Mathematics

Structure and calculation

This notation keeps very large and very small numbers short by writing them as a value that is at least 1 and below 10 multiplied by a power of ten. The index counts how far the decimal point has moved, so 4500000 is 4.5 × 10⁶, and 0.00072 is 7.2 × 10⁻⁴, where the negative index shows a number between 0 and 1. To multiply, multiply the front values and add the indices; to divide, divide the front values and subtract the indices. Then check that the front value is still at least 1 and below 10, adjusting if it is not, so 40 × 10⁵ is rewritten as 4 × 10⁶. To add or subtract, either turn both numbers into ordinary form first, or rewrite them with the same index and then combine the front values.

calculate with and interpret standard form A × 10ⁿ, where 1 ≤ A < 10 and n is an integer, calculate exactly with surds simplify surd expressions involving squares (eg √12 = √(4 × 3) = √4 × √3 = 2√3) and rationalise denominators (Higher tier only), calculate exactly with fractions +11 more

Fractions, decimals and percentages

The word of means multiply, so a fraction or a percentage acts on a number and scales it. To find 3/4 of 60, divide by 4 and multiply by 3, giving 45. A percentage does the same job once it is written as a decimal, so 15% of 40 is 0.15 × 40 = 6. The version that does the most work is the multiplier: an increase of 15% is a multiplier of 1.15 and a decrease of 15% is a multiplier of 0.85, which lets several changes run in a single line. A price of 200 pounds rising by 15% and then falling by 10% gives 200 × 1.15 × 0.9 = 207 pounds. Working backwards, divide by the multiplier instead of taking the percentage off, so a price of 207 pounds after a 15% rise started at 207 ÷ 1.15 = 180 pounds. Check each time which amount the operator is acting on.

interpret fractions and percentages as operators, identify and work with fractions in ratio problems, work interchangeably with terminating decimals and their corresponding fractions (such as 3.5 and 7/2 or 0.375 and 3/8) +1 more

Measures and accuracy

An estimate is a deliberately rough answer you can produce in your head, and its purpose is to tell you whether a calculated value is believable. The standard method is to round every number to one significant figure and work with those instead. For 38.2 × 5.1 divided by 0.198, use 40 × 5 = 200, then 200 ÷ 0.2 = 1000, so the true answer should sit near a thousand; a display reading 98 or 10000 means a key went astray. Write the estimate with the ≈ sign rather than an equals sign. Dividing a positive number by a number between 0 and 1 makes the result larger, and that is the step most often taken backwards. A calculation can also be checked by reversing it: if 17 × 23 = 391, then 391 ÷ 23 must return 17.

estimate answers check calculations using approximation and estimation, including answers obtained using technology, including upper and lower bounds (Higher tier only), use inequality notation to specify simple error intervals due to truncation or rounding +3 more

Notation, vocabulary and manipulation

Changing the subject means rewriting a formula so a different letter stands alone on one side. The moves are the ones you use to solve an equation, applied to letters instead of numbers, and whatever you do to one side you do to the other. To make r the subject of C = 2πr, divide both sides by 2π, which gives r = C/(2π). Work from the outside in, undoing whatever is applied last to the letter you want: in 3x + 2 you take the 2 away first, but in 3(x + 2) you divide by 3 first. If the letter you want appears twice, gather those terms on one side, take it out as a common factor, then divide by the bracket. A question often gives a standard formula, asks for a new subject, and then asks you to use the rearranged version with given numbers.

understand and use standard mathematical formulae rearrange formulae to change the subject, know the difference between an equation and an identity argue mathematically to show algebraic expressions are equivalent, and use algebra to support and construct arguments, to include identities +9 more

Graphs

The two axes cut the plane into four regions. A point is written as an ordered pair, across first and then up, with right and up counted as positive and left and down as negative. So the top right region holds points like (3, 4), the top left holds (−3, 4), the bottom left holds (−3, −4) and the bottom right holds (3, −4). To read a point off a grid, start at the origin, count along the horizontal axis, then count parallel to the vertical axis, keeping the sign of each direction. You also need the midpoint of a line segment, which is the average of the two horizontal values paired with the average of the two vertical values: the midpoint of (−2, 6) and (4, −2) is (1, 2). Always check the scale, because one square is often worth more than one unit.

