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Foundation Tier : Stage 1 Ma2 Number and Algebra 2 Using and applying number and algebra 2 Numbers and the number system 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; understand and use positive numbers, both as positions and translations on a number line; order integers; use the concepts and vocabulary of factor (divisor) multiple and common factor. 4AB; 14BC; 17F; 21ABCDG Fractions c Understand equivalent fractions, simplifying a fraction by cancelling all common factors; order fractions by rewriting them with a common denominator. 24ABC; 53AB; 60ABDF Decimals d Use decimal notation and recognise that each terminating decimal is a fraction; order decimals. 7ACD; 28C Percentages e Understand that ‘percentage’ means ‘number of parts per 100’ and use this to compare proportions; interpret percentage as the operator ‘so many hundredths of’. 28BEF; 47ABC; 69A Ratio f Use ratio notation, including reduction to its simplest form and its various links to fraction notation. 39EF 3 Calculations Number operations and the relationships between them a Add, subtract, multiply and divide integers and then any number.; multiply or divide any number by powers of 10, and any positive number by a number 2A; 4CD; 14D; 19ABC; 22; between 0 and 1. 26ABCDE; 27ABCD; 37C; 44ABCD b Use brackets and the hierarchy of operations. 15A c Calculate a given fraction of a given quantity, expressing the answer as a fraction; perform short division to convert a simple fraction to a decimal. 24E; 28C; 52A; 67ABC; 69D d Understand and use unit fractions as multiplicative inverses; multiply a fraction by an integer; multiply a fraction by a unit fraction. 60C; 67AF Mental methods g Recall all positive integer complements to 100. Recall all multiplication facts to 10 x 10, and use them to derive quickly the corresponding division facts; recall the fraction-to-decimal conversion of familiar simple fractions. 22; 28CF h Round to the nearest integer. 14A I Develop a range of strategies for mental calculation; derive unknown facts from those they know; mentally add and subtract 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. 19ABC; 22; 26ABCDE; 37C; 65A Written methods j Use standard column procedures for addition and subtraction and subtraction of integers and decimals. 19ABC; 65A 5 Equations, formulae and identities Use of symbols a Distinguish the different roles played by letter symbols in algebra, knowing that letter symbols represent definite unknown numbers in equations. 3B b Understand that the transformation of algebraic expressions obeys and generalises the rules of arithmetic; manipulate algebraic expressions by collecting like terms, by multiplying a single term over a bracket. 32ABC; 46C; 50ABDEFG 6 Sequences, functions and graphs Graphs of linear functions b Use the conventions for coordinates in the plane; plot points in all four quadrants 8D; 48A Ma3 Shape, space and measures 13b7d541-4e78-4717-b795-cdf2d13c6767.xls page 1 of 30 1 Using and applying shape, space and measures 2 Geometrical reasoning Angles b Distinguish between acute, obtuse, reflex and right angles; estimate the size of an angle in degrees. 23B; 38A; 53D Properties of triangles and other rectilinear shapes f Recall the essential properties of special types of quadrilateral, including square, rectangle, parallelogram, trapezium and rhombus; classify quadrilaterals by their geometric properties. 8AC; 23A; 38A Proportions of circles i Recall the definition of a circle and the meaning of related terms, including centre, radius, diameter, circumference. 29A 3 Transformations and coordinates Specifying transformations a Understand that rotations are specified by a centre and an (anticlockwise) angle; rotate a shape about the origin; measure the angle of rotation using right angles, simple fractions of a turn; understand that reflections rate specified by a mirror line, at first using a line parallel to an axis. 8D; 64ABD Properties of transformations b Recognise and visualise rotations and reflections including reflection symmetry and rotation symmetry of 2-D shape; transform triangles and other 2-D shapes by rotation and reflection, recognising that these transformations preserve length and angle, so that any figure is congruent to its image under any of these transformations. 8BCD; 53C; 64ABDG 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’; use axes and coordinates to specify points in all four quadrants; locate points with given coordinates. 8D; 48AC 4 Measures and construction Measures a Interpret scales on a range of measuring instruments, including those for time and mass; convert measurements from one unit to another; make sensible estimates of a range of measures in everyday settings. 1D; 5ABC; 11AB; 14E; 45A; 54A b Understand angle measure using the associated language. 51CD Construction d Measure and draw lines to the nearest millimetre, and angles to the nearest degree; draw triangles and other 2-D shapes using a ruler and protractor, given information about their side lengths and angles. 68BC 1 Using and applying handling data Problem solving a Carry out each of the four aspects of the handling data cycle to solve problems: 10ABCD; 30ABCDE I Specify the problem and plan; formulate questions in terms of the data needed, and consider what inferences can be drawn from the data, decide what data to collect (including sample size and data format) and what statistical analysis is needed. 10ABCD; 30ABCDE ii Collect data from a variety of suitable sources, including experiments and surveys, and primary and secondary sources. 10ABCD; 30ABCDE iii Process and represent the data; turn the raw data into usable information that gives insight into the problem. 10ABCD; 30ABCDE iv Interpret and discuss: answer the initial question by drawing conclusions from the data. 10ABCD; 30ABCDE b Identify what further information is needed to pursue a particular line of enquiry. 10ABCD; 30ABCDE c Select and organise the appropriate mathematics and resources to use for a task. 10ABCD; 30ABCDE d Review progress while working, check and evaluate solutions. 10ABCD; 30ABCDE Communicating 13b7d541-4e78-4717-b795-cdf2d13c6767.xls page 2 of 30 e Interpret, discuss and synthesise information presented in a variety of forms. 10ABCD; 30ABCDE f Communicate mathematically, including using ICT, making use of diagrams and related explanatory text. 10ABCD; 30ABCDE Reasoning h Apply mathematical reasoning, explaining inferences and deductions. 10ABCD; 30ABCDE I Explore connections in mathematics and look for cause and effect when analysing data. 10ABCD; 30ABCDE 3 Collecting data a Design and use data-collection sheets for grouped discrete data; collect data using various methods, including observation, controlled experiment, data- logging, questionnaires and surveys. 20D; 30AB b Gather discrete data from secondary sources, including printed tables and lists from ICT-based sources. 43AB 4 Processing and representing data a Draw and produce, using paper and ICT, pie charts for categorical data and 2C; 16C; 17D; 20D; 30D; 52DEF; frequency diagrams. 