Edexcel GCSE Maths: grade by grade

Every skill, from the first marks to the top grade. For each grade you also need the skills for the grades below it. Tick them off in the app's Notes section.

Grade 1

Grade 1 checklist (PDF)

Order integers and decimals Line up the decimal points and compare digits from the left, so 0.407 is smaller than 0.47. Place value, ordering and order of operations
Use the order of operations (BIDMAS) Brackets, then indices, then × and ÷, then + and −, so 3 + 4 × 2 = 11. Place value, ordering and order of operations
Find factors and multiples of a number Factors divide exactly into a number (factors of 12: 1, 2, 3, 4, 6, 12); multiples are in its times table (12, 24, 36, ...). Primes, factors, HCF and LCM
Simplify a fraction Divide the top and bottom by a common factor, e.g. \(\frac{18}{24} = \frac{3}{4}\). Fractions and mixed numbers
Recall square numbers and square roots Know the squares up to \(15^2 = 225\) and their square roots, e.g. \(\sqrt{144} = 12\). Indices and roots
Round to the nearest 10, 100 or 1000 Look at the next digit: 5 or more rounds up, so 3472 is 3500 to the nearest 100. Rounding, estimation and error intervals
Collect like terms Add or subtract terms with exactly the same letters, e.g. \(3a + 5b - a + 2b = 2a + 7b\). Algebraic notation, substitution and identities
Find the perimeter by adding the sides Add the lengths of all the edges around the outside of the shape. Area and perimeter
Describe likelihood on a 0 to 1 scale Use impossible (0), unlikely, even chance (0.5), likely and certain (1), and mark events on a probability scale. Probability and relative frequency
Read pictograms, bar charts and tables Use the key of a pictogram (for example, one symbol stands for 4 people) and read bar heights carefully against the scale. Charts, diagrams and time series
Find the mode and range of a list The mode is the most common value; the range is the largest value minus the smallest value. Averages and spread

Grade 2

Grade 2 checklist (PDF)

Calculate with negative numbers Add, subtract, multiply and divide with negatives, e.g. −3 − (−8) = 5 and −4 × −6 = 24. Place value, ordering and order of operations
Use the inequality symbols correctly Use \(=\), \(\ne\), \(\lt\), \(\gt\), \(\le\) and \(\ge\) to compare numbers, e.g. \(-7 \lt -2\). Place value, ordering and order of operations
Recognise prime numbers A prime has exactly two factors, 1 and itself: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ... Primes, factors, HCF and LCM
Find a fraction of an amount Divide by the denominator, then multiply by the numerator: \(\frac{3}{5}\) of 40 = 40 ÷ 5 × 3 = 24. Fractions and mixed numbers
Recall cube numbers and cube roots Know 1, 8, 27, 64, 125 and 1000, e.g. \(\sqrt[3]{64} = 4\). Indices and roots
Round to a number of decimal places Count digits after the decimal point: 6.4851 is 6.49 to 2 decimal places. Rounding, estimation and error intervals
List outcomes in a systematic order E.g. the two-digit numbers made from 3, 5 and 8 without repeats: 35, 38, 53, 58, 83, 85. Listing and the product rule
Write products in algebraic notation Write \(a \times b\) as \(ab\), \(y \times y\) as \(y^2\) and \(3 \times a \times a \times b\) as \(3a^2b\). Algebraic notation, substitution and identities
Solve one-step equations Use the inverse operation, e.g. \(x + 8 = 15\) gives \(x = 7\), and \(4x = 28\) gives \(x = 7\). Linear equations
Continue a sequence using a term-to-term rule e.g. 'add 4' starting at 3 gives 3, 7, 11, 15, …; 'multiply by 2' starting at 5 gives 5, 10, 20, 40, … Sequences and the nth term
Simplify a ratio using the HCF Divide every part by the highest common factor, e.g. 12 : 18 = 2 : 3. Ratio
Find percentages of amounts without a calculator Build up from 10%, 5% and 1%, e.g. 35% of £60 = £18 + £3 = £21. Percentages and reverse percentages
Convert metric units of length, mass and capacity Multiply or divide by 10, 100 or 1000, e.g. 3.4 km = 3400 m. Converting units
Convert between units of time Use 60 seconds in a minute and 60 minutes in an hour, e.g. 2.5 hours = 2 hours 30 minutes. Converting units
Use angles on a line and around a point Angles on a straight line add up to 180° and angles around a point add up to 360°. Angles, parallel lines and polygons
Find the area of rectangles and triangles Rectangle = length × width; triangle = ½ × base × perpendicular height. Area and perimeter
Count faces, edges and vertices e.g. a cuboid has 6 faces, 12 edges and 8 vertices. Volume and surface area
Draw and recognise nets of 3D shapes Know which arrangements of faces fold up into cubes, cuboids, prisms and pyramids. Plans and elevations
Find a probability for equally likely outcomes Divide the number of ways the event can happen by the total number of outcomes, e.g. P(5 on a fair six-sided dice) = \(\frac{1}{6}\). Probability and relative frequency
Draw a bar chart or vertical line chart Label both axes, start the frequency scale at 0 and make the bars the same width with equal gaps. Charts, diagrams and time series
Find the median of a list Put the values in order and find the middle one, or halfway between the middle two. Averages and spread
Plot points on a scatter graph Plot each pair of values as a cross, reading both scales carefully. Scatter graphs and correlation
Know the words population, sample and census The population is the whole group; a sample is part of it; a census collects data from every member. Sampling and capture–recapture

Grade 3

Grade 3 checklist (PDF)

