Edexcel GCSE Maths Linear Exam Topic List - HIGHER

NUMBER

Add, subtract, multiply, divide

c Write numbers in words

c Write numbers from words

c Add, subtract, multiply, divide whole numbers, integers, negatives, fractions, and decimals and numbers in index form

c Multiply and divide any number between 0 and 1.

c Divide decimals up to 2 decimal places

c Solve a problem involving division by a decimal (up to two decimal places)

c Know the fraction-to-decimal conversion of familiar fractions

Order numbers

c Put in order of size, integers, decimals and fractions

c Understand and use positive and negative numbers on a number line

Factors, multiples and primes

Understand the terms;

c Odd and even

c Factor

c Multiple

c Common factor

c Highest common factor

c Least (lowest) common multiple

c Prime number

c Be able to identify factors, multiples and primes from a list of numbers

c Express a number as a product of prime factors (factor tree)

c Find common multiples or common factors of two numbers

c Find the highest common factor (HCF) or the lowest common multiple (LCM) of two numbers.

Squares, square roots, cubes and cube roots

c Know all the square numbers from 2² = 4 up to 15² = 225

c Know all the cube numbers from 2³ = 8 up to 5³ = 125 and also 10³ = 1000

Index notation

c Use index notation for squares and cubes, eg. 5³

c Use index notation for powers of 10, eg. 106

c Understand indices in calculations

Index laws

c Multiply and divide by adding or subtracting indices

c Calculate using index laws when indices are fractions or negative

c Understand that for any number n, nº = 1

c Understand that n-1 = 1 / n

c Understand that n½ = √n

c Understand that n⅓ = 3√n

Standard form

c Understand numbers written in standard form

c Write large or small numbers in standard form

c Convert between standard form and normal form

c Understand and use standard form on a calculator

Equivalent fractions and adding and subtracting fractions

c Find equivalent fractions

c Simplify a fraction to its simplest form

c Convert between improper fractions and mixed numbers

c Add and subtract fractions

Decimals, including recurring decimals

c Know fraction to decimal conversions for simple fractions

c Convert between fractions and decimals

c Understand that all recurring decimals are exact fractions, and that some exact fractions are recurring decimals

c Convert between recurring decimals and fractions

c Know how to convert from recurring decimal to fraction using a proof

Percentages

c Understand percentages

c Convert between fractions, decimals and percentages

Using fractions, decimals and percentages

c Find a fraction of a quantity

c Find a percentage of a quantity

c Use decimals to find quantities

c Use a multiplier to increase of decrease a quantity (eg. use x 1.05 to increase by 5%, or 0.88 to decrease by 12%))

Percentages and proportional change

c Use percentages to calculate and use

o VAT

o Simple interest

o Income tax

o Compound interest

o Depreciation

o Prices after an increase or decrease

o Percentage profit and loss

c Find the original amount, given the new amount and the percentage change

c Calculate repeated proportional change

c Use a multiplier raised to a power to calculate repeated proportional change

c Use a multiplier to increase or decrease by a percentage

Direct and indirect proportion

c Calculate an unknown quantity where quantities are in direct proportion

c Calculate an unknown quantity where quantities are in inverse proportion

Fractions, decimals and percentages

c Find one number as a fraction of another number

c Find one number as a percentage of another number

c Multiply using percentages as operators

Number operations and the relationships between them, including order of operations and inverse operations

c Understand multiplying and dividing, and that one is the inverse of the other

c Use inverse operations

c Understand the use of brackets in calculations

c Understand the hierarchy of operations (BIDMAS)

c Solve word problems

c Understand and find reciprocals

c Understand that the inverse of raising to the power of n is the same as raising to the power of 1 over n

c Understand and use 1 over a number is the inverse of multiplying by that number

c Use reverse percentage calculations

Ratio

c Write a ratio in its simplest form

c Divide a quantity in a given ratio

c Solve problems using ratios

Use surds and π in exact calculations

c Use surds (roots) and π in calculations without a calculator, leaving the surd or π in the answer, eg. give an answer of 25 π

c Give an answer to a Pythagoras question as √17

c Manipulate surds in calculations, eg. (3 - √3)²

c Rationalise a denominator, ie. manipulate so that there is no longer a surd on the bottom of the fraction

