Edexcel GCSE Maths (Linear) – Foundation specification mapped to old Heinemann series
Mathematics A (Linear) (2540)UG017611Graham CummingMay 2006Issue 1
Edexcel GCSE Maths Foundation
New two-tier specification mapped to the old three-tier Heinemann series
References to relevant sections in the old books are given in the following form: F15.2 refers to the Foundation 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 reference1 / Using and Applying Number and Algebra
Students should be taught to:
Problem solving
a / select and use suitable problem-solving strategies and efficient techniques to solve numerical and algebraic problems / Questions in this section will normally be found in the Mixed exercises at the end of each chapter on Number and Algebra.
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
b / break down a complex calculation into simpler steps before attempting to solve it and justify their choice of methods
c / use algebra to formulate and solve a simple problem — identifying the variable, setting up an equation, solving the equation and interpreting the solution in the context of the problem / F21.5
d / make mental estimates of the answers to calculations
use checking procedures, including use of inverse operations
work to stated levels of accuracy
Communicating
e / interpret and discuss numerical and algebraic information presented in a variety of forms
f / use notation and symbols correctly and consistently within a given problem
g / use a range of strategies to create numerical, 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
h / present and interpret solutions in the context of the original problem
i / review and justify their choice of mathematical presentation
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 / show step-by-step deduction in solving a problem
l / understand the difference between a practical demonstration and a proof
m / recognise the importance of 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 reference
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 / F1.1, F1.2, F1.5
I1.1. I 6.1
understand and use positive numbers and negative integers, both as positions and translations on a number line / F1.3, F1.10
I1.5
order integers / F1.10, I1.3
a / use the concepts and vocabulary of factor (divisor), multiple, common factor, highest common factor, least common multiple, prime number and prime factor decomposition / F1.6, F1.7
I14.1, I14.8, I14.9, I14.10
Powers and roots
b / use the terms square, positive and negative square root, cube and cube root / F1.8, I14.2, I14.3, I14.4
use index notation for squares, cubes and powers of 10 / F1.9, I14.3, I14.7
use index laws for multiplication and division of integer powers / I14.7
express standard index form both in conventional notation and on a calculator display / I14.12
Content / Section reference
Fractions
c / understand equivalent fractions, simplifying a fraction by cancelling all common factors / F4.1, F4.2, F4.3, F4.4, F4.6
I11.1, I11.2, I11.3
order fractions by rewriting them with a common denominator / F4.7
I 11.4
Decimals
d / use decimal notation and recognise that each terminating decimal is a fraction / F6.1, F6.7
I11.4
order decimals / F6.2
I1.2
d / recognise that recurring decimals are exact fractions, and that some exact fractions are recurring decimals / I 11.4
Percentages
e / understand that ‘percentage’ means ‘number of parts per 100’ and use this to compare proportions / F14.1, F14.2, F14.3
I22.1
interpret percentage as the operator ‘so many hundredths of ’ / F14.4, F14.6, I22.2
use percentage in real-life situations / F14.5, I 22.3, I22.5, I22.6, I22.7, I22.8
Ratio
f / use ratio notation, including reduction to its simplest form and its various links to fraction notation / F17.1, F17.2, F17.3, F17.5
I25.1, I25.2, I25.3
3 / Calculations
Students should be taught to:
Number operations and the relationships between them
a / add, subtract, multiply and divide integers and then any number / F1.4, F6.4, F6.5, F6.6
I1.1, I1.4
multiply or divide any number by powers of 10, and any positive number by a number between 0 and 1 / F1.4, F6.4
I1.2
a / find the prime factor decomposition of positive integers / I14.8, I14.9, I14.10
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)
Content / Section references
multiply and divide by a negative number / F1.11, F21.3, I1.5
use index laws to simplify and calculate the value of numerical expressions involving multiplication and division of integer powers / I14.6, I14.7
use inverse operations
b / use brackets and the hierarchy of operations / F2.8, I21.4
c / calculate a given fraction of a given quantity, expressing the answer as a fraction / F4.5
I11.6
express a given number as a fraction of another / F4.2,I11.7
add and subtract fractions by writing them with a common denominator / F4.8, F4.9
I11.5
perform short division to convert a simple fraction to a decimal / F6.7, I11.4
d / understand and use unit fractions as multiplicative inverses / F4.5, 11.6
d / multiply and divide a fraction by an integer, by a unit fraction and by a general fraction / F 4.10, F4.11
I11.6
e / convert simple fractions of a whole to percentages of the whole and vice versa / F14.2, F14.6
I22.1
understand the multiplicative nature of percentages as operators
f / divide a quantity in a given ratio / F17.4
I25.4, I25.5
Mental methods
g / recall all positive integer complements to 100 / Any Number chapter can be used to reinforce the ideas behind mental methods.
