Cambridge IGCSE International Mathematics

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Number

Factors and number types

C1.1; E1.1 - Factors, multiples and primesYou can arrange 18 tiles in three different rectangles, counting a turned rectangle as the same arrangement. Why does 17 allow only one? This lesson develops the factors, multiples and prime factors in C1.1 and E1.1, so you can recognise the structure hidden inside a whole number.
C1.1, C1.5, C1.6; E1.1, E1.5, E1.6 - Integers and place valueA temperature rises from below zero to above zero: how do you describe both the temperature and the change? Number types and place value in C1.1 and E1.1 help you read quantities accurately; ordering in C1.5 and E1.5 compares them; calculations in C1.6 and E1.6 tell you how they change.
C1.1, C1.3; E1.1, E1.3 - Squares, cubes and rootsYou can build 16 dots into a square, but 16 dots do not make a complete triangular pattern with consecutive rows. What distinguishes these number patterns? C1.1 and E1.1 identify square, cube and triangle numbers; C1.3 and E1.3 use powers and roots to build quantities and work backwards.
C1.1; E1.1 - Highest common factor and lowest common multipleTwo lights flash together, then repeat every 12 seconds and every 20 seconds. When will they next flash together? Common factors and common multiples in C1.1 and E1.1 help you choose between dividing quantities into equal groups and bringing repeating events together.
C1.1; E1.1 - Rational numbers and reciprocalsA calculator displays 1.414 for a square root. Does that mean the exact root is a fraction? Rational and irrational numbers in C1.1 and E1.1 distinguish an exact value from a decimal approximation; reciprocals then find the number that multiplies a non-zero value to give 1.

Fractions and calculations

C1.4; E1.4 - Equivalent fractions and mixed numbersA measuring strip shows seven quarter-metres. How can the same length be written using whole metres and a fraction? This part of C1.4 and E1.4 develops fraction notation, equivalent fractions in simplest form, and conversion between improper fractions and mixed numbers.
C1.6; E1.6 - Adding and subtracting fractionsA bottle contains three quarters of a litre and you pour in another eighth of a litre. Why must you change the units being counted before adding the numerators? This part of C1.6 and E1.6 develops common denominators, then uses them for improper fractions, mixed numbers and negative values.
C1.6; E1.6 - Multiplying and dividing fractionsA jug contains 3/4 litre of juice. How much is 2/3 of that amount, and how many 1/8-litre cups could it fill? This lesson develops fraction multiplication and division in C1.6 and E1.6, then uses the same methods with mixed numbers and estimates.
C1.4, C1.6; E1.4, E1.6 - Decimal calculationsA bottle holds 0.6 litre and each glass holds 0.15 litre. How can dividing two numbers smaller than 1 give an answer greater than 1? This lesson connects decimal place value in C1.4 and E1.4 with signed addition, subtraction, multiplication and division in C1.6 and E1.6.
C1.4; E1.4 - Fractions, decimals and percentagesA container is 3/4 full. Another display says 0.75, and a third says 75%: could all three describe the same amount? C1.4 and E1.4 use fractions, decimals and percentages as interchangeable ways of expressing a quantity, including quantities greater than one whole.
C1.5; E1.5 - Comparing quantitiesWhich is greatest: 3/5, 0.58 or 62%? Their appearances make the comparison harder than it needs to be. C1.5 and E1.5 ask you to order quantities by size, use precise comparison symbols and extend the same reasoning to negative numbers.
C1.6; E1.6 - Order of operationsDoes 3+2×5² have the same value as (3+2)×5²? A small pair of brackets changes which calculation happens first. C1.6 and E1.6 use a common order for brackets, powers and the four operations, so integer, fraction and decimal calculations have one unambiguous value.

Indices and accuracy

C1.7; E1.7 - Integer index lawsThe powers 2³, 2² and 2¹ halve each time the index falls by 1. What values should 2⁰ and 2⁻¹ have if the pattern continues? This lesson develops the integer indices in C1.7 and E1.7, explaining how repeated multiplication leads to rules for positive, zero and negative p...
E1.7 - Fractional indices (Extended)What could it mean to multiply 64 by itself “one third of a time”? Fractional indices extend the index laws you already know so that roots can be written as powers. This Extended lesson develops reciprocal, general fractional and negative fractional indices in E1.7, using positive bases throughout.
C1.8; E1.8 - Standard formHow can 61 million and 0.000061 share the same digits without a long trail of zeros? Standard form separates a number's significant digits from its scale. C1.8 and E1.8 cover recognising this form, converting numbers and calculating with them; the hand calculation methods are Extended content.
C1.9; E1.9 - Rounding and significant figuresA distance of 6740 metres can sensibly be described as about 7000 metres, but 6.740 metres cannot. Rounding keeps a chosen level of place value while preserving the number's scale. This lesson develops the place-value, decimal-place and significant-figure rounding required by C1.9 and E1.9.
C1.9; E1.9 - Estimation and sensible accuracyA calculator reports that 48.6 kilograms at $19.70 per kilogram would cost about $96. Does the scale look right? An estimate can expose an entry error before you trust the digits. This lesson develops the calculation estimates, scale checks and sensible final accuracy required by C1.9 and E1.9.
C1.13; E1.13 - Using a graphic display calculatorWhy can two almost identical entries, (-4)² and -4², produce opposite signs? A calculator follows the expression you enter, including its grouping. This lesson develops accurate entry, retained working precision and interpretation of money and time displays in C1.13 and E1.13.

