Exploring mathematics: Tier 4 NC levels 5 and 6 (levels 5abc, 6c) Autumn 36 lessons Spring 31 lessons S4.2 Enquiry 1 Collecting, representing, interpreting data: two-way tables, bar charts for grouped discrete data, scatter graphs, pie charts, stem-and-leaf diagrams Comparing distributions using mean, median, mode or modal class 5/6 lessons A4.1 Linear sequences Integer sequences Linear sequences; general term; spreadsheet and calculator use 4 lessons N4.2 Whole numbers, decimals and fractions Decimal place value, ordering, rounding Mental and written calculations: adding/subtracting decimals; multiplying/dividing whole numbers Equivalent fractions; adding/subtracting fractions; fractions of quantities Calculator use, including fraction key 6 lessons A4.2 Expressions and formulae Simplifying linear expressions Deriving and substituting in simple formulae 4 lessons N4.3 Fractions, decimals and percentages Equivalent fractions; ordering fractions Multiplying/dividing decimals by decimals; adding/subtracting factions; fractions of quantities; multiplying/dividing by proper fraction Percentage increases and decreases Calculator use 5/6 lessons N4.4 Proportional reasoning Calculating ratios; simple scale drawings Direct proportion; unitary method Using fractions, decimals and percentages to compare proportions 5/6 lessons Summer 33 lessons G4.3 Transformations Symmetries of 2-D shapes Rotations, reflections and translations, on paper and using ICT Enlargement, given a centre of enlargement and a positive whole-number scale factor Coordinates of midpoint of line segment 6/7 lessons R4.1 Revision/support Number, algebra, geometry and measures, statistics 5 lessons A4.4 Equations and formulae Linear equations, unknowns both sides, brackets; using formulae 5/6 lessons A4.5 Expressions, equations and graphs Simplifying linear expressions Plotting graphs; graphs of real situations Using algebra to solve problems 7/8 lessons N4.5 Solving problems Number problems and investigations Using algebra to solve problems History of mathematics 2/3 lessons 100 lessons 1 | Exploring mathematics | Tier 4 (red) G4.2 Measures and mensuration Using measures in problem solving Metric and imperial equivalents Perimeter and area of triangles, rectangles, parallelograms, trapeziums Volume and surface area of cuboids 5 lessons A4.3 Functions and graphs Mappings Graphs of linear functions Graphs of real situations 5/6 lessons R4.2 Revision/support Number, algebra, geometry and measures, statistics 5 lessons S4.3 Enquiry 2 Collecting, representing, interpreting data: frequency tables and diagrams for continuous data, line graphs Calculating statistics Comparing distributions using mean, median, mode or modal class 6/7 lessons G4.1 Angles and shapes Alternate and corresponding angles Angle, side, diagonal and symmetry properties of triangles, quadrilaterals and polygons Proof of angle sum/exterior angle of triangle Finding unknown angles 6 lessons G4.4 Constructions Straight edge and compasses constructions of midpoints, perpendiculars, bisectors and triangles (SSS) Scale drawings; bearings Simple loci; exploring constructions with ICT 8/9 lessons Using and applying mathematics is integrated into each unit S4.1 Probability Mutually exclusive outcomes for single events and two successive events Estimating probabilities from experimental data 6 lessons N4.1 Properties of numbers Add, subtract, multiply and divide integers Order of operations, including brackets Squares, cubes and roots; factors, primes, prime factor decomposition, HCF, LCM; calculator use 5 lessons Units SUPPORT Number 1: Properties of numbers (5 hours) Integers, powers and roots (48–59) Calculations (86–87, 92–101, 108–109) CORE Before they start, pupils should be able to: In this unit, pupils learn to: • order, add and subtract positive and negative • identify the mathematical features of a context or problem integers in context • try out and compare mathematical representations • use simple tests of divisibility • conjecture and generalise, identifying exceptional cases • recognise square numbers to 12 12 and the • calculate accurately, selecting mental methods or a calculator as corresponding roots • use the bracket keys and memory of a calculator. appropriate • use accurate notation • refine own findings and approaches on the basis of discussion with others • record methods, solutions and conclusions and to: • add, subtract, multiply and divide integers • use the order of operations, including brackets, with more complex calculations • use multiples, factors, common factors, highest common factor, lowest common multiple and primes • find the prime factorisation of a number (e.g. 8000 = 26 53) • use squares, positive and negative square roots, cubes and cube roots, and index notation for small positive integer powers • strengthen and extend mental methods of calculation • use the function keys of a calculator for sign change, brackets, powers and roots, and interpret the display in context. 