Year 7 - Maths
NRICH nrich. problems linked to the Framework for Secondary Mathematics
N.B. This is work in progress - last updated 20-08-2012. Please email any comments to enquiries.nrich@
Ticked items (() identify problems that have detailed Teachers’ Notes suggesting how they can be integrated into lessons.
Highlighted problems are ideal for using at the start of a topic.
Asterisked problems (*) appear in two places.
|Year 7… |
|Place value, ordering and rounding |
|understand and use decimal notation |read and write positive integer |extend knowledge of integer powers of|express numbers in standard index form, |A Question of Scale ( | |
|and place value; multiply and divide |powers of 10; multiply and divide |10; recognise the equivalence of 0.1,|both in conventional notation and on a | | |
|integers and decimals by 10, 100, |integers and decimals by 0.1, 0.01 |1/10 |calculator display | | |
|1000, and explain the effect | |and 10-1; multiply and divide by any | | | |
|Dicey Operations* ( | |integer power of 10 | | | |
|Always a Multiple?* ( | | | | | |
|compare and order decimals in |order decimals | |convert between ordinary and standard |use standard index form to make sensible| |
|different contexts; know that when | | |index form representations |estimates for calculations involving | |
|comparing measurements the units must| | | |multiplication and/or division | |
|be the same | | | | | |
|Nice and Nasty ( | | | | | |
|round positive whole numbers to the |round positive numbers to any given |use rounding to make estimates and to|round to a given number of significant |understand how errors can be compounded |understand upper and lower bounds |
|nearest 10, 100 or 1000, and decimals|power of 10; round decimals to the |give solutions to problems to an |figures; use significant figures to |in calculations | |
|to the nearest whole number or one |nearest whole number or to one or two|appropriate degree of accuracy |approximate answers when multiplying or | | |
|decimal place |decimal places | |dividing large numbers | | |
|Integers powers and roots |
|understand negative numbers as |add, subtract, multiply and divide | | | | Difference Sudoku |
|positions on a number line; order, |integers | | | | |
|add and subtract integers in context |Playing Connect Three ( | | | | |
|Magic Letters ( |Weights ( | | | | |
|First Connect Three ( |Consecutive Negative Numbers( | | | | |
| |Article: Adding & Subtracting | | | | |
| |Negative Numbers | | | | |
|recognise and use multiples, factors,|use multiples, factors, common |use the prime factor decomposition of|Filling the Gaps ( | |Expenses |
|primes (less than 100), common |factors, highest common factors, |a number |American Billions ( | | |
|factors, highest common factors and |lowest common multiples and primes; |Product Sudoku ( | | | |
|lowest common multiples in simple |find the prime factor decomposition |Funny Factorisation | | | |
|cases; use simple tests of |of a number, e.g. | | | | |
|divisibility |8000 = 26 × 53 | | | | |
|Sieve of Eratosthenes ( |Counting Cogs ( | | | | |
|How much can we spend? ( |Stars ( | | | | |
|Dozens ( |Power Mad! ( | | | | |
|Factors and Multiples Game ( |14 Divisors ( | | | | |
|Factors and Multiples Puzzle( |Take Three from Five ( | | | | |
|Article: Divisibility Tests |Differences ( | | | | |
|recognise the first few triangular |use squares, positive and negative |use ICT to estimate square roots and |Generating Triples ( | | |
|numbers; recognise the squares of |square roots, cubes and cube roots, |cube roots | | | |
|numbers to at least 12 × 12 and the |and index notation for small positive| | | | |
|corresponding roots |integer powers | | | | |
| |Sissa's Reward | | | | |
| | |use index notation for integer |use index notation with negative and |use inverse operations, understanding |understand and use rational and |
| | |powers; know and use the index laws |fractional powers, recognising that the |that the inverse operation of raising a |irrational numbers |
| | |for multiplication and division of |index laws can be applied to these as well|positive number to power n is raising | |
| | |positive integer powers | |the result of this operation to power | |
| | | | |1/n | |
| | | | |Power Countdown ( | |
| | | |know that n½ = √n and | | |
| | | |n⅓= 3√n for any positive number n | | |
|Fractions, decimals, percentages, ratio and proportion |
|express a smaller whole number as a |recognise that a recurring decimal is|understand the equivalence of simple |distinguish between fractions with |use an algebraic method to convert a | |
|fraction of a larger one; simplify |a fraction; use division to convert a|algebraic fractions; know that a |denominators that have only prime factors |recurring decimal to a fraction | |
|fractions by cancelling all common |fraction to a decimal; order |recurring decimal is an exact |2 or 5 (terminating decimals), and other |Repetitiously ( | |
|factors and identify equivalent |fractions by writing them with a |fraction |fractions (recurring decimals) | | |
|fractions; convert terminating |common denominator or by converting | |Tiny nines ( | | |
|decimals to fractions, e.g. |them to decimals | | | | |
|0.23=23/100; use diagrams to compare |Farey Sequences ( | | | | |
|two or more simple fractions |Round and Round and Round | | | | |
|add and subtract simple fractions and|add and subtract fractions by writing|use efficient methods to add, |understand and apply efficient methods to | | |
|those with common denominators; |them with a common denominator; |subtract, multiply and divide |add, subtract, multiply and divide | | |
|calculate simple fractions of |calculate fractions of quantities |fractions, interpreting division as a|fractions, interpreting division as a | | |
|quantities and measurements |(fraction answers); multiply and |multiplicative inverse; cancel common|multiplicative inverse | | |
|(whole-number answers); multiply a |divide an integer by a fraction |factors before multiplying or |Twisting and Turning | | |
|fraction by an integer |Diminishing Returns ( |dividing |More Twisting and Turning | | |
|Fractions Jigsaw ( | |Ben's Game ( | | | |
|Peaches Today, Peaches Tomorrow... ( | |Fair Shares? ( | | | |
|understand percentage as the ‘number |interpret percentage as the operator |recognise when fractions or |calculate an original amount when given | | |
|of parts per 100’; calculate simple |‘so many hundredths of’ and express |percentages are needed to compare |the transformed amount after a percentage | | |
|percentages and use percentages to |one given number as a percentage of |proportions; solve problems involving|change; use calculators for reverse | | |
|compare simple proportions |another; calculate percentages and |percentage changes |percentage calculations by doing an | | |
|Matching Fractions Decimals |find the outcome of a given | |appropriate division | | |