57BC b Calculate mean, range and median of small data sets with discrete data; identify the modal class for grouped discrete data. 2CD; 20D; 30D; 57ABC c Understand and use the probability scale. 62A e List all outcomes for single events, and for two successive events, in a systematic way. 36BCD; 62C 5 Interpreting and discussing results b 2B; 7B; 16ABD; 17E; 20D; 43CDE; Interpret a wide range of graphs and diagrams and draw conclusions. 52BC g Use the vocabulary of probability to interpret results involving uncertainty and prediction. 36A j Discuss implications of findings in the context of the problem. 30E Foundation Tier : Stage 2 Ma2 Number and Algebra 2 Numbers and the number system Powers and roots b Use the terms square, positive square root, cube; use index notation for squares and cubes and powers of 10. 21EF; 41C 3 Calculations Number operations and the relationships between them c Express a given number as a fraction of another. 24D e Convert simple fractions of a whole to percentages of the whole and vice versa. 28AD; 47ADE; 52A Mental methods g Recall the cubes of 2, 3, 4, 5 and 10 21F; 41C Calculator methods o Use calculators effectively: know how to enter complex calculations and use function keys for reciprocals, squares and powers. 63BCDEFGH 4 Solving numerical problems a Draw on their knowledge of the operations and the relationships between them, and of simple integer powers and their corresponding roots, to solve problems involving ratio and proportion, a range of measures including speed, metric units, and conversion between metric and common imperial units, set in a variety of contexts. 37F; 39AB; 49CD; 54ADFG; 65BC 5 Equations, formulae and identities Index notation c Use index notation for simple integer powers. 46CD Linear equations e Solve linear equations, with integer coefficients, in which the unknown appears on either side or both sides of the equation. 3CD; 9E; 42ABCDEFG; 66A Formulae f Use formulae from mathematics and other subjects expressed initially in words 9ABC; 15BCD; 27E; 46ABD; and then using letters and symbols; substitute numbers into a formula. 61ACD 6 Sequences, functions and graphs 13b7d541-4e78-4717-b795-cdf2d13c6767.xls page 3 of 30 Sequences a Generate terms of a sequence using term-to-term and positions-to-term definitions of the sequence. 9BC; 58AB Ma3 Shape, space and measures 2 Geometrical reasoning 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. 23AB; 53E Properties of triangles and other rectilinear shapes c Understand that the angle sum of a triangle is 180 degrees. 23A; 38B; 53F d Use angle properties of equilateral, isosceles and right-angled triangles; understand congruence; explain why the angle sum of any quadrilateral is 360 degrees. 23A; 38ABC 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. 31BC; 40C g Calculate the use and sums of the interior and exterior angles of quadrilaterals, pentagons and hexagons; calculate the use and angles of regular polygons. 38CEFG Properties of circles I Understand that inscribed regular polygons can be constructed by equal division of a circle. 38D 3 Transformations and coordinates Specifying transformations a Understand that translations are specified by a distance and direction, and enlargements by a centre and positive scale factor. 59B; 64AC Properties of transformations b Recognise and visualise translations; transform triangles and other 2-D shapes by translation, recognising that these transformations preserve length and angle, so that any figure is congruent to its image under this transformation. 64ACG c Recognise, visualise and construct enlargements of objects using positive scale factors greater than one. 59AB; 64E d Recognise that enlargements preserve angle but not length; identify the scale factor of an enlargement as the ratio of the lengths of any two corresponding line segments and apply this to triangles; understand the implications of enlargement for perimeter. 59BC 4 Measures and construction Construction e Use straight edge and compasses to do standard constructions including an equilateral triangle with a given side. 25C; 38G; 68AD Mensuration f Find areas of rectangles, recalling the formula, understanding the connection to counting squares and how it extends this approach; recall and use the formulae for the area of a parallelogram and a triangle; find the surface area of simple shapes using the area formulae for triangles and rectangles; calculate perimeters 1ABCD; 17CG; 31ABCDE; and areas of shapes made from triangles and rectangles. 40ABCD; 55D; 56B g Find volumes of cuboids, recalling the formula and understanding the connection to counting cubes and how it extends this approach; calculate volumes of shapes made from cubes and cuboids. 55AB Ma4 Handling Data 2 Specifying the problem and planning a See that random processes are unpredictable. 10ABCD b Identify questions that can be addressed by statistical methods. 10ABCD; 30ABE c Discuss how data relate to a problem. 10ABCD d Identify which primary data they need to collect and in what format, including grouped data, considering appropriate equal class intervals. 10ABCD e Design an experiment of survey; decide what secondary data to use. 10ABCD; 30E 3 Collecting data 13b7d541-4e78-4717-b795-cdf2d13c6767.xls page 4 of 30 a Design and use data-collection sheets for continuous data. 57D 4 Processing and representing data a Draw and produce, using paper and ICT, diagrams for continuous data, including scatter graphs and stem-and-leaf diagrams. 20A; 33A; 57CD b Calculate mean, range and median of small data sets with continuous data; identify the modal class for grouped continuous data. 20BC; 57AD d Understand and use estimates or measures of probability from theoretical models (including equally likely outcomes). 36ACD; 62ABC f Identify different mutually exclusive outcomes and know that the sum of the probabilities of all these outcomes is 1. 62FG h Draw lines of best fit by eye, understanding what these represent. 33C 5 Interpreting and discussing results a Relate summarised data to the initial questions. 10CD; 30E b 2B; 7B; 16ABD; 17E; 20D; Interpret a wide range of graphs and diagrams and draw conclusions. 43CDEF; 52BC c Look at data to find patterns and exceptions. 10CD; 20C; 30DE d Compare distributions and make inferences, using the shapes of distributions and measures of average and range. 2B; 20C; 57A e Consider and check results and modify their approach if necessary. 10CD f Have a basic understanding of correlation as a measure of the strength of the association between two variables; identify correlation using lines of best fit. 33BC h Compare experimental data and theoretical probabilities. 62E l Understand that if they repeat an experiment, they may – and usually will - get different outcomes, and that increasing sample size generally leads to better estimates of probability and population characteristics. 62DG Foundation Tier : Stage 3 Ma2 Number and Algebra Numbers and the number system Percentages e Use percentage in real-life situations. 