Use written methods for all four operations Use column or grid multiplication and short or long division without a calculator. Place value, ordering and order of operations
Multiply and divide with decimals Work with whole numbers, then place the decimal point, e.g. 0.3 × 0.04 = 0.012 and 7.2 ÷ 0.3 = 24. Place value, ordering and order of operations
Find HCF and LCM by listing List factors for the HCF and multiples for the LCM: for 12 and 18, the HCF is 6 and the LCM is 36. Primes, factors, HCF and LCM
Convert fractions, decimals and percentages E.g. \(\frac{3}{8}\) = 3 ÷ 8 = 0.375 = 37.5%. Fractions and mixed numbers
Add and subtract fractions Use a common denominator: \(\frac{2}{3} + \frac{1}{4} = \frac{8}{12} + \frac{3}{12} = \frac{11}{12}\). Fractions and mixed numbers
Recognise powers of 2, 3, 4 and 5 E.g. \(2^5 = 32\), \(3^4 = 81\), \(4^3 = 64\) and \(5^3 = 125\). Indices and roots
Write large numbers in standard form Place the point after the first digit and count the moves: 45,000 = \(4.5 \times 10^4\). Standard form
Round to significant figures Start counting at the first non-zero digit: 0.03064 is 0.0306 to 3 significant figures. Rounding, estimation and error intervals
Estimate by rounding to 1 significant figure Round every number to 1 s.f., then calculate: 38.2 × 5.9 ≈ 40 × 6 = 240. Rounding, estimation and error intervals
List combinations from two or three sets Fix one item and run through every option for the others, then check you have them all. Listing and the product rule
Substitute positive numbers into expressions Replace each letter by its value, e.g. when \(a = 4\) and \(b = 3\), \(5a - 2b = 20 - 6 = 14\). Algebraic notation, substitution and identities
Tell expressions, equations, formulae and identities apart An expression has no equals sign, an equation can be solved, a formula links quantities and an identity is true for every value. Algebraic notation, substitution and identities
Expand a single bracket Multiply every term inside by the term outside, e.g. \(4(2x - 3) = 8x - 12\). Expanding brackets
Rearrange a one-step formula Use the inverse operation, e.g. \(P = x + 7\) gives \(x = P - 7\). Changing the subject
Solve two-step equations e.g. \(3x - 4 = 11\), so \(3x = 15\) and \(x = 5\). Linear equations
Write an inequality from a number line An open circle means the value is not included (\(\lt\) or \(\gt\)); a filled circle means it is included (\(\le\) or \(\ge\)). Inequalities
Recognise square, cube and triangular numbers Squares 1, 4, 9, 16, …; cubes 1, 8, 27, 64, …; triangular numbers 1, 3, 6, 10, … Sequences and the nth term
Plot a straight line from its equation Make a table of values, plot the points and join them with a ruler. Straight-line graphs
Complete a table of values and plot Substitute each x (with brackets for negatives), plot the points and join them with a smooth curve. Graphs of functions
Share an amount in a given ratio Find the value of one part first, e.g. £40 shared in the ratio 3 : 5 gives £15 and £25. Ratio
Write one quantity as a fraction of another Use the same units, then write the first over the second and simplify, e.g. 40 minutes out of an hour is \(\frac{2}{3}\). Ratio
Convert between fractions, decimals and percentages For example, \(\frac{3}{8}\) = 0.375 = 37.5%, and 120% = 1.2. Percentages and reverse percentages
Write one amount as a percentage of another Divide the part by the whole and multiply by 100, e.g. 18 out of 24 is 75%. Percentages and reverse percentages
Scale a recipe up or down Find the amount for one (or a scale factor) and multiply every ingredient by the same number. Direct and inverse proportion
Use speed = distance ÷ time For example, 150 km in 2.5 hours is an average speed of 60 km/h. Speed, density and pressure
Convert money using an exchange rate Multiply by the rate to change pounds into the other currency, and divide to change back. Converting units
Read and use a conversion graph Read across and down carefully, working out the value of one small square first. Converting units
Find missing angles in triangles and quadrilaterals Angles in a triangle add up to 180°, angles in a quadrilateral add up to 360°, and the base angles of an isosceles triangle are equal. Angles, parallel lines and polygons
Know the properties of special quadrilaterals Know the sides, angles, diagonals and symmetry of squares, rectangles, parallelograms, rhombuses, kites and trapezia. Angles, parallel lines and polygons
Name the parts of a circle Radius, diameter, circumference, chord, tangent, arc, sector and segment. Arcs and sectors
Find the area of parallelograms and trapezia Parallelogram = base × perpendicular height; trapezium = ½(a + b)h. Area and perimeter
Find the volume of a cuboid Volume = length × width × height (or count the centimetre cubes). Volume and surface area
Draw the plan of a 3D solid Draw the view from directly above, full size or to scale, on squared paper. Plans and elevations
Draw front and side elevations Draw the views from the front and from the side, showing the correct heights. Plans and elevations
Identify congruent shapes Congruent shapes are identical in shape and size, even if one is turned over or rotated. Congruence and geometric proof
Reflect a shape in a mirror line Reflect in lines such as x = 2, y = −1, y = x and y = −x; each image point is the same distance from the line. Transformations
Translate a shape using a column vector The top number moves right (+) or left (−); the bottom number moves up (+) or down (−). Transformations
Use a scale to find real distances e.g. on a 1 : 50 000 map, 4 cm represents 200 000 cm = 2 km. Bearings and scale drawings
Measure and draw lines and angles accurately Measure lengths to the nearest millimetre and angles to the nearest degree. Bearings and scale drawings
Construct triangles from given sides and angles Use a ruler and protractor for SAS and ASA, and a ruler and compasses for SSS. Constructions and loci
Use the fact that probabilities sum to 1 Find a missing probability in a table, or use P(not A) = 1 − P(A). Probability and relative frequency
Complete and use a frequency tree Split a total into groups and then subgroups, and read off the numbers you need for a probability. Probability and relative frequency
List all the outcomes systematically Fix the first item and list every option for the second, then change the first, e.g. HH, HT, TH, TT for two coins. Tree diagrams and combined events
Use a sample space diagram Draw a grid for two dice or spinners, then count the outcomes you want out of the total number of cells. Tree diagrams and combined events
Sort items into a two-set Venn diagram Put items in both sets in the overlap, the rest of each set in its own circle, and items in neither set outside the circles. Venn diagrams and set notation
Complete and use a two-way table Use the fact that each row and each column adds up to its total to fill in the missing values. Charts, diagrams and time series
Draw and read a stem-and-leaf diagram Put the leaves in order, include a key, and use the diagram to find the median and the range. Charts, diagrams and time series
Calculate the mean of a list Add up all the values and divide by how many values there are. Averages and spread
Name the type of correlation Positive, negative or no correlation, and whether it is strong or weak. Scatter graphs and correlation
Describe the relationship in context Say how one variable changes as the other increases, using the names of the variables. Scatter graphs and correlation
Explain why a sample may be biased Say which group is more or less likely to be chosen, so the sample does not represent the population. Sampling and capture–recapture

Grade 4

Grade 4 checklist (PDF)

