequences and growth, Functions, domain and range Leads to: Modelling with functions, Exponential and log functions Lessons: lessons coming
Modelling with technology Year 11, Standard Fitting linear and quadratic models with a graphing calculator after a first pass by hand. Builds on: Quadratic functions, Linear models and break-even Leads to: Modelling with functions Lessons: lessons coming
Reciprocal and rational functions Year 11, Extended only Graphs of 1/x and (ax + b)/(cx + d), with their asymptotes, domain and range. Builds on: Algebraic fractions, Transforming graphs Leads to: Rational functions Lessons: lessons coming
Graphs of sine, cosine and tangent Year 11, Extended only Periodic graphs with amplitude, period and shifts, used to model tides and turning wheels. Builds on: Trigonometry beyond 90°, Transforming graphs Leads to: Solving trigonometric equations, Circular functions Lessons: lessons coming

Geometry and Trigonometry

Elevation, depression and bearings Year 11, Standard and Extended Choosing the right ratio in real problems, with three-figure bearings. Builds on: Sine, cosine and tangent Leads to: Right-angled triangles in 3D, Bearings and applications Lessons: lessons coming
Right-angled triangles in 3D Year 11, Standard and Extended Finding the right-angled triangle hidden inside a solid: space diagonals and the angle between a line and a plane. Builds on: Elevation, depression and bearings, Pythagoras' theorem Leads to: Solids and angles in 3D Lessons: lessons coming
Exact trigonometric values Year 11, Standard and Extended sin, cos and tan of 30°, 45° and 60° from two special triangles, without a calculator. Builds on: Sine, cosine and tangent, Surds Leads to: Trigonometry beyond 90°, The unit circle Lessons: lessons coming
Arcs and sectors Year 11, Standard and Extended Arc length and sector area are the same fraction of the whole circle as the angle is of 360°. Builds on: Circumference and area of a circle Leads to: Arcs, sectors and segments, Radians Lessons: lessons coming
Circle theorems Year 11, Standard and Extended Angles in circles and at tangents, proved from isosceles triangles rather than memorized. Builds on: Triangles and their angles, Conjecture and proof Leads to: Perpendicular bisectors Lessons: lessons coming
Trigonometry beyond 90° Year 11, Extended only The unit circle extends sine and cosine to any angle and fixes their signs in each quadrant. Builds on: Exact trigonometric values, Coordinates in four quadrants Leads to: Graphs of sine, cosine and tangent, The unit circle Lessons: lessons coming
Sine and cosine rules Year 11, Standard and Extended Any triangle, not only right-angled ones: the sine rule, the cosine rule and area = ½ab sin C. Builds on: Sine, cosine and tangent, Pythagoras' theorem Leads to: Non-right-angled triangles, The scalar product Lessons: lessons coming
Solving trigonometric equations Year 11, Extended only Finding every angle in an interval, using the symmetry of the graph rather than one calculator answer. Builds on: Graphs of sine, cosine and tangent Leads to: Trigonometric equations Lessons: lessons coming
Networks Year 11, Extended only Vertices, edges and degree; routes that use every edge or visit every vertex; shortest paths and spanning trees. Builds on: Generalizing with algebra Leads to: Graph theory Lessons: lessons coming
Vectors Year 11, Extended only Column vectors, magnitude and direction, adding and scaling, and using vectors to prove geometric results. Builds on: Reflections, rotations and translations, Pythagoras' theorem Leads to: Vectors in 2D and 3D Lessons: lessons coming

Statistics and Probability

Describing grouped data Year 11, Standard An estimated mean, modal and median classes, quartiles from cumulative frequency, and the 1.5 × IQR rule for outliers. Builds on: Quartiles and box plots Leads to: Histograms with unequal widths, Comparing distributions, Measures of centre and spread Lessons: lessons coming
Histograms with unequal widths Year 11, Standard Frequency density makes the area of each bar, not its height, show the frequency. Builds on: Describing grouped data Leads to: Presenting data Lessons: lessons coming
Comparing distributions Year 11, Standard Comparing two groups on centre and spread, in context, and spotting graphs that mislead. Builds on: Describing grouped data, Bar charts, pie charts and line graphs Leads to: Measures of centre and spread Lessons: lessons coming
Sampling methods Year 11, Standard and Extended Random, systematic, stratified and quota samples, and the bias each one can bring in. Builds on: Correlation, causation and bias Leads to: Sampling and bias Lessons: lessons coming
Interpolation and extrapolation Year 11, Standard and Extended Predicting inside the range of the data is reasonable; predicting far outside it is a guess. Builds on: Line of best fit Leads to: Correlation and regression Lessons: lessons coming
Pearson's r and regression Year 11, Extended only Pearson's r by formula once, then a regression line by technology, and judging whether a line suits the data. Builds on: The correlation coefficient, Standard deviation Leads to: Correlation and regression Lessons: lessons coming

