GCSE Maths

Specification
AQA GCSE Mathematics Course – Higher Tier – No Coursework (4307)

Course Overview
GCSE Maths is a must if you wish to be taken seriously by employers. This course will provide you with all the knowledge you require to pass the exam.

Course Content
AO2 Number and algebra
1. Using and applying number and algebra
Problem solving
H2.1a Select and use appropriate and efficient techniques and strategies to solve problems of increasing complexity, involving numerical and algebraic manipulation
H2.1b Identify what further information may be required in order to pursue a particular line of enquiry and give reasons for following or rejecting particular approaches
H2.1c Break down a complex calculation into simpler steps before attempting to solve it and justify their choice of methods
H2.1d Make mental estimates of the answers to calculations; present answers to sensible levels of accuracy by; understand how errors are compounded in certain calculations; use of rounding appropriately in intermediate calculations and final answers
.
Communicating
H2.1e Discuss work and explain their reasoning using correct mathematical language and notation
H2.1f Use a variety of strategies and diagrams for establishing algebraic or graphical representations of a problem and its solution; move from one form of representation to another to get different perspectives on the problem
H2.1g Present and interpret solutions in the context of the original problem
H2.1h Use numerical and algebraic notation and symbols correctly and consistently within a given problem
H2.1i Examine critically, improve, then justify their choice of mathematical presentation, present a concise, reasoned argument

Reasoning
H2.1j Explore, identify, and use pattern and symmetry in algebraic contexts, investigating whether particular cases can be generalised further, understanding the importance of a counter-example and be able to find one as appropriate, identify exceptional cases when solving problems
H2.1k Understand the difference between a practical demonstration and a proof
H21.l Show step-by-step deduction in solving a problem; derive proofs using short chains of deductive reasoning
H2.1m Recognise the significance of stating constraints and assumptions when deducing results; recognise the limitations of any assumptions that are made and the effect that varying the assumptions may have on the solution to a problem

Learning Outcomes
At the end of this section the student will be able to:
1. Solve a variety of numerical and algebraic problems by selecting the most suitable techniques to do so
2. Correctly use numerical and algebraic notation

2. Numbers and the number system
Integers
H2.2a Use their previous understanding of integers and place value to deal with arbitrarily large positive numbers and round them to a given power of 10; understand and use negative integers both as positions and translations on a number line; order integers; use the concepts and vocabulary of factor (divisor), multiple, common factor, highest common factor, least common multiple, prime number and prime factor decomposition

Powers and roots
H2.2b Use the terms square, positive square root, negative square root, cube and cube root; use index notation and index laws for multiplication and division of integer powers; use standard index form, expressed in conventional notation and on a calculator display

Fractions
H2.2c Understand equivalent fractions, simplifying a fraction by cancelling all common factors; order fractions by rewriting them with a common denominator

Decimals
H2.2d Use decimal notation and recognise that each terminating decimal is a fraction; recognise that recurring decimals are exact fractions, and that some exact fractions are recurring decimals; convert recurring decimals to rational numbers; order decimals

Percentages
H2.2e Understand that ‘percentage’ means ‘number of parts per 100’ and use this to compare proportions; interpret percentage as the operator ‘so many hundredths of’; use percentage in real-life situations

Ratio
H2.2f Use ratio notation, including reduction to its simplest form and its various links to fraction notation

Learning Outcomes
At the end of this section the student will be able to:
1. Deal with arbitrarily large numbers
2. Understand and use positive numbers and negative integers
3. Use and understand the concepts factor and multiple
4. Use and understand the terms power and root
5. Understand the basics of fractions and perform calculations with fractions
6. Use decimal and ratio notation and understand their respective relationships with fractions
7. Understand the concept of percentage

