Edexcel GCSE Maths (Linear) – Higher specification mapped to the old Heinemann series
Edexcel GCSE Maths Higher
New two-tier specification mapped to the old three-tier Heinemann series
References to the relevant sections in the old books are given in the following form: H15.2 refers to theHigher tier book Chapter 15 section 2.
Page numbers are not included, so this document can be used with any of the previous versions of the textbooks.
Ma2 Number and algebra
Content / Section references1 / Using and Applying Number and Algebra
Students should be taught to:
Problem solving
a / select and use appropriate and efficient techniques and strategies to solve problems of increasing complexity, involving numerical and algebraic manipulation / Questions in this section will normally be found in the Mixed exercises at the end of each chapter on Number and Algebra.
b / 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
c / break down a complex calculation into simpler steps before attempting to solve it and justify their choice of methods
d / make mental estimates of the answers to calculations
present answers to sensible levels of accuracy
understand how errors are compounded in certain calculations
Communicating
e / discuss their work and explain their reasoning using an increasing range of mathematical language and notation
f / 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
g / present and interpret solutions in the context of the original problem
h / use notation and symbols correctly and consistently within a given problem
i / examine critically, improve, then justify their choice of mathematical presentation, present a concise, reasoned argument
Reasoning
j / explore, identify, and use pattern and symmetry in algebraic contexts, investigating whether particular cases can be generalised further, and understanding the importance of a counter-example
identify exceptional cases when solving problems
k / understand the difference between a practical demonstration and a proof
l / show step-by-step deduction in solving a problem
derive proofs using short chains of deductive reasoning
m / 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
Content / Section references
2 / Numbers and the Number System
Students should be taught to:
Integers
a / 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 / I1.1. I6.1
H12.1
understand and use negative integers both as positions and translations on a number line / I1.5
order integers / I1.3
use the concepts and vocabulary of factor (divisor), multiple, common factor, highest common factor, least common multiple, prime number and prime factor decomposition / I14.1, I14.8, I14.9, I14.10
H1.1, H1.2, H1.3, H1.4
Powers and roots
b / use the terms square, positive and negative square root, cube and cube root / I14.3, I14.7
H1.5, H1.6, H1.9
use index notation and index laws for multiplication and division of integer powers / I14.7, H1.7, H1.8
use standard index form, expressed in conventional notation and on a calculator display / I14.12, H5.10
Content / Section references
Fractions
c / understand equivalent fractions, simplifying a fraction by cancelling all common factors / I11.1, I11.2, I11.3
H1.10
order fractions by rewriting them with a common denominator / I11.4
Decimals
d / recognise that each terminating decimal is a fraction / I11.4
H23.1
recognise that recurring decimals are exact fractions, and that some exact fractions are recurring decimals / I11.4
H23.1, H23.2
order decimals / I1.2
Percentages
e / understand that ‘percentage’ means ‘number of parts per 100’ and use this to compare proportions / I22.1,
H5.1
interpret percentage as the operator ‘so many hundredths of’ / I22.2
e / use percentage in real-life situations / I22.7, I22.8
H5.1 to H5.7 inclusive
Ratio
f / use ratio notation, including reduction to its simplest form and its various links to fraction notation / I25.1, I25.2, I25.3
H5.9, Chapter H17
3 / Calculations
Students should be taught to:
Number operations and the relationships between them
a / multiply or divide any number by powers of 10, and any positive number by a number between 0 and 1 / I1.2
H1.1
find the prime factor decomposition of positive integers / I14.8, I14.9, I14.10
H1.2, H1.3, H1.4
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) / H1.7
multiply and divide by a negative number / I1.5
Content / Section references
use index laws to simplify and calculate the value of numerical expressions involving multiplication and division of integer, fractional and negative powers / I14.6, I14.7
H1.7, H1.8, H1.9
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 / H1.8
b / use brackets and the hierarchy of operations / I21.4
Chapter H10
c / calculate a given fraction of a given quantity, expressing the answer as a fraction / I11.6
express a given number as a fraction of another / I11.7
add and subtract fractions by writing them with a common denominator / I11.5
H1.10
perform short division to convert a simple fraction to a decimal / I11.4
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) / H23.1
convert a recurring decimal to a fraction / H23.2
d / understand and use unit fractions as multiplicative inverses / 11.6, H1.10
multiply and divide a given fraction by an integer, by a unit fraction and by a general fraction / I11.6
H1.10
e / convert simple fractions of a whole to percentages of the whole and vice versa / I22.1
H5.1, H5.2