47F; 69BCEF 3 Calculations Number operations and the relationships between them c Add and subtract fractions by writing them with a common denominator. 60CEG d Divide a fraction by an integer. 67DE f Divide a quantity in a given ratio. 39G Mental methods h Round to one significant figure; estimate answers to problems involving decimals. 4BE; 37ABD Written methods 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. 26ABCDE; 34AC; 65ABC 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. 60EG; 67BCDEF m Solve simple percentage problems, including increase and decrease. 69BCDEF n Solve word problems about ratio and proportion, including using informal, strategies and the unitary method of solution. 39ABCD; 49CD Calculator methods p Enter a range of calculations, including those involving measures. 49ABCD; 54DFG q Understand the calculator display, interpreting it correctly, and knowing not to round during the intermediate steps of a calculation. 49ABCD 4 Solving numerical problems 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. 41ABCDEFG c Use a variety of checking procedures, including working the problem backwards, and considering whether a result is of the right order of magnitude. 4E; 37D 13b7d541-4e78-4717-b795-cdf2d13c6767.xls page 5 of 30 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. 37E 5 Equations, formulae and identities Use of symbols a Distinguish the different roles played by letter symbols in algebra, knowing that letter symbols represent 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 know quantities. 9AB; 15BD; 32D b Manipulate algebraic expressions by taking out single common term factors. 50CDG Index notation c Substitute positive and negative numbers into expressions such as 3x2 + 4 and 2x3. 44E; 46D; 61BC Linear equations e Solve linear equations that require prior simplification of brackets, including those that have negative signs occuring anywhere in the equation, and those with a negative solution. 66BCD Formulae f Derive a formula. 9B; 32D; 35A; 42F; 46EF; 61E 6 Sequences, functions and graphs Graphs of linear functions b Plot graphs of functions in which y is given explicitly in terms of x, or implicitly. 35BCD c Construct linear functions from real-life problems and plot their corresponding graphs; discuss and interpret graphs arising from real situations. 9D; 12ABC; 18AB; 46G; 70AB Interpet graphical information e Interpret information presented in a range of linear and non-linear graphs. 12ABC; 18AB; 54E; 70A Ma3 Shape, space and measures 2 Geometrical reasoning Properties of triangles and other rectilinear shapes c Use parallel lines, alternate angles and corresponding angles; understand the properties of parallelograms and a proof that the angle sum of a triangle is 180 degrees; understand a proof that the exterior angle of a triangle is equal to the sum of the interior angles at the other two vertices. 23CDE; 38AB Properties of circles i Recall the meaning of the terms chord, tangent and arc as they relate to the circle. 29A 3-D shapes j Explore the geometry of cuboids (including cubes) and shapes made from cuboids. 25ABC 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. 25ABCD 3 Transformations and coordinates Properties of transformations b Recognise and visualise rotations, reflections and translations, including reflection symmetry of 3-D shapes. 25E; 64F c Understand from this that any two circles and any two square are mathematically similar, while, in general, two rectangles are not. 59C d Use and interpret maps and scale drawings. 6; 7D; 51ABD; 68D Coordinates e Find the coordinates of points identified by geometrical information ; find the coordinates of the midpoint of the line segment AB, given points A and B. 8D; 48AB 4 Measures and construction Measures 13b7d541-4e78-4717-b795-cdf2d13c6767.xls page 6 of 30 a Know rough metric equivalents of pounds, feet, miles, pints and gallons. 13AB; 45BC c Understand and use speed. 54DEFG Construction d Understand, from their experience of constructing them, that triangles satisfying SSS, SAS, ASA and RHS are unique, but SSA triangles are not; construct cubes, regular tetrahedra, square-based pyramids and other 3-D shapes from given information. 25CD; 55D; 68ABC Mensuration h Find circumferences of circles and areas enclosed by circles, recalling relevant formulae. 29BCD; 56ACD i Convert between area measures, including square centimetres and square metres, and volume measures, including cubic centimetres and cubic metres. 40E; 55C Ma4 Handling Data 3 Collecting data c Design and use two-way tables for discrete and grouped data. 2A; 17G; 43AB; 54BC; 62B 4 Processing and representing data a Draw and produce, using paper and ICT, line graphs for time series. 43E 5 Interpreting and discussing results k Interpret social statistics including index numbers; time series and survey data. 43EF 13b7d541-4e78-4717-b795-cdf2d13c6767.xls page 7 of 30 Intermediate Tier: Stage 1 Ma2 Number and Algebra 1 Using and applying number and algebra 2 Numbers and the number system Integers 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 a factor, least common multiple, prime number and prime factor decomposition. Powers and roots b Use the terms square, positive square root, negative square root, cube and cube root Fractions Understand equivalent fractions, simplifying a fraction by cancelling all common factors; c order fractions by rewriting them with a common denominator. Decimals Recognise that each terminating decimal is a fraction; recognise that recurring decimals d are exact fractions, and that some exact fractions are recurring decimals; order decimals. Percentages Understand that ‘percentage’ means ‘number of parts per 100’, and interpret percentage e as the operator ‘so many hundredths of’. 3 Calculations Number operations and the relationships between them 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; multiply and a divide by a negative number. b Use brackets and the hierarchy of operations. 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 c represented by recurring decimals). Understand and use unit fractions as multiplicative inverses; multiply and divide a given d fraction by an integer, by a unit fraction and by a general fraction. e Convert simple fractions of a whole to percentages of the whole and vice versa. Written methods 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 I express the exact answer. j Solve percentage problems, including percentage increase and decrease. Calculator methods 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 p calculation. 5 Equations, formulae and identities Use of symbols 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 formula, general, unspecified and independent numbers in identities, and in a functions they define new expressions or quantities by referring to known quantities. Understand that the transformation of algebraic entities obeys and generalises the well- defined rules of generalised arithmetic; manipulate algebraic expressions by collecting like b terms, multiplying a single term over a bracket and taking out common factors. c Know the meaning of and use the words ‘equation’, ‘formula’, ‘identity’ and ‘expression’. Equations Set up simple equations; solve simple equations by using inverse operations or by e transforming both sides in the same way. Linear equations 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; solve linear equations that require prior simplification of brackets, including those that have negative signs f occurring anywhere in the equation, and those with a negative solution. Formulae g Use formulae from mathematics and other subjects; substitute numbers into a formula. 