2 | Exploring mathematics | Tier 4 (red) Number 2: Whole numbers, decimals and fractions (6 hours) Before they start, pupils should be able to: In this unit, pupils learn to: • round and order whole numbers and decimals to • identify the mathematical features of a context or problem two places • use efficient written methods of addition and subtraction and of short multiplication and division Place value (36–47) Fractions, decimals, percentages of whole numbers • simplify fractions by cancelling and identify equivalent fractions (60–77) • write decimals as fractions, e.g. 0.23 = 23⁄100 Calculations • add and subtract simple fractions with the same (82–85, 88–107, 110–111) denominator • calculate simple fractions of numbers and quantities. • use accurate notation • make connections with related contexts • manipulate numbers and apply routine algorithms • calculate accurately, selecting mental methods or a calculator as appropriate • estimate, approximate and check working, giving accurate solutions appropriate to the context or problem or problem • evaluate the efficiency of alternative strategies and approaches and to: • read and write positive integer powers of 10 • round positive numbers to any given power of 10 and decimals to the nearest whole number of one or two decimal places • find equivalent fractions, and equivalent fractions and decimals • strengthen and extend mental methods of calculation, working with decimals and fractions • use efficient written methods to: – add and subtract integers and decimals of any size, including numbers with differing numbers of decimal places – multiply and divide 3-digit by 2-digit whole numbers • add and subtract fractions by writing them with a common denominator • calculate fractions of quantities • use a calculator to carry out more difficult calculations, entering numbers, including fractions, using the memory, and interpreting the display in context • select from a range of checking methods, including estimating in context and using inverse operations. 3 | Exploring mathematics | Tier 4 (red) Number 3: Decimals, fractions and percentages (6 hours) Before they start, pupils should be able to: In this unit, pupils learn to: • convert one metric unit to another • identify the mathematical features of a context or problem • convert fractions to decimals by using a calculator • make connections between related problems • add and subtract simple fractions and multiply a • use accurate notation fraction by an integer Calculations • understand percentage as the ‘number of parts (82–85, 88–107, 110–111) per 100’ and calculate simple fractions and Calculator methods percentages of numbers and quantities (108–109) Measures (228–231) Solving problems (28–29) • carry out calculations with more than one step using brackets and the memory of a calculator. • calculate accurately, selecting mental methods or a calculator as appropriate • give accurate solutions appropriate to the context or problem • evaluate the efficiency of alternative strategies and approaches and to: • multiply and divide integers and decimals by 0.1 or 0.01, and derive products such as 6 × 0.7, 8 × 0.03 • use division to convert a fraction to a decimal and recognise that a recurring decimal is a fraction • order fractions by writing them with a common denominator or by converting them to decimals • express one given number as a percentage of another • strengthen and extend mental methods of calculation, working with decimals, fractions and percentages • use efficient written methods to: – multiply and divide decimals, understanding where to position the decimal point by considering equivalent calculations – add and subtract fractions – multiply and divide fractions by integers – calculate percentages and find the outcome of a given percentage increase or decrease • use a calculator to carry out more difficult calculations, entering numbers, and interpreting the display in context • select from a range of checking methods, including estimating in context and using inverse operations. 4 | Exploring mathematics | Tier 4 (red) Number 4: Proportional reasoning (6 hours) Before they start, pupils should be able to: In this unit, pupils learn to: • understand percentage as the ‘number of parts • identify the mathematical features of a context or problem per 100’ and calculate simple fractions and percentages of quantities and measurements Fractions, decimals, percentages (60–77) Ratio and proportion (78–81) Calculations (92–101) • divide a quantity in a given ratio. • make connections with related contexts • use accurate notation • calculate accurately, selecting mental methods or a calculator as appropriate • estimate, approximate and check working, giving accurate solutions appropriate to the context or problem • refine own findings and approaches on the basis of discussion with others • evaluate the efficiency of alternative strategies and approaches Solving problems (2–35) and to: • apply understanding of the relationship between ratio and proportion • simplify ratios, including ratios expressed in different units, recognising links with fraction notation • divide a quantity into two or more parts in a given ratio • calculate fractions of quantities, using a calculator where appropriate • interpret percentage as the operator ‘so many hundredths of’ and calculate percentages • use equivalent fractions, decimals and percentages to compare proportions • use the unitary method to solve problems involving ratio and direct proportion • enter numbers and interpret the display of a calculator in different contexts. 5 | Exploring mathematics | Tier 4 (red) Number 5: Solving problems (3 hours) Solving problems (2–35) Before they start, pupils should be able to: In this unit, pupils learn to: • understand place value in whole numbers and • identify the mathematical features of a context or problem decimals • relate the current problem and structure to previous situations • understand the order of operations • conjecture and generalise • calculate with whole numbers and decimals • use logical argument to interpret the mathematics in a given context or to mentally, on paper and with a calculator, as appropriate • estimate and check results. establish the truth of a statement • calculate accurately, selecting mental methods or a calculator as appropriate • give accurate solutions appropriate to the context or problem • refine own findings and approaches on the basis of discussion with others • recognise efficiency in an approach • relate the current problem and structure to previous situations • record methods, solutions and conclusions and to: • recognise some of the historical and cultural roots of mathematics. Algebra 1: