Mathematics 0580 — specification explorer

1. Types of number Notes and examples

• C1.1.1

  Identify and use:

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

• C1.2.1Understand and use set language, notation and Venn diagrams are limited to two sets. Venn diagrams to describe sets. The following set notation will be used:
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• C1.2.2Example definition of sets:

3. Powers and roots Notes and examples

• C1.3.1

  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.
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4. Fractions, decimals and percentages Notes and examples

• C1.4.1

  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 Candidates are
  • improper fractions decimal notation.
  • mixed numbers
  • decimals
  • percentages.
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• C1.4.2Recognise equivalence and convert between Candidates are these forms. conversion of a recurring decimal to a fraction and vice versa.
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5. Ordering Notes and examples

• C1.5.1Order quantities by magnitude and demonstrate familiarity with the symbols =, ≠, >, < , ⩾ and ⩽ .

6. The four operations Notes and examples

• C1.6.1

  Use 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

• C1.7.1Understand and use indices (positive, zero and e.g. find the value of negative integers).
• C1.7.2Understand and use the rules of indices. e.g. find the value of

8. Standard form Notes and examples

• C1.8.1Use the standard form A × 10 where n is a positive or negative integer and 1 ⩽ A < 10 .
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• C1.8.2Convert numbers into and out of standard form.
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• C1.8.3Calculate with values in standard form. Core candidates are expected to calculate with standard form only on Paper 3.
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9. Estimation Notes and examples

• C1.9.1Round values to a specified degree of accuracy. Includes decimal places and significant figures.
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• C1.9.2Make 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
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• C1.9.3Round answers to a reasonable degree of figure, estimate the value of accuracy in the context of a given problem.
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10. Limits of accuracy Notes and examples

• C1.10.1Give upper and lower bounds for data rounded to a e.g. write down the upper bound of a length specified accuracy. measured correct to the nearest metre.
• C1.10.2Candidates are of the results of calculations which have used data rounded to a specified accuracy.

11. Ratio and proportion Notes and examples

• C1.11.1

  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
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12. Rates Notes and examples

• C1.12.1Use common measures of rate. e.g. calculate with:
• C1.12.2Apply other measures of rate. e.g. calculate with:
• C1.12.3Required 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?
• C1.12.4Notation used will be, e.g. m/s (metres per second),

13. Percentages Notes and examples

• C1.13.1Calculate a given percentage of a quantity.
• C1.13.2Express one quantity as a percentage of another.
• C1.13.3Calculate percentage increase or decrease.
• C1.13.4Calculate with simple and compound interest. Formulas are
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• C1.13.5Percentage calculations may include:

14. Using a calculator Notes and examples

• C1.14.1Use a calculator efficiently. e.g. know not to round values within a calculation and to only round the final answer.
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• C1.14.2Enter values appropriately on a calculator. e.g. enter 2 hours 30 minutes as 2.5 hours or
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• C1.14.3Interpret the calculator display appropriately. e.g. in money 4.8 means $4.80; in time 3.25 means 3 hours 15 minutes.
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15. Time Notes and examples

• C1.15.1Calculate with time: seconds (s), minutes (min), 1 year = 365 days. hours (h), days, weeks, months, years, including the relationship between units.
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• C1.15.2Calculate 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.
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• C1.15.3Read clocks and timetables. Includes problems involving time zones, local times and time differences.
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16. Money Notes and examples

• C1.16.1Calculate with money.
• C1.16.2Convert from one currency to another.

1. Introduction to algebra Notes and examples

• C2.1.1Know that letters can be used to represent generalised numbers.
• C2.1.2Substitute numbers into expressions and formulas.