Rounding and approximation

c Round to the nearest integer (whole number)

c Round numbers to any given power of 10

c Round to a number of decimal places

c Round to a number of significant figures

c Estimate the answer to a calculation by using rounding

Upper and lower bounds

Find the upper and lower bound of a calculation, especially in the calculation of:

c measurements

c perimeter

c area

c volume

c Give a final answer to a calculation to an appropriate degree of accuracy using upper and lower bounds

Use a calculator effectively

c Simple and complex calculations, including involving time or money

c Use the following functions

c +, -, x, ÷

c x² and √x

c memory functions

c brackets

c x to the power of y

c x to the power of 1 over y

c brackets

c trigonometrical functions

c Understand that rounding too early can causes inaccuracy

c Understand numbers shown in standard form, and be able to enter numbers in standard form

c Calculate in standard form

c Use for dividing to do reverse percentage calculations

c Use a multiplier and the power key to calculate exponential growth or decay

ALGEBRA

Algebraic notation

c Understand notation and symbols used in algebra

c Understand the difference between “expression”, “formula”, “equation” and “identity”

c Be able to select an expression, formula, equation or identity from a list

c Be able to write an expression to solve a problem

Manipulate algebraic expressions

c Simplify by collecting like terms

c Multiply out a single bracket

c Factorise a single bracket by taking out a common factor

c Expand two brackets

c Factorise quadratics into two brackets

c Factorise quadratics using the difference of two squares, eg. 4y² - 25 = (2y + 5)(2y - 5)

c Simplify algebraic expressions by cancelling, adding, subtracting and multiplying

c Use index laws, including fractional, zero and negative powers, and powers raised to another power

Solve linear equations

c Set up simple equations for a problem

c Rearrange simple equations

c Solve simple equations

c Solve equations with the unknown on either side

c Solve equations with the unknown on both sides

c Solve equations that include brackets

c Solve equations with negatives, including negative answers

c Solve equations involving fractions

Solve simultaneous equations with two unknowns

c Use elimination to solve simultaneous equations

c Use substitution to solve simultaneous equations

c Draw straight line graphs and find the solution from the intersection of the two graphs

c Write simultaneous equations for a problem

Solve quadratic equations

c Solve quadratic equations by factorisation

c Solve quadratic equations by completing the square

c Solve quadratic equations using the quadratic formula

Using formulae

c Derive formulae

c Substitute numbers (positive or negative) into a formula, including formulae with x² or x³ terms

c Change the subject of a simple formula

c Change the subject of a formula where the subject appears on both sides of the formula

c Change the subject of a formula that includes a power of the subject

Solve linear inequalities

c Solve a simple linear inequality with one variable

c Show the solution to a linear inequality with one variable on a number line

c Show the solution to several inequalities with two variables on a graph

Trial and improvement

c Use trial and improvement to find an approximate solution to an equation

Sequences

c Understand odd and even numbers

c Generate number sequences from diagrams

c Describe the rule for a number sequence

c Find a particular term in a sequence, or explain why a particular number is not in a sequence

Nth term of a sequence

c Find the nth term expression for a sequence

c Use the nth term expression to find a number in the sequence

Coordinates

c Use axes and coordinates, both positive and negative in 2D

c Understand and plot points in four quadrants

c Use axes and coordinates in 3D

c Find the coordinates of a point

c Plot a point given the coordinates, in 2D or 3D

c Find the mid-point of a line

c Calculate the length of a line using coordinates

Graphs

c Draw, label and add a scale to axes

c Understand that an equation of the form y = mx + c corresponds to a straight line graph

c Plot straight line graphs from their equations

c Plot and draw a graph of an equation in the form
y = mx + c

c Find the gradient of a straight line graph

c Find the gradient of a straight line graph from its equation

c Understand that a graph of an equation in the form
y = mx + c has gradient of m and a y intercept of c
(ie. crosses the y axis at c)

c Understand how the gradient of a real life graph relates to the relationship between the two variables