Years 12 and 13: the IB Diploma

Number and Algebra

Standard form and order of magnitude IB Diploma: AA SL and HL, AI SL and HL Writing and calculating in standard form, and comparing quantities by their power of ten. Builds on: Scientific notation, Units, prefixes and conversions Leads to: nothing later on the map yet Lessons: AA and AI 1.1: Standard form and order of magnitude
Arithmetic sequences and series IB Diploma: AA SL and HL, AI SL and HL The nth term and the sum of the first n terms, applied to steady growth. Builds on: Arithmetic sequences, Series and sigma notation Leads to: Geometric sequences and series Lessons: AA and AI 1.2: Arithmetic sequences and series
Geometric sequences and series IB Diploma: AA SL and HL, AI SL and HL The nth term and the sum of a geometric sequence, and spotting a common ratio in context. Builds on: Geometric sequences and growth, Arithmetic sequences and series Leads to: Compound interest and depreciation, Infinite geometric series Lessons: AA and AI 1.3: Geometric sequences and series
Compound interest and depreciation IB Diploma: AA SL and HL, AI SL and HL Money growing and shrinking by compound interest and depreciation, and its real value after inflation. Builds on: Geometric sequences and series, Percentage increase and decrease Leads to: Loans and annuities Lessons: AA and AI 1.4: Compound interest and depreciation
Exponents and logarithms IB Diploma: AA SL and HL, AI SL and HL The laws of exponents, and logarithms in base 10 and base e as the inverse of an exponential. Builds on: Laws of indices, Logarithms and their laws Leads to: Logarithm laws and equations, Rational exponents, Exponential and log functions Lessons: AA and AI 1.5: Exponents and logarithms
Approximation and error IB Diploma: AI SL and HL Bounds, estimation and percentage error, and how much accuracy an answer can honestly claim. Builds on: Upper and lower bounds, Significant figures and estimation Leads to: nothing later on the map yet Lessons: AI 1.6: Approximation and error
Loans and annuities IB Diploma: AI SL and HL Repaying a loan and saving by regular payments, with a finance solver and a check by hand. Builds on: Compound interest and depreciation Leads to: nothing later on the map yet Lessons: AI 1.7: Loans and annuities
Solving with technology IB Diploma: AI SL and HL Systems of linear equations and polynomial equations solved on a calculator, with the answer interpreted. Builds on: Systems of linear equations, Solving quadratic equations Leads to: nothing later on the map yet Lessons: AI 1.8: Solving with technology
Deductive proof IB Diploma: AA SL and HL Proving an identity by working from one side to the other, with every step justified. Builds on: Generalizing with algebra Leads to: Proof by induction, Contradiction and counterexample Lessons: AA 1.6: Deductive proof
Logarithm laws and equations IB Diploma: AA SL and HL, AI HL only The laws of logarithms, change of base, and solving exponential equations exactly. Builds on: Exponents and logarithms Leads to: Linearizing data Lessons: AI HL 1.9: Logarithm laws and equations; AA 1.7: Logarithm laws and equations
Infinite geometric series IB Diploma: AA SL and HL, AI HL only When the common ratio lies between −1 and 1 the sum settles at a limit; otherwise the formula gives nonsense. Builds on: Geometric sequences and series Leads to: Maclaurin series Lessons: AI HL 1.11: Infinite geometric series; AA 1.8: Infinite geometric series
The binomial theorem IB Diploma: AA SL and HL Expanding (a + b)ⁿ with Pascal's triangle and binomial coefficients. Builds on: Expanding double brackets Leads to: Differentiation from first principles Lessons: AA 1.9: The binomial theorem
Rational exponents IB Diploma: AI HL only Fractional powers as roots, and simplifying expressions that contain them. Builds on: Fractional indices, Exponents and logarithms Leads to: nothing later on the map yet Lessons: AI HL 1.10: Rational exponents
Counting principles IB Diploma: AA HL only Permutations and combinations, and deciding which one a problem needs. Builds on: Sample spaces Leads to: nothing later on the map yet Lessons: AA HL 1.10: Counting principles
Partial fractions IB Diploma: AA HL only Splitting a rational expression into simpler fractions, ready for integration. Builds on: Algebraic fractions Leads to: Further derivatives Lessons: AA HL 1.11: Partial fractions
Complex numbers IB Diploma: AA HL only, AI HL only i² = −1, complex numbers in Cartesian form on an Argand diagram, and the roots a negative discriminant hides. Builds on: The discriminant Leads to: Polar and Euler form Lessons: AI HL 1.12: Complex numbers; AA HL 1.12: Complex numbers
Polar and Euler form IB Diploma: AA HL only, AI HL only Modulus and argument, and why multiplying complex numbers multiplies the moduli and adds the arguments. Builds on: Complex numbers, The unit circle Leads to: De Moivre's theorem and roots Lessons: AI HL 1.13: Polar and Euler form; AA HL 1.13: Polar and Euler form
De Moivre's theorem and roots IB Diploma: AA HL only Powers and roots of complex numbers, with the roots spaced evenly around a circle. Builds on: Polar and Euler form Leads to: nothing later on the map yet Lessons: AA HL 1.14: De Moivre's theorem and roots
Proof by induction IB Diploma: AA HL only Prove the first case, show that each case forces the next, and the result holds for every n. Builds on: Deductive proof Leads to: nothing later on the map yet Lessons: AA HL 1.15: Proof by induction
Contradiction and counterexample IB Diploma: AA HL only Assuming the opposite and reaching something impossible, and disproving a claim with a single example. Builds on: Deductive proof Leads to: nothing later on the map yet Lessons: lesson coming
Systems of three equations IB Diploma: AA HL only Three equations in three unknowns, and what no solution or infinitely many solutions look like. Builds on: Systems of linear equations Leads to: nothing later on the map yet Lessons: AA HL 1.16: Systems of three equations
Matrices IB Diploma: AI HL only Matrix arithmetic, determinants and inverses, and solving systems of equations with them. Builds on: Systems of linear equations Leads to: Eigenvalues and eigenvectors, Matrix transformations, Adjacency matrices, Markov chains Lessons: AI HL 1.14: Matrices
Eigenvalues and eigenvectors IB Diploma: AI HL only The directions that a matrix only stretches, found from the characteristic equation. Builds on: Matrices, Solving quadratic equations Leads to: Markov chains, Coupled systems and phase portraits Lessons: AI HL 1.15: Eigenvalues and eigenvectors

