Use natural numbers, integers (positive, negative and zero), prime numbers, square and cube numbers, common factors and common multiples, rational and irrational numbers, real numbers and reciprocals
Expressing numbers as a product of prime factors
Lowest common multiple (LCM) and highest common factor (HCF) of two numbers
Venn diagrams
Squares, square roots, cubes and cube roots and other powers and roots of numbers
Use directed numbers in practical situations. e.g. temperature changes, flood levels
Simple vulgar and decimal fractions. Recognise equivalence and convert between these forms
Order quantities by magnitude and demonstrate familiarity with the symbols =, ≠, >, <, \(\geq\), \(\leq\)
Indices (fractional, negative and zero) and use the rules of indices
Standard form A x 10n where n is a positive or negative integer, and 1 \(\leq\) A < 10
Use the four rules for calculations, including correct ordering of operations and use of brackets
Estimation and approximations. Significant figures and decimal places
Upper and lower bounds
Ratio and proportion (direct and inverse proportion)
Average speed and rate
Percentage
Time
Currency converter
Personal and household finance, simple interest and compound interest
[2] Algebra and Graphs
Extract common factors. Expand products of algebraic expressions.
Use and interpret positive, negative and zero indices. Use the rules of indices.
Solve simple linear equations in one unknown. Solve simultaneous linear equations in two unknowns.
Continue a given number sequence. Recognise patterns in sequences including the term to term rule and relationships between different sequences. Recognise sequences of square, cube and triangular numbers. Find and use the nth term of sequences. Linear, simple quadratic and cubic sequences.
Draw and use graphs in practical situations including travel graphs and conversion graphs.
Draw and interpret roots of linear, quadratic and inverse graphs.
[3] Coordinate Geometry
Gradient of a straight line
Equation of a straight line graph in the form y = mx + c
Equation of a straight line parallel to a given line
[4] Geometry
Use and interpret the geometrical terms:
[point, line, parallel, bearing, right angle, acute, obtuse and reflex angles, perpendicular, similarity and congruence]
Triangles, quadrilaterals, circles, polygons and simple solid figures including nets
Construct a triangle given the three sides using a ruler and a pair of compasses only
Read and make scale drawings
Calculate lengths of similar figures
Recognise congruent shapes
Recognise rotational and line symmetry (including order of rotational symmetry) in two dimensions. Includes properties of triangles, quadrilaterals and circles directly related to their symmetries
Calculate unknown angles using the following geometrical properties:
angles at a point
angles at a point on a straight line and intersecting straight lines
angles formed within parallel lines
angle properties of triangles and quadrilaterals
angle properties of regular polygons
angle in a semicircle
angle between tangent and radius of a circle
[5] Mensuration
Convert between units including units of area and volume
Finding perimeter and area of a rectangle, triangle, parallelogram and trapezium and compound shapes derived from these
Finding the circumference and area of a circle [answers may be asked for in multiples of π]. Solve problems involving the arc length and sector area as fractions of the circumference and area of a circle. Where the sector angle is a factor of 360.
Finding the surface area and volume of a cuboid, prism and cylinder [answers may be asked for in multiples of π]. Solve problems involving the surface area and volume of a sphere, pyramid and cone.
Carry out calculations involving the areas and volumes of compound shapes [answers may be asked for in multiples of π].
[6] Trigonometry
Interpret and use three-figure bearings. Measured clockwise from the North, i.e. 000°-360°.
Apply Pythagoras' theorem and the sine, cosine and tangent ratios for acute angles to the calculation of a side or of an angle of a right-angled triangle. Angles will be quoted in degrees.
[7] Vectors and Transformations
Describe a translation by using a vector represented by e.g. \(\begin{pmatrix}
x \\
y
\end{pmatrix}\), \(\overrightarrow{AB}\) or a. Add and subtract vectors. Multiply a vector by a scalar.
Reflect simple plane figures in horizontal or vertical lines.
Rotate simple plane figures about the origin, vertices or midpoints of edges of the figures, through multiples of 90°.
Construct given translations and enlargements of simple plane figures [positive and fractional scale factors for enlargements only].
Recognise and describe reflections, rotations, translations and enlargements [positive and fractional scale factors for enlargements only].
[8] Probability
Calculate the probability of a single event as either a fraction, decimal or percentage
Understand and use the probability scale from 0 to 1
Understand that the probability of an event occurring = 1 - the probability of the event not occurring.
Understand relative frequency as an estimate of probability. Expected frequency of occurrences.
Calculate the probability of simple combined events, using possibility diagrams, tree diagrams and Venn diagrams. Venn diagrams will be limited to two sets.
[9] Statistics
Construct and interpret bar charts, pie charts, pictograms, stem-and-leaf diagrams, simple frequency distributions, histograms with equal intervals and scatter diagrams.
Calculate the mean, median, mode and range for individual and discrete data and distinguish between the purposes for which they are used.
Understand what is meant by positive, negative and zero correlation with reference to a scatter diagram.
