Natural numbers, integers, prime numbers, square numbers, cube numbers, common factors and common multiples, rational and irrational numbers, real numbers.
Expressing numbers as a product of prime factors.
Lowest Common Multiple (LCM) and Highest Common Factor (HCF).
[2] Set Language and Notation
Use language, notation and Venn diagrams to describe sets.
Definition of sets:
A = {x : x is a natural number}
B = {(x, y): y = mx + c}
C = {x : a \(\leq\) x \(\leq\) b}
D = {a, b, c…}
Number of elements in set A [n(A)]
“… is an element of …” [∈]
“… is not an element of …” [∉]
Complement of set A [A']
The empty set [∅]
Universal set [\(\xi\)]
A is a subset of B [A ⊆ B]
A is a proper subset of B [A ⊂ B]
A is not a subset of B [A ⊈ B]
A is not a proper subset of B [A ⊄ B]
Union of A and B [A \(\cup\) B]
Intersection of A and B [A ∩ B]
[3] Squares, Square Roots, Cubes and Cube Roots
Squares, square roots, cubes and cube roots of numbers.
[4] Directed Numbers
Use directed numbers in practical situations, e.g. temperature changes or flood levels.
[5] Vulgar and Decimal Fractions and Percentages
Utilize the language and notation of simple vulgar and decimal fractions, as well as percentages. Identify equivalence and convert between these different forms.
[6] Ordering
Arrange quantities in order of magnitude and demonstrate understanding of the symbols =, ≠, >, <, \(\geq\), \(\leq\).
[7] Standard Form
Use the standard form A x 10n where n is a positive or negative integer, and 1 \(\leq\) A < 10. Convert numbers into and out of standard form.
[8] The Four Operations
Perform the four arithmetic operations with whole numbers, decimals, and proper (and mixed) fractions, including correctly applying the order of operations and using brackets when necessary.
[9] Estimation
Estimate numbers, quantities, and lengths, provide approximations to specified numbers of significant figures and decimal places, and round off answers to appropriate accuracy based on the context of a given problem.
[10] Limits of Accuracy
Determine suitable upper and lower bounds for solutions to simple problems, based on given data and specified accuracy.
[11] Ratio, Proportion, Rate
Demonstrate an understanding of ratio and proportion [divide a quantity in a given ratio, direct and inverse proportion, use scales in practical situations]
Increase and decrease a quantity by a given ratio.
Use common measures of rate.
Problems involving average speed.
[12] Percentages
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].
[13] Use of an Electronic Calculator
Use a calculator efficiently.
Apply appropriate checks of accuracy.
[14] Time
Calculate times in terms of the 24-hour and 12-hour clock.
Includes problems involving time zones.
[15] Money
Solve problems involving money and convert from one currency to another.
[16] Personal and Small Business Finance
Utilize provided data to solve problems related to personal and small business finance, including calculations involving earnings, simple interest, compound interest, discounts, and profit and loss (as an amount or a percentage).
[17] Algebraic Representation and Formulae
Express generalized numbers using letters and represent arithmetic processes algebraically.
Substitute numbers for words and letters in formulas.
Use and interpret positive, negative, fractional and zero indices
[20] Solutions of Equations and Inequalities
Solve simple linear equations in one unknown
Solve fractional equations with numerical and linear algebraic denominators
Solve simultaneous linear equations in two unknowns
Solve quadratic equations by factorisation, completing the square or by use of the formula
Solve simple linear inequalities
[21] Graphical Representation of Inequalities
Represent linear inequalities graphically
[22] Sequences
Continue a given number sequence [includes linear sequences, quadratic and cubic sequences, exponential sequences and simple combinations of these]
Recognise patterns in sequences and relationships between different sequences
Generalise sequences as simple algebraic statements [including expressions for the nth term]
[23] Variation
Represent direct and inverse variation using algebraic expressions and utilize this form of expression to determine unknown quantities. This includes linear, square, square root, and cubic variation.
[24] Graphs in Practical Situations
Interpret and apply graphs in practical situations, including travel graphs and conversion graphs.
Apply the concept of rate of change to basic kinematics involving distance-time and speed-time graphs, acceleration, and deceleration.
Determine distance travelled as the area under a linear speed-time graph.
[25] Graphs of Functions
Create tables of values and plot graphs for functions of the form axn, where a is a rational constant, and n = -2, -1, 0, 1, 2, 3, as well as for simple sums of not more than three of these. Also, construct tables of values and graphs for functions of the form kax, where a is a positive integer.
Interpret graphs of linear, quadratic, cubic, reciprocal and exponential functions.
Solve associated equations approximately by graphical methods.
Estimate gradients of curves by drawing tangents.
[26] Function Notation
Use function notation, e.g. f(x) = 3x - 5, f:x ⟼ 3x - 5, to describe simple functions.
Find inverse functions f-1(x).
[27] Coordinate Geometry
Find the gradient of a straight line.
Calculate the length and the coordinates of the midpoint of a line segment from the coordinates of its end points.
