Mathematics 0580 — specification explorer

1. Types of number Notes and examples

• E1.1.1

  Identify and use: Example tasks include:

  • natural numbers
  • integers (positive, zero and negative)
  • prime numbers
  • square numbers
  • cube numbers
  • common factors
  • common multiples
  • rational and irrational numbers
  • reciprocals.

2. Sets Notes and examples

• E1.2.1Understand and use set language, notation and Venn diagrams are limited to two or three sets. Venn diagrams to describe sets and represent The following set notation will be used: relationships between sets.
• E1.2.2Example definition of sets:

3. Powers and roots Notes and examples

• E1.3.1

  Calculate with the following: Includes recall of squares and their corresponding corresponding roots of 1, 2, 3, 4, 5 and 10, e.g.:

  • squares roots from 1 to 15, and recall of cubes and their
  • square roots
  • cubes
  • cube roots
  • other powers and roots of numbers.

4. Fractions, decimals and percentages Notes and examples

• E1.4.1

  Use the language and notation of the following in Candidates are expected to be able to write appropriate contexts: fractions in their simplest form.:

  • proper fractions Recurring decimal notation
  • improper fractions
  • mixed numbers
  • decimals
  • percentages.
• E1.4.2Recognise equivalence and convert between Includes converting between recurring decimals these forms. and fractions and vice versa, e.g. write

5. Ordering Notes and examples

• E1.5.1Order quantities by magnitude and demonstrate familiarity with the symbols =, ≠, >, < , ⩾ and ⩽ .

6. The four operations Notes and examples

• E1.6.1Use the four operations for calculations with Includes: integers, fractions and decimals, including correct ordering of operations and use of brackets.

7. Indices I Notes and examples

• E1.7.1Understand and use indices (positive, zero, Examples include: negative, and fractional).
• E1.7.2Understand and use the rules of indices. e.g. find the value of

8. Standard form Notes and examples

• E1.8.1Use the standard form A × 10 where n is a positive or negative integer and 1 ⩽ A < 10.
• E1.8.2Convert numbers into and out of standard form.
• E1.8.3Calculate with values in standard form.

9. Estimation Notes and examples

• E1.9.1Round values to a specified degree of accuracy. Includes decimal places and significant figures.
• E1.9.2Make estimates for calculations involving e.g. write 5764 correct to the nearest thousand. numbers, quantities and measurements. e.g. by writing each number correct to 1 significant figure, estimate the value of
• E1.9.3Round answers to a reasonable degree of accuracy in the context of a given problem.

10. Limits of accuracy Notes and examples

• E1.10.1Give upper and lower bounds for data rounded e.g. write down the upper bound of a length to a specified accuracy. measured correct to the nearest metre.
• E1.10.2

  Find upper and lower bounds of the results of Example calculations include: calculations which have used data rounded to a

  • specified accuracy.

11. Ratio and proportion Notes and examples

• E1.11.1

  Understand and use ratio and proportion to: context. value.:

  • give ratios in their simplest form e.g. 20 : 30 : 40 in its simplest form is 2 : 3 : 4.
  • divide a quantity in a given ratio
  • use proportional reasoning and ratios in e.g. adapt recipes; use map scales; determine best

12. Rates Notes and examples

• E1.12.1Use common measures of rate. e.g. calculate with:
• E1.12.2Apply other measures of rate. e.g. calculate with:
• E1.12.3Required formulas will be given in the question. 3 Solve problems involving average speed. Knowledge of speed/distance/time formula is required. e.g. A cyclist travels 45 km in 3 hours 45 minutes. What is their average speed?
• E1.12.4Notation used will be, e.g. m/s (metres per second),

13. Percentages Notes and examples

• E1.13.1Calculate a given percentage of a quantity.
• E1.13.2Express one quantity as a percentage of another.
• E1.13.3Calculate percentage increase or decrease.
• E1.13.4Calculate with simple and compound interest. Problems may include repeated percentage change.
• E1.13.5Formulas are 5 Calculate using reverse percentages. e.g. find the cost price given the selling price and the percentage profit.
• E1.13.6Percentage calculations may include:

14. Using a calculator Notes and examples

• E1.14.1Use a calculator efficiently. e.g. know not to round values within a calculation and to only round the final answer.
• E1.14.2Enter values appropriately on a calculator. e.g. enter 2 hours 30 minutes as 2.5 hours or
• E1.14.3Interpret the calculator display appropriately. e.g. in money 4.8 means $4.80; in time 3.25 means 3 hours 15 minutes.