Functions

Equations of straight lines IB Diploma: AA SL and HL, AI SL and HL Gradient, intercepts and the forms of a line, with parallel and perpendicular lines. Builds on: Forms of a straight line, Parallel and perpendicular lines Leads to: Vector equation of a line, Tangents and normals Lessons: AA and AI 2.1: Equations of straight lines
Domain, range and inverses IB Diploma: AA SL and HL, AI SL and HL Domain and range read from a graph, and an inverse function as a reflection in y = x. Builds on: Functions, domain and range Leads to: Composite and inverse functions, Rational functions, Transformations of graphs Lessons: AA and AI 2.2: Domain, range and inverses
Graphs and key features IB Diploma: AA SL and HL, AI SL and HL Sketching compared with drawing, and finding intercepts, turning points and asymptotes with technology. Builds on: Functions, domain and range Leads to: Modelling with functions, Limits and the derivative Lessons: AA and AI 2.3: Drawing and sketching graphs; AA and AI 2.4: Key features of graphs
Composite and inverse functions IB Diploma: AA SL and HL, AI HL only Applying one function after another, why the order matters, and undoing a function algebraically. Builds on: Domain, range and inverses Leads to: Odd, even and self-inverse, Reciprocal and inverse trigonometry, Chain, product and quotient rules Lessons: AI HL 2.7: Composite and inverse functions; AA 2.5: Composite and inverse functions
Modelling with functions IB Diploma: AI SL and HL Choosing between linear, quadratic, exponential, power and sinusoidal models, and testing the choice. Builds on: Exponential growth and decay, Graphs and key features, Direct and inverse proportion, Modelling with technology Leads to: Logistic and piecewise models, Linearizing data Lessons: AI 2.5: Choosing a model; AI 2.6: The modelling process
The quadratic function IB Diploma: AA SL and HL Standard, factorized and vertex forms of a parabola, and moving between them. Builds on: Quadratic functions, Quadratic sequences Leads to: The discriminant and inequalities, Choosing how to solve, Polynomial functions Lessons: AA 2.6: The quadratic function
The discriminant and inequalities IB Diploma: AA SL and HL Using b² − 4ac to find when a quadratic has two, one or no roots, and solving the inequality that follows. Builds on: The discriminant, The quadratic function Leads to: Polynomial and rational inequalities Lessons: AA 2.7: The discriminant and inequalities
Rational functions IB Diploma: AA SL and HL Reciprocal and simple rational functions, with asymptotes read from the coefficients. Builds on: Reciprocal and rational functions, Domain, range and inverses Leads to: Oblique asymptotes Lessons: AA 2.8: Rational functions
Exponential and log functions IB Diploma: AA SL and HL The graphs of an exponential and its logarithm as reflections of each other, with growth and decay in context. Builds on: Exponents and logarithms, Exponential growth and decay Leads to: Choosing how to solve Lessons: AA 2.9: Exponential and log functions
Choosing how to solve IB Diploma: AA SL and HL Deciding between an algebraic and a graphical method, including equations that reduce to quadratics. Builds on: The quadratic function, Exponential and log functions Leads to: nothing later on the map yet Lessons: AA 2.10: Choosing how to solve
Transformations of graphs IB Diploma: AA SL and HL, AI HL only Translations, stretches and reflections of any graph, and the order in which combined ones apply. Builds on: Transforming graphs, Domain, range and inverses Leads to: Modulus functions, Circular functions Lessons: AI HL 2.8: Transformations of graphs; AA 2.11: Transformations of graphs
Logistic and piecewise models IB Diploma: AI HL only Models that level off or change rule part of the way along, and the limits of any model. Builds on: Modelling with functions Leads to: Non-linear regression Lessons: AI HL 2.9: Logistic and piecewise models
Linearizing data IB Diploma: AI HL only Taking logarithms to turn exponential and power data into a straight line, and reading the model from it. Builds on: Logarithm laws and equations, Modelling with functions Leads to: nothing later on the map yet Lessons: AI HL 2.10: Linearizing data
Polynomial functions IB Diploma: AA HL only The factor and remainder theorems, and the sum and product of the roots read from the coefficients. Builds on: The quadratic function Leads to: Oblique asymptotes, Polynomial and rational inequalities Lessons: AA HL 2.12: Polynomial functions
Oblique asymptotes IB Diploma: AA HL only Rational functions whose numerator has the higher degree, and the slanting line the graph approaches. Builds on: Rational functions, Polynomial functions Leads to: nothing later on the map yet Lessons: AA HL 2.13: Oblique asymptotes
Odd, even and self-inverse IB Diploma: AA HL only Symmetry in a function's rule: f(−x) = f(x), f(−x) = −f(x), and functions that are their own inverse. Builds on: Composite and inverse functions Leads to: nothing later on the map yet Lessons: AA HL 2.14: Odd, even and self-inverse
Polynomial and rational inequalities IB Diploma: AA HL only Solving inequalities without multiplying by something of unknown sign, using critical values and sign diagrams. Builds on: The discriminant and inequalities, Polynomial functions, Quadratic and rational inequalities Leads to: nothing later on the map yet Lessons: AA HL 2.15: Polynomial and rational inequalities
Modulus functions IB Diploma: AA HL only The graphs of |f(x)| and f(|x|), and equations and inequalities involving the modulus. Builds on: Absolute value, Transformations of graphs Leads to: nothing later on the map yet Lessons: AA HL 2.16: Modulus functions