Use one calculation to find another Given 34 × 26 = 884, adjust the place value to get 3.4 × 2.6 = 8.84 or 884 ÷ 2.6 = 340. Place value, ordering and order of operations
Find and use reciprocals The reciprocal of a number is 1 divided by it, so the reciprocal of 4 is \(\frac{1}{4}\) and of \(\frac{2}{3}\) is \(\frac{3}{2}\). Place value, ordering and order of operations
Write a number as a product of primes Use a factor tree or repeated division: \(360 = 2 \times 2 \times 2 \times 3 \times 3 \times 5 = 2^3 \times 3^2 \times 5\). Primes, factors, HCF and LCM
Multiply and divide fractions Multiply tops and bottoms; to divide, multiply by the reciprocal of the second fraction. Fractions and mixed numbers
Use the laws of indices with numbers Add indices to multiply, subtract to divide, multiply for a power of a power: \((3^4)^2 = 3^8\). Indices and roots
Write small numbers in standard form A number between 0 and 1 has a negative power: 0.00072 = \(7.2 \times 10^{-4}\). Standard form
Change standard form to ordinary numbers \(3.06 \times 10^5\) = 306,000 and \(3.06 \times 10^{-3}\) = 0.00306. Standard form
Order numbers in standard form Compare the powers of 10 first, then the values of \(A\). Standard form
Use a calculator with standard form Use the \(\times 10^x\) key, and write the answer in proper standard form, not calculator notation. Standard form
Decide if an estimate is over or under Use how each number was rounded, e.g. rounding up the numbers you multiply gives an overestimate. Rounding, estimation and error intervals
Use the product rule for two choices If there are \(m\) ways to do one thing and \(n\) ways to do another, there are \(m \times n\) ways to do both. Listing and the product rule
Substitute negative numbers into formulae Put negative values in brackets, e.g. when \(x = -3\), \(x^2 = (-3)^2 = 9\). Algebraic notation, substitution and identities
Expand and simplify two single brackets Expand each bracket, then collect like terms, e.g. \(3(x + 2) - 2(x - 5) = x + 16\). Expanding brackets
Expand with a letter outside the bracket Multiply letters as well as numbers, e.g. \(2a(3a + b) = 6a^2 + 2ab\). Expanding brackets
Take out a single common factor Find a number or letter that divides every term, e.g. \(6x + 15 = 3(2x + 5)\). Factorising
Multiply powers of the same letter Add the indices: \(a^4 \times a^3 = a^7\). Laws of indices in algebra
Divide powers of the same letter Subtract the indices: \(y^9 \div y^4 = y^5\). Laws of indices in algebra
Rearrange a two-step formula Undo the operations in reverse order, e.g. \(y = 3x - 5\) gives \(x = \frac{y + 5}{3}\). Changing the subject
Solve equations with brackets Expand first, e.g. \(2(x + 5) = 18\) gives \(2x + 10 = 18\), so \(x = 4\). Linear equations
Solve with the unknown on both sides Collect the x terms on one side, e.g. \(7x - 3 = 4x + 12\) gives \(3x = 15\), so \(x = 5\). Linear equations
Read solutions from a quadratic graph The solutions of \(x^2 + bx + c = 0\) are the x-coordinates where \(y = x^2 + bx + c\) crosses the x-axis. Solving quadratic equations
Read a solution from two line graphs The solution is the coordinates of the point where the two lines cross. Simultaneous equations
List integers that satisfy an inequality e.g. \(-2 \lt n \le 3\) gives \(n = -1, 0, 1, 2, 3\). Inequalities
Solve a linear inequality Solve it like an equation, e.g. \(4x + 1 \gt 13\) gives \(x \gt 3\). Inequalities
Generate terms from an nth term Substitute \(n = 1, 2, 3, \ldots\), e.g. \(3n + 2\) gives 5, 8, 11, … Sequences and the nth term
Find the nth term of a linear sequence The common difference multiplies n, then adjust: 5, 9, 13, 17, … has nth term \(4n + 1\). Sequences and the nth term
Find the gradient between two points Gradient = change in y ÷ change in x, e.g. from (1, 2) to (4, 11): \(9 \div 3 = 3\). Straight-line graphs
Identify gradient and intercept from an equation In \(y = 5x - 2\), the gradient is 5 and the line crosses the y-axis at (0, −2). Straight-line graphs
Recognise the shapes of standard graphs Linear, quadratic (U or ∩ shape), cubic (S shape) and reciprocal \(y = \frac{1}{x}\) (two separate curves). Graphs of functions
Interpret distance–time graphs The gradient is the speed and a horizontal section means the object is stationary. Graphs of functions
Convert between ratios and fractions In the ratio 2 : 5 the first share is \(\frac{2}{7}\) of the total, not \(\frac{2}{5}\). Ratio
Use one share or a difference Divide the known amount by the number of parts it stands for to find one part. Ratio
Write ratios in the form 1 : n Divide both parts by the first part, e.g. 4 : 10 = 1 : 2.5. Ratio
Increase or decrease by a percentage Use a multiplier, e.g. × 1.15 for a 15% increase or × 0.85 for a 15% decrease. Percentages and reverse percentages
Calculate simple interest Find one year's interest on the original amount and multiply by the number of years. Percentages and reverse percentages
Work out compound interest year by year Add each year's interest before working out the next, e.g. £500 at 4% becomes £520, then £540.80. Compound interest, growth and decay
Compare value for money using unit prices Work out the cost per item, or per 100 g, for each pack and compare. Direct and inverse proportion
Solve inverse proportion problems like workers and days Find the total work (workers × days), then divide by the new number of workers. Direct and inverse proportion
Convert between minutes and decimal hours Divide the minutes by 60, e.g. 1 hour 45 minutes = 1.75 hours. Speed, density and pressure
Calculate density, mass or volume Use density = mass ÷ volume and its rearrangements, with matching units. Speed, density and pressure
Use a given metric–imperial conversion Treat it as a ratio, e.g. with 5 miles ≈ 8 km, 40 miles ≈ 64 km. Converting units
Use alternate, corresponding and co-interior angles On parallel lines, alternate angles and corresponding angles are equal, and co-interior angles add up to 180°. Angles, parallel lines and polygons
Find interior and exterior angles of polygons Sum of interior angles = (n − 2) × 180°, and the exterior angles of any polygon add up to 360°. Angles, parallel lines and polygons
Find the hypotenuse of a right-angled triangle Square the two shorter sides, add, then square root: \(c = \sqrt{a^2 + b^2}\). Pythagoras' theorem (incl. 3D)
Find a shorter side of a right-angled triangle Square the hypotenuse, subtract the square of the other side, then square root. Pythagoras' theorem (incl. 3D)
Label the hypotenuse, opposite and adjacent sides The opposite is across from the angle, the hypotenuse is opposite the right angle, and the adjacent is the other side next to the angle. Right-angled trigonometry
Find the area and circumference of circles Area = πr²; circumference = πd = 2πr. Area and perimeter
Find areas and perimeters of compound shapes Split into rectangles, triangles and parts of circles, finding any missing lengths first. Area and perimeter
Find volumes of prisms and cylinders Volume = area of cross-section × length; for a cylinder, V = πr²h. Volume and surface area
Find the surface area of a prism Add the areas of all the faces; sketch a net so you do not miss any. Volume and surface area
Draw a solid on isometric paper Keep vertical edges vertical and draw the other edges along the sloping lines of dots. Plans and elevations
Recognise similar shapes and find scale factors Check that corresponding sides are all in the same ratio; scale factor = new length ÷ original length. Similar shapes (length, area, volume)
Rotate a shape about a given centre Use tracing paper, the angle and the direction (clockwise or anticlockwise). Transformations
Enlarge a shape by a positive scale factor Multiply the distance from the centre of enlargement to each vertex by the scale factor. Transformations
Read and draw column vectors The top number is the move across and the bottom number the move up or down. Vectors
Add and subtract column vectors Add or subtract the top numbers and the bottom numbers separately. Vectors
Measure and draw bearings Measure clockwise from north at the starting point, and write the bearing with three figures. Bearings and scale drawings
Make a scale drawing from real measurements Divide each real length by the scale, e.g. at 1 cm to 5 m, 20 m is drawn as 4 cm. Bearings and scale drawings
Construct the perpendicular bisector of a line Draw arcs of equal radius from each end, then join the two points where they cross. Constructions and loci
Construct the bisector of an angle Draw an arc from the vertex, then equal arcs from where it crosses the arms, and join to the vertex. Constructions and loci
Calculate relative frequency from experiment results Relative frequency = number of times the outcome happened ÷ total number of trials. Probability and relative frequency
Work out an expected number of outcomes Multiply the probability by the number of trials, e.g. 0.15 × 200 = 30. Probability and relative frequency
Complete a tree diagram for independent events Each pair of branches adds up to 1, and the second set of branches repeats the first when the events are independent. Tree diagrams and combined events
Understand ξ, intersection and union \(\xi\) is everything, \(A \cap B\) is the overlap, and \(A \cup B\) is everything inside either circle. Venn diagrams and set notation
Find a probability from a Venn diagram Divide the number in the region you want by the total in \(\xi\), including anything outside the circles. Venn diagrams and set notation
Draw and interpret a pie chart Work out each angle as frequency ÷ total × 360°, and each frequency as angle ÷ 360° × total. Charts, diagrams and time series
Draw and interpret a frequency polygon Plot each frequency at the midpoint of its class and join the points with straight lines. Charts, diagrams and time series
Describe trends and patterns in time series Say whether the values are generally rising or falling, and describe any pattern that repeats each year. Charts, diagrams and time series
Find averages from a frequency table Mean = total of the fx column ÷ total frequency; the mode is the value with the highest frequency. Averages and spread
Compare two data sets using average and spread Compare an average and the range, and say what each comparison means in context. Averages and spread
Draw a line of best fit Draw one straight ruled line that follows the trend, with the points spread evenly on either side. Scatter graphs and correlation
Identify and explain an outlier An outlier is a point that does not fit the general pattern; leave it out when you draw the line. Scatter graphs and correlation
Describe how to take a random sample Number every member of the population and use random numbers to choose, so each has an equal chance. Sampling and capture–recapture
Use a sample to estimate for a population Scale up the proportion in the sample: proportion × population size. Sampling and capture–recapture
Suggest how to make a sample more reliable Use a larger random sample, taken from the whole population rather than one place or time. Sampling and capture–recapture

Grade 5

Grade 5 checklist (PDF)