Ratio, percentages and rates

C1.10; E1.10 - Ratio and sharingTwo people share $90 in the ratio 2:3. Does the first person get two thirds of the money? Ratio compares parts, so you must first decide what each number refers to. This lesson develops simplifying ratios, sharing quantities and connecting part-to-part and part-to-whole comparisons in C1.10 a...
C1.10; E1.10 - Proportional reasoningA recipe serves four people, but six people are coming. Does adding the same extra amount to every ingredient preserve the taste? C1.10 and E1.10 use proportional reasoning to scale quantities, compare purchases fairly and turn a map distance into a real distance.
C1.12; E1.12 - Percentage amounts and comparisonsA price rises from $80 to $92, then falls back to $80. The two changes are both $12, but are their percentages equal? C1.12 and E1.12 use percentages to calculate amounts, compare quantities and describe change with a clearly identified starting amount.
C1.12; E1.12 - Percentage multipliersA shop raises a price by 10% and later reduces it by 10%. Does the price return to its starting point? C1.12 and E1.12 turn percentage changes into multipliers, apply them to purchases and earnings, and combine successive changes without losing track of the changing base.
C1.12; E1.12 - Simple and compound interestTwo accounts both advertise 5% interest per year. One adds $50 each year to a $1000 investment; the other adds a little more each year. C1.12 and E1.12 distinguish simple from compound interest, calculate interest and balances, and compare offers over the same period.
E1.12 - Reverse percentages (Extended)A sale price is $144 after a 20% reduction. Taking another 20% off cannot reveal the old price: the old price must be larger. E1.12 uses division by percentage multipliers to recover original amounts, including cost prices and values before successive changes.
E1.16 - Exponential growth and decay (Extended)A population grows by 5% each year. Does three years of growth mean a total increase of 15%? E1.16 builds exponential growth and decay models from repeated multipliers, uses them for populations and depreciation, and interprets predictions over sensible time periods.
C1.14; E1.14 - Time, clocks and timetablesA journey can finish on a different day even though it lasts less than two hours. Learn to convert time units, read clocks and timetables, and compare local times using C1.14 and E1.14. Keeping the date beside the time makes overnight journeys much easier to follow.
C1.15, C1.11; E1.15, E1.11 - Money and currency conversionIs a price of €72 cheaper than a price of $86? The digits alone cannot tell you: first put both prices into the same currency. This lesson combines money calculations in C1.15 and E1.15 with exchange-rate reasoning in C1.11 and E1.11, so you can calculate totals, change and fair price comparisons.
C1.11; E1.11 - Rates and average speedIf you travel one part of a journey at 20 km/h and another at 30 km/h, must the average be 25 km/h? Rates compare quantities with different units, and the time spent at each speed matters. Use C1.11 and E1.11 to calculate pay, flow, fuel consumption and average speeds for complete journeys.

Sets and exact values

C1.2; E1.2 - Sets and Venn diagramsOf 12 students, 7 swim, 6 cycle and 3 do both. How many do neither? Sets let you organise overlapping groups without counting anyone twice. Learn the shared C1.2 and E1.2 language of sets, unions, intersections, complements and counts using diagrams with two sets.
E1.2 - Three sets and subset notation (Extended)A person can belong to three clubs at once, and a whole set can sit inside another set. Extend the set language of E1.2 to distinguish elements from sets, use subset notation and organise three overlapping groups. You will also interpret a set defined by a rule for coordinate pairs.
E1.17 - Simplifying surds (Extended)Can a square with area 48 have an exact side length that looks simpler than √(48)? Surds keep irrational values exact rather than replacing them with rounded decimals. Develop the simplifying, adding and multiplying skills in E1.17 by recognising square factors and keeping track of like t...
E1.17 - Rationalising denominators (Extended)Can the exact number 1/(√(3)-1) be written without a square root underneath the fraction? E1.17 develops a way to change a fraction's appearance while keeping its value. You will rationalise single-term and two-term denominators, including a negative rational term.