recall all multiplication facts to 10 10, and use them to derive quickly the corresponding division facts
recall integer squares from 1111 to 1515 and the corresponding square roots, recall the cubes of 2, 3, 4, 5 and 10, and the fraction-to-decimal conversion of familiar simple fractions
Content / Section references
h / round to the nearest integer and to one significant figure / F1.5, F6.3
Chapter I6
estimate answers to problems involving decimals / F6.3, I6.5
i / develop a range of strategies for mental calculation / Use ideas in Chapter F6.4
Use ideas in Chapter I6
derive unknown facts from those they know
add and subtract mentally numbers with up to two decimal places
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
j / use standard column procedures for addition and subtraction of integers and decimals / F1.4, F6.4, F1.11, F21.3
I1.1,I1.4
k / use standard column procedures for multiplication of integers and decimals, understanding where to position the decimal point by considering what happens if they multiply equivalent fractions / F1.4, F6.5
I1.4
solve a problem involving division by a decimal (up to 2 decimal places) by transforming it to a problem involving division by an integer / F6.6
I1.4
l / 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 / F4.8, F4.8, F4.10, F4.11
I11.2, I11.5, I11.6
m / solve simple percentage problems, including increase and decrease / F14.4, F14.5, F14.6
I22.3, I 22.6
n / solve word problems about ratio and proportion, including using informal strategies and the unitary method of solution / F17.3, F17.4
I25.2
n / use in exact calculations, without a calculator
Content / Section references
Calculator methods
o / use calculators effectively and efficiently: know how to enter complex calculations and use function keys for reciprocals, squares and powers / F1.8, F1.9, F24.1
I14.2, I14.3, I 14.4, I 14.5, I14.6, I14.7
p / enter a range of calculations, including those involving standard index form and measures / F24.1, ideas from Chapter F13
I30.1
q / 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.
4 / Solving Numerical Problems
Students should be taught to:
a
a / draw on their knowledge of operations, inverse operations and the relationships between them, and of simple integer powers and their corresponding 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, a range of measures and compound measures, metric units, and conversion between metric and common imperial units, set in a variety of contexts / F1.8, F1.9, F13.2, F14. 4, F17.4, F19.7
I1.1, I1.2, I1 .4, I6.4,
Chapter I11
Chapter I14
Chapter I22
Chapter I25
b / select appropriate operations, methods and strategies to solve number problems, including trial and improvement where a more efficient method to find the solution is not obvious / F24.4
I14.5
b / estimate answers to problems
use a variety of checking procedures, including working the problem backwards, and considering whether a result is of the right order of magnitude / F1.5, F6.3
I6.5
d / give solutions in the context of the problem to an appropriate degree of accuracy, interpreting the solution shown on a calculator display, and recognising limitations on the accuracy of data and measurements / Ideas in this section need to be emphasised whenever questions are set in context
Content / Section references
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 / F2.1, F21.1, F21.2
b / understand that the transformation of algebraic expressions obeys and generalises the rules of generalised arithmetic / F2.2, F2.3, F2.3, F2.6
manipulate algebraic expressions by collecting like terms, by multiplying a single term over a bracket, and by taking out common factors / F2.4, F2.5, F2.9
I21.4, I21.5
distinguish in meaning between the words ‘equation’, ‘formula’, ‘identity’ and ‘expression’ / I7.1
b / expand the product of two linear expressions / I21.5
Index notation
c / use index notation for simple integer powers / F2.7, I21.3
use simple instances of index laws / F2.7, I 21.3
substitute positive and negative numbers into expressions such as 3x2 + 4 and 2x3 / F21.2, F21.4
I21.2
Content / Section references
Equations
e / set up simple equations / F21.5
solve simple equations by using inverse operations or by transforming both sides in the same way / F15.1, F15.2, F15.3
I28.3
Linear equations
e / solve linear equations, with integer coefficients, in which the unknown appears on either side or on both sides of the equation / F15.1, F15.2
I28.1, I28.2, I28.3
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 / F15.3
I28.3
Formulae
f / use formulae from mathematics and other subjects expressed initially in words and then using letters and symbols / F21.2, F21.2
I21.1, I21.2
substitute numbers into a formula / F21.2 F21.4, I21.1, I21.2, I21.6
derive a formula and change its subject / F21.5, I21.7
Inequalities
d / solve simple linear inequalities in one variable, and represent the solution set on a number line / F21.6
I28.7
Numerical methods
m / use systematic trial and improvement to find approximate solutions of equations where there is no simple analytical method of solving them / F24.4
I18.8, I30.4
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 / F2.10
Ideas in Chapter I2
use linear expressions to describe the nth term of an arithmetic sequence, justifying its form by referring to the activity or context from which it was generated / F2.12
I2.9
a / generate common integer sequences (including sequences of odd or even integers, squared integers, powers of 2, powers of 10, triangular numbers) / F2.11