then understand the multiplicative nature of percentages as operators / H5.1 to H5.5 inclusive
calculate an original amount when given the transformed amount after a percentage change / H5.6
reverse percentage problems
f / divide a quantity in a given ratio / I25.4, I25.5
H5.9
Content / Section references
Mental methods
g / recall integer squares from 2 2 to 15 15 and the corresponding square roots, the cubes of 2, 3, 4, 5 and 10, the fact that n0=1 and n–1 = for positive integers n, the corresponding rule for negative numbers, and for any positive numbern / H1.6, H1.7, H1.8
h / round to a given number of significant figures / Chapter I6
Chapter H12
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
i / develop a range of strategies for mental calculation / Use ideas in Chapter I6
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
k / division by decimal (up to 2 decimal places) by division using an integer / I1.4
understand where to position the decimal point by considering what happens if they multiply equivalent fractions
i / 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 / I11.2, I11.5, I11.6
H1.10
j / solve percentage problems, including percentage increase and decrease / I22.3, I22.6
H5.1 to H5.8
reverse percentages
n / solve word problems about ratio and proportion, including using informal strategies and the unitary method of solution / I25.2
H5.9
Content / Section references
k / represent repeated proportional change using a multiplier raised to a power / H5.7
l / calculate an unknown quantity from quantities that vary in direct or inverse proportion / Chapter H17
m / calculate with standard index form / H5.10
n / use surds and in exact calculations, without a calculator / H23.2
rationalise a denominator such as
Calculator methods
o / use calculators effectively and efficiently, knowing how to enter complex calculations / Ideas should introduced and reinforced at appropriate moments during the course
use an extended range of function keys, including trigonometrical and statistical functions relevant across this programme of study
p / enter a range of calculations, including those involving measures / I30.1
H29
p / 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 / Ideas for this section need to be emphasised in any calculations involving more than one step.
q / use calculators, or written methods, to calculate the upper and lower bounds of calculations, particularly when working with measurements / H23.5, H23.6, H23.7
r / use standard index form display and know how to enter numbers in standard index form / H5.10
s / use calculators for reverse percentage calculations by doing an appropriate division / H5.7
t / use calculators to explore exponential growth and decay, using a multiplier and the power key / H28.1
Content / Section references
4 / Solving Numerical Problems
Students should be taught to:
a / 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 / I1.1, I1.2, I1.4, I6.4,
Chapter I11
Chapter I14
Chapter I22
Chapter I25
Chapter H1
Chapter H5
Chapter H12
Chapter H23
b / check and estimate answers to problems / H23.4
select and justify appropriate degrees of accuracy for answers to problems / H16.1, H16.2
H23.6, H23.7
recognise limitations on the accuracy of data and measurements
5 / Equations, Formulae and Identities
Students should be taught to:
Use of symbols
a / 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 / Chapter H2
Chapter H10
b / understand that the transformation of algebraic entities obeys and generalises the well-defined rules of generalised arithmetic / Chapter H2
Chapter H10
expand the product of two linear expressions / I21.5, H10.2
Content / Section references
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 / I21.3, I21.4, I21.5
H10.1, H10.2, H10.3
H20.1, H20.2, H20.3
c / know the meaning of and use the words ‘equation’, ‘formula’, ‘identity’ and ‘expression’ / H20.1
Index notation
d / use index notation for simple integer powers / I21.3, H20.2, H20.3
use simple instances of index laws / I21.3, H20.4
substitute positive and negative numbers into expressions such as 3x2 + 4 and 2x3 / I21.1, I21.2
Equations
e / set up simple equations / H10.6, H10.7
solve simple equations by using inverse operations or by transforming both sides in the same way / I28.3
H2.1, H2.2, H10.5
Linear equations
f / solve linear equations in one unknown, with integer or fractional coefficients, in which the unknown appears on either side or on both sides of the equation / I28.1, I28.2, I28.3
H2.1, H2.2
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 / I28.3
H10.5
Formulae
g / use formulae from mathematics and other subjects / I21.1, I21.2
substitute numbers into a formula / I21.2, I21.2
Content / Section references
change the subject of a formula including cases where the subject occurs twice, or where a power of the subject appears / I21.7
H2.3, H14.5, H10.8
generate a formula / H10.6
Direct and inverse proportion
h / 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 / H17.5, H17.6, H17.7
Simultaneous linear equations
i / find 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 / I28.5, I28.6
H7.4, H7.5, H7.6
Inequalities
j / solve linear inequalities in one variable, and represent the solution set on a number line / I28.7
H2.4, H2.5
solve several linear inequalities in two variables and find the solution set / H7.7
Quadratic equations
