Notes

Maths - Tell your teacher that you studied the higher tier paper over the holidays so she automatically moves you there hopefully!

Topics:

1 Number
2 Algebra
3 Ratio, proportion & rates of change
4 Geometry & measures
5 Probability
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What students need to learn to pass higher maths:
1 - NUMBER:
{Structure and calculation}

N1 order positive and negative integers, decimals and fractions; use the
symbols =, ≠, <, >, ≤, ≥

N2 apply the four operations, including formal written methods, to integers,
decimals and simple fractions (proper and improper), and mixed numbers –
all both positive and negative; understand and use place value
(e.g. when working with very large or very small numbers, and when
calculating with decimals - like the use of Standard form, Significant factors and Decimal Places.

N3 recognise and use relationships between operations, including inverse
operations (e.g. cancellation to simplify calculations and expressions);
use 'conventional notation' for priority of operations, including:
- brackets
- powers
- roots
- reciprocals

N4 use the concepts and vocabulary of:
prime numbers,
factors (divisors),
multiples,
common factors,
common multiples,
highest common factor,
lowest common multiple,
prime factorisation,
including using product
notation and the unique factorisation theorem.

'N5 apply systematic listing strategies, including use of the product rule for
counting.

(i.e. if there are m ways of doing one task and for each of
these, there are n ways of doing another task, then the total number
of ways the two tasks can be done is m × n ways)'


N6 use positive integer powers and associated real roots
(square, cube and higher), recognise powers of 2, 3, 4, 5;
estimate powers and roots of any given positive number.

N7 calculate with roots, and with integer and fractional indices.

N8 calculate exactly with fractions, surds and multiples of π; simplify surd expressions involving squares
(e.g. √12 = √(4 × 3) = √4 × √3 = 2√3) and rationalise denominators.

N9 calculate with and interpret standard form A × 10n, where 1 ≤ A < 10 and n is
an integer.

{Fractions, decimals and percentages}


N10 work interchangeably with terminating decimals and their corresponding
fractions (such as 3.5 and 7/2 or 0.375 or 38); change recurring decimals
into their corresponding fractions and vice versa.

N11 identify and work with fractions in ratio problems.

N12 interpret fractions and percentages as operators.

{Measures and accuracy}

N13 use standard units of mass, length, time, money and other measures
(including standard compound measures) using decimal quantities where
appropriate.

N14 estimate answers; check calculations using approximation and estimation,
including answers obtained using technology.

N15 round numbers and measures to an appropriate degree of accuracy
(e.g. to a specified number of decimal places or significant figures); use
inequality notation to specify simple error intervals due to truncation or
rounding.

N16 apply and interpret limits of accuracy, including upper and lower bounds

{Algebra}
Notation, vocabulary and manipulation.

A1 use and interpret algebraic manipulation, including:
• ab in place of a × b
• 3y in place of y + y + y and 3 × y
• a
2 in place of a × a, a
3 in place of a × a × a, a
2
b in place of a × a × b
• a
b in place of a ÷ b
● coefficients written as fractions rather than as decimals
● brackets


A2 substitute numerical values into formulae and expressions, including
scientific formulae.

A3 understand and use the concepts and vocabulary of expressions, equations,
formulae, identities, inequalities, terms and factors.

A4 simplify and manipulate algebraic expressions (including those involving
surds and algebraic fractions) by:
● collecting like terms
● multiplying a single term over a bracket
● taking out common factors
● expanding products of two or more binomials

● factorising quadratic expressions of the form x
2 + bx + c, including the difference of two squares; factorising quadratic expressions of the
form ax
2 + bx + c

● simplifying expressions involving sums, products and powers, including
the laws of indices


A5 understand and use standard mathematical formulae; rearrange formulae to
change the subject

A6 know the difference between an equation and an identity; argue
mathematically to show algebraic expressions are equivalent, and use
algebra to support and construct arguments and proofs.

A7 where appropriate, interpret simple expressions as functions with inputs
and outputs; interpret the reverse process as the ‘inverse function’;
interpret the succession of two functions as a ‘composite function’
(the use of formal function notation is expected).


{Graphs}

A8 work with coordinates in all four quadrants

A9 plot graphs of equations that correspond to straight-line graphs in the
coordinate plane; use the form y = mx + c to identify parallel and
perpendicular lines; find the equation of the line through two given points
or through one point with a given gradient

A10 identify and interpret gradients and intercepts of linear functions graphically
and algebraically

A11 identify and interpret roots, intercepts, turning points of quadratic functions
graphically; deduce roots algebraically and turning points by completing
the square

A12 recognise, sketch and interpret graphs of linear functions, quadratic
functions, simple cubic functions, the reciprocal function 1
y= 1/x = with x ≠ 0,
exponential functions y = k(to the power of x) for positive values of k, and the
trigonometric functions (with arguments in degrees) y = sin x,
y = cos x and y = tan x for angles of any size

