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Kj%?!_~寿R6l=u… .\KjIENDB`3  @@   +recall integer squares from 11 x 11 to 15 x 15 and the corresponding square roots; recall the cubes of 2, 3, 4, 5 and 10, and the fraction-to-decimal conversion of familiar simple fractions5calculate, 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 methods40ABCDEFuse angle measure-convert measurements from one unit to another10G, 33EAunderstand and use compound measures, including speed and density 6ABCDE, 10DFdraw approximate constructions of triangles and other 2-D shapes, using a ruler and protractor, given information about side lengths and anglesunderstand, from their experience of constructing them, that triangles satisfying SSS, SAS, ASA and RHS are unique, but SSA triangles are not42AYconstruct specified cubes, regular tetrahedra, square-based pyramids and other 3-D shapesuse 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 angleKcalculate perimeters and areas of shapes made from triangles and rectangles10BFffind the surface area of simple shapes by using the formulae for the areas of triangles and rectangles10AC10EAcalculate the lengths of arcs and the areas of sectors of circles25A14ABwselect and justify a sampling scheme and a method to investigate a population, including random and stratified sampling21ABCD1D, 24Fzcollect data using various methods, including observation, controlled experiment, data logging, questionnaires and surveys24ABCDEF24ABAdeal with practical problems such as non-response or missing data24BC)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, stem-and-leaf diagrams, cumulative frequency tables and diagrams, box plots and histograms for grouped continuous data5BCD, 18DEF, 22A, 27ABCE, 59A7ABCD, 15BCE, 37Ckunderstand and use estimates or measures of probability from theoretical models, or from relative frequency17Bdraw, 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  35ABCD;use the sine and cosine rules to solve 2-D and 3-D problems 46BCDF, 47ABC25AB`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; explain why the perpendicular from the centre to a chord bisects the chord; understand that inscribed regular polygons can be constructed by equal division of a circle61D25B, 29Dvprove 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; prove and use the alternate segment theorem61ABCE29BCDEFdsolve problems involving surface areas and volumes of prisms, pyramids, cylinders, cones and spheres 10E, 25CDElsolve problems involving more complex shapes and solids, including segments of circles and frustums of cones25BDunderstand 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 of a turn or degreesWlist all outcomes for single events, and for two successive events, in a systematic way 17Cfind the median, quartiles and interquartile range for large data sets and calculate the mean for large data sets with grouped data27D, 5E 7ABC, 15D'calculate an appropriate moving average59B7D58C17CDcuse tree diagrams to represent outcomes of compound events, recognising when events are independent58D17EFG13H24F1CD5ABCD, 18BC, 22C, 27E, 59A7BC, 15E, 37AB/identify seasonality and trends in time series compare distributions and make inferences, using shapes of distributions and measures of average and spread, including median and quartiles27E15E, 37Cunderstand frequency density37ABBconsider and check results, and modify their approach if necessaryunderstand 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 parameters21C2use pi in exact calculations, without a calculatorkrecall all multiplication facts to 10 x 10, and use them to derive quickly the corresponding division factsVunderstand and use the effect of enlargement on areas and volumes of shapes and solids33CDE 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 ; use axes and coordinates to specify points in all four quadrants; locate points with given coordinates; find the coordinates of points identified by geometrical information; find the coordinates of the midpoint of the line segment AB, given the points A and B, then calculate the length AB4    "understand and use vector notationDfind the gradient of lines given by equations of the form y = mx + c (when values are given for m and c); understand that the form y = mx + c represents a straight line and that m is the gradient of the line and c is the value of the y-intercept; explore the gradients of parallel lines and lines perpendicular to each other: ;> @C D` af g      13CDEFG"Interpreting graphical information\construct linear functions and plot the corresponding graphs arising from real-life problems 21B, 35ABC 6B, 13B, 22B6discuss and interpret graphs modelling real situations6B, 13BQuadratic functions26B 26BCD, 28ADfind 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 functions26E, 34HOther functionsV 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 = sinx and y = cosx, using a spreadsheet or graph plotter as well as