Linear sequences (4 hours) Before they start, pupils should be able to: In this unit, pupils learn to: • generate terms of a simple sequence given a rule • try out and compare mathematical representations • generate a sequence given a simple practical • conjecture and generalise, identifying exceptional cases or counter- context. Sequences and functions (144–157) examples • move between the general and the particular to test the logic of an argument • select appropriate procedures and tools, including ICT • refine own findings and approaches on the basis of discussion with others • record methods, solutions and conclusions and to: • generate terms of a linear sequence using term-to-term and position-toterm rules, on paper and using a spreadsheet or graphics calculator • use linear expressions to describe the nth term of a simple arithmetic sequence, justifying its form by referring to the context from which it was generated. 6 | Exploring mathematics | Tier 4 (red) Algebra 2: Expressions and formulae (4 hours) Before they start, pupils should be able to: In this unit, pupils learn to: • use letter symbols to represent unknown numbers • try out and compare mathematical representations or variables • use accurate notation • simplify linear algebraic expressions by collecting Equations and formulae (112–119, 138–143) like terms • manipulate algebraic expressions and equations • explore the effect of varying values • multiply a constant over a bracket • refine own findings and approaches on the basis of discussion with others • substitute integers into simple linear expressions and formulae, and positive integers into expressions involving small powers, e.g. 3x + 4 2 or 2x3. and to: • understand that algebraic operations, including the use of brackets, follow the rules of arithmetic • multiply a single term over a bracket • simplify or transform linear expressions by collecting like terms • derive and substitute integers into simple formulae and expressions. Algebra 3: Functions and graphs (6 hours) Before they start, pupils should be able to: In this unit, pupils learn to: • express simple functions in words • identify the mathematical features of a context or problem • generate coordinate pairs that satisfy a simple • try out and compare mathematical representations linear rule Sequences, functions, graphs (160–177) • recognise straight-line graphs parallel to the x-axis or y-axis. • manipulate algebraic expressions and equations • visualise and manipulate dynamic images and explore the effect of varying values • select appropriate procedures and tools, including ICT • draw accurate graphs on paper and on screen • conjecture and generalise, identifying exceptional cases or counterexamples and to: • express simple functions algebraically and represent them in mappings or on a spreadsheet • generate points in all four quadrants and plot graphs of linear functions (y given explicitly in terms of x), on paper and using ICT • recognise that equations of the form y = mx + c correspond to straight-line graphs • discuss and interpret graphs arising from real situations. 7 | Exploring mathematics | Tier 4 (red) Algebra 4: Equations and formulae (6 hours) Before they start, pupils should be able to: In this unit, pupils learn to: • construct and solve simple linear equations with • identify the mathematical features of a problem or context whole-number coefficients (unknown on one side only) using a method such as inverse operations Equations and formulae (112–113, 122–125, 138–143) • substitute positive integers into simple formulae. • try out and compare mathematical representations • use accurate notation • manipulate algebraic expressions and equations • use logical argument to establish the truth of a statement • explore the effect of varying values • refine own findings and approaches on the basis of discussion with others • evaluate the efficiency of alternative strategies and approaches and to: • construct and solve linear equations with integer coefficients (unknown on one or both sides, without and with brackets), e.g. by using inverse operations or by transforming both sides in the same way • use, derive and substitute integers into simple expressions and formulae from mathematics and other subjects. 8 | Exploring mathematics | Tier 4 (red) Algebra 5: Expressions, equations and graphs (8 hours) Before they start, pupils should be able to: In this unit, pupils learn to: • simplify linear algebraic expressions by collecting • try out and compare representations in algebraic or graphical form like terms • construct and solve simple linear equations with whole-number coefficients (unknown on one side Equations and formulae (116–137) Sequences, functions and graphs (164–177) only) • recognise straight-line graphs parallel to the x-axis or y-axis. • select appropriate procedures and tools, including ICT • visualise and manipulate dynamic images and explore the effect of varying values • draw accurate graphs on paper and on screen • manipulate algebraic expressions and equations • use logical argument to interpret the mathematics in a given context or to establish the truth of a statement Solving problems (6–13, 28–29) • explain own findings and approaches after discussion with others • evaluate the efficiency of alternative strategies and approaches • relate the current problem and structure to previous situations and to: • multiply a single term over a bracket • simplify or transform linear expressions by collecting like terms • construct and solve linear equations with integer coefficients (unknown on one or both sides, without and with brackets) • derive and substitute integers into simple formulae and expressions, including examples that lead to an equation to solve • generate points in all four quadrants and plot graphs of linear functions (y given explicitly in terms of x), on paper and using ICT • construct linear functions arising from real-life problems and plot and interpret their corresponding graphs • use graphs and set up equations to solve simple problems involving direct proportion. 