NRICH www
NRICH nrich. problems linked to
the Framework for teaching mathematics in Years 7, 8 and 9
This is work in progress – it was last updated on 15th December 2008. The most recent updates are listed at the end of the document.
Ticked items (() identify problems that have Teachers’ Notes suggesting how they can be integrated into lessons.
Please email any comments to cfg21@cam.ac.uk
| |Year 7 |Year 8 |Year 9 |Year 9+ |
| |Using and Applying Mathematics |
| |Solve word problems and investigate in a range of |Solve more demanding problems and investigate in a |Solve increasingly demanding problems and evaluate | |
| |contexts: |range of contexts: number, algebra, shape, space and |solutions; explore connections in mathematics across a | |
| |number, algebra, shape, space and measures, and |measures, and handling data; compare and evaluate |range of contexts: number, algebra, shape, space, and | |
| |handling data; compare and evaluate solutions. |solutions. |measures, and handling data; generate fuller solutions.| |
| |Identify the necessary information to solve a problem; |Identify the necessary information to solve a problem; |Represent problems and synthesise information in | |
| |represent problems mathematically, making correct use |represent problems and interpret solutions in |algebraic, geometric or graphical form; move from one | |
| |of symbols, words, diagrams, tables and graphs. |algebraic, geometric or graphical form, using correct |form to another to gain a different perspective on the | |
| | |notation and appropriate diagrams. |problem. | |
| |Break a complex calculation into simpler steps, |Solve more complex problems by breaking them into |Solve substantial problems by breaking them into | |
| |choosing and using appropriate and efficient |smaller steps or tasks, choosing and using efficient |simpler tasks, using a range of efficient techniques, | |
| |operations, methods and resources, including ICT. |techniques for calculation, algebraic manipulation and |methods and resources, including ICT; use trial and | |
| | |graphical representation, and resources, including ICT.|improvement where a more efficient method is not | |
| | | |obvious. | |
| |Present and interpret solutions in the context of the |Use logical argument to establish the truth of a |Present a concise, reasoned argument, using symbols, |Recognising limitations on the accuracy of data and |
| |original problem; |statement; give solutions to an appropriate degree of |diagrams, graphs and related explanatory text; give |measurements; give reasons for choice of presentation, |
| |explain and justify methods and conclusions, orally and|accuracy in the context of the problem. |solutions to problems to an appropriate degree of |explaining selected features and showing insight into |
| |in writing. | |accuracy. |the problem’s structure. |
| |Suggest extensions to problems by asking ‘What if…?’; |Suggest extensions to problems, conjecture and |Suggest extensions to problems, conjecture and |Justify generalisations, arguments or solutions; pose |
| |begin to generalise and to understand the significance |generalise; identify exceptional cases or |generalise; identify exceptional cases or |extra constraints and investigate whether particular |
| |of a counter-example. |counter-examples. |counter-examples, explaining why; |cases can be generalised further. |
Cont.
| | Numbers and the number system |
| |Place Value |
| |Understand and use decimal notation and place value; |Read and write positive integer powers of 10; multiply |Extend knowledge of integer powers of 10; multiply and |Begin to write numbers in standard form. |
| |multiply and divide integers and decimals by 10, 100, |and divide integers and decimals by 0.1, 0.01. |divide by any integer power of 10; | |
| |1000, and explain the effect. | | | |
| |Compare and order decimals in different contexts; know |Order decimals. |Use rounding to make estimates; round numbers to the |or three decimal places; and to a given number of |
| |that when comparing measurements they must be in the | |nearest whole number or to one or two decimal places, |significant figures; understand upper and lower bounds.|
| |same units. | | | |
| |Round positive whole numbers to the nearest 10, 100 or |Round positive numbers to any given power of 10; round | | |
| |1000 and decimals to the nearest whole number or one |decimals to the nearest whole number or to one or two | | |
| |decimal place. |decimal places. | | |
| |Integers powers and Roots | | | |
| |Understand negative numbers as positions on a number |Add, subtract, multiply and divide integers. | | |
| |line; order, add and subtract positive and negative |NRICH: Playing Connect Three ( | | |
| |integers in context. |NRICH: Weights ( | | |
| |NRICH: First Connect Three ( |NRICH: Consecutive Negative Numbers ( | | |
| | |NRICH Article: | | |
| | |Adding & Subtracting Negative Numbers | | |
| |Recognise and use multiples, factors (divisors), common|Recognise and use multiples, factors (divisors), common|Use the prime factor decomposition of a number. | |
| |factor, highest common factor and lowest common |factor, highest common factor, lowest common multiple |Use ICT to estimate square roots and cube roots. | |
| |multiple in simple cases, and primes (less than 100); |and primes; find the prime factor decomposition of a |NRICH: Product Sudoku ( | |
| |use simple tests of divisibility. |number (e.g. 8000 = 26 × 53). |NRICH: Funny Factorisation | |
| |NRICH: 14 Divisors ( |NRICH: Stars ( | | |
| |NRICH: Dozens ( |NRICH: Power Mad! ( | | |
| |NRICH: Factors and Multiples Game ( |NRICH: Take Three from Five ( | | |
| |NRICH: Factors and Multiples Puzzle ( |NRICH: American Billions ( | | |
| |NRICH Article: Divisibility Tests | | | |
| |Recognise the first few triangular numbers, squares of |Use squares, positive and negative square roots, cubes |Use index notation for integer powers and simple | |
| |numbers to at least 12 × 12 and the corresponding |and cube roots, and index notation for small positive |instances of the index laws; know and use the index | |
| |roots. |integer powers. |laws for multiplication and division of positive | |
| | |NRICH: Sissa's Reward |integer powers; begin to extend understanding of index| |
| | | |notation to negative and fractional powers, recognising| |
| | | |that the index laws can be applied to these as well. | |
Cont.