Gradients of parallel and perpendicular lines

c Understand how the gradients of parallel lines are related

c Understand how the gradients of perpendicular lines are related

c Understand that if the gradient of a graph in the form y = mx + c is m, then the gradient of a line perpendicular to it will be

c Generate equations of a line parallel or perpendicular to a straight line graph

Simultaneous equations (one linear and one quadratic)

c Find the intersection of a linear and a quadratic graph to find (approximate) solutions to simultaneous equations

c Solve simultaneous equations (one linear, one quadratic in one variable) by elimination

c Solve simultaneous equations where one equation is of the form x² + y² = r²

Other graphs

c Plot, sketch or recognise graphs of cubic functions

c Plot, sketch or recognise graphs of y = 1/x

c Plot, sketch or recognise graphs of y = kx for integer values of x

c Plot, sketch or recognise graphs of y = sin x and y = cos x from -360° to +360°

c Draw or plot other mathematical functions

c Recognise or analyse other mathematical functions

Graphs of loci

c Construct the graphs of simple loci including the circle, x² + y² = r²

c Find the points of intersection of a circle and a straight line

c Apply understanding of loci to construct graphs based on circles and perpendicular lines

Graphs from quadratic and other functions

c Generate points for quadratic functions

c Plot graphs of quadratic functions

c Find (approximate) solutions to a quadratic equation from the graph of its function

c Find (approximate) solutions to simultaneous equations, one quadratic and one linear from the intersections of their graphs

Real life graphs

c Plot a linear graph

c Interpret information on linear and non-linear graphs

Direct and inverse proportion

c Set up equations to solve word problems involving direct proportion

c Set up equations to solve word problems involving indirect proportion

c Understand and use graphs of equations involving direct and indirect proportion

Transformation of functions

Apply to the graph of y = f(x) the following transformations:

c y = f(x) + a

c y = f(ax)

c y = f(x + a)

c y = a f(x)

for linear, quadratic and sine and cosine functions, f(x)

c Apply the following transformations to functions:

c reflection

c rotation

c enlargement

c translation

c Analyse transformations of functions and write them algebraically

GEOMETRY

Angles on intersecting lines, in triangles and quadrilaterals, and on parallel lines

c Angles round a point add up to 360°

c Angles on a straight line add up to 180°

c Perpendicular lines

c Know the properties of scalene, isosceles, equilateral and right-angled triangles

c Angles in a triangle add up to 180°

c Angle properties of intersecting lines, and vertically opposite angles are equal

c Be able to mark parallel lines on a diagram

c Corresponding angles in parallel lines

c Alternate angles in parallel lines

c Calculate angles and give reasons

c Explain why the angle sum of a quadrilateral is 360°

c Understand a proof that the angle sum of a triangle is 180°

c Understand the proof that the exterior angle of a triangle of a triangle is equal to the sum of the interior angles at the other two vertices

c Calculate angles in more complex problems

Interior and exterior angles of polygons

c Calculate the sum of interior angles in a polygon

c Understand the polygon names; hexagon, heptagon, octagon and decagon

c Use the angle sum of an irregular polygon in a problem

c Calculate and use the sum of the interior angles of a regular polygon

c Understand and use fact that the exterior angles of a polygon add up to 360°

c Understand and use the fact that the interior and exterior angles at one vertex of a polygon add up to 180°

c Be able to calculate the exterior angle of a regular polygon

c Be able to calculate the interior angle of a regular polygon

c Be able to deduce the number of sides of a regular polygon, given one of its angles

c Understand tessellations of regular and irregular polygons

c Tessellate combinations of polygons

c Explain why some shapes tessellate and some do not

Properties of quadrilaterals

Remember the definitions and properties (including symmetry) of special quadrilaterals, ie.