2. Algebraic manipulation Notes and examples

• C2.2.1Simplify expressions by collecting like terms. Simplify means give the answer in its simplest form, e.g.
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• C2.2.2Expand products of algebraic expressions. e.g. expand
• C2.2.3Includes products of two brackets involving one variable, e.g. expand 3 Factorise by extracting common factors. Factorise means factorise fully, e.g.
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3. Indices II Notes and examples

• C2.4.1Understand and use indices (positive, zero and e.g. negative).
• C2.4.2Understand and use the rules of indices. e.g. simplify:
• C2.4.3Knowledge of logarithms is

4. Equations Notes and examples

• C2.5.1Construct simple expressions, equations and e.g. write an expression for a number that is 2 more formulas. than
• C2.5.2Includes constructing linear simultaneous equations. 2 Solve linear equations in one unknown. Examples include:
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• C2.5.3

  Solve simultaneous linear equations in two

  • unknowns.
• C2.5.4Change the subject of simple formulas. e.g. change the subject of formulas where:

5. Inequalities Notes and examples

• C2.6.1Represent and interpret inequalities, including on a number line.
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6. Sequences Notes and examples

• C2.7.1Continue a given number sequence or pattern.
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• C2.7.2Recognise patterns in sequences, including the term-to-term rule, and relationships between different sequences.
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• C2.7.3

  Find and use the n th term of the following sequences:

  • (a) linear
  • (b) simple quadratic
  • (c) simple cubic.
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7. Graphs in practical situations Notes and examples

• C2.9.1Use and interpret graphs in practical situations including travel graphs and conversion graphs.
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• C2.9.2Draw graphs from given data.

8. Graphs of functions Notes and examples

• C2.10.1

  Construct tables of values, and draw, recognise and interpret graphs for functions of the following forms: where a and b are integer constants.:

  • ax + b
  • ± x + ax + b
  • ( x ≠ 0)
• C2.10.2Solve associated equations graphically, including e.g. find the intersection of a line and a curve. finding and interpreting roots by graphical methods.

9. Sketching curves Notes and examples

• C2.11.1

  Recognise, sketch and interpret graphs of the following functions:

  • (a) linear
  • (b) quadratic. Knowledge of symmetry and roots is required.
• C2.11.2Knowledge of turning points is

1. Coordinates Notes and examples

• C3.1.1Use and interpret Cartesian coordinates in two dimensions.
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2. Drawing linear graphs Notes and examples

• C3.2.1Draw straight-line graphs for linear equations.
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3. Gradient of linear graphs Notes and examples

• C3.3.1Find the gradient of a straight line.

4. Equations of linear graphs Notes and examples

• C3.5.1Interpret and obtain the equation of a straight-line graph in the form y = mx + c .
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5. Parallel lines Notes and examples

• C3.6.1Find the gradient and equation of a straight line parallel to a given line.
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1. Geometrical terms Notes and examples

• C4.1.1

  Use and interpret the following geometrical Candidates are terms: shapes are congruent.:

  • point
  • vertex
  • line
  • parallel
  • perpendicular
  • bearing
  • right angle
  • acute, obtuse and reflex angles
  • interior and exterior angles
  • similar
  • congruent
  • scale factor.
• C4.1.1aSimple solids: 3 Use and interpret the vocabulary of a circle. Includes the following terms:
• C4.1.2

  Use and interpret the vocabulary of: Includes the following terms:

  • triangles Triangles:
  • special quadrilaterals
  • polygons
  • nets
  • simple solids.
• C4.1.3Quadrilaterals:
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• C4.1.4Polygons:

2. Geometrical constructions Notes and examples

• C4.2.1Measure and draw lines and angles. A ruler should be used for all straight edges.
• C4.2.2Constructions 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.
• C4.2.3Draw, use and interpret nets. Examples include:

3. Scale drawings Notes and examples

• C4.3.1Draw and interpret scale drawings.
• C4.3.2Use and interpret three-figure bearings.

4. Similarity Notes and examples

• C4.4.1Calculate lengths of similar shapes.
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5. Symmetry Notes and examples

• C4.5.1Recognise line symmetry and order of rotational symmetry in two dimensions.
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6. Angles Notes and examples

• C4.6.1

  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
• C4.6.2

  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).
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• C4.6.3Know and use angle properties of regular polygons.