| |Fractions, decimals, percentages, ratio and proportion |
| |Use fraction notation to describe parts of shapes and |Know that a recurring decimal is a fraction; use |Understand the equivalence of simple algebraic |use algebraic methods to convert a recurring decimal to|
| |to express a smaller whole number as a fraction of a |division to convert a fraction to a decimal; order |fractions; know that a recurring decimal is an exact |a fraction in simple cases. |
| |larger one; simplify fractions by cancelling all common|fractions by writing them with a common denominator or |fraction; |NRICH: Repetitiously ( |
| |factors and identify equivalent fractions; convert |by converting them to decimals. | | |
| |terminating decimals to fractions, e.g. 0.23 = 23⁄100; |NRICH: Round and Round and Round | | |
| |use a diagram to compare two or more simple fractions. |NRICH: Sept 03 | | |
| |Begin to add and subtract simple fractions and those |Add and subtract fractions by writing them with a |Use efficient methods to add, subtract, multiply and | |
| |with common denominators; calculate simple fractions of|common denominator; calculate fractions of quantities |divide fractions, interpreting division as a | |
| |quantities and measurements (whole-number answers); |(fraction answers); multiply and divide an integer by a|multiplicative inverse; cancel common factors before | |
| |multiply a fraction by an integer. |fraction. |multiplying or dividing. | |
| |NRICH: Fractions Jigsaw ( |NRICH: Ben's Game ( |NRICH: Twisting and Turning | |
| |NRICH: Peaches Today, Peaches Tomorrow... ( | |NRICH: More Twisting and Turning | |
| |Understand percentage as the ‘number of parts per 100’;|Interpret percentage as the operator ‘so many |Recognise when fractions or percentages are needed to | NRICH: Fraction and Percentage Card Game |
| |recognise the equivalence of percentages, fractions and|hundredths of’ and express one given number as a |compare proportions; solve problems involving | |
| |decimals; calculate simple percentages and use |percentage of another; use the equivalence of |percentage changes. | |
| |percentages to compare simple proportions. |fractions, decimals and percentages to compare | | |
| |NRICH: |proportions; calculate percentages and find the outcome| | |
| |Matching Fractions Decimals Percentages ( |of a given percentage increase or decrease. | | |
| |Understand the relationship between ratio and |Consolidate understanding of the relationship between |Use proportional reasoning to solve a problem, choosing|understand and use proportionality and calculate the |
| |proportion; use direct proportion in simple contexts; |ratio and proportion; reduce a ratio to its simplest |the correct numbers to take as 100%, or as a whole; |result of any proportional change using multiplicative |
| |use ratio notation, reduce a ratio to its simplest form|form, including a ratio expressed in different units, |interpret and use ratio in a range of contexts, |method; understand the implications of enlargement for |
| |and divide a quantity into two parts in a given ratio; |recognising links with fraction notation; divide a |including solving word problems. |area and volume; compare two ratios. |
| |solve simple problems about ratio and proportion using |quantity into two or more parts in a given ratio; use |NRICH: Mixing Paints |NRICH: A Chance to Win? |
| |informal strategies. |the unitary method to solve simple word problems |NRICH: Mixing More Paints | |
| | |involving ratio and direct proportion. | | |
| |Number operations |
| |Understand addition, subtraction, multiplication and |Understand addition and subtraction of fractions and |Understand the effects of multiplying and dividing by |recognise and use reciprocals. |
| |division as they apply to whole numbers and decimals; |integers, and multiplication and division of integers; |numbers between 0 and 1; use the laws of arithmetic and| |
| |know how to use the laws of arithmetic and inverse |use the laws of arithmetic and inverse operations. |inverse operations; | |
| |operations. |NRICH: Egyptian Fractions | | |
| |NRICH: Make 37 ( | | | |
| |NRICH: Consecutive Numbers ( | | | |
| |NRICH: Consecutive Sums ( | | | |
| |NRICH: Consecutive Seven ( | | | |
| |Know and use the order of operations, including |Use the order of operations, including brackets, with |Understand the order of precedence and effect of | |
| |brackets. |more complex calculations. |powers. | |
Cont.