work with coordinates in all four quadrants, calculate or estimate gradients of graphs and areas under graphs (including quadratic and other non- linear graphs), and interpret results in cases such as distance-time graphs, velocity-time graphs and graphs in financial contexts (Higher tier only), plot graphs of equations that correspond to straight-line graphs in the coordinate plane +13 more

Solving equations and inequalities

These questions hand you words and expect algebra back. Name the unknown first and say what it stands for, with its unit: let c be the cost of one child ticket in pounds. Then convert the information a phrase at a time, so three more than twice a number is 2n + 3, and a rectangle whose length is 4 cm more than its width has perimeter 2w + 2(w + 4). Two facts about two unknowns give two equations to solve together. For instance, two adult and three child tickets cost £31 while one adult and four child tickets cost £23, so 2a + 3c = 31 and a + 4c = 23, which solve to a = 11 and c = 3. The final step is interpretation: reply in the language of the question, saying that an adult ticket costs £11.

translate simple situations or procedures into algebraic expressions or formulae derive an equation (or two simultaneous equations), solve the equation(s) and interpret the solution, solve linear inequalities in one or two variable(s), and quadratic inequalities in one variable represent the solution set on a number line, using set notation and on a graph (Higher tier only), solve linear inequalities in one variable represent the solution set on a number line +7 more

Sequences

This is Higher tier only. Some geometric sequences have a common ratio that is a surd, so when the first term is a whole number the terms alternate between surd form and whole numbers. Take 1, √3, 3, 3√3, 9: each term is the one before it multiplied by √3, because √3 × √3 = 3. To find the ratio, divide a term by the previous one and rationalise if you need to, for example 3 ÷ √3 becomes √3 once you multiply top and bottom by √3. You may also meet sequences built from other rules, including ones that mix a linear part with a multiplying part. Work in exact surd form the whole way through instead of switching to decimals, since exact answers are wanted, and remember that squaring a surd removes the root sign, so (√5)² = 5.

including other sequences including where r is a surd (Higher tier only), including quadratic sequences (Higher tier only), generate terms of a sequence from either a term-to-term or a position-to-term rule +3 more

Ratio, proportion and rates of change

On a real-life graph the steepness tells you how fast one quantity changes as the other changes, and its unit is the vertical axis unit per horizontal axis unit. Work it out as the change up divided by the change across, using two points that are easy to read. On a distance-time graph that number is a speed; on a graph of pay against hours worked it is the hourly rate; on a tank filling with water it is litres per minute. Example: a line from (0, 0) to (4, 30) on a distance-time graph measured in km and hours gives 30 ÷ 4, a speed of 7.5 km/h. A straight line through the origin shows the direct kind of proportion. The inverse kind gives a curve falling from the top left, approaching both axes but never meeting them.

interpret the gradient of a straight-line graph as a rate of change recognise and interpret graphs that illustrate direct and inverse proportion, use scale factors, scale diagrams and maps, solve problems involving direct and inverse proportion, including graphical and algebraic representations +18 more

Properties and constructions

Geometry has a fixed vocabulary and questions expect you to use it. A point is named with a capital letter, the line segment joining two points is written AB, and an angle is written with three letters, ABC, with the vertex in the middle. The standard triangle labelling puts side a opposite angle A, side b opposite angle B and side c opposite angle C. On a diagram, dashes mark equal sides, arrowheads mark parallel sides and a small square marks a right angle. A polygon is regular when its sides are all equal and its angles are all equal as well. Reflection symmetry is counted in lines, rotation symmetry in order, meaning how many times the shape fits onto itself in one full turn. A typical question gives a written description, such as a quadrilateral with one pair of parallel sides and no right angles, and asks you to draw it.

use conventional terms and notations: points, lines, vertices, edges, planes, parallel lines, perpendicular lines, right angles, polygons, regular polygons and polygons with reflection and/or rotation symmetries use the standard conventions for labelling and referring to the sides and angles of triangles draw diagrams from written description, apply and prove the standard circle theorems concerning angles, radii, tangents and chords, and use them to prove related results (Higher tier only), use the basic congruence criteria for triangles (SSS, SAS, ASA, RHS) +14 more