Algebra

Expressions and manipulation

C2.1, C2.5; E2.1, E2.5 - Letters and substitutionA machine triples any number you put in, then adds 4. How can you describe every possible output in one line? C2.1 and E2.1 introduce letters and substitution; C2.5 and E2.5 use them to build expressions, equations and formulas from words.
C2.2; E2.2 - Collecting like termsTwo pieces of ribbon have lengths 4x cm and 3x cm. Their total is 7x cm, but what changes if the second length is 3y cm? In C2.2 and E2.2, you will simplify expressions by combining matching terms while keeping different variable parts separate.
C2.2; E2.2 - Expanding bracketsA rectangle is 3 cm high and (x+4) cm wide. Why can its area be written as either 3(x+4) or 3x+12? C2.2 and E2.2 develop expansion from this idea, then use it to multiply two brackets and simplify quadratic expressions.
E2.2 - Several brackets and variables (Extended)Is (2x+3)² really just 4x²+9? A small numerical check says otherwise. This Extended E2.2 lesson develops complete expansions with two variables, squared brackets and products of three brackets, keeping every cross-product and collecting powers carefully.
C2.2; E2.2 - Factorising common factorsTwo adjacent rectangles have areas 12x and 18 square units. If they share a height of 6 units, what is the width of the whole rectangle? C2.2 and E2.2 reverse expansion to extract numerical and variable common factors, then check a complete factorisation by multiplying back.
E2.2 - Factorising quadratics (Extended)Which two brackets expand to x²+7x+12? The middle coefficient and the constant give two clues, and both must fit. This Extended E2.2 lesson develops quadratic factorisation, then combines it with extracting a common factor to factorise cubic expressions fully.
E2.2 - Special factorisations (Extended)Why can x²-25 be factorised immediately, while x²+25 does not use the same pattern? This Extended E2.2 lesson recognises cancelling cross-products, repeated brackets and shared brackets hidden in four-term expressions.

Equations and formulas

C2.5; E2.5 - Forming and solving linear equationsA rectangle is 5 cm longer than it is wide, and its perimeter is 50 cm. How can one unknown describe both dimensions and lead to their values? C2.5 and E2.5 connect balanced equation solving with constructing, solving and checking models of lengths, costs and quantities.
C2.5; E2.5 - Simultaneous linear equations and modellingTwo notebooks and a pen cost $11; one notebook and three pens cost $13. How can you find both prices when neither is given? C2.5 and E2.5 develop elimination and substitution, then use two independent contextual conditions to construct and solve a pair of equations.
C2.5; E2.5 - Changing the subjectA fare is calculated by C=4+3d. If the fare is known, how can you write a rule that gives the distance instead? C2.5 and E2.5 develop rearrangement by balanced inverse operations, including simple fractions; each requested subject appears once and has no power or root in this lesson.
E2.5 - Rearranging harder formulas (Extended)If a circle's area is known, rearranging A=πr² seems to give two possible values for r. Which one can be a radius? This Extended E2.5 lesson rearranges powers and roots, collects a repeated subject, and uses the original formula and context to keep only valid values.
E2.5 - Solving quadratic equations (Extended)Two numbers can satisfy the same quadratic equation. How can you find both, and what can you do when no integer factor pair works? This Extended E2.5 lesson develops factorisation and the supplied quadratic formula, including exact answers written with surds.

Fractions and indices

C2.3; E2.3 - Simplifying algebraic fractionsIs 6x/9x equal to 2/3 for every value of x? Cancelling common factors makes an algebraic fraction shorter, but it does not remove the original denominator's restrictions. This lesson develops one-step simplification in C2.3 and E2.3, including powers and the difference bet...
E2.3 - Calculating with algebraic fractions (Extended)Why can you cancel across a product but not across a sum? Algebraic fractions follow the same arithmetic as numerical fractions, provided every denominator and divisor is valid. This Extended E2.3 lesson develops sums, products and quotients, then uses factorisation to simplify rational expressions.
E2.5 - Solving fractional equations (Extended)Can a value solve a rearranged equation but fail the original one? Clearing fractions can reveal a familiar linear or quadratic equation, but a zero denominator is never allowed. This Extended E2.5 lesson solves fractional equations with numerical and linear algebraic denominators, then checks an...
C2.4; E2.4 - Algebraic index lawsDoes x³× x² mean x⁵ or x⁶? Counting equal factors explains the answer and helps you keep coefficients, signs and letters in their correct roles. This lesson develops the integer algebraic index laws in C2.4 and E2.4 and solves equations by rewriting both sides with a common b...
E2.4 - Fractional algebraic indices (Extended)How can x^(1/3)× x^(1/2) be written as one power? Fractional indices let roots take part in the same index arithmetic as integer powers. This Extended E2.4 lesson develops products, quotients and nested rational powers, then solves common-base equations whose exponents contain the unknown.

Inequalities and sequences

C2.6; E2.6 - Inequalities on a number lineA lift allows a load of at most 500 kg. Is a load of exactly 500 kg allowed, and how can one picture show every permitted load? C2.6 and E2.6 use inequalities, endpoints and number lines to describe whole ranges of values precisely.
E2.6 - Solving linear inequalities (Extended)A club has $52 available and pays $10 to book a room plus $6 per person. What is the greatest number who can attend? E2.6 turns limits into linear inequalities, solves them with balanced operations and interprets the resulting range.
C2.7; E2.7 - Sequences and term rulesA row of one square uses 4 sticks, two adjoining squares use 7, and three use 10. Why does each new square need only three more sticks? C2.7 and E2.7 connect growing pictures, number patterns and term-to-term rules, including square, cube and triangular sequences.
C2.7; E2.7 - The nth term of a linear sequenceThe sequence 5,8,11,14,… adds 3 each time. Can you find its hundredth term without writing the other ninety-nine? C2.7 and E2.7 use position-to-term rules to describe linear sequences, calculate distant terms and decide whether a number occurs.
C2.7; E2.7 - Simple quadratic and cubic sequencesThe terms 2,5,10,17,… have gaps 3,5,7,…, so adding a fixed amount will not work. What familiar pattern lies just beneath them? C2.7 and E2.7 develop simple square- and cube-based nth terms, using differences to reveal their structure.
E2.7 - Quadratic, cubic and exponential sequences (Extended)The terms 4,11,22,37,56,… are neither shifted squares nor a constant-ratio pattern. How can you uncover the hidden rule systematically? E2.7 develops quadratic and cubic difference methods, then distinguishes exponential growth and simple combinations of sequences.
E2.8 - Direct and inverse proportion (Extended)If an input becomes four times as large, must its output also become four times as large? E2.8 uses direct and inverse proportion to describe several different answers, including square, square-root and cube relationships. You will turn these descriptions into equations and use one known pair of...
E2.8 - Choosing a variation model (Extended)Two outputs can both decrease as their inputs increase, yet follow different rules. E2.8 asks you to choose a variation model from data, rather than assume the relationship in advance. You will test for constant ratios or products, distinguish exact patterns from approximate fits, and explain wha...