| |Mental methods, rapid recall | | | |
| |Consolidate the rapid recall of number facts, including|Recall known facts, including fraction to decimal |Use known facts to derive unknown facts; extend mental | |
| |positive integer complements to 100 and multiplication |conversions; use known facts to derive unknown facts, |methods of calculation, working with decimals, | |
| |facts to 10 × 10, and quickly derive associated |including products involving numbers such as 0.7 and 6,|fractions, percentages, factors, powers and roots; | |
| |division facts. |and 0.03 and 8. |solve word problems mentally. | |
| |Consolidate and extend mental methods of calculation to|Consolidate and extend mental methods of calculation, |Make and justify estimates and approximations of |Estimate calculations by rounding numbers to one |
| |include decimals, fractions and percentages, |working with decimals, fractions and percentages, |calculations; |significant figure and multiplying or dividing |
| |accompanied where appropriate by suitable jottings; |squares and square roots, cubes and cube roots; solve | |mentally. |
| |solve simple word problems mentally. |word problems mentally. | | |
| |Make and justify estimates and approximations of |Make and justify estimates and approximations of | | |
| |calculations. |calculations. | | |
| |Written methods | | | |
| |Use standard column procedures to add and subtract |Consolidate standard column procedures for addition and|Use standard column procedures to add and subtract | |
| |whole numbers and decimals with up to two places. |subtraction of integers and decimals with up to two |integers and decimals of any size, including a mixture | |
| |NRICH: Two and Two ( |places. |of large and small numbers with differing numbers of | |
| |NRICH: Legs Eleven ( | |decimal places; multiply and divide by decimals, | |
| | | |dividing by transforming to division by an integer. | |
| | | |NRICH: How Many Miles to Go? ( | |
| |Multiply and divide three-digit by two-digit whole |Use standard column procedures for multiplication and | | , |
| |numbers; extend to multiplying and dividing decimals |division of integers and decimals, including by | | |
| |with one or two places by single-digit whole numbers. |decimals such as 0.6 or 0.06; understand where to | | |
| |NRICH: Multiplying with Lines ( |position the decimal point by considering equivalent | | |
| | |calculations. | | |
| | |NRICH: Largest Product ( | | |
| |Calculator methods | | | |
| |Carry out calculations with more than one step using |Carry out more difficult calculations effectively and |Use a calculator efficiently and appropriately to |Use the reciprocal key. |
| |brackets and the memory; use the square root and sign |efficiently using the function keys for sign change, |perform complex calculations with numbers of any size, | |
| |change keys. |powers, roots and fractions; |knowing not to round during intermediate steps of a | |
| | | |calculation; use the constant, ð and sign change keys, | |
| | | |function keys for powers, roots and fractions, brackets| |
| | | |and the memory; | |
| |Enter numbers and interpret the display in different |Use brackets and the memory. |Enter numbers and interpret the display in context |(numbers in standard form) |
| |contexts (decimals, percentages, money, metric | |(negative numbers, fractions, decimals, percentages, | |
| |measures). | |money, metric measures, and time). | |
| | |Enter numbers and interpret the display in different | | |
| | |contexts (negative numbers, fractions, decimals, | | |
| | |percentages, money, metric measures, and time). | | |
Cont.