6 Sequences, functions and graphs Sequences 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 a reference to the activity or context from which it was generated. Graphs of linear functions b Use conventions for coordinates in the plane; plot points in all four quadrants. Ma3 Shape, space and measures 1 Using and applying shape, space and measures 2 Geometrical reasoning Properties of triangles and other rectilinear shapes Distinguish between lines and line segments; use parallel lines, alternate angles and corresponding angles; understand the consequent properties of parallelograms and a proof that the angle sum of a triangle is 180 degrees; understand a proof that the exterior a angle of a triangle is equal to the sum of the interior angles at the other two vertices. Use angle properties of equilateral, isosceles and right-angled triangles; explain why the b angle sum of a quadrilateral is 360 degrees. Recall the definitions of special types of quadrilateral, including square, rectangle, parallelogram, trapezium and rhombus; classify quadrilaterals by their geometric c properties. Calculate and use the sums of the interior and exterior angles of quadrilaterals, d pentagons, hexagons; calculate and use the angles of regular polygons. Properties of circles Recall the definition of a circle and the meaning of related terms, including centre, radius, h chord, diameter, circumference, tangent, arc, sector and segment. 3 Transformations and coordinates Specifying transformations Understand that rotations are specified by a centre and an (anticlockwise) angle; use any point as the centre of rotation; measure the angle of rotation, using right angles, fractions a of a turn or degrees; understand that reflections are specified by a (mirror) line. Properties of transformations Recognise and visualise rotations, reflections and translations including reflection symmetry of 2-D and 3-D shapes, and rotation symmetry of 2-D shapes; transform b triangles and other 2-D shapes by translation, rotation and reflection. Coordinates Use axes and coordinates to specify points in all four quadrants; locate points with given e coordinates; find the coordinates of points identified by geometrical information. 4 Measures and construction Measures Use angle measure; know that measurements using real numbers depend on the choice a of unit; convert measurements from one unit to another. Construction Draw approximate constructions of triangles and other 2-D shapes, using a ruler and protractor, given information about side lengths and angles; construct specified cubes, b regular tetrahedra, square-based pyramids and other 3-D shapes. Ma4 Handling data 1 Using an applying handling data Problem solving a Carry out each of the four aspects of the handling data cycle to solve problems: Specify the problem and plan: formulate questions in terms of the data needed, and consider what inferences can be drawn from the data; decide what data to collect I (including sample size and data format) and what statistical analysis is needed. Collect data from a variety of suitable sources, including experiments and surveys, and ii primary and secondary sources. Process and represent the data: turn the raw data into usable information that gives insight iii into the problem. Interpret and discuss the data: answer the initial question by drawing conclusions from the iv data. Select the problem-solving strategies to use in statistical work, and monitor their effectiveness (these strategies should address the scale and manageability of the tasks, and should consider whether the mathematics and approach used are delivering the most b appropriate solutions). Communicating Communicate mathematically, with emphasis on the use of an increasing range of diagrams and related explanatory text, on the selection of their mathematical presentation, explaining its purpose and approach, and on the use of symbols to convey statistical c meaning. Reasoning Apply mathematical reasoning, explaining and justifying inferences and deductions, d justifying arguments and solutions. e Identify exceptional or unexpected cases when solving statistical problems. Explore connections in mathematics and look for relationships between variables when f analysing data. Recognise the limitations of any assumptions and the effects that varying the assumptions g could have on the conclusions drawn from data analysis. 3 Collecting data Collect data using various methods, including observation, controlled experiment, data a logging, questionnaires and surveys. Gather data from secondary sources, including printed tables and lists from ICT-based b sources. c Design and use two-way tables for discrete and grouped data. d Deal with practical problems such as non-response or missing data. 4 Processing and representing data c List all outcomes for single events, and for two successive events, in a systematic way. Identify different mutually exclusive outcomes and know that the sum of the probabilities of d all these outcomes is 1. Intermediate Tier: Stage 2 Ma2 Number and Algebra Powers and roots b Use standard index form, expressed in conventional notation and on a calculator display. Ratio Use ratio notation, including reduction to its simplest form and its various links to fraction f notation. 3 Calculations Number operations and the relationships between them f Divide a quantity in a given ratio. Mental methods h Round to a given number of significant figures. Calculator methods Use calculators effectively and efficiently, knowing how to enter complex calculations; use an extended range of function keys, including trigonometrical and statistical functions o relevant across this programme of study. 4 Solving numerical problems Check and estimate answers to problems; select and justify appropriate degrees of b accuracy for answers to problems. 