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300 lessons, Year 1 to A level

Every lesson in Mathfinch, by school year. The year is where a topic is usually first met in England; any learner can open any lesson. Open the app to start one.

Primary Years 1 to 6 · 151 lessons

Year 1

  • Sharing into equal groups

    Share a group of objects equally and say how many are in each group.

    Number

  • Halves and quarters

    Know that a half is one of two equal parts, and a quarter is one of four equal parts.

    Fractions

  • Comparing and measuring: longer, heavier, fuller

    Compare and describe lengths, heights, masses, capacities and times using words such as longer, taller, heavier, full, half full, quicker and earlier; solve simple problems with double and half; and measure with everyday units (cubes, marbles, cups, a stopwatch), recording how many.

    Measurement

  • First, next, yesterday, tomorrow: ordering events and days

    Put events in order using first, next, before and after; use today, yesterday and tomorrow with the days of the week; say whether things happen in the morning, afternoon or evening; know the days of the week and months of the year in order, and that there are 7 days in a week and 12 months in a year.

    Measurement

  • Naming 2-D and 3-D shapes

    Recognise and name squares, rectangles (a square is a special rectangle), circles and triangles, and cubes, cuboids, pyramids, spheres, cylinders and cones, whichever way round they are drawn and whatever their size.

    Geometry

  • Where is it? Position, direction and turns

    Describe where things are (above, below, left, right, next to, between), follow movements forwards and to the left and right, and make whole, half, quarter and three-quarter turns.

    Geometry

  • Counting to 100 and beyond

    Count forwards and backwards to and across 100 from any number, count in 2s, 5s and 10s, read and write numbers to 100 in numerals, and find one more and one less.

    Number

  • Numbers to 20: showing, comparing and writing in words

    Show and find numbers to 20 with pictures and on a number line, use the words equal to, more, fewer, most and least, and read and write numbers from 1 to 20 in numerals and in words.

    Number

  • Number bonds to 10 and 20

    Know and use number bonds within 20 and the subtraction facts that go with them, including adding and subtracting 0, and write a related subtraction fact as a number sentence.

    Number

  • Adding and subtracting to 20: number sentences and problems

    Add and subtract one- and two-digit numbers to 20 (including 0), write a number sentence with +, − and = for a story, match a number sentence to its story, and solve missing-number problems such as 7 = ☐ − 9.

    Number

  • Equal groups, arrays and sharing

    Solve one-step multiplication and division problems from pictures: count equal groups, count an array by its rows, share equally, and find how many groups.

    Number

Year 2

  • Money: coins, notes, totals and change

    Know the value of each coin and note; use the symbols £ and p (100p = £1); find the total of a set of coins; find different sets of coins that make the same amount; add and subtract money of the same unit (pence with pence, pounds with pounds) and give change.

    Measurement

  • Telling the time: clocks, a.m. and p.m., 24-hour times and durations

    Read analogue clocks (numbered 1–12 or I–XII) to the hour, half hour, five minutes and nearest minute; use o'clock, past, to, a.m., p.m., noon and midnight; convert between 12-hour and 24-hour times; order intervals of time; find how long an event lasts; know the days of the week and months of the year in order.

    Measurement

  • Writing times and setting clocks

    Write the time shown on an analogue clock (numbers or Roman numerals) as hours and minutes; write times given in words as digital times; set both hands of a clock to a time; and write times on the 12-hour clock (with a.m. or p.m.) and the 24-hour clock.

    Measurement

  • Sides, vertices and symmetry of 2-D shapes

    Count the sides and vertices (corners) of 2-D shapes in any orientation, name shapes by their number of sides (triangle, quadrilateral, pentagon, hexagon, octagon), and say whether a vertical line is a line of symmetry.

    Geometry

  • Faces, edges and vertices of 3-D shapes

    Count the faces, edges and vertices of cubes, cuboids, pyramids and prisms (including the hidden ones), name the 2-D shapes of their faces, and compare and sort 3-D shapes by their properties.

    Geometry

  • Patterns, straight moves and right-angle turns

    Continue and complete repeating patterns of shapes (including a shape turning a quarter turn each time); describe movement in straight lines; and describe turns as quarter, half, three-quarter and whole turns and as right angles, clockwise and anticlockwise.

    Geometry

  • Tally charts, tables and sorting into groups

    Count objects into groups, read and make tally charts (in gates of five), read and complete simple tables, put groups in order from most to fewest, and choose questions the data can answer.

    Statistics

  • Pictograms and block diagrams

    Read and make pictograms where one symbol stands for 1 or for 2, read and make block diagrams, and answer questions about totals and differences.

    Statistics

  • Counting in 2s, 3s, 5s and 10s

    Count in steps of 2, 3 and 5 from 0, and in tens from any number, forwards and backwards; find a missing number in a counting sequence and where a number of jumps lands.

    Number

  • Tens and ones: place value to 100

    Know the value of each digit in a two-digit number, partition numbers in different ways, read, place and estimate numbers on a number line, compare with <, > and =, put numbers in order, and read and write numbers to 100 in numerals and words.

    Number

  • Addition and subtraction facts, and the inverse

    Recall addition and subtraction facts to 20, use them to work out facts with tens to 100, know that addition can be done in any order but subtraction cannot, and use adding to check subtracting (and the other way) and to find missing numbers.

    Number

  • Adding and subtracting two-digit numbers

    Add and subtract a two-digit number and ones, a two-digit number and tens, and two two-digit numbers within 100, add three one-digit numbers, and solve problems with quantities, lengths and money.

    Number

  • The 2, 5 and 10 times tables, odd and even

    Recall and use multiplication and division facts for the 2, 5 and 10 times tables, recognise odd and even numbers, write multiplication and division number sentences for arrays and sharing, know that multiplication (not division) can be done in any order, and solve problems.

    Number

  • Thirds and quarters of shapes, sets and lengths

    Recognise, find, name and write 1/3, 1/4, 2/4 and 3/4 of a shape, a set of objects, a length and a quantity; write simple fraction statements such as 1/2 of 6 = 3; and know that 2/4 is the same as 1/2.

    Fractions

Year 3

  • Naming fractions

    Write a fraction to show how many equal parts are shaded out of the whole.

    Fractions

  • A unit fraction of an amount

    Find a unit fraction of an amount, like 1/4 of 20, by dividing into equal parts.

    Fractions

  • Fractions on a number line

    Place and read fractions between 0 and 1 on a number line.

    Fractions

  • Comparing fractions: same denominator, and unit fractions

    Compare fractions with the same denominator, and compare unit fractions.

    Fractions

  • Adding and subtracting fractions with the same denominator

    Add and subtract fractions with the same denominator, like 2/7 + 3/7.

    Fractions

  • Tenths

    Count up and down in tenths, and know that dividing by 10 makes tenths.

    Decimals and percentages

  • Place value in three-digit numbers

    Know what each digit in a three-digit number is worth, read and write numbers to 1000, compare them, and find 10 or 100 more or less.

    Number

  • Times tables: 2, 3, 4, 5, 8 and 10

    Know the multiplication and division facts for the 2, 3, 4, 5, 8 and 10 times tables, and use them to solve problems.

    Number

  • Adding and subtracting with columns

    Add and subtract numbers with up to three digits using column methods, add and subtract ones, tens and hundreds in your head, and check with the inverse.

    Number

  • Multiplying a two-digit number by a one-digit number

    Multiply a two-digit number by a one-digit number, by partitioning and then with the formal written method, and use it to solve scaling problems.

    Number

  • Units, scales and comparing measures

    Choose sensible units (mm, cm, m, km; g, kg; ml, l; °C) and estimates; read rulers, measuring jugs, thermometers and weighing scales where the small steps are not 1; compare and order measures, including in different units; add and subtract measures.

    Measurement

  • Perimeter

    Find the perimeter of 2-D shapes by adding every side: simple shapes, shapes on a centimetre grid, rectangles and squares (using equal sides), and composite rectilinear shapes, working out missing sides; in cm and m.

    Measurement

  • Money: pounds and pence

    Write amounts in pounds and pence (135p = £1.35); find the total of coins and notes in pounds and pence; add and subtract amounts using both £ and p, and give change, in practical contexts.