Geometry and Trigonometry

Solids and angles in 3D IB Diploma: AA SL and HL, AI SL and HL Volume and surface area of composite solids, and the angles between lines and planes. Builds on: Right-angled triangles in 3D, Pyramids, cones and spheres, Similar shapes and solids Leads to: nothing later on the map yet Lessons: AA and AI 3.1: Solids and angles in 3D
Non-right-angled triangles IB Diploma: AA SL and HL, AI SL and HL The sine rule, the cosine rule and the area formula, and deciding which one a triangle needs. Builds on: Sine and cosine rules Leads to: Bearings and applications Lessons: AA and AI 3.2: Non-right-angled triangles
Bearings and applications IB Diploma: AA SL and HL, AI SL and HL Elevation, depression and bearings in multi-step problems that need more than one triangle. Builds on: Elevation, depression and bearings, Non-right-angled triangles Leads to: nothing later on the map yet Lessons: AA and AI 3.3: Bearings and applications
Arcs, sectors and segments IB Diploma: AA SL and HL, AI SL and HL Arc length, sector area and segment area, in degrees and in radians. Builds on: Arcs and sectors Leads to: nothing later on the map yet Lessons: AI 3.4: Arcs, sectors and segments; AA 3.4: Arcs, sectors and segments
Perpendicular bisectors IB Diploma: AI SL and HL Every point the same distance from two given points, as an equation and as a construction. Builds on: Parallel and perpendicular lines, Distance and midpoint, Ruler and compass constructions, Circle theorems Leads to: Voronoi diagrams Lessons: AI 3.5: Perpendicular bisectors
Voronoi diagrams IB Diploma: AI SL and HL Dividing a map into the regions nearest each site, and finding the point furthest from every site. Builds on: Perpendicular bisectors Leads to: nothing later on the map yet Lessons: AI 3.6: Voronoi diagrams
Radians IB Diploma: AA SL and HL, AI HL only Measuring an angle by the arc it cuts on a circle of radius 1, and converting to and from degrees. Builds on: Arcs and sectors Leads to: The unit circle Lessons: AI HL 3.7: Radians
The unit circle IB Diploma: AA SL and HL, AI HL only Sine and cosine as coordinates on a circle, the ambiguous case, and exact values in every quadrant. Builds on: Trigonometry beyond 90°, Exact trigonometric values, Radians Leads to: Polar and Euler form, Trigonometric identities, Circular functions Lessons: AI HL 3.8: The unit circle; AA 3.5: The unit circle
Trigonometric identities IB Diploma: AA SL and HL sin²θ + cos²θ = 1, tan θ = sin θ ÷ cos θ, and the double-angle identities. Builds on: The unit circle Leads to: Trigonometric equations, Compound angle identities Lessons: AA 3.6: Trigonometric identities
Circular functions IB Diploma: AA SL and HL Graphs of sin, cos and tan with amplitude, period and shifts, measured in radians. Builds on: Graphs of sine, cosine and tangent, The unit circle, Transformations of graphs Leads to: Trigonometric equations, Reciprocal and inverse trigonometry, Symmetry of trigonometric graphs Lessons: AA 3.7: Circular functions
Trigonometric equations IB Diploma: AA SL and HL Every solution in an interval, including quadratic equations in sin or cos. Builds on: Circular functions, Trigonometric identities, Solving trigonometric equations Leads to: nothing later on the map yet Lessons: AA 3.8: Trigonometric equations
Reciprocal and inverse trigonometry IB Diploma: AA HL only sec, cosec and cot, and the inverse trigonometric functions with their restricted domains. Builds on: Circular functions, Composite and inverse functions Leads to: Further derivatives Lessons: AA HL 3.9: Reciprocal and inverse trigonometry