5 Equations, formulae and identities Expand the product of two linear expressions; manipulate algebraic expressions by b factorising quadratic expressions. Index notation d Substitute positive and negative numbers into expressions such as 3x2 + 4 and 2x3. Formulae g Change the subject of a formula; generate a formula. Simultaneous linear equations Solve simple linear inequalities in one variable, and represent the solution set on a j number line. Numerical methods Use systematic trial and improvement to find approximate solutions of equations where m there is no simple analytical method of solving them. 6 Sequences, functions and graphs Graphs of linear functions Plot graphs of functions in which y is given explicitly in terms of x (as in y = 2x + 3), or b implicitly (as in x + y = 7) Interpreting graphical information Construct linear functions and plot the corresponding graphs arising from real-life d problems; discuss and interpret graphs modelling real situations. Quadratic functions Generate points and plot graphs of simple quadratic functions, then more general e quadratic functions. Ma3 Shape, space and measures 2 Geometrical reasoning Properties of triangles and other rectilinear shapes f Understand, recall and use Pythagoras’ theorem in 2-D problems. Understand, recall and use trigonometrical relationships in right-angled triangles, and use g these to solve problems, including those involving bearings. Properties of circles Understand that the tangent at any point on a circle is perpendicular to the radius at that point; understand and use the fact that tangents from an external point are equal in length; understand that inscribed regular polygons can be constructed by equal division of a h circle. 3 Transformations and coordinates Specifying transformations Understand that translations are specified by giving a distance and direction (or a vector), a and enlargements by a centre and a positive scale factor. Properties of transformations Recognise, visualise and construct enlargements of objects; understand from this that any two circles and any two squares are mathematically similar, while, in general, two c rectangles are not, then use positive fractional scale factors. Recognise that enlargements preserve angle but not length; identify the scale factor of an enlargement as the ratio of the lengths of any two corresponding line segments; understand the implications of enlargement for perimeter; use and interpret maps and d scale drawings. 4 Measures and construction Measures Recognise that measurements given to the nearest whole unit may be inaccurate by up to one half in either direction; understand and use compound measures, including speed and a density. Mensuration Find the surface area of simple shapes by using the formulae for the areas of triangles and rectangles; find volumes of cuboids, recalling the formula and understanding the connection to counting cubes and how it extends this approach; calculate volumes of right prisms and of shapes made from cubes and cuboids; find circumferences of circles and d areas enclosed by circles, recalling relevant formulae. Ma4 Handling data 2 Specifying the problem and planning a See that random processes are unpredictable. b Identify key questions that can be addressed by statistical methods. Discuss how data relate to a problem; identify possible sources of bias and plan to c minimise it. Identify which primary data they need to collect and in what format, including grouped d data, considering appropriate equal class intervals. e Design an experiment or survey; decide what primary and secondary data to use. 4 Processing and representing data Draw and produce, using paper and ICT, pie charts for categorical data, and diagrams for continuous data, including line graphs (time series), scatter graphs, frequency diagrams, a stem-and-leaf diagrams, cumulative frequency tables and diagrams, box plots. Understand and use estimates or measures of probability from theoretical models, or from b relative frequency. Find the median, quartiles and interquartile range for large data sets and calculate the e mean for large data sets with grouped data. I Draw lines of best fit by eye, understanding what these represent. 5 Interpreting and discussing results a Relate summarised data to the initial questions. b Interpret a wide range of graphs and diagrams and draw conclusions. c Look at data to find patterns and exceptions. e Consider and check results, and modify their approaches if necessary. Appreciate that correlation is a measure of the strength of the association between two f variables; distinguish between positive, negative and zero correlation using lines of best fit. g Use the vocabulary of probability to interpret results involving uncertainty and prediction. h Compare experimental data and theoretical probabilities. Understand that if they repeat an experiment they may – and usually will – get different outcomes, and that increasing sample size generally leads to better estimates of I probability and population parameters. Intermediate Tier: Stage 3 Ma2 Number and Algebra 2 Numbers and the number system Powers and roots b Use index notation and index laws for multiplication and division of integer powers. 3 Calculations Number operations and the relationships between them 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); use index laws to simplify and calculate the value of numerical expressions a involving multiplication and division of integer powers; use inverse operations. Understand the multiplicative nature of percentages as operators; calculate an original amount when given the transformed amount after a percentage change; reverse e percentage problems. Mental methods Recall integer squares from 2 x 2 to 15 x 15 and the corresponding square roots, the g cubes of 2, 3, 4, 5 and 10. 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 h and/or division. Written methods j Solve percentage problems including reverse percentages. k Represent repeated proportional change using a multiplier raised to a power. l Calculate an unknown quantity from quantities that vary in direct proportion. m Calculate with standard index form. n Use surds and π in exact calculations, without a calculator. Calculator methods r Use standard index form display and how to enter numbers in standard index form. s Use calculators for reverse percentage calculations by doing an appropriate division. 4 Solving numerical problems 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, surds, measures and conversion a between measures, and compound measures defined within a particular situation. b Recognise limitations on the accuracy of data and measurement. 5 Equations, formulae and identities Use of symbols Manipulate algebraic expressions using the difference of two squares and by cancelling b common factors in rational expressions. Index notation d Use index notation for simple integer powers, and simple instances of index laws. Formulae Change the subject of a formula, including cases where the subject occurs twice, or where g a power of the subject appears. Simultaneous linear equations Find the exact solution 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 I intersection. j Solve several linear inequalities in two variables and find the solution set. Quadratic equations k Solve quadratic equations by factorisation. Sequences, functions and graphs Graphs of linear functions Recognise (when values are given for m and c) that equations of the form y = mx + c b correspond to the straight-line graphs in the coordinate plane. 