|Percentages ( |percentage increase or decrease | | | | |
|recognise the equivalence of |use the equivalence of fractions, | | | | |
|percentages, fractions and decimals |decimals and percentages to compare | | | | |
| |proportions | | | | |
|understand the relationship between |apply understanding of the |use proportional reasoning to solve |understand and use proportionality and |calculate an unknown quantity from |understand and use direct and inverse |
|ratio and proportion; use direct |relationship between ratio and |problems, choosing the correct |calculate the result of any proportional |quantities that vary in direct |proportion; solve problems involving |
|proportion in simple contexts; use |proportion; simplify ratios, |numbers to take as 100%, or as a |change using multiplicative methods |proportion using algebraic methods where|inverse proportion (including inverse |
|ratio notation, simplify ratios and |including those expressed in |whole; compare two ratios; interpret |Ratios and Dilutions ( |appropriate |squares) using algebraic methods |
|divide a quantity into two parts in a|different units, recognising links |and use ratio in a range of contexts |A Chance to Win? |Areas and Ratios ( |Triathlon and Fitness ( |
|given ratio; solve simple problems |with fraction notation; divide a |Mixing Paints | | | |
|involving ratio and proportion using |quantity into two or more parts in a |Mixing More Paints | | | |
|informal strategies |given ratio; use the unitary method | | | | |
|Mixing Lemonade ( |to solve simple problems involving | | | | |
| |ratio and direct proportion | | | | |
|Number operations |
|understand and use the rules of |understand and use the rules of |understand the effects of multiplying|recognise and use reciprocals; understand |use a multiplier raised to a power to | |
|arithmetic and inverse operations in |arithmetic and inverse operations in |and dividing by numbers between 0 and|'reciprocal' as a multiplicative inverse; |represent and solve problems involving | |
|the context of positive integers and |the context of integers and fractions|1; consolidate use of the rules of |know that any number multiplied by its |repeated proportional change, e.g. | |
|decimals | |arithmetic and inverse operations |reciprocal is 1, and that zero has no |compound interest | |
|What Numbers Can |Keep it Simple ( | |reciprocal because division by zero is not|The Legacy | |
|We Make?* ( |Egyptian Fractions ( | |defined |Dating Made Easier | |
|Consecutive Numbers ( |The Greedy Algorithm ( | | | | |
|Where Can We Visit? ( | | | | | |
|Consecutive Seven ( | | | | | |
|use the order of operations, |use the order of operations, |understand the order of precedence of| | | |
|including brackets |including brackets, with more complex|operations, including powers | | | |
|Make 100 ( |calculations | | | | |
|Mental calculation methods |
|recall number facts, including |recall equivalent fractions, decimals| | | | |
|positive integer complements to 100 |and percentages; use known facts to | | | | |
|and multiplication facts to 10 × 10, |derive unknown facts, including | | | | |
|and quickly derive associated |products involving numbers such as | | | | |
|division facts |0.7 and 6, and 0.03 and 8 | | | | |
|Missing Multipliers ( | | | | | |
|Countdown | | | | | |
|Remainders ( | | | | | |
|The Remainders Game | | | | | |
|strengthen and extend mental methods |strengthen and extend mental methods |use known facts to derive unknown | | |use surds and π in exact calculations,|
|of calculation to include decimals, |of calculation, working with |facts; extend mental methods of | | |without a calculator; rationalise a |
|fractions and percentages, |decimals, fractions, percentages, |calculation, working with decimals, | | |denominator such as |
|accompanied where appropriate by |squares and square roots, cubes and |fractions, percentages, factors, | | |1/√3 = √3/3 |
|suitable jottings; solve simple |cube roots; solve problems mentally |powers and roots; solve problems | | |The Root of the Problem |
|problems mentally | |mentally | | | |
|Number Daisy | |Cinema Problem ( | | | |
|Got It ( | | | | | |
|make and justify estimates and |make and justify estimates and |make and justify estimates and |make and justify estimates and | | |
|approximations of calculations |approximations of calculations |approximations of calculations |approximations of calculations by rounding| | |
| |Place Your Orders* ( | |numbers to one significant figure and | | |
| |Thousands and Millions* ( | |multiplying or dividing mentally | | |
|Written Calculation Methods |
|use efficient written methods to add |use efficient written methods to add | | | | |
|and subtract whole numbers and |and subtract integers and decimals of| | | | |
|decimals with up to two places |any size, including numbers with | | | | |
|Dicey Operations* ( |differing numbers of decimal places | | | | |
|Two and Two ( | | | | | |
|multiply and divide three-digit by |use efficient written methods for |use efficient written methods to add | | | |
|two-digit whole numbers; extend to |multiplication and division of |and subtract integers and decimals of| | | |
|multiplying and dividing decimals |integers and decimals, including by |any size; multiply by decimals; | | | |
|with one or two places by |decimals such as 0.6 or 0.06; |divide by decimals by transforming to| | | |
|single-digit whole numbers |understand where to position the |division by an integer | | | |
|Method in Multiplying Madness? ( |decimal point by considering |How Many Miles to Go? ( | | | |
| |equivalent calculations | | | | |
| |Legs Eleven ( | | | | |
| |Largest Product ( | | | | |
|Calculator methods |
|carry out calculations with more than|carry out more difficult calculations|use a calculator efficiently and |use an extended range of function keys, |use calculators to explore exponential |use calculators, or written methods, |
|one step using brackets and the |effectively and efficiently using the|appropriately to perform complex |including the reciprocal and trigonometric|growth and decay, using a multiplier and|to calculate the upper and lower |
|memory; use the square root and sign |function keys for sign change, |calculations with numbers of any |functions |the power key |bounds of calculations in a range of |
|change keys |powers, roots and fractions; use |size, knowing not to round during | | |contexts, particularly when working |
| |brackets and the memory |intermediate steps of a calculation; | | |with measurements |
| | |use the constant, π and sign change | | | |
| | |keys; use the function keys for | | | |
| | |powers, roots and fractions; use | | | |
| | |brackets and the memory | | | |
|enter numbers and interpret the |enter numbers and interpret the | |use standard index form, expressed in |calculate with standard index form, | |