    Measurement

  • 3-D shapes turned round, and drawing 2-D shapes

    Recognise cubes, cuboids, pyramids, prisms, cylinders, cones and spheres however they are turned (on their side, upside down, tipped), describe them by their faces, and draw squares, rectangles and right-angled triangles on a square grid.

    Geometry

  • Right angles and turns

    Recognise an angle as the corner of a shape and as an amount of turn; identify right angles in shapes in any orientation; know that 2 right angles make a half turn, 3 a three-quarter turn and 4 a complete turn; and say whether an angle is greater or less than a right angle.

    Geometry

  • Horizontal, vertical, parallel and perpendicular lines

    Identify horizontal and vertical lines, and pairs of parallel lines (always the same distance apart, never meeting) and perpendicular lines (meeting or crossing at a right angle), in any position and in shapes.

    Geometry

  • Pictograms and bar charts with scales

    Read and make pictograms where one symbol stands for 2, 5 or 10 (including half symbols), read and make bar charts with scales going up in 2s, 5s and 10s, and record chart data in a table.

    Statistics

  • How many more? How many fewer? Problems with charts

    Answer one-step and two-step questions — how many more, how many fewer, how many altogether, how many did not — using scaled bar charts, pictograms and tables.

    Statistics

  • Showing equivalent fractions and fraction families

    Show that two fractions are equivalent by shading a diagram (3/4 of a bar in 8 parts is 6 parts), and continue families of equivalent fractions such as 1/3 = 2/6 = 3/9.

    Fractions

  • Putting fractions in order

    Put unit fractions in order (the bigger the denominator, the smaller the part) and put fractions with the same denominator in order.

    Fractions

  • Fraction problems

    Solve word problems using Year 3 fractions: fractions of a set, adding and subtracting parts of a whole, comparing fractions, equivalent fractions and tenths.

    Fractions

  • Writing multiplication and division sentences

    Write multiplication and division number sentences for arrays, for sharing and grouping, and as related facts, using the 3, 4 and 8 times tables (and 2, 5 and 10).

    Number

Year 4

  • Equivalent fractions

    Find fractions that are equal in value, like 1/2 = 2/4 = 4/8.

    Fractions

  • Dividing by 10 and 100

    Divide one- and two-digit numbers by 10 and 100, and say what each digit in the answer is worth.

    Decimals and percentages

  • Comparing and ordering decimals

    Compare and order decimals, including ones with different numbers of decimal places, like 0.5 and 0.45.

    Decimals and percentages

  • Rounding decimals

    Round decimals to the nearest whole number and to one decimal place.

    Decimals and percentages

  • Place value in four-digit numbers

    Know what each digit in a four-digit number is worth, compare and order numbers beyond 1000, find 1000 more or less, and count in 6s, 7s, 9s, 25s and 1000s.

    Number

  • Rounding to the nearest 10, 100 or 1000

    Round any number up to 9999 to the nearest 10, 100 or 1000, including numbers exactly halfway, and use rounding to describe amounts.

    Number

  • Counting back through zero

    Count backwards through zero into negative numbers, read them on a number line, and use them for temperatures and floors below ground.

    Number

  • Roman numerals to 100

    Read Roman numerals from I to C, and know that our number system is different because it has a zero and place value.

    Number

  • Column addition and subtraction with four digits

    Add and subtract numbers with up to four digits using column methods, check answers with an estimate and with the inverse, and choose the right operations for two-step problems.

    Number

  • Times tables up to 12 × 12

    Recall the multiplication and division facts for every times table up to 12 × 12, and use factor pairs and the fact that 7 × 12 = 12 × 7.

    Number

  • Multiplying and dividing in your head

    Multiply and divide in your head using place value and facts you know (60 × 3, 240 ÷ 6), multiply by 0 and 1, divide by 1, and multiply three numbers together.

    Number

  • Multiplying by a one-digit number: formal written method

    Multiply two-digit and three-digit numbers by a one-digit number using the formal written method, and use multiplication to solve scaling, correspondence and multiply-then-add problems.

    Number

  • Converting units of measure and time

    Know the time facts (60 minutes in an hour, 24 hours in a day, days in each month and in a leap year) and convert between units in whole numbers: km and m, m and cm, cm and mm, kg and g, l and ml; hours, minutes and seconds; days, weeks, months and years, including in problems.

    Measurement

  • Area: counting squares and rectangles

    Find area by counting squares; calculate the area of rectangles and squares (length × width) in cm² and m²; find a missing side from the area; compare areas; and estimate the area of an irregular shape by counting whole squares and squares at least half covered.

    Measurement

  • Classifying quadrilaterals, triangles and angles

    Name and classify quadrilaterals (square, rectangle, rhombus, parallelogram, trapezium, kite) and triangles (equilateral, isosceles, scalene, right-angled) from their properties in any orientation, compare shapes by size, identify acute and obtuse angles, and order angles up to two right angles.

    Geometry

  • Lines of symmetry, and completing symmetric figures

    Identify lines of symmetry in 2-D shapes drawn in any orientation (including slanting lines), count them, and complete a symmetric figure on a square grid by reflecting it in a given mirror line.

    Geometry

  • Coordinates and translations (first quadrant)

    Read and write the coordinates of a point in the first quadrant as (x, y), across then up; plot points; describe a translation as squares left or right and up or down; and plot the missing vertex to complete a polygon.

    Geometry

  • Bar charts and time graphs: counting and measuring

    Read bar charts with scales in 20s, 25s, 50s and 100s; read time graphs of measurements (temperature, height, water) and find changes; build both on screen; and tell counted (discrete) data from measured (continuous) data to choose a suitable graph.

    Statistics

  • Comparison, sum and difference problems with charts

    Solve comparison, sum and difference problems from bar charts, pictograms (including quarter symbols), tables and time graphs, and spot when a bar chart's axis doesn't start at 0, so bar lengths can mislead.

    Statistics

  • Adding and subtracting fractions beyond one whole

    Add and subtract fractions with the same denominator when the answer goes beyond one whole, subtract from a whole or a mixed number, and write a total greater than 1 as a mixed number (2/5 + 4/5 = 6/5 = 1 1/5).

    Fractions

  • Where hundredths come from

    Recognise that hundredths arise when one whole is divided into 100 equal parts and when a tenth is divided into 10; divide tenths by 10; and count in hundredths across a tenth.

    Decimals and percentages

  • Money and measures with fractions and decimals

    Solve simple money and measure problems with fractions and decimals to two decimal places: a fraction of £1 or 1 m, change from a note, and adding or subtracting lengths and amounts of money.

    Decimals and percentages

Year 5

  • Mixed numbers and improper fractions

    Convert between improper fractions like 11/4 and mixed numbers like 2 3/4.

    Fractions

  • Any fraction of an amount

    Find a non-unit fraction of an amount, like 3/4 of 20, and use it to solve problems.

    Fractions

  • Fractions and decimals

    Write tenths and hundredths as decimals, and know decimal equivalents like 1/4 = 0.25.

    Decimals and percentages

  • Thousandths and place value

    Read, write and use numbers with up to three decimal places, and say what each digit is worth.

    Decimals and percentages

  • Adding and subtracting decimals

    Add and subtract decimals with up to three decimal places, lining up the decimal points.

    Decimals and percentages

  • Place value up to 1,000,000

    Read, write, order and compare numbers up to 1,000,000, know what each digit is worth, and count forwards or backwards in steps of 10, 100, 1000, 10,000 and 100,000.

    Number

  • Rounding numbers up to 1,000,000

    Round any number up to 1,000,000 to the nearest 10, 100, 1000, 10,000 or 100,000, including halfway numbers and rounding past a 9.

    Number

  • Negative numbers in context

    Count forwards and backwards through zero in different steps, find the difference between a negative and a positive number, and say what negative numbers mean in temperatures, depths and money.

    Number

  • Roman numerals to 1000, and years

    Read Roman numerals up to M (1000), including D (500) and the pairs CD (400) and CM (900), and read years written in Roman numerals.

    Number

  • Adding and subtracting big numbers

    Add and subtract numbers with more than four digits using column methods, use rounding to check answers, decide how accurate an answer needs to be, and solve multi-step problems.

    Number

  • Adding and subtracting big numbers in your head

    Add and subtract increasingly large numbers mentally: multiples of 10, 100, 1000 and 10,000; nearly-round numbers like 9,999; and complements to 1000, 10,000 and 100,000.