15. Time Notes and examples

• E1.15.1Calculate with time: seconds (s), minutes (min), 1 year = 365 days. hours (h), days, weeks, months, years, including the relationship between units.
• E1.15.2Calculate times in terms of the 24-hour and In the 24-hour clock, for example, 3.15 a.m. will be 12-hour clock. denoted by 03 15 and 3.15 p.m. by 15 15.
• E1.15.3Read clocks and timetables. Includes problems involving time zones, local times and time differences.

16. Money Notes and examples

• E1.16.1Calculate with money.
• E1.16.2Convert from one currency to another.

17. Exponential growth and decay Notes and examples

• E1.17.1Use exponential growth and decay. e.g. depreciation, population change.
• E1.17.2Knowledge of

18. Surds Notes and examples

• E1.18.1Understand and use surds, including simplifying Examples include: expressions.
• E1.18.2Rationalise the denominator. Examples include:

1. Introduction to algebra Notes and examples

• E2.1.1Know that letters can be used to represent generalised numbers.
• E2.1.2Substitute numbers into expressions and formulas.

2. Algebraic manipulation Notes and examples

• E2.2.1Simplify expressions by collecting like terms. Simplify means give the answer in its simplest form, e.g.
• E2.2.2Expand products of algebraic expressions. e.g. expand
• E2.2.3Includes products of more than two brackets, e.g. expand 3 Factorise by extracting common factors. Factorise means factorise fully, e.g.
• E2.2.4

  Factorise expressions of the form:

  • ax + bx + kay + kby
  • a x − b y
  • a + 2 ab + b
  • ax + bx + c
  • ax + bx + cx .
• E2.2.5Complete the square for expressions in the form ax + bx + c .

3. Algebraic fractions Notes and examples

• E2.3.1Manipulate algebraic fractions. Examples include:
• E2.3.2Factorise and simplify rational expressions. e.g.

4. Indices II Notes and examples

• E2.4.1Understand and use indices (positive, zero, e.g. solve: negative and fractional). •
• E2.4.2Understand and use the rules of indices. e.g. simplify:
• E2.4.3Knowledge of logarithms is

5. Equations Notes and examples

• E2.5.1Construct expressions, equations and formulas. e.g. write an expression for the product of two consecutive even numbers.
• E2.5.2Includes constructing simultaneous equations. 2 Solve linear equations in one unknown. Examples include:
• E2.5.3Solve fractional equations with numerical and Examples include: linear algebraic denominators.
• E2.5.4Solve simultaneous linear equations in two unknowns.
• E2.5.5Solve simultaneous equations, involving one With powers no higher than two. linear and one non-linear.
• E2.5.6Solve quadratic equations by factorisation, Includes writing a quadratic expression in completing the square and by use of the completed square form. quadratic formula. Candidates may be expected to give solutions in surd form.
• E2.5.7The quadratic formula is given in the List of formulas. 7 Change the subject of formulas. e.g. change the subject of a formula where:

6. Inequalities Notes and examples

• E2.6.1Represent and interpret inequalities, including on When representing and interpreting inequalities on a number line. a number line: e.g.
• E2.6.2Construct, solve and interpret linear inequalities. Examples include:
• E2.6.3Represent and interpret linear inequalities in two The following conventions should be used: variables graphically. e.g.
• E2.6.4List inequalities that define a given region. Linear programming problems are

7. Sequences Notes and examples

• E2.7.1Continue a given number sequence or pattern. Subscript notation may be used, e.g. term of sequence
• E2.7.2Recognise patterns in sequences, including the Includes linear, quadratic, cubic and exponential term-to-term rule, and relationships between sequences and simple combinations of these. different sequences.
• E2.7.3Find and use the n th term of sequences.

8. Proportion Notes and examples

• E2.8.1Express direct and inverse proportion in algebraic Includes linear, square, square root, cube and cube terms and use this form of expression to find root proportion. unknown quantities. Knowledge of proportional symbol (

9. Graphs in practical situations Notes and examples

• E2.9.1Use and interpret graphs in practical situations Includes estimation and interpretation of the including travel graphs and conversion graphs. gradient of a tangent at a point.
• E2.9.2Draw graphs from given data.
• E2.9.3Apply the idea of rate of change to simple kinematics involving distance–time and speed–time graphs, acceleration and deceleration.
• E2.9.4Calculate distance travelled as area under a Areas will involve linear sections of the graph only. speed–time graph.