|display in different contexts |display in different contexts (extend| |conventional notation and on a calculator |using a calculator as appropriate | |
|(decimals, percentages, money, metric|to negative numbers, fractions, time)| |display; know how to enter numbers in | | |
|measures) | | |standard form | | |
|Going Round in Circles ( | | | | | |
|Checking results |
|check results by considering whether |select from a range of checking |check results using appropriate methods |check results using appropriate methods |check results using appropriate methods |check results using appropriate |
|they are of the right order of |methods, including estimating in | | | |methods |
|magnitude and by working problems |context and using inverse operations | | | | |
|backwards | | | | | |
|Rule of Three | | | | | |
|Algebra |
|Equations, formulae, expressions and identities |
|use letter symbols to represent |recognise that letter symbols play |distinguish the different roles played | | | |
|unknown numbers or variables; know |different roles in equations, |by letter symbols in equations, | | | |
|the meanings of the words term, |formulae and functions; know the |identities, formulae and functions | | | |
|expression and equation |meanings of the words formula and | | | | |
|Your Number Is… ( |function | | | | |
|Number Pyramids ( | | | | | |
|Crossed Ends ( | | | | | |
|understand that algebraic operations |understand that algebraic operations,|use index notation for integer powers |know and use the index laws in generalised| | |
|follow the rules of arithmetic |including the use of brackets, follow|and simple instances of the index laws |form for multiplication and division of | | |
| |the rules of arithmetic; use index | |integer powers | | |
| |notation for small positive integer | | | | |
| |powers | | | | |
|simplify linear algebraic expressions|simplify or transform linear |simplify or transform algebraic |square a linear expression; expand the |factorise quadratic expressions, |Why 24? ( |
|by collecting like terms; multiply a |expressions by collecting like terms;|expressions by taking out single-term |product of two linear expressions of the |including the difference of two squares,|2-digit square ( |
|single term over a bracket (integer |multiply a single term over a bracket|common factors; add simple algebraic |form x ± n and simplify the corresponding |e.g. x2 − 9 = (x + 3) (x − 3) and cancel|Always Perfect |
|coefficients) |Perimeter Expressions ( |fractions |quadratic expression; establish identities|common factors in rational expressions, |Perfectly Square |
|Always a Multiple?* ( |Special Numbers ( |Harmonic Triangle ( |such as a2 − b2 = (a + b) (a − b) |e.g. | |
|More Number Pyramids ( | | |Pair Products ( |2(x+1)2/(x+1) | |
| | | |What's Possible? ( |Factorising with Multilink ( | |
| | | |Plus Minus ( |Finding Factors ( | |
| | | |Multiplication Square |Odd Squares ( | |
| | | | |simplify simple algebraic fractions to | |
| | | | |produce linear expressions; use | |
| | | | |factorisation to simplify compound | |
| | | | |algebraic fractions | |
|construct and solve simple linear |construct and solve linear equations |construct and solve linear equations |solve linear equations in one unknown with|solve equations involving algebraic | |
|equations with integer coefficients |with integer coefficients (unknown on|with integer coefficients (with and |integer and fractional coefficients; solve|fractions with compound expressions as | |
|(unknown on one side only) using an |either or both sides, without and |without brackets, negative signs |linear equations that require prior |the numerators and/or denominators | |
|appropriate method (e.g. inverse |with brackets) using appropriate |anywhere in the equation, positive or |simplification of brackets, including | | |
|operations) |methods (e.g. inverse operations, |negative solution) |those with negative signs anywhere in the | | |
|Your Number Was… ( |transforming both sides in same way) | |equation | | |
| |Think of Two Numbers | |Fair Shares* ( | | |
| | | | | | |
| |use graphs and set up equations to |use algebraic methods to solve problems | | | |
| |solve simple problems involving |involving direct proportion; relate | | | |
| |direct proportion |algebraic solutions to graphs of the | | | |
| | |equations; use ICT as appropriate | | | |
| | |Introductory work on simultaneous |solve a pair of simultaneous linear |explore 'optimum' methods of solving |solve exactly, by elimination of an |
| | |equations |equations by eliminating one variable; |simultaneous equations in different |unknown, two simultaneous equations in|
| | |What's it Worth? ( |link a graph of an equation or a pair of |forms |two unknowns, where one is linear in |
| | | |equations to the algebraic solution; |CD Heaven ( |each unknown and the other is linear |
| | | |consider cases that have no solution or an|Matchless |in one unknown and quadratic in the |
| | | |infinite number of solutions |Multiplication Arithmagons ( |other or of the form x2 + y2 = r2 |
| | | |Arithmagons ( | | |
| | | |solve linear inequalities in one variable;|solve linear inequalities in one and two| |
| | | |represent the solution set on a number |variables; find and represent the | |
| | | |line |solution set | |
| | | |Which Is Cheaper? ( |Which Is Bigger? ( | |
| | |use systematic trial and improvement | |solve quadratic equations by |solve quadratic equations by |
| | |methods and ICT tools to find | |factorisation |factorisation, completing the square |
| | |approximate solutions to equations such | |How Old Am I? ( |and using the quadratic formula, |
| | |as | | |including those in which the |
| | |x2 + x = 20 | | |coefficient of the quadratic term is |
| | | | | |greater than 1 |
| | | | | |Golden Thoughts ( |
| | |explore ways of constructing models of | | | |
| | |real-life situations by drawing graphs | | | |
| | |and constructing algebraic equations and| | | |
| | |inequalities | | | |
|use simple formulae from mathematics |use formulae from mathematics and |use formulae from mathematics and other |derive and use more complex formulae; |derive and use more complex formulae; |derive relationships between different|
|and other subjects; substitute |other subjects; substitute integers |subjects; substitute numbers into |change the subject of a formula, including|change the subject of a formula, |formulae that produce equal or related|
|positive integers into linear |into simple formulae, including |expressions and formulae; derive a |cases where a power of the subject appears|including cases where the subject occurs|results |
|expressions and formulae and, in |examples that lead to an equation to |formula and, in simple cases, change its|in the question or solution, e.g. find r |twice | |
|simple cases, derive a formula |solve; substitute positive integers |subject |given that A = πr2 | | |