Draw, interpret and use lines of best fit by eye.
EXTENDED
[1] Number
Use natural numbers, integers (positive, negative and zero), prime numbers, square and cube numbers, common factors and common multiples, rational and irrational numbers, real numbers and reciprocals.
Express numbers as a product of prime factors.
Lowest common multiple (LCM) and highest common factor (HCF) of two numbers.
Use language, notation and Venn diagrams to describe sets.
Calculate with squares, square roots, cubes and cube roots and other powers and roots of numbers.
Use directed numbers in practical situations. e.g. temperature changes, flood levels.
Vulgar and decimal fractions and percentages. Recognise equivalence and convert between these forms. Includes the conversion of recurring decimals to fractions, e.g. change 0.\(\overset{.}{7}\) to a fraction.
Order quantities by magnitude and demonstrate familiarity with the symbols =, ≠, >, <, \(\geq\), \(\leq\).
Understand the meaning of indices (fractional, negative and zero) and use the rules of indices.
Use the standard form A x 10n where n is a positive or negative integer, and 1 \(\leq\) A < 10.
Make estimates of numbers, quantities and lengths, give approximations to specified numbers of significant figures and decimal places.
Give appropriate upper and lower bounds for data given to a specified accuracy.
Ratio and proportion. Direct and inverse proportion. Average speed.
Calculate a given percentage of a quantity. Express one quantity as a percentage of another. Calculate percentage increase or decrease. Carry out calculations involving reverse percentages. e.g. finding the cost price given the selling price and the percentage profit.
Use a calculator efficiently. Apply appropriate checks of accuracy.
Calculate times in terms of the 24-hour and 12-hour clock. Read clocks, dials and timetables.
Calculate using money and convert from one currency to another.
Personal and household finance involving earnings, simple interest and compound interest. Includes discount, profit and loss.
Use exponential growth and decay in relation to population and finance. e.g. depreciation, growth of bacteria.
[2] Algebra and Graphs
Use brackets and extract common factors. Expand products of algebraic expressions.
Manipulate algebraic fractions. Factorise and simplify rational expressions.
Use and interpret positive, negative and zero indices. Use and interpret fractional indices.
Derive and solve linear equations in one unknown. Derive and solve simultaneous linear equations in two unknowns. Derive and solve simultaneous equations, involving one linear and one quadratic. Derive and solve quadratic equations by factorisation, completing the square and by use of the formula. Derive and solve linear inequalities. Including representing and interpreting inequalities on a number line.
Represent inequalities graphically and use this representation to solve simple linear programming problems. The conventions of using broken lines for strict inequalities and shading unwanted regions will be expected.
Continue a given number sequence. Recognise patterns in sequences including the term to term rule and relationships between different sequences. Find and use the nth term of sequences. Linear, quadratic, cubic and exponential sequences and simple combinations of these.
Direct and inverse proportion
Use function notation, e.g. f(x) = 3x - 5, f:x ⟼ 3x - 5, to describe simple functions. Find inverse functions f-1(x). Form composite functions as defined by gf(x) = g(f(x)).
Interpret and use graphs in practical situations including travel graphs and conversion graphs. Involve distance-time and speed-time graphs, acceleration and deceleration. Calculate distance travelled as area under a speed-time graph. May include estimation and interpretation of the gradient of a tangent at a point.
Construct tables of values and draw graphs for functions of the form axn (and simple sums of these) and functions of the form abx + c [a and c are rational constants, b is a positive integer, and n = -2, -1, 0, 1, 2, 3]. Solve associated equations approximately, including finding and interpreting roots by graphical methods. Find turning points of quadratics by completing the square. Draw and interpret graphs representing exponential growth and decay problems. Recognise, sketch and interpret graphs of functions. Linear, quadratic, cubic, reciprocal and exponential. Knowledge of turning points and asymptotes is required.
Estimate gradients of curves by drawing tangents.
Understand the idea of a derived function. Use the derivatives of functions of the form axn, and simple sums of not more than three of these [a is a rational constant and n is a positive integer or 0]. Apply differentiation to gradients and turning points (stationary points). Discriminate between maxima and minima by any method.
[3] Coordinate Geometry
Find the gradient of a straight line. Calculate the gradient of a straight line from the coordinates of two points on it.
Calculate the length and the coordinates of the midpoint of a straight line from the coordinates of its end points.
Interpret and obtain the equation of a straight line graph.
Determine the equation of a straight line parallel to a given line.
Find the gradient of parallel and perpendicular lines.
[4] Geometry
Use and interpret the geometrical terms:
[point, line, parallel, bearing, right angle, acute, obtuse and reflex angles, perpendicular, similarity and congruence].
Use and interpret vocabulary of triangles, quadrilaterals, circles, polygons and simple solid figures including nets.
Measure and draw lines and angles. Construct a triangle given the three sides using a ruler and a pair of compasses only.
Read and make scale drawings.