3. Calculations
Number operations and the relationships between them
H2.3a Multiply or divide any number by powers of 10, and any positive number by a number between 0 and 1; find the prime factor decomposition of positive integers; understand 'reciprocal' as multiplicative inverse, knowing that any non-zero number multiplied by its reciprocal is 1 (and that zero has no reciprocal, because division by zero is not defined); multiply and divide by a negative number; use index laws to simplify and calculate the value of numerical expressions involving multiplication and division of integer, fractional and negative powers; use inverse operations, understanding that the inverse operation of raising a positive number to power n is raising the result of this operation to power 1/n
H2.3b Use brackets and the hierarchy of operations
H2.3c Calculate a given fraction of a given quantity expressing the answer as a fraction; express a given number as a fraction of another; add and subtract fractions by writing them with a common denominator; perform short division to convert a simple fraction to a decimal; distinguish between fractions with denominators that have only prime factors of 2 and 5 (which are represented by terminating decimals), and other fractions (which are represented by recurring decimals); convert a recurring decimal to a fraction
H2.3d Understand and use unit fractions as multiplicative inverses multiply and divide a given fraction by an integer, by a unit fraction and by a general fraction, multiply and divide mixed numbers
H2.3e Convert simple fractions of a whole to percentages of the whole and vice versa then understand the multiplicative nature of percentages as operators calculate an original amount when given the transformed amount after a percentage change; reverse percentage
problems
H2.3f Divide a quantity in a given ratio

Mental methods
H2.3g Recall integer squares from 2x?2 to 15x??15 and the corresponding square roots, the cubes of 2, 3, 4, 5 and 10, the fact that   = 1 and   =   for positive integers n the corresponding rule for negative numbers    =   and    =   for any positive number n

H2.3h Round to a given number of significant figures; develop a range of strategies for mental calculation; derive unknown facts from those they know; convert between ordinary and standard index form representations converting to standard index form to make sensible estimates for calculations involving multiplication and/or division
F2.3i Develop a range of strategies for mental calculation; add and subtract mentally numbers with up to one decimal place; multiply and divide numbers with no more than one decimal digit, using the commutative, associative, and distributive laws and factorisation where possible, or place value adjustments

Written methods
F2.3k Division by decimal (up to 2 d.p.) by division using an integer; understand where to position the decimal point by considering what happens if they multiply equivalent fractions, eg, given that…work out…
H2.3i Use efficient methods to calculate with fractions, including cancelling common factors before carrying out the calculation, recognising that, in many cases, only a fraction can express the exact answer
H2.3j Solve percentage problems including percentage increase and
decrease and reverse percentages
F2.3n Solve word problems about ratio and proportion, including using
informal strategies and the unitary method of solution
H2.3k Represent repeated proportional change using a multiplier raised to a
power
H2.3l Calculate an unknown quantity from quantities that vary in direct or
inverse proportion
H2.3m Calculate with standard index form
H2.3n Use surds and ? ?in exact calculations, without a calculator; rationalise a denominator such as   = 

Calculator methods
H2.3o Use calculators effectively and efficiently, knowing how to enter complex calculations; use an extended range of function keys, including trigonometry and statistical functions relevant across this programme of study
F2.3p Enter a range of calculations, including those involving measures

H2.3p Understand the calculator display, knowing when to interpret the display, when the display has been rounded by the calculator, and not to round during the intermediate steps of a calculation
H2.3q Use calculators, or written methods, to calculate the upper and lower bounds of calculations, particularly when working with measurements
H2.3r Use standard index form display and know how to enter numbers in standard index form
H2.3s Use calculators for reverse percentage calculations by doing an appropriate division 
H2.3t Use calculators to explore exponential growth and decay using a multiplier and the power key

Learning Outcomes
At the end of this section the student will be able to:
1. Understand and find reciprocals
2. Use index laws to simplify expressions
3. Perform calculations involving negative and fractional powers
4. Recognise and use the hierarchy of operations
5. Convert fractions to percentages and vice versa
6. Perform percentage change and ratio calculations
7. Perform mental and written calculations using a range of strategies
8. Use a calculator effectively and efficiently

4. Solving numerical problems
H2.4a Draw on their knowledge of operations and inverse operations (including powers and roots), and of methods of simplification (including factorisation and the use of the commutative, associative and distributive laws of addition, multiplication and factorisation) in order to select and use suitable strategies and techniques to solve problems and word problems, including those involving ratio and proportion, repeated proportional change, fractions, percentages and reverse percentages, inverse proportion, surds, measures and conversion between measures, and compound measures defined within a particular situation
H2.4b Check and estimate answers to problems; select and justify appropriate degrees of accuracy for answers to problems; recognise limitations on the accuracy of data and measurements