    Number

  • Factors and multiples

    Tell factors from multiples, find all the factor pairs of a number, and find common factors of two numbers.

    Number

  • Prime numbers

    Know what prime, composite and prime factor mean, decide whether a number up to 100 is prime, and recall the primes up to 19.

    Number

  • Square and cube numbers

    Recognise square numbers and cube numbers, use the notation ² (squared) and ³ (cubed), and use them to solve problems.

    Number

  • Long multiplication, and multiplying 4-digit numbers

    Multiply numbers up to 4 digits by a one-digit number with short multiplication, and by a two-digit number with long multiplication.

    Number

  • Multiplying and dividing in your head with facts you know

    Multiply and divide mentally by drawing on known facts: scale a times-table fact by 10, 100 and 1000, derive new facts from one you are given, double and halve, and adjust from a nearby easy calculation.

    Number

  • Short division, and what to do with remainders

    Divide numbers up to 4 digits by a one-digit number with short division, and decide what a remainder means in a problem: round up, round down, or give the remainder.

    Number

  • The equals sign, and problems using all four operations

    Understand that = means both sides have the same value (4 × 6 = 3 × 8), find missing numbers on either side, and solve problems that combine addition, subtraction, multiplication and division.

    Number

  • Scaling by fractions, and simple rates

    Solve multiplication and division problems that scale an amount by a simple fraction ('3/4 as tall') and that use simple rates ('18 km each hour', '8 litres a minute').

    Number

  • Metric and imperial units

    Know that inches measure length, pounds mass and pints capacity, and use approximate equivalences (1 inch ≈ 2.5 cm, 1 kg ≈ 2.2 lb, 1 pint ≈ 570 ml) to convert and compare, stating which equivalence is used.

    Measurement

  • Volume and capacity: counting cubes and estimating

    Find the volume of a cuboid built from 1 cm³ cubes by counting in layers; estimate volumes (cm³) and capacities (ml, l) of everyday objects; find the capacity of a container by reading the water it holds in a measuring jug.

    Measurement

  • Problems with measures and money (decimals and scaling)

    Solve problems with lengths, masses, capacities and money in decimal notation using all four operations, changing to the same unit first, and scale quantities (recipes) up and down.

    Measurement

  • 3-D shapes from drawings, nets and descriptions

    Identify 3-D shapes, including cubes and other cuboids, from 2-D representations: drawings in any position, nets, and descriptions of their faces.

    Geometry

  • Measuring, drawing and estimating angles in degrees

    Know angles are measured in degrees; name acute, obtuse and reflex angles; estimate and compare angles; measure angles with a protractor (choosing the right scale); and draw given angles.

    Geometry

  • Angles on a straight line and around a point

    Know that angles on a straight line add up to 180° (half a turn), that angles around a point add up to 360° (a whole turn), and multiples of 90° (a right angle is 90°, three right angles 270°), and use these to find missing angles.

    Geometry

  • Using the properties of rectangles

    Use the properties of rectangles — opposite sides equal and parallel, all angles 90°, diagonals equal — to find missing lengths (including from the perimeter) and missing angles (the two angles a diagonal makes at a corner add to 90°).

    Geometry

  • Regular and irregular polygons

    Distinguish regular polygons (all sides equal AND all angles equal) from irregular ones, reasoning about both sides and angles, in any orientation.

    Geometry

  • Reflections and translations

    Identify whether a shape has been reflected or translated, describe a translation (squares left/right and up/down) or a reflection (in which mirror line), plot where a vertex goes, and know that reflecting or translating a shape doesn't change its size or shape.

    Geometry

  • Line graphs: comparing, adding and finding differences

    Solve comparison, sum and difference problems from line graphs with one or two lines, reading each line with the key and the scale (including scales that don't start at 0).

    Statistics

  • Tables and timetables

    Read and interpret tables, including 24-hour bus timetables: find arrival times, work out journey times across the hour, choose the latest bus to arrive in time, and complete missing entries in timetables and two-way tables.

    Statistics

  • Ordering fractions and decimals; tenths and hundredths

    Put in order fractions whose denominators are multiples of the same number, write equivalent tenths and hundredths, and put decimals with up to three decimal places in order.

    Fractions

  • Multiplying fractions by whole numbers, with diagrams

    Multiply a proper fraction or a mixed number by a whole number, using a diagram of equal groups, and give the answer as a mixed number or improper fraction.

    Fractions

  • Decimal and percentage problems

    Solve problems with numbers up to three decimal places, write percentages as decimals (and decimals as percentages), and solve problems that need the equivalents of halves, quarters, fifths and fractions out of 10, 20, 25, 50 and 100.

    Decimals and percentages

Year 6

  • Simplifying fractions

    Write a fraction in its simplest form by dividing the numerator and denominator by a common factor.

    Fractions

  • Fractions, decimals and percentages

    Understand per cent as 'parts per hundred' and convert between fractions, decimals and percentages.

    Decimals and percentages

  • Comparing fractions with different denominators

    Compare fractions with different denominators, by making the denominators the same or by reasoning.

    Fractions

  • Adding and subtracting fractions with different denominators

    Add and subtract fractions with different denominators, like 1/3 + 1/4, by making the parts the same size first.

    Fractions

  • Multiplying fractions

    Multiply a fraction by a whole number, and multiply two fractions, like 1/4 × 1/2 = 1/8.

    Fractions

  • Dividing a fraction by a whole number

    Divide a proper fraction by a whole number, like 1/3 ÷ 2 = 1/6.

    Fractions

  • A fraction as a division

    Understand that 3/4 means 3 ÷ 4, and use division to write fractions as decimals, like 3/8 = 0.375.

    Fractions

  • Percentages of amounts

    Find percentages of amounts, like 15% of 360, by building them from 10%, 5% and 1%.

    Decimals and percentages

  • Multiplying and dividing by 10, 100 and 1000

    Multiply and divide numbers (including decimals) by 10, 100 and 1000, with answers up to three decimal places.

    Decimals and percentages

  • Multiplying decimals by whole numbers

    Multiply one-digit numbers with up to two decimal places by whole numbers, like 3.47 × 6.

    Decimals and percentages

  • Division with decimal answers

    Use written division when the answer has up to two decimal places, like 37 ÷ 4 = 9.25, and round answers when a problem asks for it.

    Decimals and percentages

  • Place value up to 10,000,000

    Read, write, order and compare numbers up to 10,000,000 and know what each digit is worth.

    Number

  • Rounding any whole number to a required accuracy

    Round any whole number up to 10,000,000 to the nearest 10, 100, 1000, 10,000, 100,000 or 1,000,000, and to other required accuracies such as the nearest 20, 50 or 500.

    Number

  • Negative numbers: intervals across zero

    Use negative numbers in context and calculate intervals across zero, including between two negative numbers and with larger values.

    Number

  • Long division, and what remainders mean

    Divide numbers up to 4 digits by a two-digit number with long division, and give the remainder as a whole number, as a fraction, or by rounding, whichever the context needs.

    Number

  • Short division by a two-digit number

    Divide numbers up to 4 digits by two-digit numbers such as 11, 12, 15, 20 and 25 using short division where their multiples are known, and interpret remainders according to the context.

    Number

  • Mental calculation with large numbers and mixed operations

    Calculate mentally with large numbers (3,450,000 + 2,700,000; 4,800,000 ÷ 8) and with mixed operations (25 × 12 − 100), using place value and known facts.

    Number

  • Common factors, common multiples and primes

    Identify common factors (including the highest common factor) and common multiples (including the lowest common multiple) of two numbers, and identify prime numbers up to 150.

    Number

  • Order of operations

    Use the order of operations to calculate with all four operations: brackets first, then multiplication and division, then addition and subtraction.

    Number

  • Multi-step problems with big numbers and money

    Solve multi-step problems with large numbers and money using all four operations, and decide which operations and methods to use, and why.

    Number

  • Estimating to check, and choosing a sensible accuracy

    Use estimates to check answers to calculations, and decide in a problem how accurate an answer needs to be: what a money share means in that situation, and buying enough tins to finish a job.

    Number

  • Relative sizes: for every …, there are …

    Solve problems about the relative sizes of two quantities ('for every 2 red beads there are 5 blue beads'; 2 : 5), finding missing values with whole-number multiplication and division facts.