| |into expressions involving small |Temperature ( |Training Schedule ( | | |
| |powers e.g. 3x2 + 4 or 2x3; derive | |Terminology | | |
| |simple formulae | | | | |
|Sequences, functions and graphs |
|describe integer sequences; generate |generate terms of a linear sequence |generate terms of a sequence using |A Little Light Thinking ( | | |
|terms of a simple sequence, given a |using term-to-term and |term-to-term and position-to-term rules,| | | |
|rule (e.g. finding a term from the |position-to-term rules, on paper and |on paper and using ICT | | | |
|previous term, finding a term given |using a spreadsheet or graphics |1 Step 2 Step ( | | | |
|its position in the sequence) |calculator |Tower of Hanoi ( | | | |
|Odds, Evens and More Evens( |Charlie’s Delightful Machine ( | | | | |
|Shifting Times Tables ( |Coordinate Patterns* ( | | | | |
|Triangle Numbers ( | | | | | |
|generate sequences from patterns or |use linear expressions to describe |generate sequences from practical |find the next term and the nth term of |Partially Painted Cube ( | |
|practical contexts and describe the |the nth term of a simple arithmetic |contexts and write and justify an |quadratic sequences and explore their |Double Trouble ( | |
|general term in simple cases |sequence, justifying its form by |expression to describe the nth term of |properties; deduce properties of the |Picture Story ( | |
|Summing Consecutive Numbers ( |referring to the activity or |an arithmetic sequence |sequences of triangular and square numbers| | |
|What Numbers Can |practical context from which it was |What Numbers Can We Make Now? ( |from spatial patterns | | |
|We Make?* ( |generated |Picturing Triangle Numbers ( |Attractive Tablecloths ( | | |
|Picturing Square Numbers ( |Seven Squares ( |Slick Summing ( |Painted Cube* ( | | |
|Squares in Rectangles ( |NRICH Article: Spaces for Exploration|Elevenses ( |Mystic Rose ( | | |
| | |Days and Dates ( |Steel Cables ( | | |
|express simple functions in words, |express simple functions |find the inverse of a linear function |plot the graph of the inverse of a linear | | |
|then using symbols; represent them in|algebraically and represent them in | |function | | |
|mappings |mappings or on a spreadsheet | | | | |
| |Pick's Theorem* ( | | | | |
|generate coordinate pairs that |generate points in all four quadrants|generate points and plot graphs of |understand that equations in the form |identify the equations of straight-line | |
|satisfy a simple linear rule; plot |and plot the graphs of linear |linear functions, where y is given |y = mx + c represent a straight line and |graphs that are parallel; find the | |
|the graphs of simple linear |functions, where y is given |implicitly in terms of x |that m is the gradient and c is the value |gradient and equation of a straight-line| |
|functions, where y is given |explicitly in terms of x, on paper |(e.g. ay + bx = 0, |of the y-intercept; investigate the |graph that is perpendicular to a given | |
|explicitly in terms of x, on paper |and using ICT; recognise that |y + bx + c = 0), on paper and using ICT;|gradients of parallel lines and lines |line | |
|and using ICT; recognise |equations of the form |find the gradient of lines given by |perpendicular to these lines |Doesn’t Add Up ( | |
|straight-line graphs parallel to the |y = mx + c correspond to |equations of the form y = mx + c, given |At Right Angles ( | | |
|x-axis or y-axis |straight-line graphs |values for m and c |Perpendicular Lines ( | | |
|Exploring Simple Mappings ( |How Steep Is the Slope? ( |Diamond Collector ( |Surprising Transformations ( | | |
| |Parallel Lines ( |Translating Lines ( | | | |
| | |Reflecting Lines ( | | | |
|plot and interpret the graphs of |construct linear functions arising |construct functions arising from |understand that the point of intersection |find approximate solutions of a |know and understand that the |
|simple linear functions arising from |from real-life problems and plot |real-life problems and plot their |of two different lines in the same two |quadratic equation from the graph of the|intersection points of the graphs of a|
|real-life situations, e.g. conversion|their corresponding graphs; discuss |corresponding graphs; interpret graphs |variables that simultaneously describe a |corresponding quadratic function |linear and quadratic function are the |
|graphs |and interpret graphs arising from |arising from real situations, e.g. time |real situation is the solution to the | |approximate solutions to the |
| |real situations, e.g. distance–time |series graphs |simultaneous equations represented by the | |corresponding simultaneous equations |
| |graphs |Fill Me Up ( |lines | | |
| |Walk and Ride ( |Maths Filler ( |Negatively Triangular | | |
| |Buses ( |How Far Does it Move? ( | | | |
| | |Speeding Up, Slowing Down ( | | | |
| | |Up and Across | | | |
| | |Steady Free Fall | | | |
| | | | | | |
| | | | | |construct the graphs of simple loci, |
| | | | | |including the circle x2 + y2 = r2; |
| | | | | |find graphically the intersection |
| | | | | |points of a given straight line with |
| | | | | |this circle and know this represents |
| | | | | |the solution to the corresponding two |
| | | | | |simultaneous equations |
| | | |explore simple properties of quadratic |plot graphs of more complex quadratic |plot and recognise the characteristic |
| | | |functions; plot graphs of simple quadratic|and cubic functions; estimate values at |shapes of graphs of simple cubic |
| | | |and cubic functions, e.g. y = x2, |specific points, including maxima and |functions (e.g. y = x3), reciprocal |
| | | |y = 3x2 + 4, y = x3 |minima |functions (e.g. y = 1/x, x ≠ 0), |
| | | |Exploring Quadratic Mappings ( | |exponential functions (y = kx for |
| | | |Minus One Two Three | |integer values of x and simple |
| | | | | |positive values of k) and |
| | | | | |trigonometric functions, on paper and |
| | | | | |using ICT |
| | | | | |What’s That Graph? ( |
| | | | | |Back Fitter ( |
| | | | |identify and sketch graphs of linear and|apply to the graph y = f(x) the |
| | | | |simple quadratic and cubic functions; |transformations y = f(x) + a, |
| | | | |understand the effect on the graph of |y = f(ax), y = f(x+a) and y = af(x) |
| | | | |addition of (or multiplication by) a |for linear, quadratic, sine and cosine|
| | | | |constant |functions |
| | | | | |Parabolic Patterns |
| | | | | |Cubics |
| | | | | |Tangled Trig Graphs ( |
| | |use ICT to explore the graphical | | | |
| | |representation of algebraic equations | | | |
| | |and interpret how properties of the | | | |
| | |graph are related to features of the | | | |
| | |equation, e.g. parallel and | | | |
| | |perpendicular lines | | | |
| | |interpret the meaning of various points | | | |
| | |and sections of straight-line graphs, | | | |
| | |including intercepts and intersection, | | | |
| | |e.g. solving simultaneous linear | | | |
