Maths 0607 — specification explorer

1. Types of number Notes and examples

• E1.1.1

  Identify and use: Example tasks include:

  • natural numbers (0, 1, 2, …)
  • integers (positive, zero and negative)
  • prime numbers
  • square numbers
  • cube numbers
  • triangle 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
  • improper fractions
  • mixed numbers
  • decimals
  • percentages.
• E1.4.2Recognise equivalence and convert between these forms.

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.
• E1.9.2

  Make estimates for calculations involving

  • numbers, quantities and measurements.
• E1.9.3Round answers to a reasonable degree of accuracy in the context of a given problem.

10. Ratio and proportion Notes and examples

• E1.10.1

  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

11. Rates Notes and examples

• E1.11.1Use common measures of rate. e.g. calculate with:
• E1.11.2Solve 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?
• E1.11.3Notation used will be e.g. m/s (metres per second),

12. Percentages Notes and examples

• E1.12.1Calculate a given percentage of a quantity.
• E1.12.2Express one quantity as a percentage of another.
• E1.12.3Calculate percentage increase or decrease.
• E1.12.4Calculate with simple and compound interest.
• E1.12.5Calculate using reverse percentages.

13. Using a calculator Notes and examples

• E1.13.1Use a calculator efficiently.
• E1.13.2Enter values appropriately on a calculator.
• E1.13.3Interpret the calculator display appropriately.

14. Time Notes and examples

• E1.14.1Calculate with time: seconds (s), minutes (min), hours (h), days, weeks, months, years, including the relationship between units.
• E1.14.2Calculate times in terms of the 24-hour and 12-hour clock.
• E1.14.3Read clocks and timetables.

15. Money Notes and examples

• E1.15.1Calculate with money.
• E1.15.2Convert from one currency to another.

16. Exponential growth and decay Notes and examples

• E1.16.1Use exponential growth and decay. e.g. depreciation, population change.
• E1.16.2Knowledge of e is

17. Surds Notes and examples

• E1.17.1Understand and use surds, including simplifying Examples include: expressions.
• E1.17.2Rationalise 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 .

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:

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 quadratic equations by factorisation, using Candidates may be expected to give solutions in a graphic display calculator and by use of the surd form. quadratic formula. The quadratic formula is given.
• E2.5.6Change the subject of formulas. e.g. change the subject of a formula where:
• E2.5.7Use a graphic display calculator to solve e.g. equations, including those which may be unfamiliar.

6. Inequalities Notes and examples

• E2.6.1Represent and interpret inequalities, including on a number line.
• E2.6.2Construct, solve and interpret linear inequalities.
• E2.6.3Solve inequalities using a graphic display calculator.
• E2.6.4Represent and interpret linear inequalities in two variables graphically.
• E2.6.5List inequalities that define a given region.

7. Sequences Notes and examples

• E2.7.1Continue a given number sequence or pattern.
• E2.7.2Recognise patterns in sequences, including the term-to-term rule, and relationships between 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 Includes linear, square, square root and cube algebraic terms and use this form of expression proportion. to find unknown quantities. Knowledge of proportional symbol (
• E2.8.2Identify the best variation model for given data.

1. Graphs of functions Notes and examples

• E3.1.1

  Recognise the following function types from the Includes recognising any symmetry. shape of their graphs:

  • (a) linear f( x ) = ax + b
  • (b) quadratic f( x ) = ax + bx + c
  • (c) cubic f( x ) = ax + bx + cx + d
  • (d) reciprocal f( x ) =
  • (e) exponential f( x ) = a with 0 < a < 1 or a > 1
  • (f) trigonometric f( x ) = a sin( bx ) ; a cos( bx ) ; Includes writing down the period and amplitude. tan x .
• E3.1.2Determine one or two of a , b , c or d for the Some of graphs above.
• E3.1.3Determine values in a function from its graph.

2. Sketching graphs on a calculator Notes and examples

• E3.2.1

  Use a graphic display calculator to: Includes unfamiliar functions not mentioned (a) sketch the graph of a function explicitly in this syllabus.

  • (b) produce a table of values
  • (c) plot points
  • (d) find zeros, local maxima or local minima
  • (e) find the intersection of the graphs of functions
  • (f) find the vertex of a quadratic.

3. Functions Notes and examples

• E3.3.1Understand functions, domain and range and Examples may include: use function notation. •
• E3.3.2Understand and find inverse functions f ( x ) .
• E3.3.3Form composite functions as defined by e.g.
• E3.3.4Give your answer as a fraction in its simplest form.
• E3.3.5Candidates are and range of composite functions.
• E3.3.6This topic may include mapping diagrams.

4. Finding a quadratic function using Notes and examples

• E3.4.1

  Find a quadratic function given:

  • (a) vertex and another point y
  • (b) x -intercepts and a point
  • (c) vertex or x -intercepts in the case where a = 1 .