| |Checking results | | | |
| |Check a result by considering whether it is of the |Check a result by considering whether it is of the |Check results using appropriate methods | |
| |right order of magnitude and by working the problem |right order of magnitude and by working the problem | | |
| |backwards. |backwards. | | |
| | |NRICH: Rule of Three | | |
| |Other NRICH Number Problems |
| |NRICH: GOT IT ( | |NRICH: Amazing Card Trick |NRICH: Six Times Five ( |
| |NRICH: Cinema Problem | |NRICH: Cunning Card Trick | |
| |Algebra | | | |
| |Equations, formulae and identities | | | |
| |Use letter symbols to represent unknown numbers or |Begin to distinguish the different roles played by |Distinguish the different roles played by letter | |
| |variables; know the meanings of the words term, |letter symbols in equations, formulae and functions; |symbols in equations, identities, formulae and | |
| |expression and equation. |know the meanings of the words formula and function. |functions. | |
| |NRICH: Crossed Ends ( | | | |
| |NRICH: Number Pyramids ( | | | |
| |Understand that algebraic operations follow the same |Know that algebraic operations follow the same |Use index notation for integer powers and simple |know and use the index laws in generalised form for |
| |conventions and order as arithmetic operations. |conventions and order as arithmetic operations; use |instances of the index laws; |multiplication and division of integer powers |
| | |index notation for small positive integer powers. | | |
| |Simplify linear algebraic expressions by collecting |Simplify or transform linear expressions by collecting |Simplify or transform algebraic expressions by taking |square a linear expression, expand the product of two |
| |like terms; begin to multiply a single term over a |like terms; multiply a single term over a bracket. |out single-term common factors; add simple algebraic |linear expressions of the form x ± n and simplify the |
| |bracket (integer coefficients). | |fractions. |corresponding quadratic expression; establish |
| |NRICH: More Number Pyramids ( | |NRICH: Harmonic Triangle ( |identities such as a^2 – b^2 = (a + b)(a – b). |
| | | | |NRICH: Partitioning Revisited |
| | | | |NRICH: Pair Products ( |
| | | | |NRICH: Multiplication Square |
| | | | |NRICH: Cubes Within Cubes Revisited |
| |Construct and solve simple linear equations with |Construct and solve linear equations with integer |Construct and solve linear equations with integer |Solve a pair of simultaneous linear equations by |
| |integer coefficients (unknown on one side only) using |coefficients (unknown on either or both sides, without |coefficients (with and without brackets, negative |eliminating one variable; link a graphical |
| |an appropriate method (e.g. inverse operations). |and with brackets) using appropriate methods (e.g. |signs anywhere in the equation, positive or negative |representation of an equation or a pair of equations to|
| | |inverse operations, transforming both sides in same |solution), using an appropriate method. |the algebraic solution; consider cases that have no |
| | |way). | |solution or an infinite number of solutions. |
| | |NRICH: Number Tricks | |NRICH: Sweet Shop |
| | |NRICH: Mind Reading | |NRICH: Arithmagons ( |
| | |NRICH: Think of Two Numbers | |NRICH: What's it Worth? ( |
| | | | |NRICH: Children at Large |
| | | | |NRICH: Matchless |
| |Use simple formulae from mathematics and other |Begin to use graphs and set up equations to solve | |Solve linear inequalities in one variable, and |
| |subjects; substitute positive integers into simple |simple problems involving direct proportion. | |represent the solution set on a number line; begin to |
| |linear expressions and formulae and, in simple cases, | | |solve inequalities in two variables. |
| |derive a formula. | | | |
| | |Use formulae from mathematics and other subjects; |Use systematic trial and improvement methods and ICT | |
| | |substitute integers into simple formulae, including |tools to find approximate solutions to equations such | |
| | |examples that lead to an equation to solve, and |as x³ + x = 20. | |
| | |positive integers into expressions involving small | | |
| | |powers (e.g. 3x²+4 or 2x³); derive simple formulae. | | |
| | | |Solve problems involving direct proportion using | |
| | | |algebraic methods, relating algebraic solutions to | |
| | | |graphical representations of the equations; use ICT as | |
| | | |appropriate. | |
| | | |Use formulae from mathematics and other subjects; |Derive and use more complex formulae, and change the |
| | | |substitute numbers into expressions and formulae; |subject of a formula. |
| | | |derive a formula and, in simple cases, change its |NRICH: Terminology |
| | | |subject; | |
| | | |NRICH: Temperature ( | |
| |Sequences, functions and graphs | | | |
| |Generate and describe simple integer sequences. |Generate and describe integer sequences. |Generate terms of a sequence using term-to-term and |Find the next term and the nth term of quadratic |
| |NRICH: Triangle Numbers ( |NRICH: 1 Step 2 Step ( |position-to term definitions of the sequence, on paper |sequences and functions and explore their properties. |