Geometry

Shapes, angles and bearings

C5.1; E5.1 - Geometrical languageA square is also a rectangle: how can one shape have two correct names? The vocabulary in C5.1 and E5.1 lets you describe lines, angles, triangles, quadrilaterals and polygons precisely, using their properties rather than their appearance.
C5.1; E5.1 - Solid shapes and circle vocabularyA cylinder and a cone both have a circular base, but their shapes behave differently as you move upwards. C5.1 and E5.1 give you the language to recognise solids and describe circles, so you can tell a radius from a chord and a sector from a segment.
E5.1 - Planes and further solids (Extended)A line can cross a segment at right angles without being its perpendicular bisector. E5.1 extends your geometrical vocabulary to planes, further solids and major and minor parts of circles, so each name states the full set of conditions.
C5.2; E5.2 - Measuring lines and anglesA ruler's end may be worn, and a protractor has two number scales: which marks should you trust? C5.2 and E5.2 teach you to measure and draw straight segments and acute, obtuse and reflex angles accurately, keeping the intended orientation.
C5.5; E5.5 - Angles at points and in trianglesTwo straight lines cross, and one angle is 128°. Can you find the other three without measuring? C5.5 and E5.5 give the angle facts for points, straight lines, triangles and quadrilaterals, together with the notation and reasons that make a calculation convincing.
C5.5; E5.5 - Angles in parallel linesA straight line crosses two parallel lines: which angles must match, and which must add to 180°? C5.5 and E5.5 connect corresponding, alternate and co-interior angles to the angle facts you already know, so you can calculate and give precise reasons.
C5.5; E5.5 - Angles in polygonsA regular polygon has an interior angle of 156°. Can you work out its number of sides without drawing it? C5.5 and E5.5 connect triangle angle sums, interior angles and turning angles, giving you a reliable way to solve this problem.
E5.5 - Irregular polygons (Extended)A house-shaped pentagon can have two right angles at its base and another at the roof peak. What must the two remaining angles add to? E5.5 uses angle sums and local angle facts together, including cases where the polygon's angles are unequal.
C5.2; E5.2 - Three-figure bearingsYou travel from a harbour on a bearing of 065°. Which bearing takes you directly back? C5.2 and E5.2 give directions a precise meaning: measure clockwise from north at the starting point, draw the route, then use parallel north lines to reverse it.

Similarity, symmetry and circles

C5.1, C5.3; E5.1, E5.3 - Similar shapes and scale factorsTwo photographs show the same scene, but one is twice as wide. Must it also be twice as high to keep the same shape? C5.1 and E5.1 distinguish similarity from congruence; C5.3 and E5.3 use corresponding sides to calculate missing lengths, even when one shape is turned over or rotated.
E5.3 - Explaining similar triangles (Extended)A short line parallel to one side of a triangle cuts off a smaller triangle. Why must the smaller one have the same shape, and which side lengths can you compare? E5.3 combines geometric angle reasons with matched proportions, including triangles that share a corner or cross over one another.
E5.3 - Similar areas and volumes (Extended)A model is twice as long as another similar model. Does it need twice as much paint, and can it hold twice as much liquid? E5.3 distinguishes length, surface-area and volume scale factors, then works backwards when an area or volume ratio is given.
C5.4; E5.4 - Line and rotational symmetryA sloping parallelogram fits its outline after a half-turn, yet it may have no mirror line at all. Why are these different kinds of symmetry? C5.4 and E5.4 connect folds, turns and familiar shape properties, so you can describe exactly how a shape matches itself.
E5.4 - Symmetry of solids (Extended)A cylinder can be reflected in a flat slice or turned about a line. Which slices and lines really leave the whole solid unchanged? E5.4 extends symmetry into three dimensions, distinguishing planes of reflection from axes of rotation in prisms, cylinders, pyramids and cones.
C5.6, C5.7; E5.6, E5.7 - Right angles and tangents in circlesJoin the ends of a diameter to a third point on the circle. Why is the angle at that third point always a right angle, even when the point moves? C5.6 and E5.6 give two circle facts about right angles; C5.7 and E5.7 add the equal lengths of tangents from one external point.
E5.6 - Circle angle theorems (Extended)The same two endpoints can define an acute angle on one side of a circle and an obtuse angle on the other. How do you choose the correct angle at the centre? E5.6 connects centre and circumference angles, angles in the same segment and opposite angles of cyclic quadrilaterals, always keeping trac...
E5.6, E5.7 - Chords and the alternate segment (Extended)A tangent angle at the edge of a circle can equal an angle on the far side of the circle. How can you tell which one? E5.6 supplies the alternate segment theorem; E5.7 uses circle symmetry to relate equal chords, their distances from the centre and their perpendicular bisectors.