| | |equations | | | |
|Geometry and Measures |
|Geometrical reasoning |
|use correctly the vocabulary, | |distinguish between conventions, |distinguish between practical |show step-by-step deduction in solving |understand the necessary and |
|notation and labelling conventions | |definitions and derived properties |demonstration and proof in a geometrical |more complex geometrical problems |sufficient conditions under which |
|for lines, angles and shapes | | |context |Squirty ( |generalisations, inferences and |
| | | |Circles in Quadrilaterals ( |Partly Circles ( |solutions to geometrical problems |
| | | |Areas of Parallelograms ( | |remain valid |
|identify parallel and perpendicular |identify alternate angles and |explain how to find, calculate and use: | | | |
|lines; know the sum of angles at a |corresponding angles; understand a |• the sums of the interior and exterior | | | |
|point, on a straight line and in a |proof that: the angle sum of a |angles of quadrilaterals, pentagons and | | | |
|triangle; recognise vertically |triangle is 180°and of a |hexagons | | | |
|opposite angles |quadrilateral is 360° the exterior |• the interior and exterior angles of | | | |
| |angle of a triangle is equal to the |regular polygons | | | |
| |sum of the two interior opposite |Semi-regular Tessellations ( | | | |
| |angles |Which Solids Can We Make? ( | | | |
| | |know the definition of a circle and the |know that the tangent at any point on a |prove and use the facts that: |prove and use the alternate segment |
| | |names of its parts; explain why |circle is perpendicular to the radius at |• the angle subtended by an arc at the |theorem |
| | |inscribed regular polygons can be |that point; explain why the perpendicular |centre of a circle is twice the angle | |
| | |constructed by equal divisions of a |from the centre to the chord bisects the |subtended at any point on the | |
| | |circle |chord |circumference | |
| | | |Compare Areas ( |• the angle subtended at the | |
| | | |Circle-in |circumference by a semicircle is a right| |
| | | | |angle | |
| | | | |• angles in the same segment are equal | |
| | | | |• opposite angles in a cyclic | |
| | | | |quadrilateral sum to 180° | |
| | | | |Triangles in Circles ( | |
| | | | |Cyclic Quadrilaterals ( | |
| | | | |Subtended Angles* ( | |
| | | | |Right Angles* ( | |
|identify and use angle, side and |solve geometrical problems using side|solve problems using properties of |solve multi-step problems using properties| | |
|symmetry properties of triangles and |and angle properties of equilateral, |angles, of parallel and intersecting |of angles, of parallel lines, and of | | |
|quadrilaterals; explore geometrical |isosceles and right-angled triangles |lines, and of triangles and other |triangles and other polygons, justifying | | |
|problems involving these properties, |and special quadrilaterals, |polygons, justifying inferences and |inferences and explaining reasoning with | | |
|explaining reasoning orally, using |explaining reasoning with diagrams |explaining reasoning with diagrams and |diagrams and text | | |
|step-by-step deduction supported by |and text; classify quadrilaterals by |text |Kite in a Square ( | | |
|diagrams |their geometrical properties |Triangles in Circles ( |Making Sixty ( | | |
|Property Chart ( |Square It ( |Cyclic Quadrilaterals ( |Sitting Pretty ( | | |
|Shapely Pairs ( |Eight Hidden Squares ( |Subtended Angles* ( | | | |
|Quadrilaterals Game ( |Square Coordinates ( |Right Angles* ( | | | |
| |Opposite Vertices ( | | | | |
| |know that if two 2-D shapes are |understand congruence and explore |know that if two 2-D shapes are similar, | |prove the congruence of triangles and |
| |congruent, corresponding sides and |similarity |corresponding angles are equal and | |verify standard ruler and compass |
| |angles are equal | |corresponding sides are in the same ratio;| |constructions using formal arguments |
| | | |understand from this that any two circles | |Triangle Mid Points ( |
| | | |and any two squares are mathematically | |Angle trisection( |
| | | |similar while in general any two | | |
| | | |rectangles are not | | |
| | | |Trapezium Four ( | | |
| | | |Nicely similar | | |
| | | |Two ladders | | |
| | | |Napkin ( | | |
| | |investigate Pythagoras’ theorem, using a| | | Pythagoras Proofs ( |
| | |variety of media, through its historical| | | |
| | |and cultural roots, including ‘picture’ | | | |
| | |proofs | | | |
| | |Tilted Squares ( | | | |
|use 2-D representations to visualise |visualise 3-D shapes from their nets;|visualise and use 2-D representations of|understand and apply Pythagoras' theorem |understand and use Pythagoras' theorem |Qqq..cubed |
|3-D shapes and deduce some of their |use geometric properties of cuboids |3-D objects; analyse 3-D shapes through |when solving problems in 2-D and simple |to solve 3-D problems | |
|properties |and shapes made from cuboids; use |2-D projections, including plans and |problems in 3-D |The Spider and the Fly | |
| |simple plans and elevations |elevations |Inscribed in a Circle | | |
| | |Nine Colours ( |Semi-detached | | |
| | |Marbles in a Box ( |Ladder and Cube | | |
| | |Tet-trouble ( |Where to Land | | |
| | |Triangles to Tetrahedra ( |Walking around a cube | | |
| | | | | | |
| | | |understand and use trigonometric |use trigonometric relationships in |draw, sketch and describe the graphs |
| | | |relationships in right-angled triangles, |right-angled triangles to solve 3-D |of trigonometric functions for angles |
| | | |and use these to solve problems, including|problems, including finding the angles |of any size, including transformations|
| | | |those involving bearings |between a line and a plane |involving scalings in either or both |
| | | |Where is the dot? |Far horizon |of the x and y directions |
| | | |Trigonometric Protractor ( | | |
| | | |Orbiting billiard balls | | |
| | | | | |use the sine and cosine rules to solve|
| | | | | |2-D and 3-D problems |
| | | | | |Hexy-metry ( |
| | | | | |Three by One ( |
| | | | | |Cubestick |
| | | | | |Bendy Quad( |
| | | | | |calculate the area of a triangle using|
| | | | | |the formula ½absinC |
|Transformations and co-ordinates |
|understand and use the language and | | | | | |
|notation associated with reflections,| | | | | |
|translations and rotations | | | | | |
|recognise and visualise the |identify all the symmetries of 2-D |identify reflection symmetry in 3-D | | | |
|symmetries of a 2-D shape |shapes |shapes | | | |
|Shady Symmetry ( | | | | | |
|Reflecting Squarely ( | | | | | |
|transform 2-D shapes by: • reflecting|transform 2-D shapes by rotation, |recognise that translations, rotations |transform 2-D shapes by combinations of | | |
|in given mirror lines |reflection and translation, on paper |and reflections preserve length and |translations, rotations and reflections, | | |