Find HCF and LCM using prime factors HCF: primes in both numbers with the lower power; LCM: every prime with the higher power (or use a Venn diagram). Primes, factors, HCF and LCM
Solve HCF and LCM word problems Sharing into equal groups or the largest size means HCF; things happening together again means LCM. Primes, factors, HCF and LCM
Calculate with mixed numbers Change mixed numbers to improper fractions first, e.g. \(2\frac{1}{3} = \frac{7}{3}\), then add, subtract, multiply or divide. Fractions and mixed numbers
Solve multi-step fraction problems E.g. find a fraction of what is left, or find the whole amount from a fraction of it. Fractions and mixed numbers
Write a fraction as a recurring decimal Use short division until the remainders repeat, e.g. \(\frac{5}{11} = 0.454545...\) Recurring decimals
Use dot notation for recurring decimals Dots go over the first and last digits of the repeating block: \(0.\dot{1}2\dot{3} = 0.123123...\) Recurring decimals
Use zero and negative indices \(a^0 = 1\) and \(a^{-n} = \frac{1}{a^n}\), e.g. \(2^{-3} = \frac{1}{8}\). Indices and roots
Multiply and divide in standard form Deal with the numbers and the powers of 10 separately, then adjust: \(12 \times 10^9 = 1.2 \times 10^{10}\). Standard form
Add and subtract in standard form Change to ordinary numbers first: \(3.2 \times 10^4 + 5 \times 10^3\) = 32,000 + 5000 = \(3.7 \times 10^4\). Standard form
Write error intervals for rounded values 7.3 to 1 decimal place means \(7.25 \le x \lt 7.35\). Rounding, estimation and error intervals
Write error intervals for truncated values 7.3 truncated to 1 decimal place means \(7.3 \le x \lt 7.4\). Rounding, estimation and error intervals
Find the bounds of a rounded measurement Go half a unit either side: 36 cm to the nearest cm has bounds 35.5 cm and 36.5 cm. Upper and lower bounds
Use the product rule for several choices E.g. 3 letters then 2 digits: 26 × 26 × 26 × 10 × 10 = 1,757,600 codes. Listing and the product rule
Use the identity symbol correctly Write \(\equiv\) when both sides are equal for every value of the letter, e.g. \(2(x + 3) \equiv 2x + 6\). Algebraic notation, substitution and identities
Expand the product of two binomials Multiply each term in the first bracket by each term in the second, e.g. \((x + 3)(x - 7) = x^2 - 4x - 21\). Expanding brackets
Expand a squared bracket Write it out twice: \((x - 4)^2 = (x - 4)(x - 4) = x^2 - 8x + 16\). Expanding brackets
Factorise fully using the HCF Take out the highest common factor of the numbers and letters, e.g. \(8x^2y - 12xy = 4xy(2x - 3)\). Factorising
Factorise quadratics with x² coefficient 1 Find two numbers that multiply to c and add to b, e.g. \(x^2 + 2x - 15 = (x + 5)(x - 3)\). Factorising
Factorise a difference of two squares Use \(a^2 - b^2 = (a + b)(a - b)\), e.g. \(x^2 - 49 = (x + 7)(x - 7)\). Factorising
Raise a power to a power Multiply the indices: \((x^3)^5 = x^{15}\). Laws of indices in algebra
Simplify terms with numbers and letters Deal with the numbers and each letter separately, e.g. \(4a^2b \times 3ab^5 = 12a^3b^6\). Laws of indices in algebra
Raise a whole term to a power The number is raised to the power too: \((3x^4)^2 = 9x^8\). Laws of indices in algebra
Use zero and negative indices \(x^0 = 1\) and \(x^{-n} = \frac{1}{x^n}\), e.g. \(x^{-2} = \frac{1}{x^2}\). Laws of indices in algebra
Rearrange formulae with brackets or fractions Clear the fraction or bracket first, e.g. \(P = \frac{a + b}{2}\) gives \(a = 2P - b\). Changing the subject
Rearrange formulae with squares or square roots Undo a square with a square root and a square root by squaring, e.g. \(A = \pi r^2\) gives \(r = \sqrt{\frac{A}{\pi}}\). Changing the subject
Solve equations with a fraction Multiply both sides by the denominator, e.g. \(\frac{x + 4}{3} = 5\) gives \(x = 11\). Linear equations
Form and solve an equation from context Write an equation for an angle sum, perimeter, age or cost problem, solve it and answer the question. Linear equations
Solve simple quadratics by factorising Factorise, then set each bracket equal to 0, e.g. \((x - 3)(x + 7) = 0\) gives \(x = 3\) or \(x = -7\). Solving quadratic equations
Solve quadratics with a missing term \(x^2 = 49\) gives \(x = \pm 7\); \(x^2 - 5x = 0\) gives \(x(x - 5) = 0\), so \(x = 0\) or \(x = 5\). Solving quadratic equations
Solve linear simultaneous equations by elimination Make the coefficients of one letter the same, then add or subtract the equations. Simultaneous equations
Form and solve simultaneous equations in context e.g. 3 teas and 2 coffees cost £8.10 gives \(3t + 2c = 810\) (in pence). Simultaneous equations
Solve a double inequality Do the same to all three parts, e.g. \(-3 \le 2x + 1 \lt 9\) gives \(-2 \le x \lt 4\). Inequalities
Reverse the sign when dividing by a negative e.g. \(-2x \gt -4\) gives \(x \lt 2\). Inequalities
Decide if a number is in a sequence Set the nth term equal to the number and check whether n is a positive whole number. Sequences and the nth term
Continue Fibonacci-type and geometric sequences Fibonacci-type: add the previous two terms. Geometric: multiply by the same number each time. Sequences and the nth term
Find the equation of a line Work out m, then substitute a point to find c, e.g. through (2, 7) and (4, 11) gives \(y = 2x + 3\). Straight-line graphs
Identify and find parallel lines Parallel lines have equal gradients: \(y = 3x + 1\) is parallel to \(y = 3x - 4\). Straight-line graphs
Rearrange an equation to find the gradient Make y the subject, e.g. \(2y + 6x = 5\) gives \(y = -3x + 2.5\), so the gradient is −3. Straight-line graphs
Interpret gradient and intercept in context e.g. in \(C = 45d + 30\), 45 is the cost per day and 30 is a fixed charge. Straight-line graphs
Read roots and turning points from graphs Roots are where the curve crosses the x-axis; the turning point is the minimum or maximum point. Graphs of functions
Solve equations using a graph Draw a horizontal line such as \(y = 3\) and read the x-coordinates where it meets the curve. Graphs of functions
Interpret a straight-line gradient as a rate The gradient shows how fast y changes as x increases, e.g. litres per minute. Gradients and areas under graphs
Combine two ratios with a shared quantity Make the shared quantity's parts equal, e.g. a : b = 1 : 2 and b : c = 3 : 4 give a : b : c = 3 : 6 : 8. Ratio
Work out percentage change, profit and loss Divide the change by the original amount and multiply by 100. Percentages and reverse percentages
Find the original amount (reverse percentages) Divide the new amount by the multiplier, e.g. £68 after a 15% reduction was 68 ÷ 0.85 = £80. Percentages and reverse percentages
Use a multiplier raised to a power Value after n years = start × multiplierⁿ, e.g. £2500 at 3% for 4 years is 2500 × 1.03⁴. Compound interest, growth and decay
Calculate depreciation and decay Use a multiplier less than 1, e.g. × 0.85ⁿ for a 15% fall each year. Compound interest, growth and decay
Find when a value first passes a target Work out the value year by year and show the values either side of the target. Compound interest, growth and decay
Recognise direct and inverse proportion graphs Direct proportion is a straight line through the origin; inverse proportion is a curve that gets closer and closer to both axes. Direct and inverse proportion
Calculate pressure, force or area Use pressure = force ÷ area, with the area in m² for a pressure in N/m². Speed, density and pressure