pencil and paper; recognise the characteristic shapes of all these functions? @C DE FL Ml m< p qr      57CEF28ABC, 35AB, 39EF, 41ATransformation of functionsapply 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)  2 49 :A BH IL MR SZ [a b 35D, 41ACDEF, 43Dcarry 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 (ii) collect data from a variety of suitable sources, including experiments and surveys, and primary and secondary sources (iii) process and represent the data: turn the raw data into usable information that gives insight into the problem (iv) interpret and discuss the data: answer the initial question by drawing conclusions from the data1ABCD, 24ABCDEFselect 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)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 meaninguapply mathematical reasoning, explaining and justifying inferences and deductions, justifying arguments and solutions1ABCDCidentify key questions that can be addressed by statistical methods 1D, 24DEF1D42ABCDHunderstand, recall and use Pythagoras theorem in 2-D, then 3-D problems 46A, 47ABCinvestigate the geometry of cuboids including cubes, and shapes made from cuboids, including the use of Pythagoras theorem to calculate lengths in three dimensions47Chunderstand similarity of triangles and of other plane figures, and use this to make geometric inferences33AB?understand, recall and use trigonometrical relationships in right-angled triangles, and use these to solve problems, including those involving bearings, then use these 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)42ABCD, 49ABCDEF2ABCD, 11ABCDEF, 46A, 47AB46Emanipulate 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 expressions2BCD, 4B, 19AB, 28ABCDE, 47BFG20ABD, 27ABCD, 34C, 38ABCDFXknow the meaning of and use the words  equation ,  formula ,  identity and  expression 4CD, 5A, 8AQuse index notation for simple integer powers, and simple instances of index laws P 23BCDEFGHIJ 4CD, 27A, 38AQsubstitute positive and negative numbers into expressions such as 3x2 + 4 and 2x3D EP 7B, 12AB, 28D, 33E 8A, 26A, 27Ayset up simple equations; solve simple equations by using inverse operations or by transforming both sides in the same way/ 4C, 41ABCDE5ABCDESsolve 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 occurring anywhere in the equation, and those with a negative solution 4AD, 41ABEF5ABCESuse formulae from mathematics and other subjects; substitute numbers into a formula08A, 12CD8A, 26Awchange the subject of a formula including cases where the subject occurs twice, or where a power of the subject appears 8B, 50ABCDEFH8BCD, 30ABCDE, 38Egenerate a formula21B, 50G10B, 30DDirect and inverse proportion<understand that reflections are specified by a (mirror) lineunderstand that translations are specified by giving a distance and direction (or a vector), and enlargements by a centre and a positive scale factor16ABrecognise and visualise rotations, reflections and translations including reflection symmetry of 2-D and 3-D shapes, and rotation symmetry of 2-D shapes30D, 45A, 53ABCDztransform triangles and other 2-D shapes by translation, rotation and reflection and combinations of these transformations53F16ACuse 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 transformationsrecognise, visualise and construct enlargements of objects; understand from this that any two circles and any two squares are mathematically similar, while, in general, two rectangles are not, then use positive fractional and negative scale factors 38ABCDE, 53E16B, 33Arecognise 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 33Acunderstand the difference between formulae for perimeter, area and volume by considering dimensionsgenerate terms of a sequence using term-to-term and position-to-term definitions of the sequence; use linear expressions to describe the nth term of an arithmetic sequence, justifying its form by reference to the activity or context from which it was generaed; generate common integer sequences (including sequences of odd or even integers, squared integers, powers of 2, powers of 10, triangular numbers)  14ABCDEFGH36ABCErecognise (when values are given for m and c) that equations of the form y = mx + c correspond to straight-line graphs in the coordinate plane; plot graphs of functions in which y is given explicitly in terms of x, or implicitly% &+ ,I JM OR S  13CD40CEFH12CEG-develop a range of strategies for mental calculation; add and subtract mentally numbers with up to one decimal place; multiply and divide numbers with no more than one decimal digit, using the commutative, associative, and distributive laws and factorisation where possible, or place value adjustmentsdivision by decimal (up to two decimal places) by division using an integer; understand where to position the decimal point by considering what happens if they multiply equivalent fractions11D, 15B, 17BF, 33ACD3ABDEsolve percentage problems, including percentage