Compound angle identities IB Diploma: AA HL only sin(A + B) and cos(A + B), and why sine does not distribute over addition. Builds on: Trigonometric identities Leads to: nothing later on the map yet Lessons: AA HL 3.10: Compound angle identities
Symmetry of trigonometric graphs IB Diploma: AA HL only Relationships such as sin(π − x) = sin x read from the symmetry of the graphs. Builds on: Circular functions Leads to: nothing later on the map yet Lessons: AA HL 3.11: Symmetry of trigonometric graphs
Vectors in 2D and 3D IB Diploma: AA HL only, AI HL only Components, magnitude, unit vectors and position vectors, in two and three dimensions. Builds on: Vectors Leads to: Vector equation of a line, The scalar product Lessons: AI HL 3.10: Vectors in 2D and 3D; AA HL 3.12: Vectors in 2D and 3D
Vector equation of a line IB Diploma: AA HL only, AI HL only A point and a direction describe a line, so the same line has many correct equations. Builds on: Vectors in 2D and 3D, Equations of straight lines Leads to: Intersecting and skew lines, Equations of planes, Vectors and motion Lessons: AI HL 3.11: Vector equation of a line; AA HL 3.14: Vector equation of a line
The scalar product IB Diploma: AA HL only, AI HL only Multiplying two vectors to get a number, finding the angle between them and testing for perpendicular lines. Builds on: Vectors in 2D and 3D, Sine and cosine rules Leads to: The vector product, Lines and planes meeting Lessons: AI HL 3.13: The scalar product; AA HL 3.13: The scalar product
The vector product IB Diploma: AA HL only, AI HL only A vector perpendicular to two others, whose length is the area of the parallelogram they make. Builds on: The scalar product Leads to: Equations of planes Lessons: AI HL 3.13: The vector product; AA HL 3.16: The vector product
Intersecting and skew lines IB Diploma: AA HL only Lines in three dimensions that meet, run parallel, or pass each other without meeting. Builds on: Vector equation of a line Leads to: nothing later on the map yet Lessons: AA HL 3.15: Intersecting and skew lines
Equations of planes IB Diploma: AA HL only Vector, parametric and Cartesian equations of a plane, with the normal vector read from the coefficients. Builds on: The vector product, Vector equation of a line Leads to: Lines and planes meeting Lessons: AA HL 3.17: Equations of planes
Lines and planes meeting IB Diploma: AA HL only Where lines and planes intersect, and the angles between them. Builds on: Equations of planes, The scalar product Leads to: nothing later on the map yet Lessons: AA HL 3.18: Lines and planes meeting
Vectors and motion IB Diploma: AI HL only Position and velocity as vectors, and the closest approach of two moving objects. Builds on: Vector equation of a line Leads to: nothing later on the map yet Lessons: AI HL 3.12: Vectors and motion
Matrix transformations IB Diploma: AI HL only Rotations, reflections, stretches and enlargements as matrices, with the determinant as an area factor. Builds on: Matrices, Combining transformations Leads to: nothing later on the map yet Lessons: AI HL 3.9: Matrix transformations
Graph theory IB Diploma: AI HL only Vertices, edges, degree and connectedness, and Eulerian and Hamiltonian routes. Builds on: Networks Leads to: Adjacency matrices, Network algorithms Lessons: AI HL 3.14: Graph theory
Adjacency matrices IB Diploma: AI HL only Counting the walks between vertices with powers of the adjacency matrix. Builds on: Graph theory, Matrices Leads to: nothing later on the map yet Lessons: AI HL 3.15: Adjacency matrices
Network algorithms IB Diploma: AI HL only Minimum spanning trees, the Chinese postman problem, and bounds for the travelling salesperson. Builds on: Graph theory Leads to: nothing later on the map yet Lessons: AI HL 3.16: Network algorithms