| | | |and using ICT; |NRICH: Tablecloth |
| |Generate terms of a simple sequence, given a rule (e.g.|Generate terms of a linear sequence using term-to-term |Generate sequences from practical contexts and write an|deduce properties of the sequences of triangular and |
| |finding a term from the previous term, finding a term |and position to- term definitions of the sequence, on |expression to describe the nth term of an arithmetic |square numbers from spatial patterns |
| |given its position in the sequence). |paper and using a spreadsheet or graphical calculator. |sequence | |
| | |NRICH: Coordinate Patterns* ( |NRICH: Picturing Triangle Numbers ( | |
| |Generate sequences from practical contexts and describe|Begin to use linear expressions to describe the nth |Find the inverse of a linear function | and plot its graph; know simple properties of |
| |the general term in simple cases. |term of an arithmetic sequence, justifying its form by | |quadratic functions. |
| |NRICH: Picturing Square Numbers ( |referring to the activity or practical context from | | |
| |NRICH: Squares in Rectangles ( |which it was generated. | | |
| | |NRICH: Seven Squares ( | | |
| |Express simple functions in words, then using symbols; |Express simple functions in symbols; represent mappings|Generate points and plot graphs of linear functions (y |Investigate the gradients of parallel lines and lines |
| |represent them in mappings. |expressed algebraically. |given implicitly in terms of x ), e.g. ay + bx = 0, y +|perpendicular to these lines; plot graphs of simple |
| | |NRICH: Pick's Theorem* ( |bx + c = 0, on paper and using ICT; given values for m |quadratic and cubic functions, e.g. y = x², y = 3x² + |
| | | |and c, find the gradient of lines given by equations of|4, y = x³. |
| | | |the form y = mx + c | |
| |Generate coordinate pairs that satisfy a simple linear |Generate points in all four quadrants and plot the |Construct functions arising from real-life problems and| |
| |rule; plot the graphs of simple linear functions, where|graphs of linear functions, where y is given explicitly|plot their corresponding graphs; interpret graphs | |
| |y is given explicitly in terms of x, on paper and using|in terms of x, on paper and using ICT; recognise that |arising from real situations, including distance–time | |
| |ICT; recognise straight-line graphs parallel to the |equations of the form y = mx + c correspond to |graphs. | |
| |x-axis or y-axis. |straight-line graphs. |NRICH: How Far Does it Move? ( | |
| | |NRICH: Parallel Lines ( |NRICH: Speeding Up, Slowing Down ( | |
| | |NRICH: Perpendicular Lines ( |NRICH: Up and Across | |
| | |NRICH: Diamond Collector ( | | |
| |Begin to plot and interpret the graphs of simple linear|Construct linear functions arising from real-life | | |
| |functions arising from real-life situations. |problems and plot their corresponding graphs; discuss | | |
| | |and interpret graphs arising from real situations. | | |
| | |NRICH: Walk and Ride ( | | |
| | |NRICH: Buses ( | | |
| |Space, Shape and Measures |
| |Geometrical reasoning: lines, angles and shapes |
| |Use correctly the vocabulary, notation and labelling |Identify alternate angles and corresponding angles; |Distinguish between conventions, definitions and |Distinguish between practical demonstration and proof; |
| |conventions for lines, angles and shapes. |understand a proof that: – the sum of the angles of a |derived properties; |know underlying assumptions, recognising their |
| | |triangle is 180° and of a quadrilateral is 360°; – the | |importance and limitations, and the effect of varying |
| | |exterior angle of a triangle is equal to the sum of the| |them. |
| | |two interior opposite angles. | |NRICH: Areas of Parallelograms ( |
| |Identify parallel and perpendicular lines; know the sum|Solve geometrical problems using side and angle |Explain how to find, calculate and use; – the sums of | |
| |of angles at a point, on a straight line and in a |properties of equilateral, isosceles and right-angled |the interior and exterior angles of quadrilaterals, | |
| |triangle, and recognise vertically opposite angles. |triangles and special quadrilaterals, explaining |pentagons and hexagons; – the interior and exterior | |
| | |reasoning with diagrams and text; classify |angles of regular polygons. | |
| | |quadrilaterals by their geometric properties. |NRICH: Semi-regular Tessellations ( | |
| | |NRICH: Eight Hidden Squares ( | | |
| | |NRICH: Square Coordinates ( | | |
| | |NRICH: Square It ( | | |
| |Begin to identify and use angle, side and symmetry |Know that if two 2-D shapes are congruent, |Solve problems using properties of angles, of parallel |Understand and apply Pythagoras’ theorem. |
| |properties of triangles and quadrilaterals; solve |corresponding sides and angles are equal. |and intersecting lines, and of triangles and other |NRICH: Tilted Squares ( |
| |geometrical problems involving these properties, using | |polygons, justifying inferences and explaining |NRICH: Inscribed in a Circle |
| |step-by-step deduction and explaining reasoning with | |reasoning with diagrams and text. |NRICH: Semi-detached |