Convert compound units such as km/h to m/s Convert one unit at a time, e.g. 72 km/h = 72 000 m per hour = 72 000 ÷ 3600 = 20 m/s. Speed, density and pressure
Find the average speed for a whole journey Divide the total distance by the total time, including any stops. Speed, density and pressure
Use density with volumes of 3D shapes Find the volume of the solid first, then multiply by the density to get the mass. Speed, density and pressure
Convert area and volume units Square or cube the length factor, e.g. 1 m² = 100 × 100 = 10 000 cm². Converting units
Convert between volume and capacity Use 1 cm³ = 1 ml and 1000 cm³ = 1 litre. Converting units
Give a geometric reason for every step Write the angle fact in words each time you use it, e.g. 'alternate angles are equal'. Angles, parallel lines and polygons
Form and solve equations from angle facts When angles are given in terms of x, use an angle fact to write an equation, solve it, then find the angle. Angles, parallel lines and polygons
Use Pythagoras in problems and on coordinates Spot right-angled triangles in rectangles, isosceles triangles and on grids, e.g. the distance between two points. Pythagoras' theorem (incl. 3D)
Test whether a triangle is right-angled Check whether the square of the longest side equals the sum of the squares of the other two sides. Pythagoras' theorem (incl. 3D)
Find a missing side using SOH CAH TOA Choose the ratio from the side you know and the side you want, then rearrange, e.g. x = 12 × sin 40°. Right-angled trigonometry
Find a missing angle using inverse trig Use sin⁻¹, cos⁻¹ or tan⁻¹ on your calculator, e.g. θ = tan⁻¹(5 ÷ 8). Right-angled trigonometry
Recall exact trig values Know sin and cos of 0°, 30°, 45°, 60° and 90°, and tan of 0°, 30°, 45° and 60°, without a calculator. Right-angled trigonometry
Solve angle of elevation and depression problems Draw the right-angled triangle with the angle measured from the horizontal. Right-angled trigonometry
Calculate the length of an arc Arc length = \(\frac{\theta}{360} \times 2\pi r\), a fraction of the circumference. Arcs and sectors
Calculate the area of a sector Sector area = \(\frac{\theta}{360} \times \pi r^2\), a fraction of the area of the circle. Arcs and sectors
Find the perimeter of a sector Add the two radii to the arc length. Arcs and sectors
Give answers in terms of π Leave π as a symbol and simplify, e.g. \(\frac{40}{360} \times \pi \times 9^2 = 9\pi\). Arcs and sectors
Solve area problems in context and backwards e.g. work out the cost of turf for a lawn, or the radius of a circle from its area. Area and perimeter
Find the surface area of a cylinder Two circles plus the curved surface: 2πr² + 2πrh. Volume and surface area
Use formulas for pyramids, cones and spheres Substitute into the formula, e.g. volume of a cone = ⅓πr²h. Volume and surface area
Work out a solid from its views Combine the plan and elevations to sketch the solid, then use it, e.g. to find its volume. Plans and elevations
Find missing lengths in similar shapes Multiply or divide a length by the scale factor. Similar shapes (length, area, volume)
Use similar triangles made by parallel lines Parallel lines give equal corresponding or alternate angles, so the triangles are similar. Similar shapes (length, area, volume)
Know the four congruence conditions SSS, SAS, ASA (or AAS) and RHS; three equal angles is not enough. Congruence and geometric proof
Choose the condition for two triangles Match the equal sides and angles, then name the condition that applies. Congruence and geometric proof
Describe a single transformation fully Give the type and every detail, e.g. rotation 90° clockwise about (1, 2). Transformations
Enlarge by a fractional scale factor A scale factor between 0 and 1 makes the shape smaller, still measured from the centre. Transformations
Carry out and describe combined transformations Do one transformation then the next, then describe the single transformation that has the same effect. Transformations
Multiply by a scalar and combine vectors Multiply both numbers by the scalar, e.g. work out 2a − 3b as a column vector. Vectors
Write a vector path using a and b Go along known vectors, e.g. \(\overrightarrow{AB} = -\mathbf{a} + \mathbf{b}\). Vectors
Find a back bearing Add or subtract 180°: if B is on a bearing of 070° from A, then A is on a bearing of 250° from B. Bearings and scale drawings
Calculate bearings and angles using angle facts Use parallel north lines with co-interior and alternate angles. Bearings and scale drawings
Use right-angled trigonometry in bearing problems Use Pythagoras and SOH CAH TOA when the journey makes a right-angled triangle. Bearings and scale drawings
Construct perpendiculars from and at a point Construct the perpendicular from a point to a line, or at a point on a line, with compasses. Constructions and loci
Draw loci and shade the required region Combine loci such as 'within 3 cm of A' and 'closer to B than to C', then shade where all are true. Constructions and loci
Solve loci problems on scale drawings e.g. show where a tree can be planted in a garden, using a scale drawing and constructions. Constructions and loci
Find probabilities written in terms of x Add the expressions, set the total equal to 1, solve, then substitute to find the probability asked for. Probability and relative frequency
Judge fairness and reliability from experiments Compare relative frequency with the theoretical probability, and explain that more trials give a more reliable estimate. Probability and relative frequency
Multiply for 'and', add for 'or' Multiply along each path, then add the probabilities of all the paths that give the result you want. Tree diagrams and combined events
Use tree diagrams without replacement For the second pick the total goes down by 1, and so does the count of the colour already taken. Tree diagrams and combined events
Complete a Venn diagram from given totals Start with the overlap, subtract it from each set's total to get the 'only' regions, then work out the number outside. Venn diagrams and set notation
Use complements such as A′ in set notation \(A'\) is everything not in \(A\), so \(A' \cap B\) is the part of \(B\) outside \(A\). Venn diagrams and set notation
Compare pie charts and criticise misleading diagrams Work out actual numbers when the totals are different, and spot axes that do not start at 0 or scales that are uneven. Charts, diagrams and time series
Estimate the mean of grouped data Use the midpoint of each class as the value of every item in that class. Averages and spread
Solve problems with totals and combined means Use total = mean × number of values to find a missing value or the mean of two groups together. Averages and spread
Complete a cumulative frequency table Keep a running total of the frequencies; the last value must equal the total frequency. Cumulative frequency and box plots
Make estimates using a line of best fit Read from the line inside the range of the data; estimates outside it (extrapolation) are unreliable. Scatter graphs and correlation
Explain why correlation does not prove causation Two variables can be correlated because both depend on something else. Scatter graphs and correlation
Estimate a population using capture–recapture Assume the proportion marked in the second sample equals the proportion marked in the whole population. Sampling and capture–recapture