Mensuration

Measures and areas

C6.1; E6.1 - Metric units and conversionsA metre contains 100 centimetres. Does a square metre contain just 100 square centimetres? This lesson develops the length, mass, area, volume and capacity conversions in C6.1 and E6.1, so that you can compare measurements without changing the quantity they describe.
C6.2; E6.2 - Perimeter and areaA rectangular garden needs edging around its boundary and turf over its surface. Why do those jobs require different calculations and different units? This lesson develops perimeter and the areas of rectangles, triangles, parallelograms and trapeziums in C6.2 and E6.2, including the choice of per...
C6.3; E6.3 - Circle circumference and areaA circular table needs a strip around its edge and a cover over its top. Which measurement does each job require, and what changes if the diameter is doubled? This lesson develops circle circumference and area in C6.3 and E6.3, beginning with the radius needed in the supplied formulas.
C6.3; E6.3 - Arcs and sectorsA circular cake is divided into six equal slices. How much curved edge and how much area belong to one slice, and why is its whole boundary longer than its curved edge? This lesson develops fractions of circles, arc length, sector area and sector perimeter in C6.3 and E6.3.
E6.3 - Major and general sectors (Extended)Two radii bound both a short route and a long route around a circle. Which angle and fraction belong to the region you actually want? This Extended lesson develops E6.3 for unrestricted degree angles, major sectors and calculations that recover a missing angle or radius.
C6.5; E6.5 - Compound areas and perimetersA rectangular sign has a semicircle attached at one end. The joining diameter helps you calculate area, but should it be included in the outside perimeter? This lesson develops the compound shapes and parts of shapes in C6.5 and E6.5 by choosing regions for area and tracing the actual boundary fo...

Surface areas and volumes

C6.4; E6.4 - Prisms and cylindersA block and a round column look different, but both can be built by repeating the same cross-section. C6.4 and E6.4 use that shared structure to calculate prism and cylinder volumes. You will identify the area that repeats, choose the correct length and keep radius separate from diameter.
C6.4; E6.4 - Pyramids and conesA pointed container holds less than a straight-sided container with the same base and height. C6.4 and E6.4 use the one-third relationship to calculate pyramid and cone volumes. You will choose the perpendicular height, keep exact multiples of π and work backwards to find a missing dimension.
C6.4; E6.4 - Surface area of solidsA container needs enough material to cover its faces; leaving its top open saves material even when its side dimensions stay the same. C6.4 and E6.4 develop surface areas of prisms, pyramids, cylinders and cones. You will use nets, count exposed faces and select the height or sloping edge that ea...
C6.4, C6.5; E6.4, E6.5 - Spheres and parts of solidsCutting a solid in half halves its volume, but does it halve its total surface area? C6.4 and E6.4 supply the sphere formulas; C6.5 and E6.5 use them for parts and combinations of solids. You will separate volume from exposed area, including the extra flat face made by a cut and the hidden faces...
E6.5 - Frustums and similar solids (Extended)A cone with its tip removed still has part of the original curved surface. E6.5 uses similarity to recover the missing tip, then calculates a frustum's volume and exposed area by subtraction. You will distinguish the full cone, the small removed cone and the remaining solid before using their dim...

Trigonometry

Right-angled triangles

C7.1; E7.1 - Pythagoras theoremA rectangular panel is 9 cm by 12 cm. Is a 15 cm strip long enough to run exactly between opposite corners? C7.1 and E7.1 connect the three sides of a right-angled triangle, letting you calculate a missing length and check whether an answer fits the triangle.
C7.1; E7.1 - Pythagoras in circles and coordinatesA chord lies 5 cm from the centre of a circle of radius 13 cm. Can you find its full length from one right triangle? C7.1 and E7.1 use Pythagoras' theorem inside circles and on grids, where the useful perpendicular sides first need identifying.
C7.2; E7.2 - Finding sides with trigonometryA 10 m cable makes an angle of 38° with a straight baseline in a right-angled triangle. How far does it reach across that baseline? C7.2 and E7.2 connect acute angles to side ratios, so one known side and an angle can determine another side.
C7.2; E7.2 - Finding angles with trigonometryA right triangle has perpendicular sides 6 cm and 11 cm. How can you find its acute angles without measuring a drawing? C7.2 and E7.2 use inverse trigonometric functions to recover angles from side ratios and combine these with Pythagoras in two-dimensional problems.
C7.2; E7.2 - Trigonometry with bearingsA boat travels 12 km north and then 9 km east. What bearing should it take to return directly to its starting point? C7.2 and E7.2 use right triangles to connect journey lengths, acute angles and the clockwise bearings used on a map.
E7.2 - Elevation, depression and shortest distance (Extended)An observer is 18 m above level ground, 30 m horizontally from a tower, and sees its top at an angle of elevation of 34°. Is the tower's height 3034°, or is something missing? E7.2 connects horizontal sight lines, perpendicular distances and linked right triangles.