5. Asymptotes Notes and examples

• E3.5.1Understand the concept of asymptotes and identify e.g. simple examples parallel to the axes on a graph.
• E3.5.2Excludes algebraic derivation of asymptotes.

6. Transforming graphs of functions Notes and examples

• E3.6.1Describe and identify transformations to a graph of e.g. The graph y = f( x ) when y = f( x ) + k or y = f( x + k ) . y terms of
• E3.6.2When

7. The logarithmic function Notes and examples

• E3.7.1

  Understand and use: All logs will be base 10 unless otherwise stated. exponential function compound interest problem or exponential growth x and decay problems. e.g. solve a log equation from a graph or otherwise. x log b:

  • the logarithmic function as the inverse of the e.g. solve log equations in the context of a
  • y = a as equivalent to x = log y
  • the solution to a = b as x = log .

1. Coordinates Notes and examples

• E4.1.1Use and interpret Cartesian coordinates in two dimensions.

2. Gradient of linear graphs Notes and examples

• E4.2.1Find the gradient of a straight line.
• E4.2.2Calculate the gradient of a straight line from the coordinates of two points on it.

3. Length and midpoint Notes and examples

• E4.3.1Calculate the length of a line segment.
• E4.3.2Find the coordinates of the midpoint of a line segment.

4. Equations of linear graphs Notes and examples

• E4.4.1Interpret and obtain the equation of a straight-line graph.

5. Parallel lines Notes and examples

• E4.5.1Find the gradient and equation of a straight line parallel to a given line.

6. Perpendicular lines Notes and examples

• E4.6.1Find the gradient and equation of a straight line perpendicular to a given line.

1. Geometrical terms Notes and examples

• E5.1.1

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

  • point
  • vertex
  • line
  • plane
  • parallel
  • perpendicular
  • perpendicular bisector
  • bearing
  • right angle
  • acute, obtuse and reflex angles
  • interior and exterior angles
  • similar
  • congruent
  • scale factor.
• E5.1.1a

  Polygons:

  • regular and irregular polygons
  • pentagon
  • hexagon
  • octagon
  • decagon.
• E5.1.2

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

  • triangles Triangles:
  • special quadrilaterals
  • polygons
  • solids.
• E5.1.2a

  Solids:

  • cube
  • cuboid
  • prism
  • cylinder
  • pyramid
  • tetrahedron
  • cone
  • sphere
  • hemisphere
  • frustum
  • face
  • surface
  • edge.
• E5.1.3Quadrilaterals:
• E5.1.3a

  Use and interpret the vocabulary of a circle. Includes the following terms:

  • centre
  • radius (plural radii)
  • diameter
  • circumference
  • semicircle
  • chord
  • tangent
  • major and minor arc
  • sector
  • segment.

2. Angle measurement in degrees

• E5.2.1Measure and draw lines and angles. A ruler must be used for all straight edges.
• E5.2.2Use and interpret three-figure bearings. Bearings are measured clockwise from north (000° to 360°). e.g. find the bearing of from
• E5.2.3Includes an understanding of the terms north, east, south and west. e.g. point

3. Similarity Notes and examples

• E5.3.1Calculate lengths of similar shapes.
• E5.3.2Use 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.
• E5.3.3Solve problems and give simple explanations Includes showing that two triangles are similar involving similarity. using geometric reasons.

4. Symmetry Notes and examples

• E5.4.1Recognise line symmetry and order of rotational Includes properties of triangles, quadrilaterals and symmetry in two dimensions. polygons directly related to their symmetries.
• E5.4.2Recognise symmetry properties of prisms, e.g. identify planes and axes of symmetry. cylinders, pyramids and cones.

5. Angles Notes and examples

• E5.5.1

  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
• E5.5.2

  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).
• E5.5.3Know and use angle properties of regular and irregular polygons.

6. Circle theorems I Notes and examples

• E5.6.1

  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.

7. Circle theorems II Notes and examples

• E5.7.1

  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

• E6.1.1Use metric units of mass, length, area, volume and capacity in practical situations and convert quantities into larger or smaller units.

2. Area and perimeter Notes and examples

• E6.2.1Carry out calculations involving the perimeter and area of a rectangle, triangle, parallelogram and trapezium.

3. Circles, arcs and sectors Notes and examples

• E6.3.1Carry out calculations involving the circumference and area of a circle.
• E6.3.2Carry out calculations involving arc length and sector area as fractions of the circumference and area of a circle.

4. Surface area and volume Notes and examples

• E6.4.1

  Carry out calculations and solve problems involving the surface area and volume of a:

  • cuboid
  • prism
  • cylinder
  • sphere
  • pyramid
  • cone.