increase and decreasereverse percentages48D6EKrepresent repeated proportional change using a multiplier raised to a power48C9CWcalculate an unknown quantity from quantities that vary in direct or inverse proportion22DEF"calculate with standard index form40H12G;use surds and  in exact calculations, without a calculator62ABC 44ABC, 45CDEH- rationalise a denominator such as 1/"3 = "3/3" #$ &) +, 45Fuse 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 6B, 20ABCDE, 42A, 49BE, 51D, 57E2A, 11BE, 24F, 39DAenter a range of calculations, including those involving measures32EF6Dunderstand the calculator display, knowing when to interpret the display, whe< n the display has been rounded by the calculator, and not to round during the intermediate steps of a calculationuse calculators, or written methods, to calculate the upper and lower bounds of calculations, particularly when working with measurements60BCDE31BCDE construct the graphs of simple loci including the circle x2 + y2 = r2 for a circle of radius r centred at the origin of coordinates8 : ;> ? @C D E] ^43Bfind 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 circle43CVdistinguish 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; understand a proof that the exterior angle of a triangle is equal to the sum of the interior angles at the other two vertices 25AB, 45BCD29Ause angle properties of equilateral, isosceles and right-angled triangles; explain why the angle sum of a quadrilateral is 36045CD10Arecall the definitions of special types of quadrilateral, including square, rectangle, parallelogram, trapezium and rhombus; classify quadrilaterals by their geometric propertiesunderstand and use SSS, SAS, ASA and RHS conditions to prove the congruence of triangles using formal arguments, and to verify standard ruler and compass constructionsdistinguish 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 quantitiesU4C, 5A, 8A, 41Eunderstand that the transformation of algebraic entities obeys and generalises the well-defined rules of generalised arithmetic20A 43ABCDEF, 47A 20C, 34BE(multiply and divide by a negative number 27D, 44BC7Ause index laws to simplify and calculate the value of numerical expressions involving multiplication and division of integer powers 6C, 23BCEuse inverse operations3A,use brackets and the hierarchy of operations15A, 63B20ABScalculate a given fraction of a given quantity, expressing the answer as a fraction.24E, 67ABC, 69D17AC/express a given number as a fraction of another24D11EDadd and subtract fractions by writing them with a common denominator60CEG11D@perform short division to convert a simple fraction to a decimal28C11F<understand and use unit fractions as multiplicative inverses67B 17BF, 33ACZmultiply and divide a fraction by an integer, by a unit fraction and by a general fraction 60C, 67ADEF17BCDEFG, 33ACD Higher TierHigher references 12AB, 31Ahunderstand and use negative integers both as positions and translations on a number line; order integers7A, 40C4BKuse the terms square, positive and negative square root, cube and cube rootSuse index notation and index laws for multiplication and division of integer powers6BC, 23ABCDEFGHIJ, 51E4ACDWuse standard index form, expressed in conventional notation and on a calculator display 40CDEFGHI12CDEF5recognise that each terminating decimal is a fraction45AB`understand that 'percentage' means 'number of perts per 100' and use this to compare proportions9A<interpret percentage as the operator 'so many hundredths of'22Aunderstand '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)3Cset 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 equations22BCDEFGSimultaneous linear equationsFind the exact solutions of two simultaneous equations in two unknowns by eliminating a variable and interpret the equations as lines and their common solution as the point of intersection 46ABCDEFGHI18ABCDEFsolve 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 set19ABCDLsolve several linear inequalities in two variables and find the solution set52ABCDE23ABCDhsolve simple quadratic equations by factorisation, completing the square and using the quadratic formula47CDEG20E, 34ADFG, 41B, 43A, 45G+Simultaneous linear and quadratic equations 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, 43C28AD[round to a given number of significant figures; derive unknown facts from those they know 3AC, 9DEFconvert between ordinary and standard index form representations, converting to standard index form to make sensible estimates for calculations involving multiplication and/or divisionunderstand 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!49ABCD20ABC, 26AB, 32EFSolving numerical problemsXdraw 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, fractions, percentages and measures and conversion between measures, and compound measures defined within a particular situation913AB, 14E, 37F, 39ABCD, 45BC, 49CD, 54ADFG, 65BC, 69BCDEF"15ABCDE, 16EF, 26AB, 