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 c parallel lines. Quadratic functions Find approximate solutions of a quadratic equation from the graph of the corresponding e quadratic function. Other functions Plot graphs of: simple cubic functions, the reciprocal function y = 1/x with x ≠ 0, using a spreadsheet or graph plotter as well as pencil and paper; recognise the characteristic f shapes of all these functions. Loci h Construct the graphs of simple loci. Ma3 Shape, space and measures 2 Geometrical reasoning Properties of triangles and other rectilinear shapes f Investigate the geometry of cuboids including cubes, and shapes made from cuboids. Understand similarity of triangles and of other plane figures, and use this to make g geometric inferences. Properties of circles Explain why the perpendicular from the centre to a chord bisects the chord; use the facts that the angle subtended by an arc at the centre of a circle is twice the angle subtended at any point on the circumference, the angle subtended at the circumference by a semicircle is a right angle, that angles in the same segment are equal, and that opposite angles of a h cyclic quadrilateral sum to 180 degrees. 3-D shapes Use 2-D representations of 3-D shapes and analyse 3-D shapes through 2-D projections and cross-sections, including plan and elevation; solve problems involving surface areas I and volumes of prisms. 3 Transformations and coordinates Properties of transformations Transform triangles and other 2-D shapes by translation, rotation and reflection and combinations of these transformations; distinguish properties that are preserved under b particular transformations. Understand the difference between formulae for perimeter, area and volume by d considering dimensions. Coordinates Understand that one coordinate identifies a point on a number line, that 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’; find the coordinates of the midpoint of the line segment AB, given the e points A and B, then calculate the length AB. Vectors f Understand and use vector notation. 4 Measures and construction Construction 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 c bisector of an angle. Mensuration d Convert between volume measures including cm3 and m3. Loci e Find loci, both by reasoning and by using ICT to produce shapes and paths. Ma4 Handling data 4 Processing and representing data f Calculate an appropriate moving average. Use tree diagrams to represent outcomes of compound events, recognising when events h are independent. j Use relevant statistical functions on a calculator or spreadsheet. 5 Interpreting and discussing results b Identify seasonality and trends in time series. Compare distributions and make inferences, using shapes of distributions and measures d of average and spread, including median and quartiles. Appreciate that zero correlation does not necessarily imply ‘no relationship’ but merely ‘no f linear relationship’. 6ADE; 7A; 9ABC; 40B 1C; 51ABC 11AB; 15B; 17A 11FG 13ABCDEF; 18A; 44A 3ABCDEFGH; 6DE; 7A 20AB 11CDEFG; 13CF; 17A 17BCDEFG; 33ACD 13ACF; 44A 11CD; 15B; 58A 13G; 44BCDEFG 20ABC; 26AB; 32EF 2AE; 4A; 8A; 21A; 43A 2BCD; 4B; 19AB; 28ABCDE 2A; 4A; 8A; 43A 4AC; 7C; 41CD 4AD; 7C; 19C; 41ABEF 2A; 7B; 8A; 12ACD; 33E 14ABCDEFGH 21A; 56A 25AB; 45BC 45CD 45A 45DE 10D; 61AD 53ACD 30D; 45A; 53ABCD 56A 16F; 36D 30B 24ABCD; 29CDEF 24ABCD; 29CDEF 24ABCD; 29CDEF 24ABCD; 29CDEF 24ABCD; 29CDEF 24ABCD; 29CDEF 24ABCD; 29CDEF 24ABCD; 29CDEF 24ABCD; 29CDEF 24ABCD; 29CDEF 24D 22A; 29ABCDEF 5CD 24B 34CD 34B 40ABCDEF 37ABDE; 38D 37C 9DEF; 20ACD 6B; 20ABCE; 32EF; 42A; 49BE; 51D; 57E 9G; 20D; 60A 43ABCDEF; 47AB 7B; 12AB; 28AD 8AB; 28F; 50ABCDG 31ABC; 55ABCDE 57D 21A; 39A 21BD; 32B; 35ABC 21C; 57B 1BCD 42ABCD; 49ABCDEF 61D 53ABE 38ABCD; 53E 36ABCD; 38ABCD 16E; 32ABCDEFG; 35B; 60B 10ABCDEF; 16ABC 24ABCD 24D; 29DEF 24BCD 24D 24D; 29F 5BCD; 18DEF; 22A; 27ABCE; 59A 34AB; 58B 5E; 27D 22B; 39E 24CD 5ABCD; 18BC; 22C; 27E; 59A 22ABC; 24CD; 29ABDEF 24CD 22ABC 34AB 34A 34A 6BC; 23ABCDEFGHIJ; 51E 23CEGHIJ; 33B; 51E 44CD; 48ABD 1C; 51AB 40ABCEF 48DE 48C 15ABCDE 40H 62ABC 40DGI 48D 15ACDE; 16E; 26AB; 32ABCDEFG; 44G; 48E 60ABCDE 47BFG 23CDEFGHIJ 50EFGH 46ABCDEFGHI 52ABCDE 47CDEG 21A; 39A 39ABCD 21C; 57B 57ACEF 16A; 30ABC 38CDE 61ABCDE 16ABCD; 30ABC 53F 63 1D; 56BCD 53B; 56A 54CDEF 16F 54ABCDEFG 59B 58CD 29F 59B 27E 22AC Higher Tier: Stage 1 Ma2 Number and Algebra 1 Using and applying number and algebra 2 Numbers and the number system Integers a Use the concepts and vocabulary of highest common factor, least common multiple, prime number and prime factor decomposition. 4B Powers and roots b Use index laws for multiplication and division of integer powers; use standard index form, expressed in conventional notation and on a calculator display. 4ACD; 12ABCDEF Decimals d Recognise that recurring decimals are exact fractions, and that some exact fractions are recurring decimals. 45AB Ratio f Use ratio notation, including reduction to its simplest form and its various links to fraction notation. 22A 3 Calculations Number operations and the relationship between them a Multiply or divide any number by a number between 0 and 1; find the prime factor decomposition of positive integers; multiply and divide by a negative number. 4B c 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. 45AB d Multiply and divide a given fraction by a unit fraction and by a general fraction. 3BDE e Understand the multiplicative nature of percentages as operators; calculate an original amount when given the transformed amount after a percentage change; reverse percentage problems. 9ABDE f Divide a quantity in a given ratio 22A Mental methods g Recall integer squares from 2 x 2 to 15 x 15 and the corresponding square roots, the cubes of 2, 3, 4, 5 and 10. h Round to a given number of significant figures; convert between ordinary and standard index form representations. 12CE Written methods j Solve percentage problems; reverse percentages. 9BDE k Represent repeated proportional change using a multiplier raised to a power. 9C Calculator methods r Use standard index form display and how to enter numbers in standard index form. 12DF s Use calculators for reverse percentage calculations by doing an appropriate division. 9D 4 Solving numerical problems a Draw on their knowledge of operations and inverse operations 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, measures and conversion 3E; 6ACDE; 9BCDE; between measures, and compound measures defined within a particular situation. 10DFG; 22CDF b Check and estimate answers to problems. 5 Equations, formulae and identities 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 formula, general, unspecified and independent numbers in identities, and in functions they define new expressions or quantities by referring to known quantities. 4C; 5A; 8A; 41E b Understand that the transformation of algebraic entities obeys and generalises the well-defined rules of generalised arithmetic; manipulate algebraic expressions by collecting like terms, multiplying a single term over a bracket, taking out common factors. 20AB; 27C c Know the meaning of and use the words ‘equation’, ‘formula’, ‘identity’ and ‘expression’. 4CD; 5A; 8A Index notation d Use index notation for simple instances of index laws. 4CD; 27AB; 38A Equations e Set up simple equations; solve simple equations by using inverse operations or by transforming both sides in the same way. 5ABCDE 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. 