Learning Outcomes
At the end of this section the student will be able to:
1. Select and use suitable strategies and techniques to solve problems and word problems
2. Select appropriate levels of accuracy and recognise accuracy limitations

5. Equations, formulae and identities
Use of symbols
H2.5a Distinguish the different roles played by letter symbols in algebra, using the correct notational conventions for multiplying or dividing by a given number, and knowing that letter symbols represent definite unknown numbers in equations, defined quantities or variables in formulae, general, unspecified and independent numbers in identities, and in functions they define new expressions or
quantities by referring to known quantities.
H2.5b Understand that the transformation of algebraic entities obeys and generalises the well-defined rules of generalised arithmetic, expand the product of two linear expressions; manipulate algebraic expressions by collecting like terms, multiplying a single term over a bracket, taking out common factors; factorising quadratic expressions including the difference of two squares and cancelling common factors in rational expressions.
H2.5c Know the meaning of and use the words ‘equation’, ‘formula’, ‘identity’ and ‘expression’

Index notation
H2.5d Use index notation for simple integer powers, and simple instances of index laws; substitute positive and negative numbers into expressions such as  3 ?? + 4 and 2

Equations
H2.5e Set up simple equations and solve simple equations by using inverse
operations or by transforming both sides in the same way

Linear Equations
H2.5f Solve linear equations with one unknown, with integer or fractional coefficients, in which the unknown appears on either side or on both; solve linear equations that require prior simplification of brackets, including those that have negative signs occurring anywhere in the equation, and those with a negative solution

Formulae
H2.5g Use formulae from mathematics and other subjects; substitute numbers into a formula; change the subject of a formula including cases where the subject occurs twice, or where a power of the subject appears; generate a formula

Direct and inverse proportion
H2.5h Set up and use equations to solve word and other problems involving direct proportion or inverse proportion and relate algebraic solutions to graphical representation of the equations

Simultaneous linear equations
H2.5i Find the exact solutions of two simultaneous equations in two unknowns by eliminating a variable and interpret the equations as lines and their common solution as the point of intersection

Inequalities
H2.5j Solve linear inequalities in one variable, and represent the solution set on a number line; solve several linear inequalities in two variables and find the solution set

Quadratic equations
H2.5k Solve quadratic equations by factorisation, completing the square and using the quadratic formula

Simultaneous linear and quadratic equations
H2.5l Solve exactly, by elimination of an unknown, two simultaneous equations in two unknowns, one of which is linear in each unknown, and the other is linear in one unknown and quadratic in the other or where the second is of the form   +   = 

Numerical Methods
H2.5m Use systematic trial and improvement to find approximate solutions of equations where there is no simple analytical method of solving them

Learning Outcomes
At the end of this section the student will be able to:
1. Distinguish the different roles played by letter symbols in algebra
2  Use a variety of methods to manipulate algebraic equations
3. Understand and use equation, formula, identity and expression
4. Manipulate and solve linear, quadratic equations and inequalities

6. Sequences, functions and graphs
Sequences
H2.6a Generate common integer sequences (including sequences of odd or even integers, squared integers, powers of 2, powers of 10, triangular numbers); generate terms of a sequence using term-to-term and position-to-term definitions of the sequence; use linear expressions to describe the nth term of an arithmetic sequence, justifying its form by reference to the activity or context from which
it was generated

Graphs of linear functions
H2.6b Use conventions for coordinates in the plane; plot points in all four
quadrants; recognise (when values are given for m and c) that equations of the form y = mx + c correspond to straight-line graphs in the coordinate plane; plot graphs of functions in which y is given explicitly in terms of x (for example, y = 2x + 3), or implicitly (for example, x + y = 7); no table or axes given
H2.6c Find the gradient of lines given by equations of the form y = mx + c (when values are given for m and c); understand that the form y = mx + c represents a straight line and that m is the gradient of the line and c is the value of the y – intercept; explore the gradients of parallel lines and lines perpendicular to each other; find the equation of a straight line when the graph is given.