c Square

c Rectangle

c Parallelogram

c Trapezium

c Rhombus

c Kite

c List or classify quadrilaterals by their properties

Reflection and rotation symmetry in 2D shapes

c Recognise reflection symmetry and be able to draw lines of symmetry on a shape

c Recognise rotation symmetry of 2D shapes

c Identify the order of rotational symmetry of a shape

c Complete a diagram given the line or lines of symmetry

c State a line of symmetry on a grid as a simple algebraic equation, eg. x = 2 or y = x

c Complete diagrams with a given order of rotational symmetry

Congruence and similarity

c Understand that angles in similar shapes are the same

c Prove the congruence of triangles using SSS, SAS, ASA and RHS and formal argument

c Understand SSS, SAS, ASA and RHS in ruler and compass constructions

c Understand similarity of triangles and other 2D shapes,

c Use understanding of similar figures in problems

c Prove formally that two triangles are similar

Pythagoras’ theorem

c Understand and use Pythagoras’ theorem in triangles

c Understand and use Pythagoras’ theorem in 3D problems

c Understand the language associated with 3D shapes, including diagonals of a cuboid

c Use Pythagoras’ theorem to calculate the length of a diagonal of a cuboid

Trigonometry

c Understand and remember trigonometric relationships in right angled triangles

c Use trigonometry in 2D problems

c Use trigonometry in 3D problems

c Use trigonometry to find the angle between a line and a plane

c Find angle of elevation and angle of depression

c Use the sine rule to solve 2D and 3D problems

c Use the cosine rule to solve 2D and 3D problems

Parts of a circle

c Draw a circle with compasses, given either the diameter or radius

Understand and remember parts of a circle:

c Centre

c Radius

c Diameter

c Chord

c Circumference

c Tangent

c Arc

c Sector

c Segment

Circle theorems and their proofs

c Prove and use each of the circle theorems:

c Tangent is perpendicular to the radius at the point the tangent meets the circle

c Two tangents from a point are equal in length

c Angle subtended from an arc at the centre is twice the angle at the circumference

c Angle in a semicircle is a right angle

c Angles in the same segment are equal

c Opposite angles of a cyclic quadrilateral add up to 180°

c Alternate segment theorem

c Perpendicular from the centre to a chord bisects the chord

Using 2D diagrams to represent 3D shapes

c Use isometric grids

c Draw nets and show how they fold to make a 3D solid shape

c Understand and draw front and side elevations and plans of simple solids

c Draw a sketch of a 3D solid shape given the front and side elevations and plan of the solid

Transformations

Rotations

c Rotate a 2D shape around the origin or other point

c Understand that a rotation is defined by an angle, direction and a centre of rotation

c Find the centre of rotation

c Understand that a rotation produces a shape congruent to the original

Reflections

c Understand and describe reflections

c Identify the mirror line for a reflection, and find its equation

c Understand that a reflection produces a shape congruent to the original

Translations

c Understand and use translations

c Understand that translations are defined by a distance and a direction using vector notation

c Translate a shape by a given vector

c Understand that a translation produces a shape congruent to the original

Enlargements

c Understand that an enlargement is defined by a centre of enlargement and a scale factor

c Understand that angles remain the same in an enlargement

c Enlarge a shape using (0, 0) or any other point as the centre

c Enlarge a shape by a positive scale factor

c Enlarge a shape by a fractional scale factor

c Enlarge a shape by a negative scale factor

c Find the centre of a given enlargement

c Identify the scale factor of a given enlargement

Combined transformations

c Describe a transformation using a combination of rotation, reflection, translation or enlargements.

Straight edge and compass constructions

c Construct a given triangle

c Construct an equilateral triangle

c Understand that SSS, SAS, ASA and RHS triangles are unique but ASS ones are not

c Construct a perpendicular bisector of a line

c Construct a perpendicular from a point to a line

c Construct a perpendicular from a point on a line

c Bisect an angle

c Construct angles of 60°, 90°, 30° and 45°

c Construct parallel lines

c Draw circles and arcs of a given radius

c Construct a regular hexagon inside a circle

c Construct diagrams involving any of the above

c Construct diagrams from given information

Loci

c Construct a region bounded by a circle and an intersecting line

c Construct a loci of a given distance from a point and a given distance from a line

c Construct a loci of equal distances from two points

c Construct a loci of equal distances from two lines

c Identify regions defined by “nearer to” or “greater than”

c Find or describe regions satisfying a combination of loci

Perimeter and area

c Measure shapes to find perimeter or area

c Find the perimeter of a rectangle or triangle

c Use a formula to find the area of a rectangle

c Use a formula to find the area of a triangle

c Use a formula to find the area of a parallelogram

c Use a formula to find the area of a trapezium

c Calculate the perimeter and area of compound shapes made from triangles, rectangles and other shapes

c Find the surface area of shapes such as prisms or pyramids by using the formulae for triangles, rectangles and other shapes