    Ratio and proportion

  • Percentages of measures, and comparing with percentages

    Calculate percentages of measures (15% of 360 m; 35% of 240 ml; 25% of 360° for a pie chart) and use percentages to compare, such as test scores out of different totals.

    Ratio and proportion

  • Similar shapes and scale factors

    Solve problems with similar rectangles and scale drawings: use a known scale factor to find a length, or find the scale factor from two matching lengths and use it.

    Ratio and proportion

  • Unequal sharing and grouping

    Solve problems where amounts are shared or grouped unequally ('Ali gets twice as much as Beth'; '3/5 of the class are boys'; 'for every egg, 3 spoons of flour'), using fractions and multiples.

    Ratio and proportion

  • Similar triangles and other shapes

    Solve problems with similar triangles and other polygons: use a known scale factor to find a side or a perimeter, or find the scale factor from one pair of matching sides and use it for the others.

    Ratio and proportion

  • Letters for numbers: expressions and equations

    Know that a letter stands for a number (an unknown, or one that can change), that 3n means 3 × n, how to find the value of an expression like 3n + 2 for a given n, and the difference between an expression (3n + 2) and an equation (3n + 2 = 17).

    Algebra

  • Using simple formulae

    Use simple formulae written in letters or in words: find the output for a given input, and work backwards with inverse operations to find the input from the output.

    Algebra

  • Missing numbers as equations

    Express a missing-number problem as an equation with a letter (4n + 3 = 31) by choosing or completing it, solve one- and two-step equations with inverse operations, and check a solution by substituting it back.

    Algebra

  • Linear number sequences

    Generate the terms of a linear sequence (going up or down by the same amount each time), describe it by its step and recognise its term-to-term rule, and use a rule for the nth term, such as 3n + 2, to find any term.

    Algebra

  • Pairs of numbers that fit an equation

    Find pairs of whole numbers that make an equation with two unknowns true (a + b = 12, 2a + b = 20, a × b = 36): find b when a is given, and recognise which pairs work.

    Algebra

  • Finding every possibility for two variables

    Enumerate all the possibilities for two variables (every pair of whole numbers in a stated range that fits a rule): count them, recognise a complete systematic list, and count combinations in context (coins, meals).

    Algebra

  • Converting units with decimals (up to three decimal places)

    Read, write and convert lengths, masses, volumes and times between smaller and larger units, both ways, using decimals up to three decimal places (2450 g = 2.45 kg, 0.045 l = 45 ml, 90 minutes = 1.5 hours), and compare measures given in different units.

    Measurement

  • Miles and kilometres

    Convert between miles and kilometres both ways using 5 miles ≈ 8 km (1 mile ≈ 1.6 km), in tables, road signs and journeys, and compare distances given in the two units.

    Measurement

  • Problems with measures: calculating and converting

    Solve multi-step problems that mix units of length, mass, volume and time, converting as needed and giving answers in the unit asked for, with decimals up to three decimal places.

    Measurement

  • Same area, different perimeter (and the other way round)

    Recognise that shapes with the same area can have different perimeters, and shapes with the same perimeter can have different areas: compare pairs of rectangles, find a rectangle with the same area or perimeter as another, and find the greatest and least perimeter (or area) for whole-number sides.

    Measurement

  • Area of parallelograms and triangles

    Calculate the area of parallelograms (base × perpendicular height) and triangles (½ × base × perpendicular height), including right-angled triangles, find a missing base or height, find the area of a compound rectilinear shape by splitting it into rectangles, and recognise when a formula can and cannot be used to find an area.

    Measurement

  • Volume of cubes and cuboids

    Calculate the volume of cubes and cuboids (length × width × height) in cm³ and m³, find a missing length from the volume, compare and estimate volumes, convert between cm³ and mm³, m³ and cm³, km³ and m³, and recognise when the volume formula can be used.

    Measurement

  • Drawing shapes to given sizes and angles

    Draw 2-D shapes from given dimensions and angles: complete rectangles, right-angled triangles and parallelograms of given sizes on a centimetre grid, and draw a given angle at the corner of a shape with a protractor.

    Geometry

  • Nets of 3-D shapes

    Recognise which arrangements of squares are nets of a cube (and which are not), find which faces will be opposite when a net is folded, and describe 3-D shapes by their faces, edges and vertices.

    Geometry

  • Missing angles in shapes and where lines meet

    Find unknown angles in triangles (angles add to 180°), including isosceles and right-angled triangles, in quadrilaterals (360°) and in regular polygons; classify shapes by their properties and sizes; and find missing angles at a point (360°), on a straight line (180°) and vertically opposite (equal).

    Geometry

  • Parts of a circle

    Name the parts of a circle — centre, radius, diameter and circumference — in any position, know that the diameter is twice the radius (so the radius is half the diameter), and mark a diameter by plotting its other end.

    Geometry

  • Coordinates in four quadrants, translations and reflections

    Read and plot coordinates in all four quadrants (including negatives); draw simple shapes by plotting a missing vertex; translate shapes; and reflect them in the x-axis and the y-axis.

    Geometry

  • Pie charts

    Interpret pie charts as fractions of a stated total (from the sector's size, its angle or its percentage), work out the angle for each part of the data, build a pie chart on screen, and know that pie charts of different totals can't be compared by sector size.

    Statistics

  • Line graphs and conversion graphs

    Interpret line graphs, including a miles–kilometres conversion graph, plot a line graph from a table, and use line graphs to solve problems.

    Statistics

  • The mean

    Calculate the mean (the total shared equally: total ÷ how many), find a missing value from a known mean, find the mean of values read from a chart, and interpret it: compare groups of different sizes by their means, and notice when one very different value pulls the mean.

    Statistics

  • Comparing and ordering fractions, including fractions greater than 1

    Compare and order fractions with different denominators, including improper fractions and mixed numbers greater than 1.

    Fractions

  • Rounding answers to problems; fractions, decimals and percentages in context

    Solve problems whose answers must be rounded to a given accuracy (the nearest penny, 1 or 2 decimal places, the nearest whole number, 10 or 100), and use equivalent fractions, decimals and percentages to solve problems in context.

    Decimals and percentages

  • Choosing short division where it's appropriate

    Judge when short division is a sensible choice for dividing by a two-digit number, knowing that short and long division always give the same answer: short division is quickest when you know the divisor's multiples, writing the multiples down (long division) helps when you don't, and a known fact can make a written method unnecessary.

    Number

Key Stage 3 Years 7 to 9 · 50 lessons

Year 7

  • Place value: decimals, measures and integers of any size

    Find the value of any digit in a decimal or a very large integer, multiply and divide by 10, 100 and 1000 by moving digits, use place value inside measures, and read numbers such as 3.2 million.

    Number

  • Primes, HCF, LCM and prime factorisation

    Write a number as a product of prime factors in index form, find the highest common factor and lowest common multiple of two numbers, and use prime factorisations to find them.

    Number

  • Powers and roots

    Work out integer powers (including zero and negative powers), square, cube and higher roots, recognise powers of 2, 3, 4 and 5, and tell exact roots from decimal approximations.

    Number

  • Calculating with negative integers and decimals

    Add, subtract, multiply and divide positive and negative integers and decimals, including with formal written methods, and solve multi-step problems with them.

    Number

  • Order of operations with powers, roots and reciprocals; inverse operations

    Use the order of operations with brackets, powers, roots and reciprocals; use the reciprocal to divide by a fraction; and use inverse operations to undo a chain of operations.

    Number

  • Terminating decimals and fractions

    Change a terminating decimal to a fraction in its simplest form and a fraction to its decimal, know which fractions give terminating decimals, and use this to compare numbers.

    Number

  • Ordering numbers and using ≠, ≤ and ≥

    Order positive and negative integers, decimals and fractions, use the number line to compare them, and write statements with =, ≠, <, >, ≤ and ≥.

    Number

  • Four operations with fractions and mixed numbers, including negatives

    Add, subtract, multiply and divide proper and improper fractions and mixed numbers, positive and negative, giving answers in their simplest form.

    Fractions

  • Algebraic notation and vocabulary

    Write and read algebra in its usual notation (ab, 3y, a², a²b, a/b, ½x, 4(n + 3)), and use the words expression, equation, inequality, term, coefficient and factor correctly.