|• rotating about a given point |and using ICT |angle, and map objects on to congruent |on paper and using ICT; use congruence to | | |
|• translating |Transformation Game ( |images |show that translations, rotations and | | |
|Mirror, Mirror… ( | | |reflections preserve length and angle | | |
|...on the Wall ( | | | | | |
|Attractive Rotations ( | | | | | |
|explore these transformations and |try out mathematical representations |explore and compare mathematical |use any point as the centre of rotation; |understand and use vector notation to |calculate and represent graphically |
|symmetries using ICT |of simple combinations of these |representations of combinations of |measure the angle of rotation, using |describe transformation of 2-D shapes by|the sum of two vectors, the difference|
| |transformations |translations, rotations and reflections |fractions of a turn or degrees; understand|combinations of translations; calculate |of two vectors and a scalar multiple |
| | |of 2-D shapes, on paper and using ICT |that translations are specified by a |and represent graphically the sum of two|of a vector; calculate the resultant |
| | | |vector |vectors |of two vectors |
| | | | |Spotting the Loophole | |
| | | | |Vector Journeys ( | |
| | |devise instructions for a computer to | | |understand and use the commutative and|
| | |generate and transform shapes | | |associative properties of vector |
| | | | | |addition |
| | | | | |solve simple geometrical problems in |
| | | | | |2-D using vectors |
| |understand and use the language and |enlarge 2-D shapes, given a centre of |enlarge 2-D shapes using positive, |understand and use the effects of | |
| |notation associated with enlargement;|enlargement and a positive integer scale|fractional and negative scale factors, on |enlargement on areas and volumes of | |
| |enlarge 2-D shapes, given a centre of|factor, on paper and using ICT; identify|paper and using ICT; recognise the |shapes and solids | |
| |enlargement and a positive integer |the scale factor of an enlargement as |similarity of the resulting shapes; |Growing Rectangles ( | |
| |scale factor; explore enlargement |the ratio of the lengths of any two |understand and use the effects of |Fit for Photocopying ( | |
| |using ICT |corresponding line segments; recognise |enlargement on perimeter | | |
| | |that enlargements preserve angle but not|Who Is the Fairest of Them All? ( | | |
| | |length, and understand the implications | | | |
| | |of enlargement for perimeter | | | |
| |make scale drawings |use and interpret maps and scale | | | |
| | |drawings in the context of mathematics | | | |
| | |and other subjects | | | |
|use conventions and notation for 2-D |find the midpoint of the line segment|use the coordinate grid to solve |find the points that divide a line in a | | |
|coordinates in all four quadrants; |AB, given the coordinates of points A|problems involving translations, |given ratio, using the properties of | | |
|find coordinates of points determined|and B |rotations, reflections and enlargements |similar triangles; calculate the length of| | |
|by geometric information | | |AB, given the coordinates of points A and | | |
|Cops and Robbers ( | | |B | | |
|Coordinate Patterns* ( | | |Beelines ( | | |
|Route to Infinity ( | | | | | |
|Construction and loci |
|use a ruler and protractor to: |use straight edge and compasses to |use straight edge and compasses to |understand from experience of constructing| | |
|• measure and draw lines to the |construct: • the midpoint and |construct triangles, given right angle, |them that triangles given SSS, SAS, ASA or| | |
|nearest millimetre and angles, |perpendicular bisector of a line |hypotenuse and side (RHS) |RHS are unique, but that triangles given | | |
|including reflex angles, to the |segment | |SSA or AAA are not | | |
|nearest degree |• the bisector of an angle • the | | | | |
|• construct a triangle, given two |perpendicular from a point to a line | | | | |
|sides and the included angle (SAS) or|• the perpendicular from a point on a| | | | |
|two angles and the included side |line | | | | |
|(ASA) |• a triangle, given three sides (SSS)| | | | |
| | | | | | |
| |Constructing Triangles ( | | | | |
|use ICT to explore constructions |use ICT to explore these |use ICT to explore constructions of | | | |
| |constructions |triangles and other 2-D shapes | | | |
|use ruler and protractor to construct|find simple loci, both by reasoning |find the locus of a point that moves |find the locus of a point that moves | | |
|simple nets of 3-D shapes, e.g. |and by using ICT, to produce shapes |according to a simple rule, both by |according to a more complex rule, both by | | |
|cuboid, regular tetrahedron, |and paths, e.g. an equilateral |reasoning and by using ICT |reasoning and by using ICT | | |
|square-based pyramid, triangular |triangle |Roundabout |Rollin’ Rollin’ Rollin’ ( | | |
|prism | | | | | |
|Measures and Mensuration |
|choose and use units of measurement |choose and use units of measurement |solve problems involving measurements in|understand and use measures of speed (and |apply knowledge that measurements given |recognise limitations in the accuracy |
|to measure, estimate, calculate and |to measure, estimate, calculate and |a variety of contexts; convert between |other compound measures such as density or|to the nearest whole unit may be |of measurements and judge the |
|solve problems in everyday contexts; |solve problems in a range of |area measures (e.g. mm2 to cm2, cm2 to |pressure); solve problems involving |inaccurate by up to one half of the unit|proportional effect on solutions |
|convert one metric unit to another, |contexts; know rough metric |m2, and vice versa) and between volume |constant or average rates of change |in either direction and use this to | |
|e.g. grams to kilograms; read and |equivalents of imperial measures in |measures (e.g. mm3 to cm3, cm3 to m3, |Speed-time Problems at the Olympics ( |understand how errors can be compounded | |
|interpret scales on a range of |common use, such as miles, pounds |and vice versa) |An Unhappy End ( |in calculations | |
|measuring instruments |(lb) and pints |Nutrition and Cycling ( |Speeding Boats ( | | |
|Place Your Orders* ( |All in a Jumble ( | | | | |
|Thousands and Millions* ( |Olympic Measures ( | | | | |
|distinguish between and estimate the |use bearings to specify direction | | | | |
|size of acute, obtuse and reflex | | | | | |
|angles | | | | | |
|Estimating Angles ( | | | | | |
| | |interpret and explore combining measures| | | |
| | |into rates of change in everyday | | | |
| | |contexts (e.g. km per hour, pence per | | | |
| | |metre); use compound measures to compare| | | |
| | |in real-life contexts (e.g. travel | | | |
| | |graphs and value for money), using ICT | | | |
| | |as appropriate | | | |