| |diagrams and text. | |NRICH: Triangles in Circles ( |NRICH: Where Is the Dot? |
| |NRICH: Property Chart ( | |NRICH: Subtended Angles ( | |
| |NRICH: Shapely Pairs ( | |NRICH: Right Angles ( | |
| |NRICH: Quadrilaterals Game ( | | | |
| |Use 2-D representations to visualise 3-D shapes and |Know and use geometric properties of cuboids and shapes|Understand congruence; |Apply the conditions SSS, SAS, ASA or RHS to establish |
| |deduce some of their properties. |made from cuboids; begin to use plans and elevations. | |the congruence of triangles. |
| |NRICH: Marbles in a Box ( | | | |
| | | | |Know that if two 2-D shapes are similar, corresponding |
| | | | |angles are equal and corresponding sides are in the |
| | | | |same ratio. |
| | | |Know the definition of a circle and the names of its |Know that the tangent at any point on a circle is |
| | | |parts; explain why inscribed regular polygons can be |perpendicular to the radius at that point; explain why |
| | | |constructed by equal divisions of a circle; |the perpendicular from the centre to the chord bisects |
| | | | |the chord. |
| | | |Visualise and use 2-D representations of 3-D objects; | |
| | | |analyse 3-D shapes through 2-D projections, including | |
| | | |plans and elevations. | |
| | | |NRICH: Nine Colours ( | |
| | | |NRICH: Triangles to Tetrahedra ( | |
| |Transformations | | | |
| |Understand and use the language and notation associated|Transform 2-D shapes by simple combinations of |Transform 2-D shapes by combinations of translations, | |
| |with reflections, translations and rotations. |rotations, reflections and translations, on paper and |rotations and reflections, on paper and using ICT; know| |
| |NRICH: Mirror, Mirror… ( |using ICT; identify all the symmetries of 2-D shapes. |that translations, rotations and reflections preserve | |
| |NRICH: ...on the Wall ( |NRICH: Transformation Game ( |length and angle and map objects on to congruent | |
| | | |images; identify reflection symmetry in 3-D shapes. | |
| |Recognise and visualise the transformation and symmetry|Understand and use the language and notation associated|Enlarge 2-D shapes, given a centre of enlargement and a|Extend to enlarging 2-D shapes, given a fractional |
| |of a 2-D shape: – reflection in given mirror lines, and|with enlargement; enlarge 2-D shapes, given a centre of|whole number scale factor, on paper and using ICT; |scale factor; recognise the similarity of the resulting|
| |line symmetry; – rotation about a given point, and |enlargement and a positive whole-number scale factor; |identify the scale factor of an enlargement as the |shapes. |
| |rotation symmetry; – translation; explore these |explore enlargement using ICT. |ratio of the lengths of any two corresponding line |NRICH: Who Is the Fairest of Them All? ( |
| |transformations and symmetries using ICT. | |segments; recognise that enlargements preserve angle | |
| |NRICH: Reflecting Squarely ( | |but not length, and understand the implications of | |
| |NRICH: Isometrically ( | |enlargement for perimeter, area and volume. | |
| | |Make simple scale drawings. |Use and interpret maps and scale drawings. | |
| |Coordinates | | | |
| |Use conventions and notation for 2-D coordinates in all|Given the coordinates of points A and B, find the | |Find points that divide a line in a given ratio, using |
| |four quadrants; find coordinates of points determined |mid-point of the line segment AB. | |the properties of similar triangles; given the |
| |by geometric information. | | |coordinates of points A and B, calculate the length of |
| |NRICH: Cops and Robbers ( | | |AB. |
| |NRICH: Coordinate Patterns* ( | | | |
| |NRICH: Route to Infinity ( | | | |
| |Construction | | | |
| |Use a ruler and protractor to: – measure and draw lines|Use straight edge and compasses to construct: – the |Use straight edge and compasses to construct a |Know from experience of constructing them that |
| |to the nearest millimetre and angles, including reflex |mid-point and perpendicular bisector of a line segment;|triangle, given right angle, hypotenuse and side (RHS);|triangles given SSS, SAS, ASA or RHS are unique, but |
| |angles, to the nearest degree; – construct a triangle |– the bisector of an angle; – the perpendicular from a |use ICT to explore constructions of triangles and other|that triangles given SSA or AAA are not. |
| |given two sides and the included angle (SAS) or two |point to a line; – the perpendicular from a point on a |2-D shapes; | |
| |angles and the included side (ASA); explore these |line; construct a triangle, given three sides (SSS); | | |
| |constructions using ICT. |use ICT to explore these constructions. | | |
| |Use ruler and protractor to construct simple nets of |Find simple loci, both by reasoning and by using ICT, |Find the locus of a point that moves according to a | |
| |3-D shapes, e.g. cuboid, regular tetrahedron, |to produce shapes and paths, e.g. an equilateral |simple rule, both by reasoning and by using ICT; extend| |
| |square-based pyramid, triangular prism. |triangle. |to more complex rules involving loci and simple | |
| | | |constructions. | |
| | | |NRICH: Roundabout | |
| | | |NRICH: Rollin' Rollin' Rollin' | |
Cont.