5ABCE Formulae g Use formulae from mathematics and other subjects; substitute numbers info a formula; generate a formula. 8A; 10B; 27A; 30D Numerical methods m Use systematic trial and improvement to find approximate solutions of equations where there is no simple analytical method of solving them. 28AD 6 Sequences, function and graphs Sequences a Generate common integer sequences (including sequences of odd or even integers, squared integers, powers of 2, powers of 10, triangular numbers); 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. 36ABCE Graphs of linear functions b 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. 13CD Interpreting graphical information d Construct linear functions and plot the corresponding graphs arising from real-life problems; discuss and interpret graphs modelling real situations. 6B; 13B; 22B Quadratic functions e Generate points and plot graphs of simple quadratic functions, then more general quadratic functions. 26AB; 28A Ma3 Shape, space and measures 1 Using and applying shape, space and measures 2 Geometrical reasoning Properties of triangles and other rectilinear shapes a Distinguish between lines and line segments. 14A f Understand recall and use Pythagoras’ theorem in 2-D problems. 46A g Understand, recall and use trigonometrical relationships in right angled triangles, and 2ABCD; 11ABCDEF; use these to solve problems, including those using bearings. 46A Properties of circles h Recall the definition of a circle and the meaning of related terms, including sector and segment; understand that the tangent at any point on a circle is perpendicular to the radius at that point; understand and use the fact that tangents from an external point are equal in length. 25AB; 29D 3 Transformations and coordinates Specifying transformations a Use any point as the centre of rotation; measure the angle of rotation, using fractions of a turn or degrees; understand that translations are specified by a vector. 16A Properties of transformations c Recognise, visualise and construct enlargements of objects using positive fractional and negative scale factors. 16B; 33A Coordinates e Given the coordinates of the points A and B, calculate the length AB. 11F 4 Measures and construction Measures a Know that measurements using real numbers depend on the choice of unit. 31A Mensuration d Find the surface area of simple shapes by using the formulae for the areas of triangles and rectangles; find volumes of cuboids, recalling the formula and understanding the connection to counting cubes and how it extends this approach; calculate volumes of right prisms. 10ACE Ma4 Handling data 1 Using and applying handling data Problem solving a Carry out each of the four aspects of the handling data cycle to solve problems: I Specify the problem and plan: formulate questions in terms of the data needed, and consider what inferences can be drawn from the data; decide what data to collect (including sample size and data format) and what statistical analysis is needed. 1ABCD; 24ABCDEF ii Collect data from a variety of suitable sources, including experiments and surveys, and primary and secondary sources. 1ABCD; 24ABCDEF iii Process and represent the data: turn the raw data into usable information that gives insight into the problem. 1ABCD; 24ABCDEF iv Interpret and discuss the data: answer the initial question by drawing conclusions from the data. 1ABCD; 24ABCDEF b Select the problem-solving strategies to use in statistical work, and monitor their effectiveness (these strategies should address the scale and manageability of the tasks, and should consider whether the mathematics and approach used are delivering the most appropriate solutions). 1ABCD; 24ABCDEF Communicating c Communicate mathematically, with emphasis on the use of an increasing range of diagrams and related explanatory text, on the selection of their mathematical presentation, explaining its purpose and approach, and on the use of symbols to convey statistical meaning. 1ABCD; 24ABCDEF Reasoning d Apply mathematical reasoning, explaining and justifying inferences and deductions, justifying arguments and solutions. 1ABCD; 24ABCDEF e Identify exceptional or unexpected cases when solving statistical problems. 1ABCD; 24ABCDEF f Explore connections in mathematics and look for relationships between variables when analysing data. 1ABCD; 24ABCDEF g Recognise the limitations of any assumptions and the effects that varying the assumptions could have on the conclusions drawn from data analysis. 1ABCD; 24ABCDEF 2 Specifying the problem and planning c Identify possible sources of bias and plan to minimise it. 1BCD e Decide what primary and secondary data to use. 1D; 24F 3 Collecting data d Deal with practical problems such as non-response or missing data. 1D; 24F 4 Processing and representing data a Draw and produce, using paper and ICT, cumulative frequency tables and diagrams, box plots and histograms for grouped continuous data. 15ABCE; 37C e Find the median, quartiles and interquartile range for large data sets and calculate the mean for large data sets with grouped data. 7ABC; 15DE f Calculate an appropriate moving average. 7D I Draw lines of best fit by eye, understanding what these represent. 13H j Use relevant statistical functions on a calculator or spreadsheet. 24F 5 Interpreting and discussing results b Identify seasonality and trends in time series. 7D d Compare distributions and make inferences, using shapes of distributions and measures of average and spread, including median and quartiles. 15DE f Appreciate that correlation is a measure of the strength of the association between two variables; distinguish between positive, negative and zero correlation using lines of best fit; appreciate that zero correlation does not necessarily imply ‘no relationship’ but merely ‘no linear relationship’. 13H; 24D Higher Tier: Stage 2 Ma2 Number and algebra 3 Calculations Number operations and the relationship between them a 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); use index laws to simplify and calculate the value of numerical expressions involving multiplication and division of integer, fractional and negative powers. 3C; 4CEF; 39ABC Mental methods g Recall the fact that no = 1 and n-1 = ? For positive integers n and the corresponding rule for negative numbers; n? = ?n and n? = ?n for any positive number n. 4E; 39AB h Converting to standard index form to make sensible estimates for calculations involving multiplication and/or division. 12DEG l Calculate an unknown quantity from quantities that vary in direct or inverse proportion. 22DEF m Calculate with standard index form. 12G n Use surds and π in exact calculations, without a calculator. 44ABC; 45CDEGH Calculator methods q Use calculators, or written methods, to calculate the upper and lower bounds of calculations, particularly when working with measurements. 31BCDE t Use calculators to explore exponential growth and decay, using a multiplier and the power key. 39EFG 4 Solving numerical problems a Draw on their knowledge of operations and inverse operations (including powers and roots), and of methods of simplification including surds, defined within a particular situation. 