k / solve simple quadratic equations by factorisation, completing the square and using the quadratic formula / I28.8
H21.1 to H21.5
Simultaneous linear and quadratic equations
l / 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 x2+y2=r2 / H21.6
Numerical methods
m / use systematic trial and improvement to find approximate solutions of equations where there is no simple analytical method of solving them / I18.8, I30.4
H18.6
Content / Section references
6 / Sequences, Functions and Graphs
Students should be taught to:
Sequences
a / generate terms of a sequence using term-to-term and position-to-term definitions of the sequence / Ideas in Chapter I2
H14.1, H14.2,
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 / I2.9
H14.2, H14.3
generate common integer sequences (including sequences of odd or even integers, squared integers, powers of 2, powers of 10, triangular numbers) / I2.5
Chapter H11A
Graphs of linear functions
b / use the conventions for coordinates in the plane
plot points in all four quadrants
recognise (when values are given for m andc) that equations of the form y = mx + c correspond to straight-line graphs in the coordinate plane / I7.1, I7.3
H7.1, H7.2
plot graphs of functions in which y is given explicitly in terms of x, or implicitly / I7.3
H7.2
c / find the gradient of lines given by equations of the form y = mx + c (when values are given for m and c) / I7.3, I7.4
H7.1, H7.2
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 / H7.2
explore the gradients of parallel lines and lines perpendicular to each other / H7.2, H7.3
Interpreting graphical information
d / construct linear functions and plot the corresponding graphs arising from real-life problems / H18.7
Content / Section references
discuss and interpret graphs modelling real situations / I18.9, H18.7
Quadratic functions
e / generate points and plot graphs of simple quadratic functions, then more general quadratic functions / I18.1 to I18.4
H18.1
find approximate solutions of a quadratic equation from the graph of the corresponding quadratic function / I18.5
H18.1
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 / H21.6
Other functions
f / plot graphs of simple cubic functions, the reciprocal function y =with x0, theexponential function y = kxfor integer values of x and simple positive values ofk,the circular functions y=sinx and y=cosx, using a spreadsheet or graph plotter as well as pencil and paper / I18.6, I18.7
H18.2, H18.3, H18.4
recognise the characteristic shapes of all these functions
Transformation of functions
g / 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) / All Chapter H24
Loci
h / construct the graphs of simple loci including the circle x2 + y2 = r2 for a circle of radius r centred at the origin of coordinates / H21.6
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 / H21.6
1
Edexcel GCSE Maths (Linear) – Higher specification mapped to the old Heinemann series
Ma3 Shape, space and measures
Content / Section references1 / Using and Applying Shape, Space and Measures
Students should be taught to:
Problem solving
a / select the problem-solving strategies to use in geometrical work, and consider and explain the extent to which the selections they made were appropriate / Questions in this section will normally be found in the Mixed exercises at the end of each chapter on Shape, Space and Measures
b / select and combine known facts and problem-solving strategies to solve more complex geometrical problems
c / develop and follow alternative lines of enquiry, justifying their decisions to follow or reject particular approaches
Communicating
d / communicate mathematically, with emphasis on a critical examination of the presentation and organisation of results, and on effective use of symbols and geometrical diagrams
e / use precise formal language and exact methods for analysing geometrical configurations
g / review and justify their choices of mathematics presentation
Reasoning
h / distinguish between practical demonstrations and proofs
f / apply mathematical reasoning, progressing from brief mathematical explanations towards full justifications in more complex contexts
g / explore connections in geometry
pose conditional constraints of the type ‘If… then…’
ask questions ‘What if…?’ or ‘Why?’
h / show step-by-step deduction in solving a geometrical problem
i / state constraints and give starting points when making deductions
j / understand the necessary and sufficient conditions under which generalisations, inferences and solutions to geometrical problems remain valid
Content / Section references
2 / Geometrical Reasoning
Students should be taught to:
Properties of triangles and other rectilinear shapes
a / distinguish between lines and line segments
use parallel lines, alternate angles and corresponding angles / I10.3,H3.1
understand the consequent properties of parallelograms and a proof that the angle sum of a triangle is 180 degrees / I4.1, I10.3
H3.2, H3.3, H3.4
understand a proof that the exterior angle of a triangle is equal to the sum of the interior angles at the other two vertices / I 10.3
b / use angle properties of equilateral, isosceles and right-angled triangles / I10.1
H3.3
explain why the angle sum of a quadrilateral is 360 degrees / I10.1,H3.5
e / use their knowledge of rectangles, parallelograms and triangles to deduce formulae for the area of a parallelogram, and a triangle, from the formula for the area of a rectangle / I20.1