Interpret graphical information
H2.6d Construct linear functions and plot the corresponding graphs arising
from real-life problems discuss and interpret graphs modelling real situations

Quadratic functions
H2.6e Generate points and plot graphs of simple quadratic functions then more general quadratic functions; find approximate solutions of a quadratic equation from the graph of the corresponding quadratic function; find the intersection points of the graphs of a linear and quadratic function, knowing that these are the approximate solutions of the corresponding simultaneous equations representing the linear
and quadratic functions

Other functions
H2.6f Plot graphs of simple cubic functions, the reciprocal function y =  with x ??0, the exponential function y =    for integer values of x and simple positive values of k, the circular functions y = sinx and y = cosx, using a spreadsheet or graph plotter as well as pencil and paper; recognise the characteristic shapes of all these functions

Transformation of functions
H2.6g Apply to the graph of y = f(x) the transformations y = f(x) + a, y = f(ax), y = f(x + a), y = af(x) for linear, quadratic, sine and cosine functions f(x)

Loci
H2.6h Construct the graphs of simple loci including the circle   +   =  for a circle of radius r centred at the origin of coordinates; find graphically the intersection points of a given straight line with this circle and know that this corresponds to solving the two simultaneous equations representing the line and the circle
 

Learning Outcomes
At the end of this section the student will be able to:
1. Generate sequences and terms of a sequence
2. Interpret the formula of and plot a straight line graph
3. Plot quadratic and a number of other functions
4. Apply transformation of functions to the graph y=f(x)

Study Time
To complete this course it will take in the region of 100 study hours which can be spread over a 12 month period to suit the student.

Pre-requisites
No previous study is required to access any of our GCSE courses, but they do require basic literacy and numeracy skills.

Support and Benefits
Full tutor support is available via email by fully qualified professionals.

Exam Dates and Information
Exams are in June of each year and the latest dates for enrolment is December.

If you start your course after December then it is unlikely you can take your exam in June unless your tutor agrees and you can find a centre, they will require late registration fees.

Please note: If you are wishing to take your exams next June (2010) if you enrol after 31st of October it is important that you check with the examination board that you will be able to sit the exam before enrolling on the course.

Qualifications
Learning at Home offers a range of GCSEs and all the course materials map to the very latest criteria laid down by the awarding bodies.

When you have taken your exams you will be awarded a grade which ranges from A – G with G being the lowest.

You will need to take GCSEs to progress onto A Levels and employers look for Maths and English as a minimum requirement for most jobs.

Taking Exams
Learning at Home in conjunction with the awarding body will give whatever help we can with examination information and finding an exam centre, but entering examinations is entirely the responsibility of the student, and the contract for sitting examinations is between the student and the exam centre. We are not an exam centre so you must read all the information in your course pack carefully and be prepared to travel to a centre which is willing to accept external candidates.

Funding
Learning at Home does not have access to any funding so if you require a subsidised course please contact your Learn Direct Centre.

Frequently Asked Questions
Q. Do your courses meet the latest syllabus changes?
A. yes, all our course materials meet any changes and will be updated free of charge if further changes are made.

Q. Why do I have to find a centre myself?
A. We have students all over the UK and Europe and it is impossible for us to arrange dates and times for individual students.

Q. What if I cannot find an examination centre?
A. If you wish to gain the qualification then be prepared to travel it is worth it!

Q. How much are exam fees?
A. These vary from centre to centre so you should check with them directly as we do not have access to the information.

Q. Are the courses paper based or on-line?
A. All our courses are paper based and come in attractive sturdy folders.

Q. How do I contact my tutor?
A. Tutors are all working Teachers or Lecturers so contact is by email only.

Q. Why can I not take my exams at when I have completed the course and have to wait?
A. Exams are taken at the same times as schools and colleges and are not flexible.

Q. I want to take my exams but there are only a few months to study, is this possible?
A. Depending on the time of year, it is sometimes impossible to complete your studies in a short space of time as your work has to be marked and checked. More importantly the examination boards have cut off times which are not flexible.