Obtain the equation of a straight line graph in the form y = mx + c.
Determine the equation of a straight line parallel to a given line.
Find the gradient of parallel and perpendicular lines.
[28] Geometrical Terms
Use and interpret the geometrical terms: [point, line, plane, parallel, perpendicular, bearing, right angle, acute, obtuse and reflex angles, interior and exterior angles, similarity and congruence]
Use and interpret vocabulary of:
Triangles: equilateral, isosceles and scalene (including right-angled triangles) Quadrilaterals: square, rectangle, kite, rhombus, parallelogram, trapezium Polygons: regular and irregular polygons, pentagon, hexagon, octagon, decagon Simple solid figures: cube, cuboid, prism, cylinder, pyramid, cone, sphere, face, surface, edge, vertex and net Circles: centre, radius, chord, diameter, circumference, tangent, arc, sector and segment
[29] Geometrical Constructions
Construct a triangle, given the three sides, and other simple geometrical figures using a ruler and pair of compasses only.
Construct angle bisectors and perpendicular bisectors.
Read and make scale drawings.
Use and interpret nets.
[30] Similarity and Congruence
Solve problems and provide simple explanations involving similarity and congruence. This includes demonstrating that two triangles are similar or congruent using the correct congruence conditions such as SSS, SAS, ASA, and RHS.
Calculate lengths of similar figures.
Apply the relationships between the areas of similar triangles, extending these principles to similar figures, and further to volumes and surface areas of similar solids. This includes using the scale factor to relate corresponding measurements.
[31] Symmetry
Identify rotational and line symmetry, including the order of rotational symmetry, in two dimensions. This includes understanding the properties of triangles, quadrilaterals, and circles directly related to their symmetries.
Recognize symmetry properties of prisms (including cylinders) and pyramids (including cones).
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
[32] Angles
Calculate unknown angles and give simple explanations 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 polygons includes angle sum]
angle properties of regular and irregular polygons
angle in a semi-circle
angle between tangent and radius of a circle
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
[33] Loci
Use the following loci and the method of intersecting loci for sets of points in two dimensions which are:
at a given distance from a given point
at a given distance from a given straight line
equidistant from two given points
equidistant from two given intersecting straight lines
[34] Measures
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]
[35] Mensuration
Solve problems involving:
the perimeter and area of rectangle, triangle, parallelogram and trapezium
the circumference and area of a circle
arc length and sector area as fractions of the circumference and area of a circle
the surface area and volume of a cuboid, cylinder, prism, sphere, pyramid and cone
the areas and volumes of compound shapes
[36] 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 trigonometrical problems in two dimensions involving angles of elevation and depression.
Extend sine and cosine functions to angles between 90° and 180°.
Solve problems using the sine and cosine rules for any triangle and the formula: area of triangle = \(\frac{1}{2}\) ab sin C
Solve simple trigonometrical problems in three dimensions.
[37] Vectors in Two Dimensions
Describe a translation by using a vector represented by \(\begin{pmatrix}
x \\
y
\end{pmatrix}\), \(\overrightarrow{AB}\) or a and their magnitudes denoted by modulus signs, e.g. |\(\overrightarrow{AB}\)| or |a| = \(\sqrt{x^2 + y^2}\)
Add and subtract vectors.
Multiply a vector by a scalar.
Express given vectors in terms of two coplanar vectors using the sum and difference of two vectors.
Use position vectors.
[38] Matrices
Solve problems involving the calculation of the sum and product of two matrices.
Calculate the product of a matrix and a scalar quantity.
Use the algebra of 2 x 2 matrices including the zero and identity 2 x 2 matrices.
Calculate the determinant |A| and inverse A-1 of a non-singular matrix A.
[39] Transformations
Use the following transformations of the plane: reflection (M), rotation (R), translation (T), enlargement (E) and their combinations [If M(a) = b and R(b) = c the notation RM(a) = c will be used. Invariants under these transformations may be assumed]
Describe transformations using coordinates and matrices [singular matrices are excluded]
[40] Probability
Calculate the probability of a single event as either a fraction or a decimal [probabilities should not be given as ratios].
Understand that the probability of an event occurring = 1 - the probability of the event not occurring.
Understand relative frequency as an estimate of probability.
Calculate the probability of simple combined events using possibility diagrams and tree diagrams where appropriate.
[41] Categorical, Numerical and Grouped Data
Collect, classify and tabulate statistical data.
Read, interpret and draw simple inferences from tables and statistical diagrams.
Calculate the mean, median, mode and range for individual and discrete data.
Calculate an estimate of the mean for grouped and continuous data.
Identify the modal class from a grouped frequency distribution.
[42] Statistical Diagrams
Construct and interpret bar charts, pie charts, pictograms, simple frequency distributions, frequency polygons, 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']
Construct and use cumulative frequency diagrams.
Estimate and interpret the median, percentiles, quartiles and interquartile range for cumulative frequency diagrams.
Calculate with frequency density.
Understand what is meant by positive, negative and zero correlation with reference to a scatter diagram.