32ACDEFG, 44GFoundation TierFoundation referencesIntermediate referencesMa2 Number and algebra%Using and applying number and algebraNumbers and the number systemIntegersause their previous understanding of integers and place value to deal with arbitrarily large positive numbers and round them to a given power of 104AB, 17F 9AB, 40AB~understand and use positive numbers and negative integers, both as positions and translations on a number line; order integers4A, 27A3B, 7Ause the concepts and vocabulary of factor (divisor), multiple, common factor, highest common factor, least common multiple, prime number and prime factor decomposition21ABCDG6ADEPowers and rootsb_use the terms  square ,  positive square root ,  negative square root ,  cube and  cube root 21EF, 41C 1C, 51ABC6use index notation for squares, cubes and powers of 10 6B, 40A, 51BCrTuse standard index form display and know how to enter numbers in standard index form40DGI12DFsTuse calculators for reverse percentage calculations by doing an appropriate division9DEt]use calculators to explore exponential growth and decay, using a multiplier and the power key739EFGgdraw 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 con< version between measures, and compound measures defined within a particular situation*15ABCDE, 16EF, 26AB, 32ACDEFG, 44G, 48BCDE%3E, 6ACDE, 9BCDE, 10DFG, 22CDF, 45DEHcheck and estimate answers to problems; select and justify appropriate degrees of accuracy for answers to problems; recognise limitations on the accuracy of data and measurements9DG, 20D, 60AB31AB4Number operations and the relationships between them?add, subtract, multiply and divide integers and then any number&2A, 19ABC, 22, 26ABCDE, 27ABCD, 44ABCD 3BEFGH, 7Abmultiply or divide any number by powers of 10, and any positive number by a number between 0 and 14CD, 14D, 26B, 37C3ABCD8find the prime factor decomposition of positive integers6DEunderstand  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)33B8A!substitute numbers into a formula$9ABC, 15BCD, 27E, 44E, 46ABD, 61ABCD12ABCDderive a formula9B, 32D, 35A, 42F, 46EF, 61E21Band change its subject 8B, 50ABC Inequalitiesasolve simple linear inequalities in one variable, and represent the solution set on a number line31ABC, 55ABCDENumerical methodsuse systematic trial and improvement to find approximate solutions of equations where there is no simple analytical method of solving them57DSequences, functions and graphs Sequences`generate terms of a sequence using term-to-term and position-to-term definitions of the sequence 9BC, 58AB14BCuse 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& (58AB14DNconvert simple fractions of a whole to percentages of the whole and vice versa 28ACD, 47ADE 13ACF, 44AEthen understand the multiplicative nature of percentages as operators 44CDEF, 48AB"divide a quantity in a given ratio39G37CMental methodsg.recall all positive integer complements to 100. 22, 26AC 21EF, 28CDF 1C, 11F, 51ABh:round to the nearest integer and to one significant figure4B, 14ABC, 37ABE9CDE/estimate answers to problems involving decimals37D9G, 20Didevelop a range of strategies for mental calculation; derive unknown facts from those they know; add and subtract mentally numbers with up to two decimal places 19AB, 22, 65Amultiply 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 26ABCD, 37CWritten methodsjuse index laws to simplify and calculate the value of numerical expressions involving multiplication and division of integer, fractional and negative powers6C, 23CEGHIJ, 51E 4CEF, 39ABCuse inverse operations, understanding that the inverse operation of raising a positive number to power n is raising the result of this operation to power 1/n.51E39ABCG,use brackets and the heirarchy of operationsScalculate a given fraction of a given quantity, expressing the answer as a fractionBperform a short division to convert a simple fraction to a decimaldistinguish 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 fraction45B=understand and use unit fractions as multiplicative inverses <`multiply and divide a given fraction by an integer, by a unit fraction and by a general fraction3BD9ABXcalculate an original amount when given the transformed amount after a percentage change44D9Dreverse percentage problems1C, 23BH, 51ABE4E, 39ABYenter a range of calculations, including those involving standard index form and measures 49ABCD, 54DFG 32EF, 40DGq23A, 38BG, 53Funderstand congruence64G4explain why the angle sum of a quadrilateral is 36038C45Duse 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 rectangle31C, 40C10ABrecall the essential properties and definitions of special types of quadrilateral, including square, rectangle, parallelogram, trapezium and rhombus; classify quadrilaterals by their geometric properties8AC, 23A, 38AC45Acalculate and use the sums of the interior and exterior angles of quadrilaterals, pentagons and hexagons; calculate and use the angles of regular