Area of a triangle

c Calculate the area of a triangle using the formulae

A = ½ ab sinC

Circumference and area of a circle

c Find circumference of a circle using C = πd or C = 2πr

c Find the area of a circle using A = πr²

c Use π = 3.142 or the π button on a calculator

c Find the perimeter and area of semcircles and quarter circles

c Calculate the length of an arc

c Calculate the area of a sector

c Give answers in terms of π if required

c Find the surface area of a cylinder

Volumes of prisms

c Use the formula to calculate the volume of a cuboid

c Calculate volume of a prism, such as a triangular prism

c Calculate the volume of a prism made from cuboids

c Find the volume of a cylinder

Complex shapes and solids

c Find the surface area of cubes, cuboids, cones, pyramids, spheres and hemispheres

c Find the volumes of cones, pyramids, spheres and hemispheres, frustrums

c Find the surface area or volume of a compound solid made up of other solid shapes, eg. a cuboid with pyramid on top, or cyclinder with cone on top.

c Use volumes in complex problems

c Find the area of a segment of a circle given the radius and length of the chord

Vectors

c Understand and use vector notation

c Add or subtract two vectors

c Multiply a vector by a number

c Calculate the result of two vectors

c Solve problems using vectors

c Use vectors in geometrical proofs

MEASURES

Maps and scale drawings

c Use, interpret and construct maps and scale drawings

c Draw lines and shapes to scale

c Estimate lengths using a scale diagram

Enlargement of shapes, including solids

c Understand the effect of enlargement on perimeter, area and volume

c Understand and use the fact that area and volume are affected differently by an enlargement

c Know the relationship between linear, area and volume scale factors when one 2D or solid shape is an enlargement of another

Interpretation and accuracy

c Read and interpret scales on measuring equipment

c Know the relationships between seconds, minutes, hours, days, weeks, months and years

c Use 12 and 24 hour clock times

c Calculate time intervals

c Recognise inaccuracy of measurement, and choose appropriate units of measurement

c Understand that choice of unit affects accuracy

c Understand that measurements given to a whole unit may be up to half a unit inaccurate in either direction

Converting measurements

c Know conversion factors between different metric units

c Convert between metric units

c Convert between imperial units, given the conversion factor

c Know rough imperial/metric equivalents as follows

c 1 kg = 2.2 pounds

c 1 litre = 1¾ pints

c 4.5 litres = 1 gallon

c 8 km = 5 miles

c 30 cm = 1 foot

c Convert between imperial and metric measures using the above conversion factors

c Convert between metric measurements of area

c Convert between metric measurements of volume

c Convert between different metric units of speed, eg. metres per second and km per hour

c Convert between metric units of volume and metric units of capacity, eg. 1 cm³ = 1 ml

Estimation of measures

c Make estimates of measurements

c Choose appropriate units for estimates of measurements

Bearings

c Use 3 figure bearings to specify direction

c Mark a point on a diagram, given a bearing and distance from another point

c Measure or draw a bearing on a map or scale plan

c Given a bearing of one point from another, find the bearing of the first point from the second

Compound measures

c Understand and use compound measures, including speed and density

Measure and draw lines and angles

c Measure and draw straight lines to the nearest mm

c Measure and draw angles to the nearest degree

Drawing using a ruler and protractor

c Make accurate drawings of triangles and other 2D shapes using ruler and protractor