    Algebra

  • Substituting into expressions and formulae

    Find the value of an expression or formula by substituting numbers (including negatives and decimals) for its letters, using the order of operations: powers before multiplying, and brackets first.

    Algebra

  • Simplifying by collecting like terms

    Simplify expressions by collecting like terms (including terms with powers, and ab with ba), and simplify products and quotients of terms such as 3a × 4b and 12ab ÷ 3a.

    Algebra

  • Multiplying out a bracket and taking out common factors

    Multiply a single term over a bracket (including negative terms and letters), expand and simplify, and factorise fully by taking out the highest common factor.

    Algebra

  • Coordinates in four quadrants: problems

    Solve problems with coordinates in all four quadrants: midpoints and end points, the missing corner of a parallelogram, lengths and areas from coordinates, and the equations of lines such as x = 3, y = −2, y = x and y = −x.

    Algebra

  • Ratio notation, simplest form, and fractions of a quantity

    Write ratios in their simplest form and as 1 : n, write a ratio from a situation, express one quantity as a fraction of another (less than or greater than 1), move between a ratio and the fractions it gives, and use those fractions to find a missing quantity.

    Ratio and proportion

  • Dividing in a ratio

    Divide a quantity into parts in a part : part ratio (two or three shares) or from a part : whole ratio, work back from the difference between two shares, and write how a quantity has been split as a ratio.

    Ratio and proportion

  • Area and perimeter: triangles, parallelograms, trapezia and composite shapes

    Find the areas of triangles (including obtuse-angled ones), parallelograms and trapezia with decimal lengths, find a missing base or height from an area, find perimeters, and find the areas of composite shapes by adding or taking away.

    Geometry

  • Translations, rotations and reflections

    Reflect shapes in vertical, horizontal and diagonal mirror lines, rotate them 90° or 180° about a given centre, and translate them on a four-quadrant grid; describe a reflection by its mirror line, a translation by its move and a rotation by its angle, direction and centre; and know what each transformation keeps.

    Geometry

  • Enlargement on a coordinate grid

    Enlarge a shape by a scale factor (including ½) from a centre on a coordinate grid, find the scale factor and centre of a drawn enlargement, work out lengths and perimeters after an enlargement, and know that an enlargement keeps angles but multiplies lengths.

    Geometry

Year 8

  • Rounding to decimal places and significant figures; estimating; error intervals

    Round numbers and measures to a number of decimal places or significant figures, choose an accuracy that suits the context, estimate answers by rounding to 1 significant figure, and write the error interval of a rounded value.

    Number

  • Percentages: multipliers, percentage change, original value and simple interest

    Write percentages as fractions and decimals, use multipliers for percentage changes, express one quantity as a percentage of another (including over 100%), compare with percentages, use fractions and percentages as operators, and solve percentage increase, decrease, original value and simple interest problems.

    Decimals and percentages

  • Converting units, scale diagrams and maps

    Change between related units of length, mass, capacity, area, volume and time with decimal quantities, work out costs when measures are in different units, and use scale factors, scale drawings and map scales.

    Ratio and proportion

  • Conjectures, counter-examples and simple proofs

    Test a conjecture with examples and say what a test shows, find a counter-example to a false conjecture, put the steps of a short algebraic proof in order, and write the general statement behind a numerical pattern.

    Number

  • Solving linear equations

    Solve linear equations with the unknown on one or both sides, with brackets and with fractions, including negative and non-whole solutions; form an equation from a situation and solve it; write an expression for a situation.

    Algebra

  • Sequences: term-to-term, position-to-term and the nth term

    Generate terms from a term-to-term rule and from a position-to-term (nth term) rule, recognise arithmetic sequences, find the nth term of an arithmetic sequence (increasing or decreasing), and find which term a given number is.

    Algebra

  • Linear graphs, gradient and intercept

    Plot straight-line graphs from their equations and from real situations, find and interpret the gradient and y-intercept (from a graph and from an equation), rearrange an equation into y = mx + c, read values from a graph, and solve simultaneous equations where two lines cross.

    Algebra

  • Direct and inverse proportion

    Solve direct and inverse proportion problems, find and use the formulas y = kx and y = k/x, read values from a direct-proportion graph, recognise from a graph which kind of proportion it shows, and write a ratio between two quantities as a linear function.

    Ratio and proportion

  • Speed, unit pricing and density

    Use compound units to solve problems: speed, distance and time (with times in hours and minutes), changing between m/s and km/h, unit prices and best buys, and density, mass and volume.

    Ratio and proportion

  • Circumference and area of circles

    Find the circumference and area of a circle from its radius or diameter, to 1 decimal place or in terms of π; work back from a circumference or area to the radius or diameter; and find the area and perimeter of semicircles and quarter circles.

    Geometry

  • Volume of prisms and cylinders

    Find the volume of cuboids, prisms and cylinders (area of the cross-section × length), find a missing length from a volume, and solve capacity problems using 1 litre = 1000 cm³.

    Geometry

  • Angle facts, parallel lines and polygons

    Find angles at a point, on a straight line and vertically opposite; use alternate, corresponding and co-interior angles on parallel lines; reason through a chain of angle facts, giving each reason; find angles in triangles (written as angle ABC) and polygons, including regular polygons; and form and solve equations from angle diagrams.

    Geometry

  • Measuring, scale drawings and constructions

    Read angles on a protractor from either end of its zero line, interpret scale drawings, use the perpendicular distance as the shortest distance from a point to a line, find the lines of symmetry and order of rotational symmetry of polygons, and use conventional notation; construct perpendicular bisectors, perpendiculars and angle bisectors with compasses and a ruler.

    Geometry

  • Averages and spread: comparing distributions

    Find the mean, median, mode and range (from lists and frequency tables), spot outliers and their effect, choose a suitable average, compare two distributions using an average and the range, and interpret grouped and continuous data.

    Statistics

  • Frequency tables and charts, including grouped data

    Build frequency tables (ungrouped and grouped) from raw data, build bar charts for categories and vertical line or bar charts for numerical data, build pie charts, and read pictograms, bar charts and pie charts.

    Statistics

  • Connecting numbers, algebra and graphs; precise language

    Place decimals, fractions, powers and roots on one number line, move between diagrams, numbers, equations and graphs, write the rule connecting numbers in a table, identify the variables in a situation, and use mathematical language and properties precisely.

    Number

Year 9

  • Standard form

    Write and interpret numbers in standard form with positive, negative and zero powers of 10, calculate and estimate with them, and compare them.

    Number

  • Interpreting calculator results; the number system

    Interpret what a calculator shows (decimal hours, money, E notation, long decimals) in the context of the question, and know that there is no largest whole number and that between any two numbers there is always another.

    Number

  • Expanding products of binomials

    Expand and simplify products of two or more binomials, including squares such as (x + 5)² and the difference of two squares, with negative and non-unit coefficients.

    Algebra

  • Using and rearranging formulae

    Use standard formulae (area of a trapezium and a circle, volume of a prism, distance = speed × time), work backwards from a formula, change the subject of a formula (including one with a square), and write a formula for a situation.

    Algebra

  • Geometric and other sequences

    Recognise and continue geometric sequences (multiplying by the same number each time, including ½ and negative ratios), find a later term from the first term and the ratio, and recognise square, cube, triangular and Fibonacci-type sequences.

    Algebra

  • Quadratic and other graphs: estimating and solving from graphs

    Plot quadratic graphs, read values and turning points from them, recognise linear, quadratic, reciprocal and exponential graphs, and find approximate answers to real problems from journey (piece-wise linear), growth and decay (exponential) and sharing (reciprocal) graphs.

    Algebra

  • Congruence and similarity

    Know the conditions for congruent triangles (SSS, SAS, ASA, RHS) and use them to identify congruent triangles; find missing sides of similar triangles; use the properties of quadrilaterals and circles to find angles; and put simple geometric proofs in order.

    Geometry

  • Pythagoras' theorem

    Use Pythagoras' theorem to find the hypotenuse or a shorter side of a right-angled triangle, decide whether a triangle is right-angled, find the distance between two points, and solve problems.

    Geometry

  • Trigonometry in right-angled triangles

    Use sin, cos and tan in right-angled triangles to find a missing side or angle, choose the ratio that links the sides given, and solve problems with angles of elevation.