10. Graphs of functions Notes and examples

• E2.10.1

  Construct tables of values, and draw, recognise Examples include: and interpret graphs for functions of the following

  • forms: these):
  • ax (includes sums of no more than three of
  • ab + c
• E2.10.1a1 , where n = –2 , –1 , − , 0 , 1 , 2 , 3 ; a and c are
• E2.10.22 rational numbers; and b is a positive integer. 2 Solve associated equations graphically, including e.g. finding the intersection of a line and a curve. finding and interpreting roots by graphical methods.
• E2.10.3Draw and interpret graphs representing exponential growth and decay problems.

11. Sketching curves Notes and examples

• E2.11.1

  Recognise, sketch and interpret graphs of the Functions will be equivalent to: following functions:

  • (a) linear
  • (b) quadratic
  • (c) cubic
  • (d) reciprocal(e) exponential. where rational, positive number.
• E2.11.2Knowledge of turning points, roots and symmetry is required.
• E2.11.3Knowledge of vertical and horizontal asymptotes is required.
• E2.11.4Finding turning points of quadratics by completing the square is required.

12. Differentiation Notes and examples

• E2.12.1Estimate gradients of curves by drawing tangents.
• E2.12.2Use the derivatives of functions of the form ax , where a is a rational constant and n is a positive integer or zero, and simple sums of not more than three of these.
• E2.12.3Apply differentiation to gradients and stationary points (turning points).
• E2.12.4Discriminate between maxima and minima by Maximum and minimum points may be identified by: any method. •
• E2.12.5Candidates are inflection.

13. Functions Notes and examples

• E2.13.1Understand functions, domain and range and use function notation.
• E2.13.2Understand and find inverse functions f ( x ) .
• E2.13.3Form composite functions as defined by

1. Coordinates Notes and examples

• E3.1.1Use and interpret Cartesian coordinates in two dimensions.

2. Drawing linear graphs Notes and examples

• E3.2.1Draw straight-line graphs for linear equations. Examples include:

3. Gradient of linear graphs Notes and examples

• E3.3.1Find the gradient of a straight line.
• E3.3.2Calculate the gradient of a straight line from the coordinates of two points on it.

4. Length and midpoint Notes and examples

• E3.4.1Calculate the length of a line segment.
• E3.4.2Find the coordinates of the midpoint of a line segment.

5. Equations of linear graphs Notes and examples

• E3.5.1Interpret and obtain the equation of a straight-line Questions may: graph.
• E3.5.2Candidates are expected to give equations of a line in a fully simplified form.

6. Parallel lines Notes and examples

• E3.6.1Find the gradient and equation of a straight line parallel to a given line.

7. Perpendicular lines Notes and examples

• E3.7.1Find the gradient and equation of a straight line perpendicular to a given line.

1. Geometrical terms Notes and examples

• E4.1.1

  Use and interpret the following geometrical terms:

  • point
  • vertex
  • line
  • plane
  • parallel
  • perpendicular
  • perpendicular bisector
  • bearing
  • right angle
  • acute, obtuse and reflex angles
  • interior and exterior angles
  • similar
  • congruent
  • scale factor.
• E4.1.2

  Use and interpret the vocabulary of:

  • triangles
  • special quadrilaterals
  • polygons
  • nets
  • solids.
• E4.1.3Use and interpret the vocabulary of a circle.

2. Geometrical constructions Notes and examples

• E4.2.1Measure and draw lines and angles. A ruler should be used for all straight edges.
• E4.2.2Constructions of perpendicular bisectors and angle bisectors are 2 Construct a triangle, given the lengths of all e.g. construct a rhombus by drawing two triangles. sides, using a ruler and pair of compasses only. Construction arcs must be shown.
• E4.2.3Draw, use and interpret nets. Examples include:

3. Scale drawings Notes and examples

• E4.3.1Draw and interpret scale drawings. A ruler must be used for all straight edges.
• E4.3.2Use and interpret three-figure bearings. Bearings are measured clockwise from north (000° to 360°). e.g. find the bearing of from
• E4.3.3Includes an understanding of the terms north, east, south and west. e.g. point

4. Similarity Notes and examples

• E4.4.1Calculate lengths of similar shapes.
• E4.4.2Use the relationships between lengths and areas Includes use of scale factor, e.g. of similar shapes and lengths, surface areas and volumes of similar solids.
• E4.4.3Solve problems and give simple explanations Includes showing that two triangles are similar involving similarity. using geometric reasons.