Further trigonometry

E7.3 - Exact trigonometric values (Extended)A calculator displays a decimal for the height of a triangle, but can you give the height without rounding at all? Special triangles give exact sine, cosine and tangent values. You will derive the values, recognise their patterns and use them in calculations that keep square roots exact.
E7.5 - The sine rule (Extended)Two sides and an angle can sometimes describe two different triangles. The sine rule lets you calculate in triangles without a right angle, but you must match each side to its opposite angle and check which triangles are possible. You will find sides, find acute or obtuse angles and resolve the a...
E7.5 - The cosine rule (Extended)Two paths of lengths 7 m and 10 m start at the same point. How far apart are their endpoints when the angle between the paths is 50 degrees, and what changes if it is 120 degrees? The cosine rule connects two sides and their included angle, or three sides and an unknown angle. You will also decid...
E7.5 - Area using two sides and an angle (Extended)Can two triangles have the same two side lengths and the same area, but different included angles? A perpendicular height connects the familiar triangle-area formula to sine. You will use that connection for acute and obtuse triangles, reverse it to find missing information and combine it with th...
E7.6 - Pythagoras and trigonometry in 3D (Extended)An 8 cm by 6 cm by 5 cm box has a 10 cm diagonal across its base. How long is the diagonal through its interior, and how steeply does it rise above the base? You will find useful plane triangles inside solids, link Pythagoras calculations and measure the angle between a line and a plane using a p...

Coordinate geometry

Points and straight lines

C4.1; E4.1 - Cartesian coordinatesDoes (-3,2) name the same place as (2,-3)? Coordinates give a precise address on a flat grid, but the order and scale both matter. C4.1 and E4.1 develop reading and placing points in all four quadrants, including grids with fractional and decimal scales.
C4.2; E4.2 - Gradient from a graphTwo lines can rise by the same amount and still have different steepness. What else must you compare? C4.2 and E4.2 measure gradient by comparing vertical change with horizontal change on a grid, then use its sign and a well-chosen pair of points to describe a straight line accurately.
E4.2 - Gradient from two points (Extended)A line passes through (-2,-1) and (4,2), but no grid is supplied. Can you still measure its gradient? E4.2 turns the rise-and-run idea into a coordinate calculation, explains consistent subtraction and uses a known gradient to recover a missing coordinate.
C4.4; E4.4 - Equations of straight linesA line crosses the y-axis at 1 and rises 2 units for every unit to the right. How can one equation describe every point on it? C4.4 and E4.4 connect a straight-line graph's gradient and intercept to its equation, including horizontal and vertical lines.
E4.4 - Different forms of a line (Extended)Do 3x+2y=8 and y=-3/2x+4 describe different lines? E4.4 uses rearrangement to reveal a line's gradient and intercept, then builds an equation from a point and a gradient or from two points. You will give a complete, simplified equation and check the original information.
C4.3; E4.3 - Midpoints and lengthsThe endpoints of a segment are (-4,1) and (2,9). Where is its centre, and how long is it? C4.3 and E4.3 use coordinate averages for a midpoint and Pythagoras' theorem for a length, then work backwards when an endpoint is missing.
C4.5; E4.5 - Parallel straight linesTwo straight tracks have equations y=2x+1 and y=2x-5. Can they ever meet? C4.5 and E4.5 connect parallel lines with equal gradients, then use a point to locate the particular parallel line you need.
E4.6 - Perpendicular lines and bisectors (Extended)A line rises 2 units for every 3 units to the right. What direction would cross it at a right angle? E4.6 develops perpendicular gradients and equations, then combines them with a midpoint to find a perpendicular bisector.
E2.6 - Inequalities in two variables (Extended)A point must lie to the right of x=1, above y=0 and below x+y=6. How can one picture communicate all three conditions, including which edges count? This part of E2.6 turns linear inequalities into regions and reads inequalities back from a diagram.