    Geometry

  • 3-D solids: faces, surfaces, edges and vertices

    Use the faces, surfaces, edges and vertices of cubes, cuboids, prisms, pyramids, cylinders, cones and spheres to solve problems: count them (also for any prism or pyramid), use Euler's formula, find surface areas, and find the total length of a frame's edges.

    Geometry

  • Probability: the 0–1 scale, outcomes summing to 1, relative frequency

    Describe likelihood with words and the 0–1 probability scale, judge fairness from results, find relative frequencies and expected frequencies, and use the fact that the probabilities of all possible outcomes add up to 1.

    Probability

  • Sample spaces, Venn diagrams and theoretical probability

    List and count the outcomes of single and combined events systematically, use sample-space tables and two-set Venn diagrams (unions and intersections), and calculate theoretical probabilities from equally likely outcomes.

    Probability

  • Scatter graphs and correlation

    Plot data on a scatter graph, describe the relationship between two variables (positive, negative or no correlation), know what a correlation does and doesn't show, spot a point that doesn't fit, and estimate using a line of best fit.

    Statistics

  • Choosing a method: structure, proportion, proof and money

    Choose a plan for a multi-step problem, decide whether a problem's structure is additive or multiplicative, decide whether quantities are proportional (or shapes similar) before using a multiplier, put the steps of a number argument in order, and decide which of two offers is better.

    Number

  • Algebra and graph fluency: substitute, rearrange, simplify, solve

    Carry out the core algebra procedures fluently when asked — substitute, rearrange, simplify and solve — and move between a function's table, equation and graph for straight lines and simple quadratics.

    Algebra

  • Solving multi-step problems, including money

    Solve problems that need several steps, deciding the steps yourself — including rounding up when only whole items can be bought, forming and solving an equation from a situation, and working out bills.

    Number

GCSE Foundation and Higher · 55 lessons

Year 10

  • Error intervals and limits of accuracy (GCSE)

    Write the error interval of a number that was truncated or rounded, and use limits of accuracy: the least and greatest a measurement could be, whether a value is possible, and whether something is certain to fit.

    Number

  • Laws of indices (GCSE)

    Multiply and divide powers of the same number by adding and subtracting indices, raise a power to a power by multiplying indices, use zero and negative indices in a calculation, and simplify algebraic expressions with the same laws.

    Number

  • Factorising and solving quadratics (GCSE)

    Factorise x² + bx + c and the difference of two squares, solve a quadratic equation by factorising (giving every solution), and find where a quadratic's graph crosses the axes, by reading the graph or by factorising.

    Algebra

  • Simultaneous equations (GCSE)

    Solve two linear equations in two unknowns together, by elimination or substitution, and set up and solve a pair of equations from a situation, saying what the solution means.

    Algebra

  • Solving linear inequalities (GCSE)

    Solve linear inequalities in one unknown, including dividing by a negative number, unknowns on both sides and inequalities with two parts, and write the solution with the right sign at each boundary.

    Algebra

  • Parts of a circle; arcs and sectors (GCSE)

    Name the parts of a circle (centre, radius, diameter, chord, tangent, arc, sector, segment), and find arc lengths, sector areas and sector angles, to 1 decimal place or in terms of π.

    Geometry

  • Spheres, cones, pyramids and composite solids (GCSE)

    Find the volume and surface area of spheres, cones and pyramids, and of solids made from them with cylinders and cuboids, to 1 decimal place or in terms of π.

    Geometry

  • Bearings (GCSE)

    Give a bearing as an angle measured clockwise from north, written with three figures; read bearings from diagrams, give compass directions as bearings, and find back bearings and new bearings after a turn.

    Geometry

  • Exact values of sin, cos and tan (GCSE)

    Know the exact values of sin θ and cos θ for 0°, 30°, 45°, 60° and 90°, and of tan θ for 0°, 30°, 45° and 60°, and use them to find sides and angles exactly, without a calculator.

    Geometry

  • Vectors (GCSE)

    Describe a translation as a column vector, read a vector from a diagram, and add, subtract and multiply column vectors by a number.

    Geometry

  • Frequency trees and tree diagrams (GCSE)

    Complete frequency trees and probability tree diagrams, with and without replacement, and use them: multiply along a path, add the paths that give the same event, and know when events are independent.

    Probability

  • Pressure, rates of pay and changing compound units (GCSE)

    Use pressure = force ÷ area, work with rates of pay and prices, change compound units (density, pressure, pay, prices) including with letters, and know what inverse proportion means.

    Ratio and proportion

  • Compound interest, growth and decay (GCSE)

    Set up, solve and interpret growth and decay problems with a percentage multiplier: compound interest, depreciation and population change, including how many years until a value passes a target.

    Ratio and proportion

  • Comparing lengths, areas and volumes with ratio (GCSE)

    Compare lengths, areas and volumes of squares and cubes using ratio notation, link a ratio of lengths to a scale factor, and explain why similar right-angled triangles have the same trigonometric ratios.

    Ratio and proportion

  • Ratio in context: conversion, concentration and comparison (GCSE)

    Use ratio to convert between currencies both ways, find the amount of one part in a mix of a given total (a concentration), and compare two mixes by the fraction of the whole each part makes.

    Ratio and proportion

  • Equations of straight lines (GCSE)

    Find the equation of a straight line through two points, or through a point with a given gradient, and use y = mx + c to tell when lines are parallel.

    Algebra

  • Identities, arguments and functions (GCSE)

    Tell expressions, equations, formulae and identities apart; show two expressions are equivalent by simplifying; write an expression in a form that proves a claim (such as a multiple of 3); and use simple functions with inputs and outputs.

    Algebra

  • Cubic and reciprocal graphs; plotting; speed and acceleration (GCSE)

    Recognise and read cubic and reciprocal graphs, plot a reciprocal graph and a graph of a real-life function point by point, and find speed and acceleration from distance–time and velocity–time graphs.

    Algebra

  • Cancelling and working with surds (GCSE)

    Cancel common factors to simplify calculations and algebraic fractions, and work with surds: collect like surds, use √a × √a = a, and multiply out brackets with surds.

    Algebra

  • Fibonacci-type and quadratic sequences (GCSE)

    Continue Fibonacci-type sequences and find missing terms in them, and continue and recognise quadratic sequences using the second difference.

    Algebra

  • Samples, populations, interpolation and extrapolation (GCSE)

    Use a sample to estimate something about a whole population, say why a sample might not represent it, describe a population with statistics, and know when an estimate from a line of best fit can be trusted.

    Statistics

  • Time series and pictograms (GCSE)

    Draw a line graph of time series data and describe its trend, choose the right kind of chart for the data, and build a pictogram (with part symbols) from a table.

    Statistics

  • Solutions on graphs and number lines (GCSE)

    Estimate the solutions of a quadratic equation from its graph, and show an inequality's solution set on a number line with the right circles and arrows.

    Algebra

  • Plans, elevations and 3-D words (GCSE)

    Interpret a plan view of a shape made of cubes (with the number of cubes on each square) to find how many cubes there are and what the front and side elevations look like, and use the words point, vertex, edge, face and plane.

    Geometry

  • Measuring lines and fractional enlargements (GCSE)

    Measure a line segment on a ruler in centimetres and millimetres, even when it doesn't start at 0, and enlarge shapes by fractional scale factors, or find a fractional scale factor.

    Geometry

  • Prime numbers (GCSE)

    Know what a prime number is, recognise primes, count the primes in a range, and find a number's smallest prime factor.

    Number

Year 11

  • Simplifying surds and rationalising (GCSE Higher)

    Simplify surds by taking out square factors, collect and expand surds exactly, and rationalise denominators, including ones of the form p + √n.

    Number

  • Fractional indices and estimating roots (GCSE Higher)

    Work out powers with fractional and negative indices, solve equations such as 8^x = 4, and estimate square and cube roots to 1 decimal place.

    Number

  • Recurring decimals and fractions (GCSE Higher)

    Change a recurring decimal into a fraction by multiplying by a power of 10 and subtracting, and change a fraction into its recurring decimal.

    Number

  • Upper and lower bounds in calculations (GCSE Higher)

    Work out the upper and lower bounds of sums, differences, products and quotients of rounded measurements, and give an answer to a suitable degree of accuracy.