45DEH b Select and justify appropriate degrees of accuracy for answers to problems; recognise limitations on the accuracy of data and measurements. 31AB 5 Equation, formulae and identities Use of symbols b Expand the product of two linear expressions; manipulate algebraic expressions by factorising quadratic expressions including the difference of two squares and 20CD; 27CD; 34BCE; cancelling common factors irrational expressions. 38ABCDF Formulae g Change the subject of a formula, including cases where the subject occurs twice, or where a power of the subject appears. 8BCD; 30ABCDE; 38E Simultaneous linear equations i Find the exact solution 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. 18ABCDEF j Solve simple 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. 19ABCD; 23ABCD Quadratic equations k Solve quadratic equations by factorisation, completing the square and using the 20E; 34ADFG; 41B; quadratic formula. 43A; 45G 6 Sequences, functions and graphs Graphs of linear functions c 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. 13CDFG Quadratic functions e Find approximate solutions of a quadratic equation from the graph of the corresponding quadratic function. 26BCD; 28AD Other functions f Plot graphs of: simple cubic functions, the reciprocal function y = 1/x with x ≠ 0, the exponential function y = kx for integer values of x and simple positive values of k, the circular functions y = sin x and y = cos x, using a spreadsheet or graph plotter as well 28ABC; 35AB; 39EF; as pencil and paper; recognise the characteristic shapes of all these functions. 41A Ma3 Shape, space and measures 2 Geometrical reasoning Properties of triangles and other rectilinear shapes e Understand and use SSS, SAS, ASA and RHS conditions to prove the congruence of triangles using formal arguments, and to verify standard ruler and compass constructions. 42ABCD g Calculate the area of a triangle using ?ab sin C. 46E Properties of circles h Explain why the perpendicular from the centre to a chord bisects the chord. 25B 3-D shapes i Solve problems involving surface areas and volumes of prisms, pyramids, cylinders, cones and spheres. 10E; 25CDE 3 Pupils should be taught to: Properties of transformations b Transform triangles and other 2-D shapes by combinations of transformations; use congruence to show that translations, rotations and reflections preserve length and angle, so that any figure is congruent to its image under any of these transformations; distinguish properties that are preserved under particular transformations. 16AC Vectors f Understand and use vector notation; calculate, and represent graphically the sum of two vectors, the difference of two vectors and a scalar multiple of a vector; calculate the resultant of two vectors; understand and use the commutative and associative properties of vector addition; solve simple geometrical problems in 2-D using vector methods. 40ABCDEF 4 Measures and construction Measures a Recognise that measurements given to the nearest whole unit may be inaccurate by up to one half in either direction; understand and use compound measures, including speed and density. 6ABCDE; 10DF; 31AB Construction c 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. 14A Mensuration d Calculate the lengths of arcs and the areas of sectors of circles. 25A Loci e Find loci, both by reasoning and by using ICT to produce shapes and paths. 14AB Ma4 Handling data 4 Processing and representing data b Understand and use estimates or measures of probability from theoretical models, or from relative frequency. 17B g Know when to add or multiply two probabilities: if A and B are mutually exclusive, then the probability of A or B occurring is P(A) + P(B), whereas if A and B are independent events, the probability of A and B occurring is P(A) x P(B). 17CD h Use tree diagrams to represent outcomes of compound events, recognising when events are independent. 17EFG 5 Interpreting and discussing results d Understand frequency density. 37ABC Higher Tier: Stage 3 Ma2 Number and algebra 3 Calculations Number operations and the relationships between them a 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 ?. 39ABCG Written methods a Rationalise a denominator such as ??? 45F Calculator methods o Use calculators effectively and efficiently, knowing how to enter complex calculations; use an extended range of function keys, including trigonometrical and statistical functions relevant across this programme of study. 2A; 11BE; 24F; 39D 5 Equations, formulae and identities 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. 22CDEFG 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. 34H; 43C 6 Sequences, functions and graphs Graphs of linear functions c Explore the gradients of parallel lines and lines perpendicular to these lines. 13CEF Quadratic functions e 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. 26E; 34H 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). 35D; 41ACDEF 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; 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. 43BCD Ma3 Shape, space and measures 2 Geometrical reasoning f Understand, recall and use Pythagoras’ theorem in 3-D problems; investigate the geometry of cuboids including cubes, and shapes made from cuboids, including the use of Pythagoras’ theorem to calculate lengths in three dimensions. 47ABC g Understand similarity of triangles and of other plane figures, and use this to make geometric inferences; use trigonometrical relationships in 3-D contexts, including finding the angles between a line and a plane (but not the angle between two planes or between two skew lines); draw, sketch and describe the graphs of trigonometric functions for angles of any size, including transformations involving scalings in either or both the x and y directions; use the sine and cosine rules to solve 2-D and 3-D 33AB; 35ABCD; problems. 46BCDF; 47ABC Properties of circles h Prove and use the facts that the angle subtended by an arc at the centre of a circle is twice the angle subtended at any point on the circumference, the angle subtended at the circumference by a semicircle is a right angle, that angles in the same segment are equal, and that opposite angles of a cyclic quadrilateral sum to 180 degrees; prove and use the alternate segment theorem. 29BCDEF 3-D shapes i Solve problems involving more complex shapes and solids, including segments of circles and frustums of cones. 25BD 3 Transformations and coordinates d Understand the difference between formulae for perimeter, area and volume by considering dimensions; understand and use the effect of enlargement on areas and volumes of shapes and solids. 32; 33CDE 4 Measures and construction Mensuration d Convert between volume measures including cm3 and m3 10G; 33E Ma4 Handling data 2 Specifying the problem and planning d Select and justify a sampling scheme and a method to investigate a population, including random and stratified sampling. 21ABCD