Statistics and Probability

Sampling and bias IB Diploma: AA SL and HL, AI SL and HL Populations and samples, the sampling techniques and the bias each can bring, and treating outliers honestly. Builds on: Sampling methods Leads to: Designing data collection Lessons: AA and AI 4.1: Sampling and bias
Presenting data IB Diploma: AA SL and HL, AI SL and HL Histograms, cumulative frequency graphs and box plots, and choosing between them. Builds on: Grouped data and cumulative frequency, Histograms with unequal widths Leads to: nothing later on the map yet Lessons: AA and AI 4.2: Presenting data
Measures of centre and spread IB Diploma: AA SL and HL, AI SL and HL Mean, median, quartiles and standard deviation, including from grouped data, and the effect of changing the data. Builds on: Standard deviation, Describing grouped data, Comparing distributions Leads to: The normal distribution Lessons: AA and AI 4.3: Measures of centre and spread
Correlation and regression IB Diploma: AA SL and HL, AI SL and HL Pearson's r, the regression line of y on x, and when a prediction from it can be trusted. Builds on: Pearson's r and regression, Interpolation and extrapolation Leads to: Spearman's rank correlation, Regression of x on y, Non-linear regression Lessons: AA and AI 4.4: Correlation and regression
Probability and expected frequency IB Diploma: AA SL and HL, AI SL and HL Probability from sample spaces and from experiments, and how many times to expect an outcome. Builds on: Probability from sample spaces, Experimental probability Leads to: Venn and tree diagrams, Discrete random variables, Chi-squared tests Lessons: AA and AI 4.5: Probability and expected frequency
Venn and tree diagrams IB Diploma: AA SL and HL, AI SL and HL Combined events, conditional probability from diagrams and tables, and independence. Builds on: Tree diagrams, Venn diagrams, Probability and expected frequency Leads to: Conditional probability, formally, Chi-squared tests, Markov chains Lessons: AA and AI 4.6: Venn and tree diagrams
Conditional probability, formally IB Diploma: AA SL and HL P(A | B) = P(A ∩ B) ÷ P(B) as a definition, and testing for independence with it. Builds on: Venn and tree diagrams, Independence and conditional probability Leads to: Bayes' theorem Lessons: AA 4.11: Conditional probability, formally
Discrete random variables IB Diploma: AA SL and HL, AI SL and HL Probability distributions, expected value and fair games. Builds on: Probability and expected frequency Leads to: The binomial distribution, Continuous random variables, Combining random variables, The Poisson distribution Lessons: AA and AI 4.7: Discrete random variables
The binomial distribution IB Diploma: AA SL and HL, AI SL and HL A fixed number of independent trials with the same chance of success, and its mean and variance. Builds on: Discrete random variables, Tree diagrams Leads to: Hypothesis tests and errors Lessons: AA and AI 4.8: The binomial distribution
The normal distribution IB Diploma: AA SL and HL, AI SL and HL Probabilities from a normal model with technology, and finding a value from a probability. Builds on: The bell curve, Measures of centre and spread Leads to: Standardizing and z-scores, The central limit theorem Lessons: AA and AI 4.9: The normal distribution
Spearman's rank correlation IB Diploma: AI SL and HL Correlating ranks, for relationships that are consistent but not linear. Builds on: Correlation and regression Leads to: nothing later on the map yet Lessons: AI 4.10: Spearman's rank correlation
Chi-squared tests IB Diploma: AI SL and HL Testing independence and goodness of fit by comparing observed frequencies with expected ones. Builds on: Venn and tree diagrams, Probability and expected frequency Leads to: Hypothesis tests and errors Lessons: AI 4.11: Chi-squared tests
Regression of x on y IB Diploma: AA SL and HL Why predicting x from y needs its own line, not the y on x line rearranged. Builds on: Correlation and regression Leads to: nothing later on the map yet Lessons: AA 4.10: Regression of x on y
Standardizing and z-scores IB Diploma: AA SL and HL z-values put every normal distribution on one scale, which is how an unknown mean or standard deviation is found. Builds on: The normal distribution Leads to: nothing later on the map yet Lessons: AA 4.12: Standardizing and z-scores
Bayes' theorem IB Diploma: AA HL only Reversing a conditional probability, and why a positive test can still leave a condition unlikely. Builds on: Conditional probability, formally Leads to: nothing later on the map yet Lessons: AA HL 4.13: Bayes' theorem
Continuous random variables IB Diploma: AA HL only Probability as the area under a density function, with the mean, median and mode found by integration. Builds on: Discrete random variables, Areas between curves Leads to: nothing later on the map yet Lessons: AA HL 4.14: Continuous random variables
Designing data collection IB Diploma: AI HL only Reliability and validity, and planning the collection so that the data can answer the question. Builds on: Sampling and bias Leads to: nothing later on the map yet Lessons: AI HL 4.12: Designing data collection
Non-linear regression IB Diploma: AI HL only Fitting exponential, power and other models, and comparing them by the sum of squared residuals. Builds on: Correlation and regression, Logistic and piecewise models Leads to: nothing later on the map yet Lessons: AI HL 4.13: Non-linear regression
Combining random variables IB Diploma: AI HL only The mean and variance of aX + b and of sums of independent random variables. Builds on: Discrete random variables Leads to: The central limit theorem Lessons: AI HL 4.14: Combining random variables
The central limit theorem IB Diploma: AI HL only Means of large samples are approximately normal, whatever the shape of the population they come from. Builds on: Combining random variables, The normal distribution Leads to: Confidence intervals Lessons: AI HL 4.15: The central limit theorem
Confidence intervals IB Diploma: AI HL only An interval for an unknown mean, and what the confidence level does and does not promise. Builds on: The central limit theorem Leads to: nothing later on the map yet Lessons: AI HL 4.16: Confidence intervals
The Poisson distribution IB Diploma: AI HL only Counting events in a fixed interval when they happen at a steady average rate. Builds on: Discrete random variables Leads to: Hypothesis tests and errors Lessons: AI HL 4.17: The Poisson distribution
Hypothesis tests and errors IB Diploma: AI HL only Type I and Type II errors, and what the significance level does and does not control. Builds on: Chi-squared tests, The binomial distribution, The Poisson distribution Leads to: nothing later on the map yet Lessons: AI HL 4.18: Hypothesis tests and errors
Markov chains IB Diploma: AI HL only Transition matrices, long-run behaviour and steady states. Builds on: Matrices, Eigenvalues and eigenvectors, Venn and tree diagrams Leads to: nothing later on the map yet Lessons: AI HL 4.19: Markov chains