Grade 6

Grade 6 checklist (PDF)

Use prime factor form to reason E.g. find the smallest number to multiply by to make a square number, using even powers. Primes, factors, HCF and LCM
Tell whether a fraction terminates or recurs In its simplest form, a fraction terminates only if the denominator has no prime factors other than 2 and 5. Recurring decimals
Convert a pure recurring decimal to a fraction E.g. \(x = 0.\dot{7}\dot{2}\): \(100x - x = 72\), so \(x = \frac{72}{99} = \frac{8}{11}\). Recurring decimals
Estimate powers and roots Use nearby squares or cubes: \(\sqrt{40}\) is between 6 and 7 (36 and 49), about 6.3. Indices and roots
Evaluate unit fractional indices \(a^{\frac{1}{n}}\) is the nth root of \(a\), so \(64^{\frac{1}{3}} = 4\). Indices and roots
Simplify a surd Take out the largest square factor: \(\sqrt{48} = \sqrt{16} \times \sqrt{3} = 4\sqrt{3}\). Surds
Multiply and divide surds \(\sqrt{a} \times \sqrt{b} = \sqrt{ab}\), e.g. \(\sqrt{6} \times \sqrt{10} = \sqrt{60} = 2\sqrt{15}\), and \(\sqrt{20} \div \sqrt{5} = \sqrt{4} = 2\). Surds
Add and subtract surds Simplify first, then collect like surds: \(\sqrt{50} + \sqrt{8} = 5\sqrt{2} + 2\sqrt{2} = 7\sqrt{2}\). Surds
Solve problems in standard form E.g. find how many times bigger one quantity is, or a total from a very small unit amount. Standard form
Find the bounds of a sum or product Upper bound: use both upper bounds; lower bound: use both lower bounds. Upper and lower bounds
Count when repeats are not allowed Each choice has one fewer option: 4-digit codes with different digits: 10 × 9 × 8 × 7 = 5040. Listing and the product rule
Find unknown constants in an identity Expand and simplify one side, then match the coefficients of each power of x and the number terms. Algebraic notation, substitution and identities
Factorise quadratics with x² coefficient above 1 Split the middle term using two numbers with product ac, e.g. \(3x^2 + 10x - 8 = (3x - 2)(x + 4)\). Factorising
Factorise fully in two stages Take out a common factor, then factorise the bracket, e.g. \(3x^2 - 12 = 3(x + 2)(x - 2)\). Factorising
Simplify algebraic fractions by factorising Factorise top and bottom, then cancel common brackets, e.g. \(\frac{3x + 6}{x^2 - 4} = \frac{3}{x - 2}\). Algebraic fractions
Add fractions with number denominators Use a common denominator, e.g. \(\frac{x}{3} + \frac{x}{4} = \frac{7x}{12}\). Algebraic fractions
Rearrange multi-step formulae with powers e.g. \(v^2 = u^2 + 2as\) gives \(u = \sqrt{v^2 - 2as}\). Changing the subject
Solve equations with several fractions Multiply every term by the LCM of the denominators, e.g. \(\frac{x}{2} + \frac{x}{3} = 10\) gives \(x = 12\). Linear equations
Factorise harder quadratics to solve them e.g. \(2x^2 + x - 6 = 0\) gives \((2x - 3)(x + 2) = 0\), so \(x = 1.5\) or \(x = -2\). Solving quadratic equations
Use the quadratic formula Substitute a, b and c into \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) and round as instructed. Solving quadratic equations
Rearrange an equation into a quadratic Expand brackets or clear fractions to get \(ax^2 + bx + c = 0\) before solving. Solving quadratic equations
Complete the square for simple quadratics Halve the coefficient of x: \(x^2 + 6x + 1 = (x + 3)^2 - 9 + 1 = (x + 3)^2 - 8\). Completing the square
Show inequalities as a region on a graph Draw each boundary line (dashed for \(\lt\) or \(\gt\), solid for \(\le\) or \(\ge\)) and shade the region that fits all of them. Inequalities
Sketch and interpret exponential graphs \(y = k^x\) with \(k \gt 1\) passes through (0, 1), rises more and more steeply and never touches the x-axis. Graphs of functions
Translate a graph vertically \(y = f(x) + a\) moves the graph up a units (down if a is negative). Transforming graphs
Evaluate a function for a given input Substitute the value for x, e.g. \(f(x) = 3x^2 - 1\) gives \(f(-2) = 11\). Composite and inverse functions
Solve an equation such as f(x) = k Set the function equal to the value and solve, e.g. \(2x + 7 = 15\) gives \(x = 4\). Composite and inverse functions
Show a solution lies between two values Substitute both values into the equation written as ... = 0; a change of sign shows there is a solution between them. Iteration
Write expressions for odd, even and consecutive numbers Even \(2n\), odd \(2n + 1\), consecutive integers \(n\), \(n + 1\), \(n + 2\). Algebraic proof
Disprove a statement with a counterexample Find one example where the statement is false and show why it is false. Algebraic proof
Show that two expressions are identical Expand and simplify one side until it matches the other exactly. Algebraic proof
Find acceleration from a velocity–time graph Acceleration is the gradient; a negative gradient is a deceleration. Gradients and areas under graphs
Find distance from a velocity–time graph Split the area under the graph into triangles, rectangles and trapezia and add them. Gradients and areas under graphs
Find the overall percentage change over time Work out multiplierⁿ, e.g. 0.9³ = 0.729 is a 27.1% decrease. Compound interest, growth and decay
Form and use equations like y = kx Substitute a known pair of values to find k, then use the formula. Direct and inverse proportion
Find the density of a mixture Divide the total mass by the total volume, working out each part separately first. Speed, density and pressure
Convert rates and algebraic quantities between units Change one unit at a time, e.g. 2.5 litres per second = 9000 litres per hour = 9 m³ per hour. Converting units
Prove angle results using known facts Use letters and a chain of reasons to show a general result, e.g. that the exterior angle of a triangle equals the sum of the two interior opposite angles. Angles, parallel lines and polygons
Give exact answers as simplified surds On a non-calculator Higher paper, an exact length such as \(\sqrt{45}\) is written as \(3\sqrt{5}\). Pythagoras' theorem (incl. 3D)
Find a triangle's area using ½ab sin C Use two sides and the angle between them: area = ½ × a × b × sin C. Sine rule, cosine rule and ½ab sin C
Use the tangent and radius facts A tangent meets a radius at 90°, and the two tangents from a point outside a circle are equal in length. Circle theorems
Use the angle in a semicircle An angle at the circumference subtended by a diameter is 90°. Circle theorems
Find the angle or radius of a sector Substitute the given arc length or area into the formula and solve for θ or r. Arcs and sectors
Form and solve equations from areas Write the area in terms of x, set it equal to the given value and solve, possibly as a quadratic. Area and perimeter
Solve problems with composite solids e.g. a cone on a hemisphere, or finding a length when the volume is given. Volume and surface area
Prove that two triangles are congruent Give three pairs of equal sides or angles with reasons, then state the condition. Congruence and geometric proof
Prove angle results using known facts Use a chain of angle facts with reasons to prove a general result, often using letters. Congruence and geometric proof
Solve problems with three events Multiply three probabilities along each path, count the paths you need, and use 1 − P(none) for 'at least one'. Tree diagrams and combined events
Complete a three-set Venn diagram Fill the centre (in all three sets) first, then the regions in exactly two sets, then each 'only' region and the outside. Venn diagrams and set notation
Find 'given that' probabilities from two-way tables Use only the row or column you are given, e.g. P(walks given Year 10) = Year 10 walkers ÷ all Year 10 students. Conditional probability
Find quartiles and IQR from a list In an ordered list of n values, the quartiles are the \(\frac{n+1}{4}\)th and \(\frac{3(n+1)}{4}\)th values. Cumulative frequency and box plots
Draw a cumulative frequency graph Plot each running total at the upper end of its class and join the points with a smooth curve or straight lines. Cumulative frequency and box plots
Estimate the median and quartiles from a graph Read across from \(\frac{n}{2}\), \(\frac{n}{4}\) and \(\frac{3n}{4}\) on the cumulative frequency axis to the curve, then down. Cumulative frequency and box plots
Draw and interpret a box plot Mark the minimum, lower quartile, median, upper quartile and maximum against a scale. Cumulative frequency and box plots
Calculate frequency density for each class Frequency density = frequency ÷ class width, so a wider class needs a lower bar for the same frequency. Histograms
Draw a histogram with unequal class widths Draw touching bars whose heights are the frequency densities, against a continuous scale. Histograms
Interpret the gradient of a line of best fit The gradient is the change in y for each increase of 1 in x, described in context. Scatter graphs and correlation
State the assumptions of capture–recapture The population does not change between samples, marked animals mix evenly, and marks are not lost. Sampling and capture–recapture

Grade 7

Grade 7 checklist (PDF)