|know and use the formula for the area|derive and use formulae for the area |know and use the formulae for the |solve problems involving lengths of |understand and use the formulae for the |Of All the Areas ( |
|of a rectangle; calculate the |of a triangle, parallelogram and |circumference and area of a circle |circular arcs and areas of sectors |length of a circular arc and area and |On the Edge ( |
|perimeter and area of shapes made |trapezium; calculate areas of |An Unusual Shape ( |Curvy Areas ( |perimeter of a sector | |
|from rectangles |compound shapes | |Salinon ( |Track Design ( | |
|Changing Areas, Changing Perimeters (|Isosceles Triangles ( | |Arclets |Triangles and Petals ( | |
|Can They Be Equal? ( |Pick's Theorem* ( | | | | |
|Fence It ( | | | | | |
|Warmsnug Double Glazing ( | | | | | |
|calculate the surface area of cubes |know and use the formula for the |calculate the surface area and volume of|solve problems involving surface areas and|solve problems involving surface areas |solve problems involving more complex |
|and cuboids |volume of a cuboid; calculate volumes|right prisms |volumes of cylinders |and volumes of cylinders, pyramids, |shapes and solids, including segments |
|Cuboids ( |and surface areas of cuboids and |Changing Areas, Changing Volumes ( |Efficient Cutting ( |cones and spheres |of circles and frustums of cones |
| |shapes made from cuboids | |Cola Can |Funnel |Fill Me up Too ( |
| |Cuboid Challenge ( | | | |Immersion ( |
| |Painted Cube* ( | | | |Gutter |
| |Sending a Parcel | | | | |
| | | | |consider the dimensions of a formula and|understand the difference between |
| | | | |begin to recognise the difference |formulae for perimeter, area and |
| | | | |between formulae for perimeter, area and|volume by considering dimensions |
| | | | |volume in simple contexts | |
|Statistics |
|Specifying a problem, planning and collecting data |
|suggest possible answers, given a |discuss a problem that can be |suggest a problem to explore using |independently devise a suitable plan for a|consider possible difficulties with |select and justify a sampling scheme |
|question that can be addressed by |addressed by statistical methods and |statistical methods, frame questions |substantial statistical project and |planned approaches, including practical |and a method to investigate a |
|statistical methods |identify related questions to explore|and raise conjectures |justify the decisions made |problems; adjust the project plan |population, including random and |
|Statistical Shorts ( |Reaction Timer ( | | |accordingly |stratified sampling |
|decide which data would be relevant |decide which data to collect to |discuss how different sets of data |identify possible sources of bias and plan|deal with practical problems such as |understand how different methods of |
|to an enquiry and possible sources |answer a question, and the degree of |relate to the problem; identify |how to minimise it |non-response or missing data |sampling and different sample sizes |
| |accuracy needed; identify possible |possible primary or secondary | | |may affect the reliability of |
| |sources; consider appropriate sample |sources; determine the sample size | | |conclusions drawn |
| |size |and most appropriate degree of | | | |
| |Who’s the Best? ( |accuracy | | | |
| | |Retiring to Paradise ( | | | |
|plan how to collect and organise |plan how to collect the data; |design a survey or experiment to |break a task down into an appropriate |identify what extra information may be | |
|small sets of data from surveys and |construct frequency tables with equal|capture the necessary data from one |series of key statements (hypotheses), and|required to pursue a further line of | |
|experiments: |class intervals for gathering |or more sources; design, trial and if|decide upon the best methods for testing |enquiry | |
|• design data collection sheets or |continuous data and two-way tables |necessary refine data collection |these | | |
|questionnaires to use in a simple |for recording discrete data |sheets; construct tables for | | | |
|survey • construct frequency tables | |gathering large discrete and | | | |
|for gathering discrete data, grouped | |continuous sets of raw data, choosing| | | |
|where appropriate in equal class | |suitable class intervals; design and | | | |
|intervals | |use two-way tables | | | |
|collect small sets of data from |collect data using a suitable method |gather data from specified secondary |gather data from primary and secondary | | |
|surveys and experiments, as planned |(e.g. observation, controlled |sources, including printed tables and|sources, using ICT and other methods, | | |
| |experiment, data logging using ICT) |lists, and ICT-based sources, |including data from observation, | | |
| | |including the internet |controlled experiment, data logging, | | |
| | | |printed tables and lists | | |
|Processing and representing data |
|calculate statistics for small sets |calculate statistics for sets of |calculate statistics and select those|use an appropriate range of statistical |use an appropriate range of statistical | |
|of discrete data: |discrete and continuous data, |most appropriate to the problem or |methods to explore and summarise data; |methods to explore and summarise data; | |
|• find the mode, median and range, |including with a calculator and |which address the questions posed |including estimating and finding the mean,|including calculating an appropriate | |
|and the modal class for grouped data |spreadsheet; recognise when it is |Top Coach ( |median, quartiles and interquartile range |moving average for a time series | |
|• calculate the mean, including from |appropriate to use the range, mean, | |for large data sets (by calculation or | | |
|a simple frequency table, using a |median and mode and, for grouped | |using a cumulative frequency diagram) | | |
|calculator for a larger number of |data, the modal class | |Olympic Triathlon ( | | |
|items |How Would You Score It? ( | | | | |
|M, M and M ( | | | | | |
|Searching for Mean(ing) ( | | | | | |
|Litov's Mean Value Theorem ( | | | | | |
| | | | |use a moving average to identify | |
| | | | |seasonality and trends in time series | |
| | | | |data, using them to make predictions | |
|construct, on paper and using ICT, |construct graphical representations, |select, construct and modify, on |select, construct and modify, on paper and|select, construct and modify, on paper |construct histograms, including those |
|graphs and diagrams to represent |on paper and using ICT, and identify |paper and using ICT, suitable |using ICT, suitable graphical |and using ICT, suitable graphical |with unequal class intervals |
|data, including: |which are most useful in the context |graphical representations to progress|representation to progress an enquiry and |representation to progress an enquiry, | |
|• bar-line graphs |of the problem. Include: |an enquiry and identify key features |identify key features present in the data.|including histograms for grouped | |