    Number

  • The product rule for counting (GCSE Higher)

    Count the ways of making several choices by multiplying the numbers of options, with and without repeats, and with conditions such as no leading zero.

    Number

  • Quadratics: ax² + bx + c, completing the square and the formula (GCSE Higher)

    Factorise ax² + bx + c, solve quadratic equations (rearranging first when needed) by factorising, by completing the square and with the quadratic formula, and find a turning point by completing the square.

    Algebra

  • Inverse and composite functions (GCSE Higher)

    Find the inverse of a function by reversing its steps, and work out composite functions such as fg(x), doing g first.

    Algebra

  • Algebraic fractions (GCSE Higher)

    Simplify algebraic fractions by factorising and cancelling, add, subtract, multiply and divide them, and solve equations that contain them.

    Algebra

  • Solving equations by iteration (GCSE Higher)

    Show that an equation has a solution in an interval by a change of sign, rearrange it into an iterative formula, and use the formula to find a solution to a given accuracy.

    Algebra

  • Quadratic nth terms and other sequences (GCSE Higher)

    Find the nth term of a quadratic sequence, work with geometric progressions whose ratio is a surd, and continue other sequences.

    Algebra

  • Perpendicular lines, circles and tangents (GCSE Higher)

    Find the gradient and equation of a perpendicular line, use the equation x² + y² = r² of a circle centred at the origin, and find the equation of the tangent to it at a point.

    Algebra

  • Gradients of curves and areas under graphs (GCSE Higher)

    Estimate the gradient of a curve from a tangent and interpret it as a rate of change, find an average rate of change from a chord, and find or estimate the area under a graph and say what it means.

    Algebra

  • Exponential graphs (GCSE Higher)

    Plot and recognise graphs of y = kˣ and exponential decay, find k from a point on the graph, and use exponential graphs and formulas for growth and decay.

    Algebra

  • Quadratic inequalities (GCSE Higher)

    Solve quadratic inequalities by finding the roots and using the shape of the graph, and write a solution in two parts with 'or' or in set notation.

    Algebra

  • Linear and quadratic simultaneous equations (GCSE Higher)

    Solve a linear and a quadratic equation simultaneously by substituting, including a line and a circle, giving every pair of solutions.

    Algebra

  • The sine rule, the cosine rule and ½ab sin C (GCSE Higher)

    Use the sine rule and the cosine rule to find sides and angles in any triangle, and Area = ½ab sin C to find an area, a side or an angle.

    Geometry

  • Pythagoras and trigonometry in 3-D (GCSE Higher)

    Use Pythagoras' theorem and trigonometry in three dimensions: the diagonal of a cuboid, the angle between a line and a plane, and a pyramid's edges.

    Geometry

  • Areas and volumes of similar shapes (GCSE Higher)

    Use the length scale factor k of similar shapes to find areas (× k²) and volumes (× k³), and work back from an area or volume ratio to lengths.

    Geometry

  • Negative enlargements and combined transformations (GCSE Higher)

    Enlarge a shape by a negative scale factor, find a negative scale factor, and describe a combination of reflections, rotations and translations as one transformation, saying which points stay where they are.

    Geometry

  • Histograms with unequal class widths (GCSE Higher)

    Work out frequency densities, draw a histogram with equal or unequal class widths, read frequencies from a histogram, and estimate how many values lie in part of a class.

    Statistics

  • Cumulative frequency graphs (GCSE Higher)

    Work out cumulative frequencies, plot a cumulative frequency graph at the upper class boundaries, and read the median, quartiles, inter-quartile range and counts above or below a value from it.

    Statistics

  • Quartiles, inter-quartile range and box plots (GCSE Higher)

    Find the quartiles and inter-quartile range of a list, read a box plot, and compare two distributions using their medians and inter-quartile ranges.

    Statistics

  • Conditional probability (GCSE Higher)

    Work out conditional probabilities — the probability of one thing given that another has happened — from two-way tables, Venn diagrams, tree diagrams and expected frequencies.

    Statistics

  • Direct and inverse proportion equations (GCSE Higher)

    Write an equation for direct or inverse proportion (y = kxⁿ or y = k/xⁿ), find k from one pair of values, and use the equation forwards and backwards.

    Ratio and proportion

  • Iterative processes: growth with regular changes (GCSE Higher)

    Use a rule such as uₙ₊₁ = 1.05uₙ + 100 step by step to find later values, find how many steps it takes to pass a target, and match a situation to its rule.

    Algebra

  • Graphs of sin, cos and tan (GCSE Higher)

    Recognise the graphs of y = sin x, y = cos x and y = tan x, know their key features, and use their symmetry to find every solution between 0° and 360°.

    Algebra

  • Translating and reflecting graphs (GCSE Higher)

    Know how y = f(x) + a, y = f(x + a), y = −f(x) and y = f(−x) move a graph, find where a key point goes, and write the equation of a translated curve.

    Algebra

  • Circle theorems (GCSE Higher)

    Use the circle theorems to find missing angles, and say which theorem gives each one.

    Geometry

  • Inequalities in two variables and regions (GCSE Higher)

    Show the region that satisfies several inequalities on a graph, write the inequality a boundary line shows, list the whole-number points in a region, and write a solution set in set notation.

    Algebra

A level AS and A level · 44 lessons

Year 12 (AS)

  • Indices: all rational exponents (AS)

    Use the laws of indices with fractional and negative exponents: evaluate powers, raise algebraic terms to rational powers, write expressions as a single power of x, and solve aˣ = b by using a common base.

    Algebra

  • Surds: rationalising and solving (AS)

    Manipulate surds exactly: rationalise denominators with surds on top and bottom, use the difference of two squares, solve linear equations whose solution is a surd, and use surds in context.

    Number

  • Quadratics: the discriminant and completing the square (AS)

    Use the discriminant b² − 4ac to decide how many real roots a quadratic has and to find unknown coefficients; complete the square for ax² + bx + c; solve quadratics in a function of the unknown; use quadratic models.

    Algebra

  • Simultaneous equations: one linear, one quadratic (AS)

    Solve a linear and a quadratic equation in two variables simultaneously by substitution, giving every solution as a pair; find when a line touches a curve; use simultaneous equations in a model.

    Algebra

  • Inequalities: fractions, set notation and 'and' (AS)

    Solve linear inequalities with brackets and fractions, quadratic inequalities with fractional roots, and a linear and a quadratic inequality together; write solutions with 'and', 'or' and set notation; use inequalities in a model.

    Algebra

  • Polynomials: the factor theorem and algebraic division (AS)

    Expand and collect three brackets; use the factor theorem to find factors and unknown coefficients; divide a cubic by (x − a); factorise a cubic fully.

    Algebra

  • Graphs of functions: y = a/x, intersections and proportion (AS)

    Know the shapes and asymptotes of y = a/x and y = a/x², find how many times a line meets them, read which equation the intersections of two graphs solve, and use proportional relationships and their graphs.

    Algebra

  • Transformations of graphs: y = af(x) and y = f(ax) (AS)

    Know the effect of y = f(x) + a, y = f(x + a), y = af(x) and y = f(ax) on a graph: find where points, turning points and asymptotes go, and describe each transformation.

    Algebra

  • Coordinate geometry: lines and circles (AS)

    Find a straight line's equation in the forms y − y₁ = m(x − x₁) and ax + by + c = 0; use the gradient conditions for parallel and perpendicular lines; find a circle's centre and radius by completing the square; use the circle properties to find tangents and centres.

    Geometry

  • Trigonometric functions: all angles, periods and identities (AS)

    Use the definitions of sine, cosine and tangent for any angle, their graphs' symmetries and periods (including sin kx), and the identities tan θ = sin θ/cos θ and sin²θ + cos²θ = 1.

    Geometry

  • Solving trigonometric equations (AS)

    Solve trigonometric equations in a given interval, giving every solution: simple equations, equations in a multiple of the angle, equations in a shifted angle, and quadratic equations in sin, cos or tan.

    Geometry

  • Exponentials and logarithms: laws and equations (AS)

    Know aˣ and eˣ and their graphs; use logₐ x as the inverse of aˣ and ln x as the inverse of eˣ; use the laws of logarithms; solve aˣ = b and equations in eˣ and ln x.