Find numbers from a given HCF and LCM Work backwards from the HCF and LCM to find possible pairs of numbers. Primes, factors, HCF and LCM
Convert decimals with a non-recurring part E.g. \(x = 0.4\dot{1}\): \(100x - 10x = 41.11... - 4.11... = 37\), so \(x = \frac{37}{90}\). Recurring decimals
Prove a recurring decimal equals a fraction Show the full algebraic method, from 'let x = ...' to the simplified fraction. Recurring decimals
Evaluate any fractional or negative fractional index Root, then power, then flip if negative: \(8^{-\frac{2}{3}} = \frac{1}{2^2} = \frac{1}{4}\). Indices and roots
Expand brackets containing surds Multiply every term by every term: \((3 + \sqrt{2})(1 - \sqrt{2}) = 1 - 2\sqrt{2}\). Surds
Rationalise a denominator with a single surd For \(\frac{6}{\sqrt{3}}\), multiply top and bottom by \(\sqrt{3}\): \(\frac{6\sqrt{3}}{3} = 2\sqrt{3}\). Surds
Find the bounds of a difference or quotient For the upper bound of \(a - b\) or \(a \div b\), use the upper bound of \(a\) and the lower bound of \(b\). Upper and lower bounds
Use bounds with compound measures Upper bound of speed = upper bound of distance ÷ lower bound of time. Upper and lower bounds
Count pairs when order doesn't matter From \(n\) people, the number of pairs is \(\frac{n(n-1)}{2}\), e.g. 8 people make 28 pairs. Listing and the product rule
Expand the product of three binomials Expand two brackets first, then multiply by the third, e.g. \((x + 1)(x + 2)(x - 3) = x^3 - 7x - 6\). Expanding brackets
Multiply and divide algebraic fractions Factorise, flip the second fraction if dividing, then cancel and multiply. Algebraic fractions
Add fractions with algebraic denominators Multiply the denominators to get a common denominator, e.g. for \(\frac{2}{x + 1} + \frac{3}{x - 2}\) use \((x + 1)(x - 2)\). Algebraic fractions
Simplify fractions involving harder quadratics Factorise quadratics such as \(2x^2 - 5x - 3\) before cancelling. Algebraic fractions
Use fractional indices in algebra The denominator is a root and the numerator a power, e.g. \((16x^8)^{\frac{1}{2}} = 4x^4\). Laws of indices in algebra
Rearrange when the subject appears twice Collect the subject terms on one side and factorise, e.g. \(ax + b = cx + d\) gives \(x = \frac{d - b}{a - c}\). Changing the subject
Form and solve a quadratic in context e.g. from an area; reject any solution that is impossible, such as a negative length. Solving quadratic equations
Solve a quadratic by completing the square Rearrange to \((x + p)^2 = k\) and square root both sides, e.g. \((x + 3)^2 = 8\) gives \(x = -3 \pm 2\sqrt{2}\). Completing the square
Find the turning point from completed square form \(y = (x + p)^2 + q\) has its minimum point at \((-p, q)\). Completing the square
Find unknowns by comparing forms Expand the completed square and match coefficients, e.g. \(x^2 + bx + 11 \equiv (x + 4)^2 + c\) gives \(b = 8\), \(c = -5\). Completing the square
Solve linear and quadratic simultaneous equations Substitute the linear equation into the quadratic, solve, then find the matching values. Simultaneous equations
Solve a quadratic inequality Find the roots, sketch the graph and choose the part above or below the x-axis, e.g. \(x^2 - x - 6 \lt 0\) gives \(-2 \lt x \lt 3\). Inequalities
Find the nth term of a quadratic sequence Halve the second difference to get the coefficient of \(n^2\), subtract that \(n^2\) term, then find the linear part. Sequences and the nth term
Find perpendicular gradients and lines Perpendicular gradients multiply to −1, so a line perpendicular to \(y = 2x + 1\) has gradient \(-\frac{1}{2}\). Straight-line graphs
Sketch and use trigonometric graphs Know \(y = \sin x\) and \(y = \cos x\) (repeat every 360°) and \(y = \tan x\) (repeats every 180°), and use symmetry to find angles. Graphs of functions
Translate a graph horizontally \(y = f(x + a)\) moves the graph a units to the left; \(y = f(x - a)\) moves it a units to the right. Transforming graphs
Reflect a graph in either axis \(y = -f(x)\) reflects it in the x-axis; \(y = f(-x)\) reflects it in the y-axis. Transforming graphs
Find where a point moves to Apply the transformation to the coordinates, e.g. under \(y = f(x - 3)\), (2, 5) moves to (5, 5). Transforming graphs
Find a composite function fg(x) means apply g first, then f: substitute g(x) into f. Composite and inverse functions
Find an inverse function Write \(y = f(x)\), make x the subject, then write \(f^{-1}(x)\) in terms of x. Composite and inverse functions
Use an iterative formula Substitute \(x_0\) to get \(x_1\), then \(x_1\) to get \(x_2\), and so on, using ANS on your calculator. Iteration
Find a solution to a given accuracy Keep iterating until successive values round to the same answer, e.g. to 3 decimal places. Iteration
Show an equation rearranges to a given form e.g. show that \(x^3 + 4x - 1 = 0\) can be written as \(x = \frac{1 - x^3}{4}\). Iteration
Prove results about odd and even numbers e.g. the sum of two odd numbers: \((2m + 1) + (2n + 1) = 2(m + n + 1)\), which is even. Algebraic proof
Prove an expression is a multiple Factorise to show it is k × an integer, e.g. \(6n + 6 = 6(n + 1)\). Algebraic proof
Find the radius from a circle's equation \(x^2 + y^2 = 36\) is a circle with centre (0, 0) and radius 6. Equation of a circle and tangents
Write the equation of a circle Radius 5 gives \(x^2 + y^2 = 25\); through (3, −4) gives \(r^2 = 9 + 16 = 25\), the same circle. Equation of a circle and tangents
Check whether a point lies on a circle Substitute the coordinates: the point is on the circle only if \(x^2 + y^2\) equals \(r^2\). Equation of a circle and tangents
Estimate a gradient by drawing a tangent Draw a tangent at the point, make a large right-angled triangle and work out rise ÷ run. Gradients and areas under graphs
Find an average rate of change Work out the gradient of the chord joining two points on the curve. Gradients and areas under graphs
Estimate the area under a curve Split it into strips of equal width and add the areas of the trapezia. Gradients and areas under graphs
Form an equation when a ratio changes Write the amounts as multiples of x, apply the change and solve the equation given by the new ratio. Ratio
Combine percentage changes using multipliers Multiply the multipliers, e.g. if the length and width of a rectangle both go up by 10%, its area is multiplied by 1.1 × 1.1 = 1.21. Percentages and reverse percentages
Work backwards to find the original value Divide the final value by multiplierⁿ. Compound interest, growth and decay
Use an iterative formula for growth or decay Apply a formula such as \(u_{n+1} = 1.04u_n\) one step at a time. Compound interest, growth and decay
Form equations involving squares, cubes and roots Write, for example, \(y = kx^2\) or \(y = \frac{k}{\sqrt{x}}\) and find k from given values. Direct and inverse proportion
Use Pythagoras in 3D shapes Find the space diagonal of a cuboid or the height of a pyramid using two right-angled triangles. Pythagoras' theorem (incl. 3D)
Find lengths and angles in 3D shapes Find the angle between a line and a plane using a right-angled triangle inside the shape. Right-angled trigonometry
Find a missing side with the sine rule Use a ÷ sin A = b ÷ sin B when you know an angle and the side opposite it. Sine rule, cosine rule and ½ab sin C
Find a missing angle with the sine rule Use sin A ÷ a = sin B ÷ b, then sin⁻¹. Sine rule, cosine rule and ½ab sin C
Find a missing side with the cosine rule Use a² = b² + c² − 2bc cos A when you know two sides and the angle between them. Sine rule, cosine rule and ½ab sin C
Find a missing angle with the cosine rule Use cos A = (b² + c² − a²) ÷ 2bc when you know all three sides. Sine rule, cosine rule and ½ab sin C
Use the angle at the centre theorem The angle at the centre is twice the angle at the circumference subtended by the same arc. Circle theorems
Use same-segment and cyclic quadrilateral facts Angles in the same segment are equal, and opposite angles of a cyclic quadrilateral add up to 180°. Circle theorems
Use the perpendicular bisector of a chord The perpendicular from the centre to a chord bisects the chord. Circle theorems
Find the area of a segment Sector area minus the area of the triangle, using ½ab sin C for the triangle. Arcs and sectors
Find the volume of a frustum Volume of the large cone minus the small cone, using similar triangles for any missing lengths. Volume and surface area
Use the area scale factor If lengths are multiplied by k, areas are multiplied by k². Similar shapes (length, area, volume)
Use the volume scale factor If lengths are multiplied by k, volumes (and masses of the same material) are multiplied by k³. Similar shapes (length, area, volume)
Use congruence to prove further results After proving triangles congruent, deduce that other sides or angles are equal. Congruence and geometric proof
Enlarge by a negative scale factor The image is on the opposite side of the centre and upside down. Transformations
Use ratios to find vectors along lines If P divides AB in the ratio 1 : 3, then \(\overrightarrow{AP} = \frac{1}{4}\overrightarrow{AB}\). Vectors
Use sine and cosine rules with bearings When the triangle is not right-angled, use the sine rule or the cosine rule. Bearings and scale drawings
Use algebra to find missing Venn numbers Write each region in terms of \(x\), add them all to make the total and solve the equation. Venn diagrams and set notation
Find conditional probabilities from a Venn diagram For 'given that A', divide the number in the region you want by the number in \(A\), not by the grand total. Venn diagrams and set notation
Find conditional probabilities from a Venn diagram Divide the number in the part you need by the number in the whole of the given set. Conditional probability
Use conditional probabilities on tree diagrams The second set of branches gives the probability of the second event given what happened first, so the branches can differ. Conditional probability
Use expected frequencies to find conditional probabilities Turn the probabilities into numbers out of a convenient total, such as 100 or 1000, then read off the fraction you need. Conditional probability
Find an unknown frequency from a mean Write the mean as an algebraic fraction, set it equal to the given mean and solve the equation. Averages and spread
Estimate how many are above a value Read up from the value to the curve and across; for 'more than', subtract this reading from the total. Cumulative frequency and box plots
Compare distributions using median and IQR Compare the medians (on average) and the interquartile ranges (consistency), in context. Cumulative frequency and box plots
Find frequencies from a histogram Frequency = frequency density × class width, which is the area of the bar. Histograms
Decide if an estimate is too high or too low Work out how a broken assumption changes the number of marked animals recaptured, and so the estimate. Sampling and capture–recapture