5. Symmetry Notes and examples

• E4.5.1Recognise line symmetry and order of rotational Includes properties of triangles, quadrilaterals and symmetry in two dimensions. polygons directly related to their symmetries.
• E4.5.2Recognise symmetry properties of prisms, e.g. identify planes and axes of symmetry. cylinders, pyramids and cones.

6. Angles Notes and examples

• E4.6.1

  Calculate unknown angles and give simple explanations using the following geometrical properties: of a quadrilateral = 360°.:

  • sum of angles at a point = 360°
  • sum of angles at a point on a straight line = 180°
  • vertically opposite angles are equal
  • angle sum of a triangle = 180° and angle sum
• E4.6.2

  Calculate unknown angles and give geometric explanations for angles formed within parallel lines:

  • corresponding angles are equal
  • alternate angles are equal
  • co-interior angles sum to 180° (supplementary).
• E4.6.3Know and use angle properties of regular and irregular polygons.

7. Circle theorems I Notes and examples

• E4.7.1

  Calculate unknown angles and give explanations using the following geometrical properties of circles: circumference 180° (supplementary):

  • angle in a semicircle = 90°
  • angle between tangent and radius = 90°
  • angle at the centre is twice the angle at the
  • angles in the same segment are equal
  • opposite angles of a cyclic quadrilateral sum to
  • alternate segment theorem.

8. Circle theorems II Notes and examples

• E4.8.1

  Use the following symmetry properties of circles: through the centre length.:

  • equal chords are equidistant from the centre
  • the perpendicular bisector of a chord passes
  • tangents from an external point are equal in

1. Units of measure Notes and examples

• E5.1.1

  Use metric units of mass, length, area, volume Units include: and capacity in practical situations and convert

  • quantities into larger or smaller units.
• E5.1.2Conversion between units includes:

2. Area and perimeter Notes and examples

• E5.2.1Carry out calculations involving the perimeter and Except for the area of a triangle, formulas are area of a rectangle, triangle, parallelogram and given. trapezium.

3. Circles, arcs and sectors Notes and examples

• E5.3.1Carry out calculations involving the Answers may be asked for in terms of circumference and area of a circle. Formulas are given in the List of formulas.
• E5.3.2Carry out calculations involving arc length and Includes minor and major sectors. sector area as fractions of the circumference and area of a circle.

4. Surface area and volume Notes and examples

• E5.4.1

  Carry out calculations and solve problems involving Answers may be asked for in terms of the surface area and volume of a: The following formulas are given in the List of:

  • cuboid formulas:
  • prism
  • cylinder
  • sphere
  • pyramid
  • cone.
• E5.4.2The term prism refers to any solid with a uniform cross-section, e.g. a cylindrical sector.

5. Compound shapes and parts of shapes Notes and examples

• E5.5.1

  Carry out calculations and solve problems Answers may be asked for in terms of involving perimeters and areas of:

  • compound shapes
  • parts of shapes.
• E5.5.2

  Carry out calculations and solve problems involving surface areas and volumes of:

  • compound solids
  • parts of solids. e.g. find the surface area and volume of a frustum.

1. Pythagoras’ theorem Notes and examples

• E6.1.1Know and use Pythagoras’ theorem.

2. Right-angled triangles Notes and examples

• E6.2.1Know and use the sine, cosine and tangent Angles will be given in degrees and answers should ratios for acute angles in calculations involving be written in degrees, with decimals correct to one sides and angles of a right-angled triangle. decimal place.
• E6.2.2Solve problems in two dimensions using Knowledge of bearings may be required. Pythagoras’ theorem and trigonometry.
• E6.2.3Know that the perpendicular distance from a point to a line is the shortest distance to the line.
• E6.2.4Carry out calculations involving angles of elevation and depression.

3. Exact trigonometric values Notes and examples

• E6.3.1Know the exact values of:
• E6.3.1asin x and cos x for x = 0°, 30°, 45°, 60° and 90° .
• E6.3.2tan x for x = 0°, 30°, 45° and 60° .

4. Trigonometric functions Notes and examples

• E6.4.1

  Recognise, sketch and interpret the following graphs for 0° ⩽ x ⩽ 360° :

  • y = sin x
  • y = cos x
  • y = tan x .
• E6.4.2Solve trigonometric equations involving sin x , e.g. solve: cos x or tan x , for 0° ⩽ x ⩽ 360° .