Calculus

Limits and the derivative IB Diploma: AA SL and HL, AI SL and HL The gradient of a curve as the limit of the gradients of chords, and the derivative as a rate of change. Builds on: Gradient and y = mx + c, Graphs and key features Leads to: Increasing and decreasing, Differentiating powers of x, Differentiation from first principles Lessons: AA and AI 5.1: Limits and the derivative
Increasing and decreasing IB Diploma: AA SL and HL, AI SL and HL Reading where a function rises and falls from the sign of its derivative. Builds on: Limits and the derivative Leads to: Stationary points Lessons: AA and AI 5.2: Increasing and decreasing
Differentiating powers of x IB Diploma: AA SL and HL, AI SL and HL The power rule for integer powers, and differentiating a sum term by term. Builds on: Limits and the derivative, Zero and negative indices Leads to: Tangents and normals, Integration and area, Stationary points, Chain, product and quotient rules Lessons: AA and AI 5.3: Differentiating powers of x
Tangents and normals IB Diploma: AA SL and HL, AI SL and HL The equations of the tangent and the normal at a point, from the derivative and a straight line. Builds on: Differentiating powers of x, Equations of straight lines Leads to: Euler's method Lessons: AA and AI 5.4: Tangents and normals
Integration and area IB Diploma: AA SL and HL, AI SL and HL Integration as differentiation in reverse, and the definite integral as the area under a curve. Builds on: Differentiating powers of x Leads to: The trapezoidal rule, Kinematics, Integration by inspection, Areas between curves, Volumes of revolution Lessons: AA and AI 5.5: Integration and area
Stationary points IB Diploma: AA SL and HL, AI SL and HL Where the derivative is zero, telling maxima from minima, and local from global. Builds on: Increasing and decreasing, Differentiating powers of x Leads to: Maxima and minima in context, The second derivative Lessons: AI 5.6: Stationary points
Maxima and minima in context IB Diploma: AA SL and HL, AI SL and HL Building a function from a situation, then using its stationary points to find the best value. Builds on: Stationary points Leads to: nothing later on the map yet Lessons: AI 5.7: Maxima and minima in context; AA 5.8: Maxima and minima in context
The trapezoidal rule IB Diploma: AI SL and HL Estimating an area with trapeziums, and deciding whether the estimate is too high or too low. Builds on: Integration and area, Trapeziums and regular polygons Leads to: nothing later on the map yet Lessons: AI 5.8: The trapezoidal rule
Chain, product and quotient rules IB Diploma: AA SL and HL, AI HL only Differentiating composite functions, products and quotients. Builds on: Differentiating powers of x, Composite and inverse functions Leads to: Integration by inspection, L'Hôpital's rule, Implicit differentiation, Further derivatives Lessons: AI HL 5.9: Chain, product and quotient rules; AA 5.6: Chain, product and quotient rules
The second derivative IB Diploma: AA SL and HL, AI HL only Concavity, points of inflexion, and reading the graphs of f, f′ and f″ together. Builds on: Stationary points Leads to: Kinematics Lessons: AI HL 5.10: The second derivative; AA 5.7: The second derivative
Kinematics IB Diploma: AA SL and HL, AI HL only Displacement, velocity and acceleration, linked by differentiating and integrating. Builds on: Integration and area, The second derivative Leads to: nothing later on the map yet Lessons: AI HL 5.13: Kinematics; AA 5.9: Kinematics
Integration by inspection IB Diploma: AA SL and HL, AI HL only Indefinite integrals of the standard functions, and spotting the reverse of the chain rule. Builds on: Integration and area, Chain, product and quotient rules Leads to: Differential equations, Integration by parts Lessons: AI HL 5.11: Integration by inspection; AA 5.10: Integration by inspection
Areas between curves IB Diploma: AA SL and HL Definite integrals for regions below the axis and between two curves. Builds on: Integration and area Leads to: Continuous random variables Lessons: AA 5.11: Areas between curves