7. Circle theorems Notes and examples

• C4.7.1

  Calculate unknown angles and give explanations using the following geometrical properties of circles:

  • angle in a semicircle = 90°
  • angle between tangent and radius = 90°.
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1. Units of measure Notes and examples

• C5.1.1

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

  • quantities into larger or smaller units.
• C5.1.2Conversion between units includes:

2. Area and perimeter Notes and examples

• C5.2.1Carry out calculations involving the perimeter and Except for area of a triangle, formulas are area of a rectangle, triangle, parallelogram and given. trapezium.
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3. Circles, arcs and sectors Notes and examples

• C5.3.1Carry 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.
• C5.3.2Carry out calculations involving arc length and sector area as fractions of the circumference and area of a circle, where the sector angle is a factor of 360°.

4. Surface area and volume Notes and examples

• C5.4.1

  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.
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• C5.4.2The term prism refers to any solid with a uniform cross-section, e.g. a cylindrical sector.
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5. Compound shapes and parts of shapes Notes and examples

• C5.5.1

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

  • compound shapes
  • parts of shapes.
• C5.5.2

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

  • compound solids
  • parts of solids. e.g. find the volume of half of a sphere.

1. Pythagoras’ theorem Notes and examples

• C6.1.1Know and use Pythagoras’ theorem.
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2. Right-angled triangles

• C6.2.1Know 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.
• C6.2.2Solve problems in two dimensions using Knowledge of bearings may be required. Pythagoras’ theorem and trigonometry.

1. Transformations Notes and examples

• C7.1.1Recognise, describe and draw the following transformations:
• C7.1.1aReflection of a shape in a vertical or horizontal line.
• C7.1.2Rotation of a shape about the origin, vertices or midpoints of edges of the shape, through multiples of 90°.
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• C7.1.3Enlargement of a shape from a centre by a scale factor. KK x OO
• C7.1.4Translation of a shape by a vector .

1. Introduction to probability Notes and examples

• C8.1.1Understand and use the probability scale from Probability notation is 0 to 1. Probabilities should be given as a fraction, decimal or percentage. Problems may require using information from tables, graphs or Venn diagrams (limited to two sets).
• C8.1.2Calculate the probability of a single event.
• C8.1.3Understand that the probability of an event e.g. The probability that a counter is blue is 0.8. not occurring = 1 – the probability of the event What is the probability that it is not blue? occurring.

2. Relative and expected frequencies Notes and examples

• C8.2.1Understand relative frequency as an estimate of e.g. use results of experiments with a spinner to probability. estimate the probability of a given outcome.
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• C8.2.2Calculate expected frequencies. e.g. use probability to estimate an expected value from a population.
• C8.2.3Includes understanding what is meant by fair, bias and random.

3. Probability of combined events Notes and examples

• C8.3.1

  Calculate the probability of combined events using, Combined events will only be with replacement. where appropriate: end of the branches and probabilities by the side of the branches.:

  • sample space diagrams
  • Venn diagrams Venn diagrams will be limited to two sets.
  • tree diagrams. In tree diagrams, outcomes will be written at the

1. Classifying statistical data Notes and examples

• C9.1.1Classify and tabulate statistical data. e.g. tally tables, two-way tables.
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2. Interpreting statistical data Notes and examples

• C9.2.1Read, interpret and draw inferences from tables and statistical diagrams.
• C9.2.2Compare sets of data using tables, graphs and e.g. compare averages and ranges between two statistical measures. data sets.
• C9.2.3Appreciate restrictions on drawing conclusions from given data.

3. Averages and range Notes and examples

• C9.3.1Calculate the mean, median, mode and range Data may be in a list or frequency table, but will not for individual data and distinguish between the be grouped. purposes for which these are used.
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4. Statistical charts and diagrams Notes and examples

• C9.4.1

  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.
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5. Scatter diagrams Notes and examples

• C9.5.1Draw and interpret scatter diagrams.
• C9.5.2Understand what is meant by positive, negative and zero correlation.
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• C9.5.3Draw by eye, interpret and use a straight line of best fit.
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