    Algebra

  • Exponential models and logarithmic graphs (AS)

    Know that the gradient of e^(kx) is ke^(kx); use exponential growth and decay models in context, with their limitations; use logarithmic graphs to estimate the parameters of y = axⁿ and y = kbˣ.

    Algebra

  • Differentiation: xⁿ, gradients and rates of change (AS)

    Understand the derivative as the gradient of the tangent and the limit of a chord's gradient; differentiate xⁿ for rational n, with multiples, sums and differences; find gradients, rates of change and second derivatives.

    Algebra

  • Tangents, normals and stationary points (AS)

    Use differentiation to find tangents and normals, stationary points and their nature, where a function is increasing or decreasing, and maximum or minimum values in a problem.

    Algebra

  • Integration: xⁿ, definite integrals and areas (AS)

    Integrate xⁿ (n ≠ −1) with multiples, sums and differences, including the constant of integration; find a curve from its gradient and a point; evaluate definite integrals; find the area under a curve; use the Fundamental Theorem of Calculus.

    Algebra

  • The binomial expansion (AS)

    Expand (a + bx)ⁿ for a positive whole number n using ⁿCᵣ; find a particular coefficient or an unknown from a coefficient; use n! and ⁿCᵣ, and link them to counting outcomes for binomial probabilities.

    Algebra

  • Vectors in two dimensions (AS)

    Use vectors in i, j and column form: find magnitude and direction, convert from magnitude and direction to components, add and scale vectors, find distances between points from their position vectors, and solve problems with forces.

    Geometry

  • Sampling, standard deviation, outliers and correlation (AS)

    Use the terms population and sample; choose and critique sampling methods; calculate and interpret standard deviation, including from Σx and Σx²; identify outliers; interpret scatter diagrams and correlation, knowing correlation does not imply causation.

    Statistics

  • Probability and the binomial distribution (AS)

    Use mutually exclusive and independent events; complete a discrete probability distribution; use the binomial distribution B(n, p) as a model and calculate P(X = r) and P(X ≤ r).

    Statistics

  • Hypothesis testing with the binomial distribution (AS)

    Write null and alternative hypotheses for a proportion (one- and two-tailed); find the p-value of an observed result and the critical region at a significance level; decide whether to reject H₀ and conclude in context; know the significance level is the probability of wrongly rejecting H₀.

    Statistics

  • Kinematics: units, graphs, suvat and calculus (AS)

    Use S.I. units; tell displacement from distance and velocity from speed; interpret displacement–time and velocity–time graphs; use the constant acceleration formulae; use calculus for motion in a straight line.

    Measurement

  • Forces and Newton's laws (AS)

    Use Newton's three laws for motion in a straight line: resultant force F = ma, weight W = mg, equilibrium, a lift's tension, and two particles connected over a smooth pulley.

    Measurement

  • Mathematical language, sets, arguments and counter-examples (AS)

    Use precise mathematical language (identity, equation, expression, function, coefficient…); read and write set notation for solution sets; find the error in a written argument; disprove a statement with a counter-example.

    Algebra

  • Modelling and problem solving (AS)

    Turn a situation into a mathematical model (a straight line from data), use it to predict, interpret its parameters in context, judge where it is reliable and how to refine it, name its assumptions, and make reasoned estimates.

    Algebra

  • Interpreting statistics: significance, regression and clean data (AS)

    Find the actual significance level of a test; interpret regression lines and scatter diagrams with distinct groups; choose and critique a way to present data; clean data by spotting errors and missing values; link data to discrete and continuous distributions.

    Statistics

Year 13

  • Sequences and series (A level)

    Find terms from a formula or a recurrence; say whether a sequence is increasing, decreasing or periodic; use sigma notation; find the nth term and the sum of arithmetic and geometric series; find a sum to infinity when |r| < 1; use series in models.

    Algebra

  • Radians, arcs, sectors and exact values (A level)

    Convert between degrees and radians; find arc lengths and sector areas with s = rθ and A = ½r²θ; use the small angle approximations; give the exact values of sin, cos and tan at multiples of π/6 and π/4.

    Geometry

  • Sec, cosec, cot and inverse trigonometric functions (A level)

    Use sec θ = 1/cos θ, cosec θ = 1/sin θ and cot θ = 1/tan θ, with exact values; give principal values of arcsin, arccos and arctan; know the graphs, domains and ranges; use sec²θ = 1 + tan²θ and cosec²θ = 1 + cot²θ.

    Geometry

  • Compound and double angles, and R cos(θ − α) (A level)

    Use sin(A ± B), cos(A ± B) and tan(A ± B) for exact values; use the double angle formulae; write a cos θ + b sin θ as R cos(θ − α) or R sin(θ + α) and find its maximum; use trigonometric models in context.

    Geometry

  • Differentiation: exponentials, logs, trig, and the product, quotient and chain rules (A level)

    Differentiate eᵏˣ, aᵏˣ, sin kx, cos kx, tan kx and ln x; use the chain, product and quotient rules; find gradients at a point and rates of change in context.

    Algebra

  • Implicit, parametric and connected rates; inflection; differential equations (A level)

    Find dy/dx for curves defined implicitly or parametrically and for inverse functions; use connected rates of change; find points of inflection and say where a curve is convex or concave; write a differential equation from a description.

    Algebra

  • Integration: exponentials, trig, substitution, parts and partial fractions (A level)

    Integrate eᵏˣ, 1/x, sin kx and cos kx; evaluate exact definite integrals; find the area between two curves; understand integration as the limit of a sum; integrate by substitution, by parts and with partial fractions.

    Algebra

  • Differential equations: separating the variables (A level)

    Separate the variables in a first order differential equation; find general and particular solutions; use solutions in context, including long-term behaviour and the limitations of the model.

    Algebra

  • Numerical methods: roots, iteration, Newton–Raphson and the trapezium rule (A level)

    Locate a root by a change of sign and know when that fails; use an iteration xₙ₊₁ = g(xₙ) and the Newton–Raphson method, and know when they fail; estimate an area with the trapezium rule and say whether it is too big or too small; use numerical methods in context and judge their accuracy.

    Algebra

  • Functions: domain, range, composites, inverses, modulus and transformations (A level)

    Use the definition, domain and range of a function; find composite and inverse functions; solve equations with the modulus of a linear function; combine transformations; simplify rational expressions and divide by a linear expression; choose sensible domains in models.

    Algebra

  • Parametric equations (A level)

    Find points on a curve given parametrically and the parameter at a point; convert between parametric and Cartesian forms; use parametric equations to model motion.

    Algebra

  • The binomial expansion for any rational n (A level)

    Expand (1 + kx)ⁿ and (a + bx)ⁿ for negative and fractional n; find coefficients; know the expansion is valid only for |bx/a| < 1; use it to approximate.

    Algebra

  • Vectors in three dimensions (A level)

    Use vectors in three dimensions with i, j, k: magnitudes, the distance between two points, the vector from one point to another, parallel vectors, and position vectors of points on a line.

    Geometry

  • Conditional probability (A level)

    Use set notation in probability; find conditional probabilities from two-way tables, Venn diagrams and tree diagrams; use P(A|B) = P(A ∩ B)/P(B); test for independence; critique the assumptions in a probability model.

    Probability

  • The Normal distribution (A level)

    Use the Normal distribution as a model; find Normal probabilities and values from probabilities; link the Normal distribution to histograms, mean, standard deviation, points of inflection and the binomial distribution; choose a suitable distribution for a context.

    Statistics

  • Hypothesis tests: correlation and the Normal mean (A level)

    Interpret a correlation coefficient using a given critical value or p-value; test a Normal mean with known variance using X̄ ~ N(μ, σ²/n): find p-values and critical regions, and conclude in context.

    Statistics

  • Motion in two dimensions and projectiles (A level)

    Use vectors for velocity and displacement; apply v = u + at and s = ut + ½at² as vectors; differentiate and integrate r, v and a given as vectors in t; find speeds; model projectiles: time of flight, range, greatest height.

    Measurement

  • Forces in two dimensions: resolving, friction and moments (A level)

    Resolve forces into components; find resultants and use F = ma as vectors; solve equilibrium problems; use Newton's second law on slopes; use F ≤ μR for friction; take moments for beams in equilibrium.

    Measurement

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