Volumes of revolution IB Diploma: AA HL only, AI HL only Spinning a region about an axis and adding up the circular slices. Builds on: Integration and area, Circumference and area of a circle Leads to: nothing later on the map yet Lessons: AI HL 5.12: Volumes of revolution; AA HL 5.17: Volumes of revolution
Differential equations IB Diploma: AA HL only, AI HL only Equations that involve a derivative, solved by separating the variables. Builds on: Integration by inspection Leads to: Slope fields, Euler's method, Coupled systems and phase portraits Lessons: AI HL 5.14: Differential equations; AA HL 5.18: Differential equations
Slope fields IB Diploma: AI HL only Seeing every solution of a differential equation at once from the gradient at each point. Builds on: Differential equations Leads to: nothing later on the map yet Lessons: AI HL 5.15: Slope fields
Euler's method IB Diploma: AA HL only, AI HL only Stepping along tangents to approximate a solution, and how the step size affects the error. Builds on: Differential equations, Tangents and normals Leads to: Second-order equations Lessons: AI HL 5.16: Euler's method
Coupled systems and phase portraits IB Diploma: AI HL only Two linked rates of change, with eigenvalues deciding the long-run behaviour. Builds on: Eigenvalues and eigenvectors, Differential equations Leads to: Second-order equations Lessons: AI HL 5.17: Coupled systems and phase portraits
Second-order equations IB Diploma: AI HL only Rewriting a second-order equation as a coupled system and solving it step by step. Builds on: Coupled systems and phase portraits, Euler's method Leads to: nothing later on the map yet Lessons: AI HL 5.18: Second-order equations
Differentiation from first principles IB Diploma: AA HL only Finding a derivative directly from the limit definition. Builds on: Limits and the derivative, The binomial theorem Leads to: L'Hôpital's rule Lessons: AA HL 5.12: Differentiation from first principles
L'Hôpital's rule IB Diploma: AA HL only Limits of the form 0/0 or ∞/∞ found by differentiating the top and the bottom. Builds on: Differentiation from first principles, Chain, product and quotient rules Leads to: nothing later on the map yet Lessons: AA HL 5.13: L'Hôpital's rule
Implicit differentiation IB Diploma: AA HL only Differentiating relations that are not written as y = f(x), and related rates of change. Builds on: Chain, product and quotient rules Leads to: nothing later on the map yet Lessons: AA HL 5.14: Implicit differentiation
Further derivatives IB Diploma: AA HL only Derivatives of the reciprocal and inverse trigonometric functions and of exponentials and logarithms in any base. Builds on: Reciprocal and inverse trigonometry, Chain, product and quotient rules, Partial fractions Leads to: Maclaurin series Lessons: AA HL 5.15: Further derivatives
Integration by parts IB Diploma: AA HL only Integrating a product by reversing the product rule, and harder substitutions. Builds on: Integration by inspection Leads to: nothing later on the map yet Lessons: AA HL 5.16: Integration by parts
Maclaurin series IB Diploma: AA HL only Approximating a function by a polynomial built from its derivatives at zero. Builds on: Further derivatives, Infinite geometric series Leads to: nothing later on the map yet Lessons: AA HL 5.19: Maclaurin series

How the maths connects, Year 7 to Year 13

Year 7

Number and Algebra

Functions

Geometry and Trigonometry

Statistics and Probability

Year 8

Number and Algebra

Functions

Geometry and Trigonometry

Statistics and Probability

Year 9

Number and Algebra

Functions

Geometry and Trigonometry

Statistics and Probability

Year 10

Number and Algebra

Functions

Geometry and Trigonometry

Statistics and Probability

Year 11

Number and Algebra

Functions

Geometry and Trigonometry

Statistics and Probability

Years 12 and 13: the IB Diploma

Number and Algebra

Functions

Geometry and Trigonometry

Statistics and Probability

Calculus

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