polygons.38CEFG45DE.understand, recall and use Pythagoras theorem1BCDProperties of circlesrecall the definition of a circle and the meaning of related terms, including centre, radius, chord, diameter, circumference, tangent, arc, sector and segment29AB 10D, 61ADselect 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 41ABCDEFGestimate 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 magnitude4E, 37D9DG, 20Dgive 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, 49BCD60A"Equations, formulae and identitiesUse of symbolsdistinguish 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 quantitiesU3B, 9AB, 15BD, 32D2A, 4A, 8A, 21A, 43Auunderstand that the transformation of algebraic expressions obeys and generalises the rules of generalised arithmeticu 9A, 32A, 50B2BCD@use index laws for multiplication and division of integer powers6C, 23CEUexpress standard index form both in conventional notation and on a calculator display40CF FractionscXunderstand equivalent fractions, simplifying a fraction by cancelling all common factors24ABC, 53AB, 60A 11AC, 15B;order fractions by rewriting them with a common denominator60BDF11BDecimalsdOuse decimal notation and recognise that each terminating decimal is a fraction N 7ACD, 28BC3B, 11Forder decimals 7AC, 28CDkrecognise that recurring decimals are exact fractions, and that some exact fractions are recurring decimals11G Percentagese`understand that  percentage means  number of parts per 100 and use this to compare proportions 28BEF, 47ABDE 13ABCD, 18A<interpret percentage as the operator  so many hundredths of 47C, 69A13E, 44A&use percentage in real-life situations69F 26B, 44CDRatiofguse ratio notation, including reduction to its simplest form and its various links to fraction notation39EF 37ABDE, 38D Calculationspuse formulae from mathematics and other subjects expressed initially in words and then using letters and symbols09BC, 15CD, 46F understand that one coordinate identifies a point on a number line, two coordinates identify a point in a plane and three coordinates identify a po< int 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; 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, then calculate the length AB/u    8D, 48ABC56ABCDVectors3understand and use vector notation for translations53B, 56AMeasures and constructionMeasuresWinterpret scales on a range of measuring instruments, including those for time and mass11B, 54Aknow 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 direction60B.convert measurements from one unit to another 1D, 5ABC, 14E, 45A16Fgenerate common integer sequences (including sequences of odd or even integers, squared integers, powers of 2, powers of 10, triangular numbers)14AGraphs of linear functionsSuse the conventions for coordinates in the plane; plot points in all four quadrants8D, 48A56Arecognise (when values are given for m and c) that equations of the form y = mx + c correspond to straight-line graphs in the coordinate plane; plot graphs of functions in which y is given explicitly in terms of x, or implicitly$ '* ,H JM OR S  35BCD21A, 39Aconstruct linear functions from real-life problems and plot their corresponding graphs; discuss and interpret graphs modelling real situations9D, 18AB, 46G, 70AB21BD, 32B, 35ABCunderstand 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 lines46HTdraw line of best fit through a set of linearly related points and find its equation39E Gradientsh find the gradient of lines given by equations of the form y = mx + c (when values are given for m and c): ;> @C D` af g39ABCDTuse standard column procedures for addition and subtraction of integers and decimals 19ABC, 65Akuse 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, 34ABC, 65ABC3EFGsolve a problem involving division by a decimal (up to two decimal places) by transforming it to a problem involving division by an integer37C, 65B3GHluse 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, 67BCDEF11D, 15B, 17B, 33ACDmAsolve simple percentage problems, including increase and decrease69BCEF 13G, 44BCDEFGnvsolve word problems about ratio and proportion, including using informal strategies and the unitary method of solution 39ABCD, 49CD 15ABCDE, 37BC62ACalculator methodsouse calculators effectively and efficiently; know how to enter complex calculations and use function keys for reciprocals, squares and powers 63BCDEFGH6B, 20ABCDE, 51D, 57Ep4Properties of triangles and other rectilinear shapes,distinguish between lines and line segments =use parallel lines, alternate angles and corresponding angles23CDE 25AB, 45Bunderstand the consequent properties of parallelograms and a proof that the angle sum of a triangle is 180; understand a proof that the exterior angle of a triangle is equal to the sum of the interior angles at the other two vertices38B45CIuse angle properties of equilateral, isosceles and right-angled trianglesmidentify what further information is needed to pursue a particular line