A13 sketch translations and reflections of a given function

A14 plot and interpret graphs (including reciprocal graphs and exponential
graphs) and graphs of non-standard functions in real contexts to find
approximate solutions to problems such as simple kinematic problems
involving distance, speed and acceleration


A15 calculate or estimate gradients of graphs and areas under graphs
(including quadratic and other non-linear graphs), and interpret
results in cases such as distance-time graphs, velocity-time graphs
and graphs in financial contexts (this does not include calculus)


A16 recognise and use the equation of a circle with centre at the origin;
find the equation of a tangent to a circle at a given point

{Solving equations and inequalities}

A17 solve linear equations in one unknown algebraically (including those with the
unknown on both sides of the equation); find approximate solutions using a
graph

A18 solve quadratic equations (including those that require rearrangement)
algebraically by factorising, by completing the square and by using the
quadratic formula; find approximate solutions using a graph

A19 solve two simultaneous equations in two variables (linear/linear or
linear/quadratic) algebraically; find approximate solutions using a graph

A20 find approximate solutions to equations numerically using iteration

A21 translate simple situations or procedures into algebraic expressions or
formulae; derive an equation (or two simultaneous equations), solve the
equation(s) and interpret the solution

A22 solve linear inequalities in one or two variable(s), and quadratic
inequalities in one variable; represent the solution set on a number line,
using set notation and on a graph

{Sequences}

A23 generate terms of a sequence from either a term-to-term or a position-to-term rule

A24 recognise and use sequences of triangular, square and cube numbers, simple
arithmetic progressions, Fibonacci type sequences, quadratic sequences, and
simple geometric progressions (r [to the power of n]where n is an integer, and r is a rational
number > 0 or a surd) and other sequences.

A25 deduce expressions to calculate the nth term of linear and quadratic
sequences

{Ratio, proportion and rates of change}

R1 change freely between related standard units (e.g. time, length, area,
volume/capacity, mass) and compound units (e.g. speed, rates of pay,
prices, density, pressure) in numerical and algebraic contexts.

R2 use scale factors, scale diagrams and maps

R3 express one quantity as a fraction of another, where the fraction is less than
1 or greater than 1

R4 use ratio notation, including reduction to simplest form

R5 divide a given quantity into two parts in a given part:part or part:whole
ratio; express the division of a quantity into two parts as a ratio; apply ratio
to real contexts and problems (such as those involving conversion,
comparison, scaling, mixing, concentrations)

R6 express a multiplicative relationship between two quantities as a ratio or a
fraction

R7 understand and use proportion as equality of ratios

R8 relate ratios to fractions and to linear functions

R9 define percentage as ‘number of parts per hundred’; interpret percentages
and percentage changes as a fraction or a decimal, and interpret these
multiplicatively; express one quantity as a percentage of another; compare
two quantities using percentages; work with percentages greater than 100%;
solve problems involving percentage change, including percentage
increase/decrease and original value problems, and simple interest including
in financial mathematics

R10 solve problems involving direct and inverse proportion, including graphical
and algebraic representations

R11 use compound units such as speed, rates of pay, unit pricing, density and
pressure

R12 compare lengths, areas and volumes using ratio notation; make links to
similarity (including trigonometric ratios) and scale factors

R13 understand that X is inversely proportional to Y is equivalent to X is
proportional to 1/Y
; construct and interpret equations that describe direct
and inverse proportion

R14 interpret the gradient of a straight line graph as a rate of change; recognise
and interpret graphs that illustrate direct and inverse proportion

R15 interpret the gradient at a point on a curve as the instantaneous rate
of change; apply the concepts of average and instantaneous rate of
change (gradients of chords and tangents) in numerical, algebraic
and graphical contexts (this does not include calculus)

R16 set up, solve and interpret the answers in growth and decay problems,
including compound interest and work with general iterative processes

{Geometry and measures}

{Properties and constructions}


G1 use conventional terms and notations: points, lines, vertices, edges, planes,
parallel lines, perpendicular lines, right angles, polygons, regular polygons
and polygons with reflection and/or rotation symmetries; use the standard
conventions for labelling and referring to the sides and angles of triangles;
draw diagrams from written description

G2 use the standard ruler and compass constructions (perpendicular bisector of a
line segment, constructing a perpendicular to a given line from/at a given
point, bisecting a given angle); use these to construct given figures and solve
loci problems; know that the perpendicular distance from a point to a line is
the shortest distance to the line

G3 apply the properties of angles at a point, angles at a point on a straight line,
vertically opposite angles; understand and use alternate and corresponding
angles on parallel lines; derive and use the sum of angles in a triangle (e.g.
to deduce and use the angle sum in any polygon, and to derive properties of
regular polygons)

G4 derive and apply the properties and definitions of: special types of
quadrilaterals, including square, rectangle, parallelogram, trapezium, kite
and rhombus; and triangles and other plane figures using appropriate
language

G5 use the basic congruence criteria for triangles (SSS, SAS, ASA, RHS)