I2.5
Graphs of linear functions
b / use the conventions for coordinates in the plane / F9.1
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 / F9.5
I7.1, I7.3
plot graphs of functions in which y is given explicitly in terms of x, or implicitly / F9.5, I7.3
c / construct linear functions from real-life problems and plot their corresponding graphs / F9.2, F9.3, F9.4
I7.2
discuss and interpret graphs modelling real situations / I7.2, I7.6, I18.9
understand that the point of intersection of two different lines in the same two variables that simultaneously describe a real situation is the solution to the simultaneous equations represented by the lines / I28.4
draw line of best fit through a set of linearly related points and find its equation / I2.7, I2.8
Gradients
d / find the gradient of lines given by equations of the form y = mx + c (when values are given for m and c) / I7.4, I7.5
investigate the gradients of parallel lines / I7.4
Content / Section references
Interpret graphical information
e / interpret information presented in a range of linear and non-linear graphs / I7.2, I18.9
Quadratic equations
generate points and plot graphs of simple quadratic functions, then more general quadratic functions / F9.5
I18.1, I18.2
find approximate solutions of a quadratic equation from the graph of the corresponding quadratic function / I18.5
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Edexcel GCSE Maths (Linear) – Foundation specification mapped to 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 problem-solving strategies and resources, including ICT tools, to use in geometrical work, and monitor their effectiveness / Questions in this section will normally be found in the Mixed exercises at the end of each chapter on Shape, Space and Measures
a / consider and explain the extent to which the selections they made were appropriate
b / select and combine known facts and problem-solving strategies to solve complex problems
c / identify what further information is needed to solve a geometrical problem
break complex problems down into a series of tasks
c / develop and follow alternative lines of enquiry
Communicating
d / interpret, discuss and synthesise geometrical information presented in a variety of forms
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
f / use geometrical language appropriately
g / review and justify their choices of mathematics presentation
Reasoning
h / distinguish between practical demonstrations and proofs
i / apply mathematical reasoning, explaining and justifying inferences and deductions
j / show step-by-step deduction in solving a geometrical problem
k / state constraints and give starting points when making deductions
l / recognise the limitations of any assumptions that are made
understand the effects that varying the assumptions may have on the solution
Content / Section references
m / identify exceptional cases when solving geometrical problems
2 / Geometrical Reasoning
Students should be taught to:
Angles
a / recall and use properties of angles at a point, angles on a straight line (including right angles), perpendicular lines, and opposite angles at a vertex / F3.1, F3.3, F3.6, F3.7
I10 (introduction)
b / distinguish between acute, obtuse, reflex and right angles / F3.2
estimate the size of an angle in degrees / F3.2
Properties of triangles and other rectilinear shapes
a / distinguish between lines and line segments
c / use parallel lines, alternate angles and corresponding angles / F3.7, I10.3
understand the consequent properties of parallelograms and a proof that the angle sum of a triangle is 180 degrees / F5.1, F3.10, I4.1, I10.3
understand a proof that the exterior angle of a triangle is equal to the sum of the interior angles at the other two vertices / F3.10, I10.3
d / use angle properties of equilateral, isosceles and right-angled triangles / F3.8
I4.1
understand congruence / F5.4, F5.5, I4.2
explain why the angle sum of a quadrilateral is 360 degrees / F3.8, I10.1
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 / F19.4
I20.1
f / recall the essential properties and definitions of special types of quadrilateral, including square, rectangle, parallelogram, trapezium and rhombus / F5.1
I4.1
classify quadrilaterals by their geometric properties / F5.1, I4.1
g / calculate and use the sums of the interior and exterior angles of quadrilaterals, pentagons and hexagons / F5.6
I10.1, I10.2
calculate and use the angles of regular polygons / F5.6, I10.2
Content / Section references
h / understand, recall and use Pythagoras’ theorem / I15.1, I15.2
Properties of circles
i / recall the definition of a circle and the meaning of related terms, including centre, radius, chord, diameter, circumference, tangent, arc, sector and segment / F19.1, F19.4
I10.4
understand that inscribed regular polygons can be constructed by equal division of a circle / F5.6
3-D shapes
j / explore the geometry of cuboids (including cubes), and shapes made from cuboids / F11.1, F11.2, F11.3, F11.4
k / use 2-D representations of 3-D shapes and analyse 3-D shapes through 2-D projections and cross-sections, including plan and elevation / F11.5, F11.6
I4.6
i / solve problems involving surface areas and volumes of prisms / F19.4, F19.5
I20.4
3 / Transformations and Coordinates
Students should be taught to:
Specifying transformations
a / understand that rotations are specified by a centre and an (anticlockwise) angle / F18.3, F22.2
I23.3