Mensuration and calculation

A sphere of radius r has volume ⁴⁄₃πr³ and surface area 4πr². A cone with radius r, perpendicular height h and slant height l has volume ⅓πr²h and curved surface area πrl, with an extra πr² added when the flat base counts. A pyramid has volume ⅓ × base area × perpendicular height. For a solid made of several pieces, work each piece out on its own line and then combine, taking care not to count a face that is hidden where two pieces meet. Example: a hemisphere of radius 3 cm has volume ½ × ⁴⁄₃ × π × 3³ = 18π cm³. Slant height and perpendicular height are different lengths, and when only one is given you get the other from a² + b² = c² inside the cone.

surface area and volume of spheres, pyramids, cones and composite solids, know the exact values of sin θ and cos θ for θ = 0°, 30°, 45°, 60° and 90° know the exact value of tan θ for θ = 0°, 30°, 45°, 60°, measure line segments and angles in geometric figures, including interpreting maps and scale drawings and use of bearings +10 more

Vectors

Column vectors combine one row at a time: add the top numbers together and the bottom numbers together, and take them away the same way. Multiplying by a scalar multiplies both entries, so doubling a vector keeps its direction and makes it twice as long, while multiplying by −1 keeps the length and reverses the direction. On a diagram, adding means placing vectors nose to tail, so a followed by b takes you along the route a + b, and taking b away means travelling the reverse of b instead. Vectors are printed in bold and underlined when written by hand, and the vector from A to B is written as AB with an arrow above it. Two vectors are equal when they have the same length and the same direction, wherever they sit on the page.

apply addition and subtraction of vectors, multiplication of vectors by a scalar, and diagrammatic and column representations of vectors, use vectors to construct geometric arguments and proofs (Higher tier only), describe translations as 2D vectors

Probability

A possibility space is the full set of what can happen, best drawn as a grid when two things happen together. Put the outcomes of one experiment along the top, the other down the side, and fill each cell with the combined result. For the total of two fair dice the grid has 36 cells, and the totals inside them run from 2 up to 12. Because the cells are equally likely, a probability is the number of cells that fit over the number of cells altogether: P(total of 5) = 4/36 = 1/9. Equally likely is the condition that makes this work, so it applies to fair dice, coins and spinners. Check whether the cells should hold sums, differences, products or ordered pairs, since the entries change completely.

construct theoretical possibility spaces for single and combined experiments with equally likely outcomes and use these to calculate theoretical probabilities, relate relative expected frequencies to theoretical probability, using appropriate language and the 0 to 1 probability scale, including using tree diagrams +7 more

Statistics

A time series records the same measurement at repeated times, so time always goes on the horizontal axis and the measured quantity on the vertical axis. Plot each reading at its time, then join consecutive points with straight lines; those lines guide your eye between readings, they are not readings of their own. To read a trend, look at the overall direction across the whole graph rather than at one step, because sales that dip every winter and rise every summer can still be growing year on year. A question gives a table of readings and asks you to finish the graph, describe the trend, or compare two periods. Read the scale carefully, since this kind of vertical axis often starts above zero.

including tables and line graphs for time series data, use and interpret scatter graphs of bivariate data recognise correlation, know that it does not indicate causation draw estimated lines of best fit make predictions interpolate and extrapolate apparent trends whilst knowing the dangers of so doing +6 more

Mathematics

This subtopic covers fundamental number concepts and algebraic manipulation, including arithmetic operations, fractions, decimals, percentages, and the use of algebraic expressions, equations, and inequalities. It forms the basis for problem-solving in various mathematical contexts, enabling students to model real-world situations and develop logical reasoning skills.

Number and Algebra

Algebra

Algebra involves the use of symbols and notation to represent mathematical relationships, expressions, and functions. Students learn to manipulate algebraic expressions, solve various types of equations and inequalities, and interpret graphical representations of linear, quadratic, and other functions.

Number

Algebra involves the use of symbols and notation to represent mathematical relationships, expressions, and functions. Students learn to manipulate algebraic expressions, solve various types of equations and inequalities, and interpret graphical representations of linear, quadratic, and other functions.

Geometry and measures

Algebra involves the use of symbols and notation to represent mathematical relationships, expressions, and functions. Students learn to manipulate algebraic expressions, solve various types of equations and inequalities, and interpret graphical representations of linear, quadratic, and other functions.