5. Non-right-angled triangles Notes and examples

• E6.5.1Use the sine and cosine rules in calculations Includes problems involving obtuse angles and the involving lengths and angles for any triangle. ambiguous case.
• E6.5.1aab area of triangle = 2 sin C . of a triangle are given in the List of formulas.
• E6.5.2Use the formula The sine and cosine rules and the formula for area

6. Pythagoras’ theorem and trigonometry Notes and examples

• E6.6.1Carry out calculations and solve problems in three dimensions using Pythagoras’ theorem and trigonometry, including calculating the angle between a line and a plane.

1. Transformations Notes and examples

• E7.1.1Recognise, describe and draw the following transformations:
• E7.1.1aReflection of a shape in a straight line.
• E7.1.2Rotation of a shape about a centre through multiples of 90°.
• E7.1.3Enlargement of a shape from a centre by a scale factor. KK x OO
• E7.1.4Translation of a shape by a vector .

2. Vectors in two dimensions Notes and examples

• E7.2.1Describe a translation using a vector represented
• E7.2.2KK x OO by , AB or a . 2 Add and subtract vectors.
• E7.2.3Multiply a vector by a scalar.

3. Magnitude of a vector Notes and examples

• E7.3.1KK x OO Calculate the magnitude of a vector as
• E7.3.22

4. Vector geometry Notes and examples

• E7.4.1Represent vectors by directed line segments.
• E7.4.2Use position vectors.
• E7.4.3Use the sum and difference of two or more vectors to express given vectors in terms of two coplanar vectors.
• E7.4.4Use vectors to reason and to solve geometric problems.

1. Introduction to probability Notes and examples

• E8.1.1Understand and use the probability scale from P( 0 to 1. P(
• E8.1.2Understand and use probability notation.
• E8.1.3Calculate the probability of a single event. Probabilities should be given as a fraction, decimal or percentage. Problems may require using information from tables, graphs or Venn diagrams.
• E8.1.4Understand that the probability of an event e.g. not occurring = 1 – the probability of the event occurring.

2. Relative and expected frequencies Notes and examples

• E8.2.1Understand relative frequency as an estimate of e.g. use results of experiments with a spinner to probability. estimate the probability of a given outcome.
• E8.2.2Calculate expected frequencies. e.g. use probability to estimate an expected value from a population.
• E8.2.3Includes understanding what is meant by fair, bias and random.

3. Probability of combined events Notes and examples

• E8.3.1

  Calculate the probability of combined events using, Combined events could be with or without where appropriate: replacement. in the context of Venn diagrams. end of branches and probabilities by the side of the branches.:

  • sample space diagrams
  • Venn diagrams The notation
  • tree diagrams. On tree diagrams outcomes will be written at the

4. Conditional probability Notes and examples

• E8.4.1Calculate conditional probability using Venn Knowledge of notation, diagrams, tree diagrams and tables. relating to conditional probability is

1. Classifying statistical data Notes and examples

• E9.1.1Classify and tabulate statistical data. e.g. tally tables, two-way tables.

2. Interpreting statistical data Notes and examples

• E9.2.1Read, interpret and draw inferences from tables and statistical diagrams.
• E9.2.2Compare sets of data using tables, graphs and e.g. compare averages and measures of spread statistical measures. between two data sets.
• E9.2.3Appreciate restrictions on drawing conclusions from given data.

3. Averages and measures of spread Notes and examples

• E9.3.1Calculate the mean, median, mode, quartiles, range and interquartile range for individual data and distinguish between the purposes for which these are used.
• E9.3.2Calculate an estimate of the mean for grouped discrete or grouped continuous data.
• E9.3.3Identify the modal class from a grouped frequency distribution.

4. Statistical charts and diagrams Notes and examples

• E9.4.1

  Draw and interpret:

  • (a) bar charts Includes composite (stacked) and dual (side-by-
  • (b) pie charts side) bar charts.
  • (c) pictograms
  • (d) stem-and-leaf diagrams Stem-and-leaf diagrams should have ordered data with a key.
  • (e) simple frequency distributions.

5. Scatter diagrams Notes and examples

• E9.5.1Draw and interpret scatter diagrams.
• E9.5.2Understand what is meant by positive, negative and zero correlation.
• E9.5.3Draw by eye, interpret and use a straight line of best fit.

6. Cumulative frequency diagrams Notes and examples

• E9.6.1Draw and interpret cumulative frequency tables and diagrams.
• E9.6.2Estimate and interpret the median, percentiles, quartiles and interquartile range from cumulative frequency diagrams.

7. Histograms Notes and examples

• E9.7.1Draw and interpret histograms.
• E9.7.2Calculate with frequency density.