| |Measures and Mensuration |
| |Use names and abbreviations of units of measurement to |Use units of measurement to estimate, calculate and |Use units of measurement to calculate, estimate, |Recognise that measurements given to the nearest whole |
| |measure, estimate, calculate and solve problems in |solve problems in everyday contexts involving length, |measure and solve problems in a variety of contexts; |unit may be inaccurate by up to one half of the unit in|
| |everyday contexts involving length, area, mass, |area, volume, capacity, mass, time, angle and bearings;|convert between area measures (mm² to cm², cm² to m², |either direction. |
| |capacity, time and angle; convert one metric unit to |know rough metric equivalents of imperial measures in |and vice versa) and between volume measures (mm³ to | |
| |another (e.g. grams to kilograms); read and interpret |daily use (feet, miles, pounds, pints, gallons). |cm³, cm³ to m³, and vice versa); | |
| |scales on a range of measuring instruments. |NRICH: All in a Jumble ( | | |
| |Use angle measure; distinguish between and estimate the|Use bearings to specify direction. | |Understand and use measures of speed (and other |
| |size of acute, obtuse and reflex angles. | | |compound measures such as density or pressure) to solve|
| |NRICH: Estimating Angles ( | | |problems; solve problems involving constant or average |
| | | | |rates of change. |
| | | | |NRICH: An Unhappy End ( |
| |Know and use the formula for the area of a rectangle; |Deduce and use formulae for the area of a triangle, |Know and use the formulae for the circumference and |…and arcs and sectors of circles. |
| |calculate the perimeter and area of shapes made from |parallelogram and trapezium; calculate areas of |area of a circle, |NRICH: Semi-circles |
| |rectangles. |compound shapes made from rectangles and triangles. |NRICH: An Unusual Shape | |
| |NRICH: On the Edge ( |NRICH: Isosceles Triangles ( | | |
| |NRICH: Hidden Dimensions ( |NRICH: Pick's Theorem* ( | | |
| |NRICH: Fence It ( | | | |
| |NRICH: Warmsnug Double Glazing ( | | | |
| |Calculate the surface area of cubes and cuboids |Know and use the formula for the volume of a cuboid; |Calculate the surface area and volume of right prisms; |Calculate lengths, areas and volumes in right prisms, |
| |NRICH: Cuboids ( |calculate volumes and surface areas of cuboids and | |including cylinders. |
| | |shapes made from cuboids. | |NRICH: Efficient Cutting |
| | |NRICH: Painted Cube ( | | |
| | |NRICH: Cuboid Challenge ( | | |
| | |NRICH: Sending a Parcel | | |
| | | | |Begin to use sine, cosine and tangent in right-angled |
| | | | |triangles to solve problems in two dimensions. |
| |Data Handling |
| |Specifying a problem, planning and collecting data |
| |Given a problem that can be addressed by statistical |Discuss a problem that can be addressed by statistical |Suggest a problem to explore using statistical methods,| |
| |methods, suggest possible answers. |methods and identify related questions to explore. |frame questions and raise conjectures. | |
| | |NRICH: Reaction Timer ( | | |
| |Decide which data would be relevant to an enquiry and |Decide which data to collect to answer a question, and |Discuss how data relate to a problem; identify possible|Identify possible sources of bias and plan how to |
| |possible sources. |the degree of accuracy needed; identify possible |sources, including primary and secondary sources; |minimise it. |
| | |sources. | | |
| |Plan how to collect and organise small sets of data; |Plan how to collect the data, including sample size; |Design a survey or experiment to capture the necessary | |
| |design a data collection sheet or questionnaire to use |construct frequency tables with given equal class |data from one or more sources; determine the sample | |
| |in a simple survey; construct frequency tables for |intervals for sets of continuous data; design and use |size and degree of accuracy needed; design, trial and | |
| |discrete data, grouped where appropriate in equal class|two-way tables for discrete data. |if necessary refine data collection sheets; construct | |
| |intervals. | |tables for large discrete and continuous sets of raw | |
| | | |data, choosing suitable class intervals; design and use| |
| | | |two-way tables. | |
| |Collect small sets of data from surveys and |Collect data using a suitable method, such as |Gather data from specified secondary sources, including|Identify what extra information may be required to |
| |experiments, as planned. |observation, controlled experiment, including data |printed tables and lists from ICT-based sources; |pursue a further line of enquiry. |
| | |logging using ICT, or questionnaire. | | |
| |Processing and representing data, using ICT as appropriate |
| |Calculate statistics for small sets of discrete data: –|Calculate statistics, including with a calculator; |Find summary values that represent the raw data, and |Find the median and quartiles for large data sets; |
| |find the mode, median and range, and the modal class |recognise when it is appropriate to use the range, |select the statistics most appropriate to the problem. |estimate the mean, median and interquartile range of a |
| |for grouped data; – calculate the mean, including from |mean, median and mode and, for grouped data, the modal |NRICH: Top Coach ( |large set of grouped data. |