Grade 8

Grade 8 checklist (PDF)

Rationalise a denominator containing a bracket For \(\frac{1}{3 + \sqrt{2}}\), multiply top and bottom by \(3 - \sqrt{2}\), since \((3 + \sqrt{2})(3 - \sqrt{2}) = 7\), giving \(\frac{3 - \sqrt{2}}{7}\). Surds
Give exact answers in geometry problems Keep surds in Pythagoras and trigonometry, e.g. a hypotenuse of \(\sqrt{72} = 6\sqrt{2}\) cm. Surds
Choose a suitable degree of accuracy Round both bounds; give the most accurate value that they both round to. Upper and lower bounds
Use bounds in formulas and trigonometry Decide which bound of each value makes the answer as large (or small) as possible. Upper and lower bounds
Count with restrictions on positions Fill the restricted places first and split into cases if needed, e.g. even numbers greater than 3000. Listing and the product rule
Solve equations with algebraic fractions Multiply every term by the common denominator, usually giving a quadratic to solve. Algebraic fractions
Simplify with negative fractional indices Root, then power, then take the reciprocal, e.g. \((27x^6)^{-\frac{2}{3}} = \frac{1}{9x^4}\). Laws of indices in algebra
Rearrange fractions with the subject twice e.g. \(y = \frac{x + 2}{x - 3}\) gives \(x = \frac{3y + 2}{y - 1}\). Changing the subject
Complete the square when a is not 1 Take out a factor of a first, e.g. \(2x^2 - 8x + 3 = 2(x - 2)^2 - 5\). Completing the square
Prove a quadratic is always positive Complete the square and use the fact that a square is never negative. Completing the square
Find where a line meets a circle Substitute into \(x^2 + y^2 = r^2\), e.g. \(y = x + 1\) and \(x^2 + y^2 = 13\) meet at \((2, 3)\) and \((-3, -2)\). Simultaneous equations
Continue geometric sequences involving surds e.g. 2, \(2\sqrt{3}\), 6, \(6\sqrt{3}\), …: multiply by \(\sqrt{3}\) each time. Sequences and the nth term
Solve equations by drawing a suitable line e.g. use the graph of \(y = x^2 - 3x\) and the line \(y = x + 2\) to solve \(x^2 - 4x - 2 = 0\). Graphs of functions
Describe a combined translation as a vector \(y = f(x + 2) - 1\) is a translation by \(\begin{pmatrix} -2 \\ -1 \end{pmatrix}\). Transforming graphs
Transform trigonometric graphs e.g. \(y = \cos(x - 90°)\) is the cos graph moved 90° to the right, which gives the same graph as \(y = \sin x\). Transforming graphs
Solve equations involving composite functions e.g. form and solve an equation from \(gf(x) = 25\). Composite and inverse functions
Explain what the iterated values represent The values get closer to a solution of the original equation. Iteration
Prove results about consecutive and square numbers e.g. the difference between the squares of two consecutive odd numbers is a multiple of 8. Algebraic proof
Find the equation of a tangent Find the gradient of the radius, take the negative reciprocal, then substitute the point to find c. Equation of a circle and tangents
Find where a line meets a circle Substitute the line into \(x^2 + y^2 = r^2\) and solve the quadratic. Equation of a circle and tangents
Tell whether trapezia overestimate or underestimate Trapezia overestimate if the curve bends upwards, and underestimate if it bends downwards. Gradients and areas under graphs
Interpret gradients and areas in context Say what the value means, with units, e.g. an area under a velocity–time graph is a distance in metres. Gradients and areas under graphs
Find the rate from start and end values Work out final ÷ start, take the nth root to get the multiplier, then turn it into a percentage. Compound interest, growth and decay
Work out the effect of changing a variable If \(y = kx^2\) and x is tripled, y is multiplied by 9. Direct and inverse proportion
Choose and combine rules in multi-step problems Pick the right rule at each stage, e.g. cosine rule for a side, then ½ab sin C for an area, including bearings problems. Sine rule, cosine rule and ½ab sin C
Use the alternate segment theorem The angle between a tangent and a chord equals the angle in the alternate segment. Circle theorems
Solve multi-step problems with full reasons Combine several theorems and angle facts, stating the reason for each step. Circle theorems
Find length scale factors from areas or volumes Square root the area factor or cube root the volume factor to get k, then use it. Similar shapes (length, area, volume)
Prove lines parallel or points collinear Show one vector is a multiple of another; for collinear points they must also share a point. Vectors
Form and solve an equation from probabilities Write the branch probabilities in terms of an unknown, set up an equation (often a quadratic) and solve it. Tree diagrams and combined events
Test whether two events are independent Compare P(A and B) with P(A) × P(B), or P(A given B) with P(A): if they are equal, the events are independent. Conditional probability
Check a box plot against a graph Test whether the median, quartiles and extreme values of a box plot agree with a cumulative frequency graph or table. Cumulative frequency and box plots
Work out a missing frequency density scale Use a bar whose frequency you know to find what each unit of height, or each square, represents. Histograms
Estimate a frequency within part of a class Assume the values are spread evenly through the class, and use frequency density × the width of the part you need. Histograms

Grade 9

Grade 9 checklist (PDF)

Solve multi-step surd problems E.g. find integers \(a\) and \(b\) when an area or expression is written as \(a + b\sqrt{c}\). Surds
Find inverses of fractional functions When x appears twice after rearranging, collect the x terms and factorise, e.g. for \(f(x) = \frac{3x + 2}{x - 4}\). Composite and inverse functions
Construct harder multi-step proofs Rearrange into a useful form, e.g. \((2n + 1)(2n + 3) = 4(n^2 + 2n + 1) - 1\), one less than a multiple of 4. Algebraic proof
Solve problems involving tangents and axes e.g. find where a tangent meets the axes and the area of the triangle it makes. Equation of a circle and tangents
Work backwards and handle obtuse angles Find an angle or side from a given area, and use 180° minus the sin⁻¹ value when the angle is obtuse. Sine rule, cosine rule and ½ab sin C
Prove circle theorems and related results Use isosceles triangles made by radii, and algebra, to prove a result in general. Circle theorems
Find the first event given the second Divide P(first and second) by the total P(second), e.g. P(rain given late) = P(rain and late) ÷ P(late). Conditional probability
Estimate the median from a histogram Find the class containing the \(\frac{n}{2}\)th value and work out how far through that class it lies. Histograms

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