|• frequency diagrams for grouped |• pie charts for categorical data |present in the data. Include: |Include: |continuous data with equal class | |
|discrete data |• bar charts and frequency diagrams |• line graphs for time series |• cumulative frequency tables and diagrams|intervals | |
|• simple pie charts |for discrete and continuous data |• scatter graphs to develop further | | | |
| |• simple line graphs for time series |understanding of correlation |• box plots | | |
| |• simple scatter graphs | |• scatter graphs and lines of best fit (by| | |
| |• stem-and-leaf diagrams | |eye) | | |
| | |work through the entire handling data| | | |
| | |cycle to explore relationships within| | | |
| | |bi-variate data, including | | | |
| | |applications to global citizenship, | | | |
| | |e.g. how fair is our society? | | | |
|Interpreting and discussing results |
|interpret diagrams and graphs |interpret tables, graphs and diagrams|interpret graphs and diagrams and |analyse data to find patterns and |interpret and use cumulative frequency |use, interpret and compare histograms,|
|(including pie charts), and draw |for discrete and continuous data, |make inferences to support or cast |exceptions, and try to explain anomalies; |diagrams to solve problems |including those with unequal class |
|simple conclusions based on the shape|relating summary statistics and |doubt on initial conjectures; have a |include social statistics such as index | |intervals |
|of graphs and simple statistics for a|findings to the questions being |basic understanding of correlation |numbers, time series and survey data | | |
|single distribution |explored | |Olympic Records ( | | |
| |Charting Success ( | |Substitution Cipher ( | | |
| | | |appreciate that correlation is a measure | | |
| | | |of the strength of 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' | | |
|compare two simple distributions |compare two distributions using the |compare two or more distributions and| |compare two or more distributions and | |
|using the range and one of the mode, |range and one or more of the mode, |make inferences, using the shape of | |make inferences, using the shape of the | |
|median or mean |median and mean |the distributions and appropriate | |distributions and measures of average | |
| | |statistics | |and spread, including median and | |
| | |Which List Is Which? ( | |quartiles | |
|write a short report of a statistical|write about and discuss the results |review interpretations and results of|examine critically the results of a |recognise the limitations of any | |
|enquiry, including appropriate |of a statistical enquiry using ICT as|a statistical enquiry on the basis of|statistical enquiry; justify choice of |assumptions and the effects that varying| |
|diagrams, graphs and charts, using |appropriate; justify the methods used|discussions; communicate these |statistical representations and relate |the assumptions could have on the | |
|ICT as appropriate; justify the | |interpretations and results using |summarised data to the questions being |conclusions drawn from data analysis | |
|choice of presentation | |selected tables, graphs and diagrams |explored | | |
|Probability |
|use vocabulary and ideas of |interpret the results of an |interpret results involving |use tree diagrams to represent outcomes of|use tree diagrams to represent outcomes | |
|probability, drawing on experience |experiment using the language of |uncertainty and prediction |two or more events and to calculate |of compound events, recognising when | |
| |probability; appreciate that random |What Does Random Look Like? ( |probabilities of combinations of |events are independent and | |
| |processes are unpredictable | |independent events |distinguishing between contexts | |
| |Sociable Cards ( | |Last One Standing ( |involving selection both with and | |
| | | | |without replacement | |
| | | | |Who’s the Winner? ( | |
| | | | |Chances Are ( | |
|understand and use the probability |know that if the probability of an |identify all the mutually exclusive |know when to add or multiply two | |recognise when and how to work with |
|scale from 0 to 1; find and justify |event occurring is p then the |outcomes of an experiment; know that |probabilities: if A and B are mutually | |probabilities associated with |
|probabilities based on equally likely|probability of it not occurring is 1 |the sum of probabilities of all |exclusive, then the probability of A or B | |independent and mutually exclusive |
|outcomes in simple contexts; identify|− p; use diagrams and tables to |mutually exclusive outcomes is 1 and |occurring is P(A) + P(B), whereas if A and| |events when interpreting data |
|all the possible mutually exclusive |record in a systematic way all |use this when solving problems |B are independent events, the probability | | |
|outcomes of a single event |possible mutually exclusive outcomes |Odds and Evens* ( |of A and B occurring is P(A) × P(B) | | |
| |for single events and for two |In a Box ( |Mathsland National Lottery ( | | |
| |successive events | |Same Number! ( | | |
| |Non-transitive Dice ( | | | | |
| |At Least One… ( | | | | |
| |Interactive Spinners ( | | | | |
|estimate probabilities by collecting |compare estimated experimental |compare experimental and theoretical |understand relative frequency as an |understand that if an experiment is | |
|data from a simple experiment and |probabilities with theoretical |probabilities in a range of contexts;|estimate of probability and use this to |repeated, the outcome may – and usually | |
|recording it in a frequency table; |probabilities, recognising that: • if|appreciate the difference between |compare outcomes of experiments |will – be different, and that increasing| |
|compare experimental and theoretical |an experiment is repeated the outcome|mathematical explanation and |Which Spinners? ( |the sample size generally leads to | |
|probabilities in simple contexts |may, and usually will, be different •|experimental evidence | |better estimates of probability and | |
|Odds and Evens* ( |increasing the number of times an |Do You Feel Lucky? ( | |population parameters | |
| |experiment is repeated generally |Two's Company ( | |The Better Bet | |
| |leads to better estimates of |Cosy Corner ( | | | |
| |probability | | | | |
| |Flippin' Discs ( | | | | |
|Latest Additions |January 2012 |March 2012: |June 2012 |
| |Triathlon and Fitness |Magic Letters |Olympic Measures |
| |Speed-time Problems at the Olympics |Double Trouble |Nutrition and Cycling |
| | |Slick Summing |Olympic Triathlon |
| |February 2012 | | |
| |Substitution Cipher |May 2012: |July 2012 |
| | |Always a Multiple? |Constructing Triangles |
| | |What Numbers Can We Make Now? |Kite in a Square |
| | |Factorising with Multilink? | |
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