| |a simple frequency table, using a calculator for a |class; calculate a mean using an assumed mean; | | |
| |larger number of items. |construct and use stem-and-leaf diagrams. | | |
| |NRICH: M, M and M ( | | | |
| |NRICH: Searching for (Mean)ing ( | | | |
| |NRICH: Litov's Mean Value Theorem ( | | | |
| |Construct, on paper and using ICT, graphs and diagrams |Construct, on paper and using ICT: – pie charts for |Select, construct and modify, on paper and using ICT, |– frequency polygons |
| |to represent data, including: – bar-line graphs; – |categorical data; – bar charts and frequency diagrams |suitable graphical representation to progress an |– lines of best fit by eye, understanding what they |
| |frequency diagrams for grouped discrete data; use ICT |for discrete and continuous data; – simple line graphs |enquiry, including: ; – line graphs for time series; – |represent; |
| |to generate pie charts. |for time series; – simple scatter graphs; identify |scatter graphs to develop further understanding of | |
| | |which are most useful in the context of the problem. |correlation; identify key features present in the | |
| | | |data. | |
| |Interpreting and discussing results | | | |
| |Interpret diagrams and graphs (including pie charts), |Interpret tables, graphs and diagrams for both discrete|Interpret graphs and diagrams and draw inferences to |Analyse data to find patterns and exceptions, look for |
| |and draw simple conclusions based on the shape of |and continuous data, and draw inferences that relate to|support or cast doubt on initial conjectures; have a |cause and effect and try to explain anomalies. |
| |graphs and simple statistics for a single distribution.|the problem being discussed; relate summarised data to |basic understanding of correlation; | |
| | |the questions being explored. | | |
| |Compare two simple distributions using the range and |Compare two distributions using the range and one or |Compare two or more distributions and make inferences, | |
| |one of the mode, median or mean. |more of the mode, median and mean. |using the shape of the distributions, the range of data| |
| | | |and appropriate statistics. | |
| |Write a short report of a statistical enquiry and |Communicate orally and on paper the results of a |Communicate interpretations and results of a |Examine critically the results of a statistical |
| |illustrate with appropriate diagrams, graphs and |statistical enquiry and the methods used, using ICT as |statistical enquiry using selected tables, graphs and |enquiry, and justify choice of statistical |
| |charts, using ICT as appropriate; justify the choice of|appropriate; justify the choice of what is presented. |diagrams in support, using ICT as appropriate; |representation in written presentations, recognising |
| |what is presented. | | |the limitations of any assumptions and their effect on |
| | | | |conclusions drawn. |
| |Probability | | | |
| |Use vocabulary and ideas of probability, drawing on |Use the vocabulary of probability when interpreting the|Use the vocabulary of probability in interpreting | |
| |experience. |results of an experiment; appreciate that random |results involving uncertainty and prediction. | |
| | |processes are unpredictable. | | |
| |Understand and use the probability scale from 0 to 1; |Know that if the probability of an event occurring is |Identify all the mutually exclusive outcomes of an | |
| |find and justify probabilities based on equally likely |p, then the probability of it not occurring is 1 – p ; |experiment; know that the sum of probabilities of all | |
| |outcomes in simple contexts; identify all the possible |find and record all possible mutually exclusive |mutually exclusive outcomes is 1 and use this when | |
| |mutually exclusive outcomes of a singe event. |outcomes for single events and two successive events in|solving problems. | |
| | |a systematic way, using diagrams and tables. |NRICH: In a Box ( | |
| | |NRICH: Interactive Spinners ( | | |
| |Collect data from a simple experiment and record in a |Estimate probabilities from experimental data; |Estimate probabilities from experimental data; |Understand relative frequency as an estimate of |
| |frequency table; estimate probabilities based on this |understand that: – if an experiment is repeated there | |probability and use this to compare outcomes of |
| |data. |may be, and usually will be, different outcomes; | |experiments. |
| | | | |NRICH: Which Spinners? ( |
| |Compare experimental and theoretical probabilities in |– increasing the number of times an experiment is |Compare experimental and theoretical probabilities in a| |
| |simple contexts. |repeated generally leads to better estimates of |range of contexts; appreciate the difference between | |
| |NRICH: Odds and Evens ( |probability. |mathematical explanation and experimental evidence. | |
| | | |NRICH: Two's Company ( | |
| | | |NRICH: Cosy Corner ( | |
| | |Compare experimental and theoretical probabilities in | | |
| | |different contexts. | | |
| | |NRICH: Flippin' Discs ( | | |
* indicates that a problem appears in more than one cell
December 2008 updates
The following problems have been added to the document:
Power Mad!
Make 37
Cops and Robbers
Hidden Dimensions
Cuboid Challenge
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