5. Compound shapes and parts of shapes Notes and examples

• E6.5.1

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

  • compound shapes
  • parts of shapes.
• E6.5.2

  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

• E7.1.1Know and use Pythagoras’ theorem. Includes finding:

2. Right-angled triangles Notes and examples

• E7.2.1Know 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.
• E7.2.2Solve problems in two dimensions using Knowledge of bearings may be required. Pythagoras’ theorem and trigonometry.
• E7.2.3Know that the perpendicular distance from a point to a line is the shortest distance to the line.
• E7.2.4Carry out calculations involving angles of elevation and depression.

3. Exact trigonometric values Notes and examples

• E7.3.1Know the exact values of:
• E7.3.1asin x and cos x for x = 0°, 30°, 45°, 60° and 90° .
• E7.3.2tan x for x = 0°, 30°, 45° and 60° .

4. Trigonometric functions Notes and examples

• E7.4.1

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

  • y = sin x
  • y = cos x
  • y = tan x .
• E7.4.2Solve 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

• E7.5.1Use the sine and cosine rules in calculations Includes problems involving obtuse angles and the involving lengths and angles for any triangle. ambiguous case.
• E7.5.1aab area of triangle = 2 sin C . of a triangle are given.
• E7.5.2Use the formula The sine and cosine rules and the formula for area

6. Pythagoras’ theorem and trigonometry Notes and examples

• E7.6.1Carry 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

• E8.1.1Recognise, describe and draw the following transformations:
• E8.1.1aReflection of a shape in a straight line.
• E8.1.2Rotation of a shape about a centre through multiples of 90°.
• E8.1.3Enlargement of a shape from a centre by a scale factor.
• E8.1.4KK x OO 4 Translation of a shape by a vector .

2. Vectors in two dimensions Notes and examples

• E8.2.1Describe a translation using a vector represented
• E8.2.2KK x OO by , AB or a . 2 Add and subtract vectors.
• E8.2.3Multiply a vector by a scalar.

3. Magnitude of a vector Notes and examples

• E8.3.1KK x OO Calculate the magnitude of a vector as
• E8.3.22

1. Introduction to probability Notes and examples

• E9.1.1Understand and use the probability scale from 0 to 1.
• E9.1.2Understand and use probability notation. P(
• E9.1.3Calculate the probability of a single event. Probabilities should be given as a fraction, decimal or percentage.
• E9.1.4Problems may require using information from tables, graphs or Venn diagrams. 4 Understand 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

• E9.2.1Understand relative frequency as an estimate of e.g. use results of experiments with a spinner to probability. estimate the probability of a given outcome.
• E9.2.2Calculate expected frequencies. e.g. use probability to estimate an expected value from a population.
• E9.2.3Includes understanding what is meant by fair, bias and random.

3. Probability of combined events Notes and examples

• E9.3.1

  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 the 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
• E9.3.2The notation exclusive events and may be used.

1. Classifying statistical data Notes and examples

• E10.1.1Classify and tabulate statistical data. e.g. tally tables, two-way tables.

2. Interpreting statistical data Notes and examples

• E10.2.1Read, interpret and draw inferences from tables and statistical diagrams.
• E10.2.2Compare sets of data using tables, graphs and e.g. compare averages and measures of spread statistical measures. between two data sets.
• E10.2.3Appreciate restrictions on drawing conclusions from given data.

3. Discrete and continuous data Notes and examples

• E10.3.1Distinguish between discrete and continuous data.

4. Averages and measures of spread Notes and examples

• E10.4.1Calculate the mean, median, mode, quartiles, range and interquartile range for individual data and distinguish between the purposes for which these are used.
• E10.4.2Calculate an estimate of the mean for grouped discrete or grouped continuous data.
• E10.4.3Identify the modal class from a grouped frequency distribution.

5. Averages on a calculator Notes and examples

• E10.5.1Use a graphic display calculator to calculate:
• E10.5.1amean, median and quartiles for discrete data
• E10.5.2mean for grouped data.

6. Statistical charts and diagrams Notes and examples

• E10.6.1

  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.

7. Scatter diagrams Notes and examples

• E10.7.1Draw and interpret scatter diagrams. Plotted points should be clearly marked, for example as small crosses (×).
• E10.7.2Understand what is meant by positive, negative and zero correlation.
• E10.7.3Draw by eye, interpret and use a straight line of A line of best fit: best fit.
• E10.7.4Use a graphic display calculator to find and use the equation of linear regression.

8. Cumulative frequency diagrams Notes and examples

• E10.8.1Draw and interpret cumulative frequency tables Plotted points on a cumulative frequency diagram and diagrams. should be clearly marked, for example as small crosses (×), and be joined with a smooth curve.
• E10.8.2Estimate and interpret the median, percentiles, quartiles and interquartile range from cumulative frequency diagrams.