of enquiry; 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)Oselect and organise the appropriate mathematics and resources to use for a task;review progress while working; check and evaluate solutions CommunicatingMinterpret, discuss and synthesise information presented in a variety of formsdcommunicate mathematically, including using ICT, making use of diagrams and related explanatory text[understand that inscribed regular polygons can be constructed by equal division of a circle38D 3-D shapesOexplore the geometry of cuboids (including cubes), and shapes made from cuboids25BC30ABCuse 2-D representations of 3-D shapes and analyse 3-D shapes through 2-D projections and cross-sections, including plan and elevation25ABCD<solve problems involving surface areas and volumes of prisms55D16BCDTransformations and coordinatesSpecifying transformationsunderstand that rotations are specified by a centre and an (anticlockwise) angle; rotate a shape about the origin, or any other point; measure the angle of rotation using right angles, simple fractions of a turn or degrees 64ADF53ADunderstand that reflections are specified by a mirror line, at first using a line parallel to an axis, then a mirror line such as y = x or y = -x64ABF53ACunderstand that translations are specified by a distance and direction(or a vector), and enlargements by a centre and positive scale factor 59AB, 64ACEF 38AB, 53ABEProperties of transformationsmanipulate algebraic expressions by collecting like terms, by multiplying a single term over a bracket, and by taking out common factors32ABC, 46C, 50ABCDEFG2BCD, 4B, 19AB, 28ABCDE[distinguish in meaning between the words  equation ,  formula ,  identity and  expression 2A, 4A, 8A, 43A,expand the product of two linear expressions43ABCDEIndex notationPuse index notation for simple integer powers, and simple instances of index laws46CD23BCEQsubstitute positive and negative numbers into expressions such as 3x2 + 4 and 2x3C D EO P 44E, 46D, 61B 7B, 12AB, 28D Equations set up simple equations 3D, 9E, 42F, 61E, 66D4C, 41CD`solve simple equations by using inverse operations or by transforming both sides in the same way 3C, 9E, 42ABC4ALinear equationssolve linear equations, with integer coefficients, in which the unknown appears on either side or on both sides of the equation3CD, 42ABCDE, 66Asolve 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 42G, 66BCD 4D, 41ABEFormulaerecognise that enlargements prese      !"#$%&'()*+,-./2rve 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 perimeter59BC38ABC)use and interpret maps and scale drawings6, 7D, 51ABCD, 68D36ABCDBunderstand the implications of enlargement for area and for volumeUdistinguish between formulae for perimeter, area and volume by considering dimensionsuunderstand and use simple examples of the relationship between enlargement and areas and volumes of shapes and solids Coordinates22B, 39EAuse relevant statistical functions on a calculator or spreadsheet29F#Interpreting and discussing results/relate summarised data to the initial questions 10CD, 30E24CDBinterpret a wide range of graphs and diagrams and draw conclusions%2B, 7B, 16ABD< , 17E, 20CD, 43CDE, 52BC5ABCD, 18BC, 22C, 59A,look at data to find patterns and exceptions10CD, 20C, 30DE, 33C22ABC, 24CD, 29ABDEFncompare distributions and make inferences, using the shapes of distributions and measures of average and range 2B, 20C, 57A5BAconsider and check results and modify their approach if necessary10CDGknow rough metric equivalents of pounds, feet, miles, pints and gallons 13AB, 45BCCmake sensible estimates of a range of measures in everyday settings11A, 45A6understand angle measure using the associated language51CD36DAunderstand and use compound measures, including speed and density554DEFG16E, 32ABCDEFG, 35B ConstructionSmeasure and draw lines to the nearest millimetre, and angles to the nearest degree R5A, 38Ddraw triangles and other 2-D shapes using a ruler and protractor, given information about their side lengths and angles; understand, from their experience of constructing them, that triangles satisfying SSS, SAS, ASA and RHS are unique, but SSA triangles are not68ABC54Cfconstruct cubes, regular tetrahedra, square-based pyramids and other 3-D shapes from given information 17C, 25CD30Buse 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 angle54BCDEFG Mensurationfind areas of rectangles, recalling the formula, understanding the connection to counting squares and how it extends this approach1ABD, 17C, 31A, 56BJrecall and use the formulae for the area of a parallelogram and a triangle 31BCDE, 40ABC+investigate the gradients of parallel lines39ABCInterpret graphical informationJinterpret information presented in a range of linear and non-linear graphs12ABC, 17E, 54E, 70A21D, 32B, 35BCQuadratic equationsdgenerate points and plot graphs of simple quadratic functions, then more general quadratic functions= 21C, 57ABifind approximate solutions