G6 apply angle facts, triangle congruence, similarity and properties of
quadrilaterals to conjecture and derive results about angles and sides,
including Pythagoras’ theorem and the fact that the base angles of an
isosceles triangle are equal, and use known results to obtain simple proofs

G7 identify, describe and construct congruent and similar shapes, including on
coordinate axes, by considering rotation, reflection, translation and
enlargement (including fractional and negative scale factors)

G8 describe the changes and invariance achieved by combinations of
rotations, reflections and translations

G9 identify and apply circle definitions and properties, including: centre, radius,
chord, diameter, circumference, tangent, arc, sector and segment

G10 apply and prove the standard circle theorems concerning angles,
radii, tangents and chords, and use them to prove related results

G11 solve geometrical problems on coordinate axes

G12 identify properties of the faces, surfaces, edges and vertices of: cubes,
cuboids, prisms, cylinders, pyramids, cones and spheres

G13 construct and interpret plans and elevations of 3D shapes

Mensuration and calculation

G14 use standard units of measure and related concepts (length, area,
volume/capacity, mass, time, money, etc.)

G15 measure line segments and angles in geometric figures, including
interpreting maps and scale drawings and use of bearings

G16 know and apply formulae to calculate: area of triangles, parallelograms,
trapezia; volume of cuboids and other right prisms (including cylinders)

G17 know the formulae: circumference of a circle = 2πr = πd , area of a
circle = πr squared
; calculate: perimeters of 2D shapes, including circles; areas of
circles and composite shapes; surface area and volume of spheres,
pyramids, cones and composite solids

G18 calculate arc lengths, angles and areas of sectors of circles

G19 apply the concepts of congruence and similarity, including the relationships
between lengths, areas and volumes in similar figures

G20 know the formulae for: Pythagoras’ theorem a squared + b squared= c squared
, and the trigonometric ratios, sin θ = opposite/hypotenuse , cos θ = adjacent/hypotenuse
and tan θ = opposite/adjacent ; apply them to find angles and lengths in right-angled triangles
and, where possible, general triangles in two and three dimensional figures.

Soh,Cah,Toa

G21 know the exact values of sin θ and cos θ for θ = 0°, 30°, 45°, 60° and 90°;
know the exact value of tan θ for θ = 0°, 30°, 45° and 60°

G22 know and apply the sine rule
a/sin A = b/sin B = c/sin C


and cosine rule a squared = b squared + c squared – 2bc cos A, to find unknown lengths and angles

G23 know and apply Area = ab sin C2

G23 know and apply Area = 1/2 ab sin C to calculate the area, sides or angles of any triangle.

{Vectors}

G24 describe translations as 2D vectors

G25 apply addition and subtraction of vectors, multiplication of vectors by a scalar, and diagrammatic and column representations of vectors; use vectors to construct geometric arguments and proofs.

{Probability}

P1 record, describe and analyse the frequency of outcomes of probability
experiments using tables and frequency trees.

P2 apply ideas of randomness, fairness and equally likely events to calculate
expected outcomes of multiple future experiments.

P3 relate relative expected frequencies to theoretical probability, using
appropriate language and the 0-1 probability scale.

P4 apply the property that the probabilities of an exhaustive set of outcomes
sum to one; apply the property that the probabilities of an exhaustive set of
mutually exclusive events sum to one.

P5 understand that empirical unbiased samples tend towards theoretical
probability distributions, with increasing sample size.

P6 enumerate sets and combinations of sets systematically, using tables, grids,
Venn diagrams and tree diagrams.

P7 construct theoretical possibility spaces for single and combined experiments
with equally likely outcomes and use these to calculate theoretical
probabilities.

P8 calculate the probability of independent and dependent combined events,
including using tree diagrams and other representations, and know the
underlying assumptions.

P9 calculate and interpret conditional probabilities through
representation using expected frequencies with two-way tables,
tree diagrams and Venn diagrams.

{Statistics}

S1 infer properties of populations or distributions from a sample, while knowing
the limitations of sampling

S2 interpret and construct tables, charts and diagrams, including frequency
tables, bar charts, pie charts and pictograms for categorical data, vertical
line charts for ungrouped discrete numerical data, tables and line graphs for
time series data and know their appropriate use

S3 construct and interpret diagrams for grouped discrete data and
continuous data, i.e. histograms with equal and unequal class
intervals and cumulative frequency graphs, and know their
appropriate use

S4 interpret, analyse and compare the distributions of data sets from univariate
empirical distributions through:

● appropriate graphical representation involving discrete, continuous and
grouped data, including box plots

● appropriate measures of central tendency (median, mean, mode and
modal class) and spread (range, including consideration of outliers,
quartiles and inter-quartile range)

S5 apply statistics to describe a population

S6 use and interpret scatter graphs of bivariate data; recognise correlation and
know that it does not indicate causation; draw estimated lines of best fit;
make predictions; interpolate and extrapolate apparent trends while
knowing the dangers of so doing.