Functions

Function notation and graphs

C3.3; E3.3 - Function notationA rule squares its input. Both 3 and -3 produce 9: can it still be a function? C3.3 and E3.3 introduce functions, mapping diagrams and notation so you can read a rule and evaluate its output accurately.
C3.1; E3.1 - Linear and quadratic graph shapesTwo graphs pass through (0,1). Must they have the same shape? C3.1 and E3.1 distinguish straight-line and quadratic functions, then use a graph to read an output or find the inputs giving a specified output.
C3.2; E3.2 - Graphs, tables and windows on a calculatorA calculator screen looks empty after you enter y=(x-12)²-4. Is there really no graph? C3.2 and E3.2 use graph entry, tables and point plotting to choose a useful view and transfer the mathematics clearly to paper.
C3.2; E3.2 - Zeros, vertices and intersectionsA graph can reveal where a function is zero, where it turns and where it meets another graph. Learn to locate these features with a graphic display calculator, then check that its coordinates answer the question. These are the shared C3.2 and E3.2 graph skills.
C2.5, C3.2; E2.5, E3.2 - Solving equations graphicallyAn equation asks where two expressions have equal values. C2.5 and E2.5 connect this question to the C3.2 and E3.2 calculator graph tools: use intersections or zeros, including for unfamiliar functions. Learn to show the graphical method and check that every answer belongs to the stated domain.
E2.6 - Solving inequalities with graphs (Extended)An inequality usually has an interval of solutions rather than a few isolated answers. Use permitted GDC graph features to find equality boundaries, decide which graph is higher, and include or exclude endpoints correctly. Check restrictions carefully when a graph has separate branches.
E3.3 - Domain and range (Extended)A function needs an allowed set of inputs as well as a rule. Learn to describe its domain, read the outputs actually reached, and track included or excluded endpoints. The same rule can have different ranges when its domain changes.
E3.1, E3.5 - Cubic and reciprocal graphs (Extended)A cubic curve can flatten or turn; a reciprocal graph has separate branches. Recognise these E3.1 families, use labelled points to determine unknown values, and identify the simple axis-parallel asymptotes in E3.5 directly from graphs. The shape helps you decide whether a calculator display makes...
E3.1 - Exponential graphs (Extended)Equal input steps give equal output factors on an exponential graph. Recognise growth and decay, recover a base or coefficient from labelled points, and describe the graph's long-term behaviour. Reciprocal bases also reveal a useful reflection symmetry.

Trigonometric and transformed functions

E3.1, E3.5, E7.4 - Trigonometric graphs (Extended)A rotating arm returns to its starting position after one full turn. How do its height and horizontal position change during that turn? E3.1 and E7.4 connect those changing positions to sine, cosine and tangent graphs, including amplitude and period. E3.5 explains the breaks in the tangent graph.
E7.4 - Trigonometric equations (Extended)A calculator gives 30° when you reverse x=1/2, but the sine graph reaches the same height twice in one turn. Where is the other solution? You will use graph symmetry, rearrangement and endpoint checks to find every solution between 0 and 360 degrees.
E3.3 - Inverse functions (Extended)A function multiplies an input by 3 and subtracts 4. If its output is 11, how can you recover the input without guessing? You will reverse input-output pairs, find inverse formulas and check that the inverse really undoes the original function.
E3.3 - Composite functions (Extended)Start with 4. Does “add 3, then double” give the same result as “double, then add 3”? Combining two function rules makes their order visible. You will interpret composite notation, substitute a whole expression into another rule, and simplify composites that contain fractions.
E3.4 - Finding a quadratic function (Extended)Many parabolas have the same lowest point. What extra information picks out just one of them? You will choose a useful quadratic form from a vertex or roots, use another point to find its coefficient, and recognise when a leading coefficient of 1 already fixes the rule.
E3.6 - Translating function graphs (Extended)Why does adding 2 outside a function move its graph up, while adding 2 inside moves it left? You will track individual points to explain vertical and horizontal translations, describe their vectors, and write equations for translated graphs.
E3.7 - Logarithms and exponential equations (Extended)How long does an investment take to grow by half when the same percentage is added each year? E3.7 introduces logarithms as the inverse of exponentials, then uses them to find unknown powers and interpret growth and decay times. Graphs provide a second way to see and check the solutions.

Transformations and vectors

Transforming shapes

C8.1; E8.1 - ReflectionsA point is 3 squares from a mirror line. How do you place its image so that folding along the line would match the two points? C8.1 and E8.1 use equal perpendicular distances to reflect points and shapes, then describe a single reflection precisely using its mirror line.
E8.1 - Reflections in oblique lines (Extended)A sloping mirror still puts each image directly across a perpendicular, rather than directly left, right, up or down. E8.1 extends reflection to any straight mirror line. You will construct images, join their vertices accurately and use midpoint and right-angle checks to distinguish a reflection...
C8.1; E8.1 - RotationsA triangle turns through a quarter-turn, but one corner remains fixed. Where must the turn have been centred? C8.1 and E8.1 teach rotations through multiples of 90 degrees and the centre, angle and direction needed to describe them. Core uses the origin, shape vertices or edge midpoints as centre...
C8.1; E8.1 - EnlargementsCan an enlargement make a triangle smaller? C8.1 and E8.1 use a centre and a positive scale factor to place every image vertex, including factors between zero and one. You will draw accurate images and recover the centre and scale factor from a pair of shapes.
E8.1 - Negative enlargements (Extended)An image can be twice as large yet appear across the centre from its object. E8.1 uses a negative scale factor to combine a change of size with this reversal of centre rays. You will draw negative enlargements, work with negative fractions and recover the signed factor and centre from a mapping.
C8.1; E8.1, E8.2 - Translations and column vectorsHow can one instruction move every corner of a shape without turning or resizing it? C8.1 and E8.1 describe a translation using a column vector; E8.2 uses the same displacement idea in vector notation. You will read the movement, draw an image and describe a complete translation with the signs in...
E8.1 - Combining and reversing transformations (Extended)Reflecting a shape and then turning it can give a different result from turning it first. E8.1 develops combinations and reverse transformations. You will keep track of intermediate images, undo a sequence in the correct order and decide when the final effect has one complete transformation descr...