rotate a shape about the origin, or any other point / F22.2, I23.3
measure the angle of rotation using right angles, simple fractions of a turn or degrees / F22.2, I23.3
understand that reflections are specified by a mirror line, at first using a line parallel to anaxis, then a mirror line such as y = x or y= –x / F18.1, F18.2, F22.3
I23.2
understand that translations are specified by a distance and direction (or a vector), and enlargements by a centre and positive scale factor / F22.1, F22.4
I23.1, I23.4
Properties of transformations
b / recognise and visualise rotations, reflections and translations, including reflection symmetry of 2-D and 3-D shapes, and rotation symmetry of 2-D shapes / All Chapters F18 and F22
I4.3, I4.4
Content / Section references
transform triangles and other 2-D shapes by translation, rotation and reflection and combinations of these transformations, recognising that these transformations preserve length and angle, so that any figure is congruent to its image under any of these transformations / I23.1, I23.2, I23.3
distinguish properties that are preserved under particular transformations
c / recognise, visualise and construct enlargements of objects using positive scale factors greater than one, then positive scale factors less than one / I23.4
understand from this that any two circles and any two squares are mathematically similar, while, in general, two rectangles are not / I4.2
d / recognise that enlargements preserve angle but not length / F22.4
I23.4
identify the scale factor of an enlargement as the ratio of the lengths of any two corresponding line segments and apply this to triangles / F22.4
I26.4
understand the implications of enlargement for perimeter
use and interpret maps and scale drawings / F17.5
understand the implications of enlargement for area and for volume / I23.4
distinguish between formulae for perimeter, area and volume by considering dimensions / I20.5
understand and use simple examples of the relationship between enlargement and areas and volumes of shapes and solids
Coordinates
e / understand that one coordinate identifies a point on a number line, two coordinates identify a point in a plane and three coordinates identify a point in space, using the terms ‘1-D’, ‘2-D’ and ‘3-D’ / F9.1, F9.6
I26.5
use axes and coordinates to specify points in all four quadrants / I26.5
locate points with given coordinates / F9.1
Content / Section references
find the coordinates of points identified by geometrical information / F9.1
find the coordinates of the midpoint of the line segment AB, given points A and B, then calculate the length AB / I26.5
Vectors
f / understand and use vector notation for translations / I23.1
4 / Measures and Construction
Students should be taught to:
Measures
a / interpret scales on a range of measuring instruments, including those for time and mass / Chapter F7
Chapter I5
know that measurements using real numbers depend on the choice of unit
recognise that measurements given to the nearest whole unit may be inaccurate by up to one half in either direction / I6.6
convert measurements from one unit to another / F13.1, I12.1, I12.2
know rough metric equivalents of pounds, feet, miles, pints and gallons / F13.2, I12.2
make sensible estimates of a range of measures in everyday settings / Chapter F7
Chapter I5
b / understand angle measure using the associated language / F3.2, F3.4, F3.5, F3.11
Chapter I10
c / understand and use compound measures, including speed and density / F9.4, F19.7
I7.6, I12.7
Construction
d / measure and draw lines to the nearest millimetre, and angles to the nearest degree / F7.7, F3.4, F3.5
I5.8, I26.1
draw triangles and other 2-D shapes using a ruler and protractor, given information about their side lengths and angles / F5.2, F5.3
I26.1
understand, from their experience of constructing them, that triangles satisfying SSS, SAS, ASA and RHS are unique, but SSA triangles are not / F5.3
I26.1
Content / Section references
construct cubes, regular tetrahedra, square-based pyramids and other 3-D shapes from given information / F11.5
I4.5
e / use straight edge and compasses to do standard constructions, including an equilateral triangle with a given side, the midpoint and perpendicular bisector of a line segment, the perpendicular from a point to a line, the perpendicular from a point on a line, and the bisector of an angle / I26.1
Mensuration
f / find areas of rectangles, recalling the formula, understanding the connection to counting squares and how it extends this approach / F19.2, F19.4
I20.1, I20.2, I20.3
recall and use the formulae for the area of a parallelogram and a triangle / F19.4
I20.1
find the surface area of simple shapes using the area formulae for triangles and rectangles / F19.4
I20.4
calculate perimeters and areas of shapes made from triangles and rectangles / F19.4
I20.1
g / find volumes of cuboids, recalling the formula and understanding the connection to counting cubes and how it extends this approach / F19.3, F19.5
I20.4
calculate volumes of right prisms and of shapes made from cubes and cuboids / I20.4
h / find circumferences of circles and areas enclosed by circles, recalling relevant formulae / F19.1, F19.4
I20.2, I20.3
i / convert between area measures, including square centimetres and square metres, and volume measures, including cubic centimetres and cubic metres / F19.6
I12.4
Loci
j / find loci, both by reasoning and by using ICT to produce shapes and paths / I26.3
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