of a quadratic equation from the graph of the corresponding quadratic function57BMa3 Shape, space and measures,Using and applying shape, space and measuresGeometrical reasoningAnglesrecall 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, 53E25A, 45B:distinguish between acute, obtuse, reflex and right angles23B, 53D(estimate the size of an angle in degreescarry 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 (ii) collect data from a variety of suitable sources, including experiments and surveys, and primary and secondary sources (iii) process and represent the data: turn the raw data into usable information that gives insight into the problem (iv) interpret and discuss the data: answer the initial question by drawing conclusions from the data * 810ABCD, 30ABCDE24ABCD, 29CDEF)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 33BC22ABC[use the vocabulary of probability to interpret results involving uncertainty and prediction36A, 62D34AB7compare experimental data and theoretical probabilities62E34Aunderstand 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 30C, 62DG>discuss implications of findings in the context of the problem30EQinterpret social statistics including index numbers; time series; and survey data3@43F59Agexamine critically, and justify, their choices of mathematical presentation of problems involving data ReasoningQapply mathematical reasoning, explaining and justifying inferences and deductionsJidentify exceptional or unexpected cases when solving statistical problemscexplore connections in mathematics and look for relationships between variables when analysing datarecognise the limitations of any assumptions and the effects that varying the assumptions could have on the conclusions drawn from data analysis#Specifying the problem and planning+see that random processes are unpredictable10ABCD24ABCDCIdentify key questions that can be addressed by statistical methods 10ABCD, 30ABE 24D, 29DEF_discuss how data relate to a problem, identify possible sources of bias and plan to minimise it 10ABCD, 30B24BCDidentify which primary data they need to collect and in what format, including grouped data, considering appropriate equal class intervalsMdesign an experiment or survey; decide what primary and secondary data to use 10ABCD, 30E24D, 29FCollecting datarecognise and visualise rotations, reflections and translations, including reflection symmetry of 2-D and 3-D shapes, and rotation symmetry of 2-D shapes; 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; distinguish properties that are preserved under particular transformations8BCD, 25E, 53C, 64ABCDFG30D, 45A, 53ABCDFrecognise, visualise and construct enlargements of objects using positive scale factors greater than one, then positive scale factors less than one 59AB, 64E 38AB, 53Eunderstand from this that any two circles and any two squares are mathematically similar, while, in general, two rectangles are not59C38CDdesign and use data-collection sheets for grouped discrete and continuous data; collect data using various methods, including observation, controlled experiment, data logging, questionnaires and surveys 20D, 30B, 57D5C, 24D]gather data from secondary sources, including printed tables and lists from ICT-based sources2A, 43AB 22A, 29ABCDEF;design and use two-way tables for discrete and grouped data2A, 17G, 43AB, 54BC, 62B5CD, 29B Processing and representing datadraw and produce, using paper and ICT, pie charts for categorical data, and diagrams for continuous data, including line graphs for time series, scatter graphs, frequency diagrams and stem-and-leaf diagrams22C, 16C, 17D, 20AD, 30D, 33ABC, 43CE, 52DEF, 57BCD5BCD, 18DEF, 22A, 59AVcalculate mean, range and median of small data sets with discrete then continuous data2CD, 20BC, 57ABC5A)identify the modal class for grouped data 20D, 57CD5CD(understand and use the probability scaleunderstand and use estimates or measures of probability from theoretical models (including equally-likely outcomes), or from relative frequency36ACD, 62ABCDE 34AB, 58BWlist all outcomes for single events, and for two successive events, in a systematic way  36BCD, 62C34CDtidentify different mutually exclusive outcomes and know that the sum of the probabilities of all these outcomes is 162FG34Bofind the median for large data sets and< calculate an estimate of the mean for large data sets with grouped data5BEAdraw lines of best fit by eye, understanding what these represent33C 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, the fact that n0 = 1 and n-1 = 1/n for positive integers n, the corresponding rule for negative numbers, n_ = "n and n_ = 3"n for any positive number n           2calculate the area of a triangle using (1/2)absinC, .1 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)3 4j lp q        find the surface area of simple shapes using the area formulae for triangles and rectangles; 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