Probability

Combined events

C9.3; E9.3 - Combined events and sample spacesTwo coins can give no heads, one head or two heads. Are those three descriptions equally likely? C9.3 and E9.3 use systematic sample spaces to keep track of individual combined outcomes, count favourable results and understand repeated selections with replacement.
C9.3; E9.3 - Probability tree diagramsA bag contains 6 red and 4 blue counters. If you draw twice with replacement, is “one of each colour” the same event as “red then blue”? C9.3 and E9.3 organise the stages in a tree, calculate one path and combine all paths belonging to an event.
E9.3 - Events without replacement (Extended)After a red counter is taken out and kept, is another red counter more or less likely? E9.3 develops two-stage selections without replacement: update the collection on each branch, multiply the correct path probabilities and combine the paths that answer the question.
E9.1, E9.3 - Probability from Venn diagrams (Extended)If two club lists each include the same student, can you add their sizes to find the chance of selecting a club member? E9.1 and E9.3 use probability notation with Venn diagrams, account for overlapping events and distinguish independent events from mutually exclusive ones.

Statistics

Data and displays

C10.1, C10.3; E10.1, E10.3 - Classifying and tabulating dataA club records how many people attend and how long each person stays. Both records are numbers, but they describe different kinds of quantity. C10.3 and E10.3 distinguish these data types; C10.1 and E10.1 organise observations into frequency and two-way tables so patterns can be read without losi...
C10.2, C10.6; E10.2, E10.6 - Bar charts and pictogramsA chart should make a comparison easier, but a missing scale or key can change its meaning. C10.6 and E10.6 develop ordinary, composite and dual bar charts and pictograms. C10.2 and E10.2 use their labels, scales and keys to recover counts and make supported comparisons.
C10.6; E10.6 - Pie charts and stem-and-leaf diagramsA pie chart makes each category's share of a total visible; a stem-and-leaf diagram keeps the individual numerical values visible. C10.6 and E10.6 use both displays, including calculating pie-chart angles, recovering frequencies and ordering leaves with a clear key.
C10.6; E10.6 - Frequency distributionsA list shows individual observations; a frequency distribution shows where observations are concentrated. C10.6 and E10.6 develop tables and simple diagrams for separate values and clearly defined groups. You will read frequencies using the actual scale and explain what grouping does and does not...

Averages and spread

C10.4; E10.4 - Mean, median and modeTwo groups can have the same total but quite different typical values. C10.4 and E10.4 develop the mean, median and mode for lists and ungrouped frequency tables, then distinguish what each average tells you. These hand methods use exact individual values rather than grouped intervals.
C10.4; E10.4 - Quartiles and measures of spreadTwo sets of times can have the same median while one set is much less consistent. C10.4 and E10.4 use the range, quartiles and interquartile range to describe spread as well as centre. The hand calculations here use individual values or ungrouped frequency tables.
E10.4 - Grouped averages (Extended)A table says that eight journeys lasted between 10 and 20 minutes, but it does not give their individual times. How can you estimate their contribution to an average? E10.4 uses class midpoints to estimate a grouped mean and identifies the modal class without pretending the original measurements...
C10.5; E10.5 - Statistics on a calculatorA calculator can give a precise-looking mean for the wrong list. How can you tell whether it has understood your frequency table? C10.5 and E10.5 connect data entry to mean, median and quartiles, then use midpoints to estimate a grouped mean on a graphic display calculator.

Relationships and conclusions

C10.7; E10.7 - Scatter diagrams and lines of best fitTwo quantities can move together without following an exact rule. Can a line through scattered observations still make a useful prediction? C10.7 and E10.7 connect paired points, correlation, a line of best fit through the mean point and the limits of predictions.
E10.7 - Linear regression on a calculator (Extended)Two sensible hand-drawn best-fit lines can give slightly different predictions. How can a calculator use every paired observation to produce one specified line? E10.7 introduces permitted linear regression on a graphic display calculator, then interprets its equation and predictions in context.
E10.8 - Cumulative frequency, quartiles and percentiles (Extended)A frequency table counts observations in separate intervals. How can you find the time below which three quarters of all journeys lie? E10.8 accumulates those counts, draws a smooth cumulative frequency curve and uses it to estimate the median, quartiles, interquartile range and percentiles.
C10.2; E10.2 - Comparing data and judging conclusionsTwo groups can have the same average while behaving very differently. What evidence would make “more reliable” a fair description of one group? C10.2 and E10.2 combine averages, spread, tables and graphs to make supported comparisons and recognise restrictions on conclusions.

Investigation and modelling

Investigating patterns

AO1, AO2 - Systematic investigationsTwenty-four square tiles can make several rectangles with the same area. Do they need the same length of edging? AO1 and AO2 connect organised cases, tables and graphs to a precise conjecture, with enough working for someone else to follow your investigation.
C2.7; E2.7; AO2 - From patterns to general rulesThe totals 7,10,15,22,31,… do not have a constant gap. Can a picture explain a rule that differences only suggest? C2.7 and E2.7 connect related sequences, while AO2 develops a general statement, tests it and repairs it when a counterexample appears.
AO2 - Explaining an investigationA border around a square gets longer as the central square grows. How can you explain its area without drawing every stage? AO2 brings together choosing mathematics, reasoning beyond examples and communicating a conclusion whose certainty matches the evidence.