Calculate lengths of similar figures. Use the relationships between areas of similar triangles, with corresponding results for similar figures and extension to volumes and surface areas of similar solids.
Use the basic congruence criteria for triangles (SSS, ASA, SAS, RHS).
Recognise rotational and line symmetry (including order of rotational symmetry) in two dimensions [includes properties of triangles, quadrilaterals and circles directly related to their symmetries]. Recognise symmetry properties of the prism (including cylinder) and the pyramid (including cone).
Use the following symmetry properties of circles:
equal chords are equidistant from the centre
the perpendicular bisector of a chord passes through the centre
tangents from an external point are equal in length
Calculate unknown angles using the following geometrical properties:
angles at a point
angles at a point on a straight line and intersecting straight lines
angles formed within parallel lines
angle properties of triangles and quadrilaterals
angle properties of regular polygons
angle in a semicircle
angle between tangent and radius of a circle
angle properties of irregular polygons
angle at the centre of a circle is twice the angle at the circumference
angles in the same segment are equal
angles in opposite segments are supplementary; cyclic quadrilaterals
alternate segment theorem
[5] Mensuration
Use current units of mass, length, area, volume and capacity in practical situations and express quantities in terms of larger or smaller units. Convert between units including units of area and volume.
Carry out calculations involving the perimeter and area of a rectangle, triangle, parallelogram and trapezium and compound shapes derived from these.
Carry out calculations involving the circumference and area of a circle. Solve problems involving the arc length and sector area as fractions of the circumference and area of a circle [answers may be asked for in multiples of π].
Carry out calculations involving the surface area and volume of a cuboid, prism and cylinder [answers may be asked for in multiples of π]. Carry out calculations involving the surface area and volume of a sphere, pyramid and cone [formulae will be given for the surface area and volume of the sphere, pyramid and cone in the question].
Carry out calculations involving the areas and volumes of compound shapes [answers may be asked for in multiples of π].
[6] Trigonometry
Interpret and use three-figure bearings. Measured clockwise from the North, i.e. 000°-360°.
Apply Pythagoras' theorem and the sine, cosine and tangent ratios for acute angles to the calculation of a side or of an angle of a right angled triangle. Solve trigonometric problems in two dimensions involving angles of elevation and depression. Know that the perpendicular distance from a point to a line is the shortest distance to the line [angles will be quoted in degrees. Answers should be written in degrees and decimals to one decimal place].
Recognise, sketch and interpret graphs of simple trigonometric functions. Graph and know the properties of trigonometric functions. Solve simple trigonometric equations for values between 0° and 360°.
Solve problems using the sine and cosine rules for any triangle [includes problems involving obtuse angles].
Solve simple trigonometrical problems in three dimensions including angle between a line and a plane.
[7] Vectors and Transformations
Describe a translation by using a vector represented by e.g. \(\begin{pmatrix}
x \\
y
\end{pmatrix}\), \(\overrightarrow{AB}\) or a. Add and subtract vectors. Multiply a vector by a scalar.
Reflect simple plane figures.
Rotate simple plane figures through multiples of 90°.
Construct given translations and enlargements of simple plane figures [positive, fractional and negative scale factors for enlargements].
Recognise and describe reflections, rotations, translations and enlargements [positive, fractional and negative scale factors for enlargements].
Calculate the magnitude of a vector \(\begin{pmatrix}
x \\
y
\end{pmatrix}\) as \(\sqrt{x^2 + y^2}\) [vectors will be printed as \(\overrightarrow{AB}\) or a and their magnitudes denoted by modulus signs, e.g. |\(\overrightarrow{AB}\)| or |a|]. Represent vectors by directed line segments. Use the sum and difference of two vectors to express given vectors in terms of two coplanar vectors. Use position vectors.
[8] Probability
Calculate the probability of a single event as either a fraction, decimal or percentage.
Understand and use the probability scale from 0 to 1.
Understand that the probability of an event occurring = 1 - the probability of the event not occurring.
Understand relative frequency as an estimate of probability. Expected frequency of occurrences.
Calculate the probability of simple combined events, using possibility diagrams, tree diagrams and Venn diagrams.
Calculate conditional probability using Venn diagrams, tree diagrams and tables.
[9] Statistics
Construct and interpret bar charts, pie charts, pictograms, stem-and-leaf diagrams, simple frequency distributions, histograms with equal and unequal intervals and scatter diagrams. For unequal intervals on histograms, areas are proportional to frequencies and the vertical axis is labelled 'frequency density'.
Calculate the mean, median, mode and range for individual and discrete data and distinguish between the purposes for which they are used.
Calculate an estimate of the mean for grouped and continuous data. Identify the modal class from a grouped frequency distribution.
Construct and use cumulative frequency diagrams. Estimate and interpret the median, percentiles, quartiles and interquartile range. Construct and interpret box-and-whisker plots.
Understand what is meant by positive, negative and zero correlation withreference to a scatter diagram.