c Make an accurate scale drawing from a diagram

c Use accurate drawing to solve bearings problems

STATISTICS

Data handling

c Decide on what data and analysis may be required for a problem

c Data collection

c Presenting data

c Discuss data

Bias

c Identify why data may be biased, and know how to minimise bias

c Understand the implications of different sizes of samples

Designing a survey

c Identify what data is needed

c Consider fairness of a survey

c Understand sample and population

c Design a question for a survey

c Criticise questions for a survey

c Understand random sampling

c Understand stratified sampling

c Calculate numbers needed for stratified sampling

Design data collection methods

c Design and use a data collection sheet, including one for continuous data

c Sort and classify data, and put data into a table

c Group data into class intervals with equal width

Tables and lists

c Extract data from tables and lists

Two-way tables

c Design two-way tables

c Complete a two-way table

Charts and diagrams

Draw the following charts or diagrams

c Bar chart

c Dual bar chart

c Pie chart

c Histogram with equal class intervals

c Frequency polygon

c Frequency diagram for grouped discrete data

c Scatter graph

c Line graph

c Frequency polygon for grouped data

c Grouped frequency table for continuous data

c Stem and leaf diagram

c Two-sided stem and leaf diagram

c Cumulative frequency table

c Cumulative frequency graph

c Box plots (from raw data, or when given the median and quartiles)

c Histograms with unequal class intervals, using frequency density

Types of average and range

Calculate the following

c Mean

c Mode

c Modal class

c Median

c Interval containing the median

c Range

c Estimate the mean of grouped data using mid-points of intervals

c Find median, quartiles and interquartile range for grouped data

c Estimate the mean for grouped data

c Find median, quartiles and interquartile range from a cumulative frequency graph

c Find median, quartiles and interquartile range from a box plot

Interpreting graphs and diagrams

Understand and find information from

c pie charts

c stem and leaf diagrams

c scatter graphs

c frequency polygons

c box plots

c cumulative frequency diagrams

c histograms

c Find the median or other information from a histogram, for example the number of people in a particular group

c Find information from line graphs, frequency polygons and frequency diagrams

c Find information from pie charts

c Find median, mode, range and interquartile range from stem and leaf diagrams

c Estimate values and find median, quartiles and interquartile range from a cumulative frequency graph

c Complete a frequency table from a histogram

c Understand and define frequency density

Patterns in data

c Find patterns in data

c Find exceptions in data

c Explain an isolated point on a scatter graph

Lines of best fit

c Draw a line of best fit

c Understand positve, negative and no correlation

c Understand that correlation does not always imply one thing causes the other

c Predict values using a line of best fit

c Understand that “no correlation” does not necessarily mean no relationship between the values, just no linear relationship

Comparing data

c Compare two sets of data using shapes of distributions

c Compare two sets of data using averages and spread, such as median, range and quartiles

c Compare spread using box plots or cumulative frequency graphs

c Compare two pie charts

c Compare data from dual bar charts

c Understand the advantages and disadvantages of different types of average

Using calculators

c Calculate mean using the correct key on a scientific calculator

c Σx and Σfx or calculation of the line of best fit

PROBABILITY

Probability language and the probability scale

c Impossible, unlikely, even chance, likely and certain events

c Mark events or probabilities on a 0 to 1 probability scale

c Write probabilities as fractions, decimals or percentages

Estimates of probability and relative frequency

c Find probabilities of events using dice, spinners, coins

c Understand and use relative frequency as estimates of probability

c Calculate an estimate of how many times an event will occur, given its probability and the number of trials

Listing events

c List the outcomes for one or two events

c Use and draw diagrams to show all possibilities

Mutually exclusive outcomes

c Understand that the sum of all the mutually exclusive outcomes is 1

c Know that if P is a probability of an outcome occurring, then 1 - P is the probability of the same outcome not occurring

c Fill in a missing probability in a table

c Know and use the fact that, for mutually exclusive events, P(A OR B) = P(A) + P(B)

Independent events

c Know that, for independent events,

P(A AND B) = P(A) x P(B)

c Understand the difference in calculation for selection of an object with or without replacement

Tree diagrams

c Draw a probability tree diagram

c Calculate probability of compound events from a tree diagram

Experimental data and theoretical probability

c Compare experimental data with theoretical probability

c Understand that the same experiment repeated can have different results, and that increasing sample size increases accuracy

c Compare results from different sample sizes

	Created by Mel Hughes about 8 years ago

GCSE Maths Note on Maths GCSE Checklist, created by Mel Hughes on 17/02/2016.