9 | Exploring mathematics | Tier 4 (red) Geometry and measures 1: Angles and shapes (5 hours) Geometrical reasoning: lines, angles and shapes Before they start, pupils should be able to: In this unit, pupils learn to: • use correct notation and labelling conventions for • make accurate mathematical diagrams and constructions, on paper and lines, angles and shapes • use accurate notation • know the sum of angles at a point, on a straight • select appropriate procedures and tools, including ICT line and in a triangle (178–191) • recognise vertically opposite angles Solving problems • estimate, measure and draw acute, obtuse and (14–17, 26–27, 30–31) on screen • identify parallel and perpendicular lines reflex angles. • visualise and manipulate dynamic images and explore the effect of varying values • use logical argument to establish the truth of a statement • refine own findings and approaches on the basis of discussion with others • evaluate the efficiency of alternative strategies and approaches • record methods, solutions and conclusions and to: • identify alternate angles and corresponding angles • solve geometrical problems using side and angle properties of triangles and special quadrilaterals, explaining reasoning with diagrams and text • classify quadrilaterals by their geometrical properties • know that if two 2D shapes are congruent, corresponding sides and angles are equal • understand a proof that: – the angle sum of a triangle is 180 and of a quadrilateral is 360 – the exterior angle of a triangle is equal to the sum of the two interior opposite angles. 10 | Exploring mathematics | Tier 4 (red) Geometry and measures 2: Measures and mensuration (6 hours) Before they start, pupils should be able to: In this unit, pupils learn to: • convert one metric unit of length, mass or • identify the mathematical features of a context or problem capacity to another • interpret scales on a range of measuring Measures and mensuration (228–231, 234–241) Solving problems (18–21) instruments • understand and use the formula for the area of a rectangle • calculate perimeters and areas of shapes made from rectangles. • move between the general and the particular to test the logic of an argument • calculate accurately, selecting mental methods or a calculator as appropriate • select appropriate procedures and tools • estimate, approximate and check working, giving accurate solutions appropriate to the context or problem • refine own findings and approaches on the basis of discussion with others and to: • choose and use units of measurement to measure, estimate, calculate and solve problems in a range of contexts • know rough metric equivalents of imperial measures in common use • visualise 3D shapes from their nets, and use geometric properties of cuboids and shapes made from cuboids • recognise simple plans and elevations • derive and use formulae for the area of a triangle, parallelogram and trapezium and the volume of a cuboid • calculate areas of compound shapes and volumes and surface areas of cuboids and shapes made from cuboids. 11 | Exploring mathematics | Tier 4 (red) Geometry and measures 3: Transformations (7 hours) Before they start, pupils should be able to: In this unit, pupils learn to: • identify and visualise transformations and • conjecture and generalise symmetries of 2D shapes: reflection and line symmetry, rotation and rotation symmetry, and Transformations (202–215) Ratio and proportion (78–81) translation • recognise and visualise symmetries and simple transformations of 2D shapes • solve simple problems involving ratio and proportion. • visualise and manipulate dynamic images and explore the effect of varying values • make accurate mathematical diagrams on paper and on screen • use accurate notation • select appropriate procedures and tools, including ICT • refine own findings and approaches on the basis of discussion with others and to: • find the midpoint of the line segment AB, given the coordinates of points A and B • identify all the symmetries of 2D shapes • transform 2D shapes by rotation, reflection and translation, on paper and using ICT, and try out mathematical representations of simple combinations of these transformations • enlarge 2D shapes, given a centre of enlargement and a positive integer scale factor, and explore enlargement using ICT. 12 | Exploring mathematics | Tier 4 (red) Geometry and measures 4: Constructions (9 hours) Before they start, pupils should be able to: In this unit, pupils learn to: • measure and draw lines to the nearest millimetre • identify the mathematical features of a problem or context • measure and draw angles to the nearest degree • make connections with related contexts • construct a triangle given two sides and the • select appropriate procedures and tools, including ICT Geometrical reasoning: lines, angles included angle (SAS) or two angles and the and shapes included side (ASA) (198–201) • construct simple nets of 3D shapes Transformation • use coordinates in all four quadrants. • make accurate mathematical constructions on paper and on screen • visualise and manipulate dynamic images • use logical argument to establish the truth of a statement (216–217) • record methods, solutions and conclusions Coordinates and to: (218–219) • use straight edge and compasses to construct: Construction and loci – the midpoint and perpendicular bisector of a line segment (220–227) – the bisector of an angle Mensuration (228–233) – the perpendicular from a point to a line • use ruler and compasses to construct a triangle, given the lengths of the three sides (SSS) • use ICT to explore constructions • make scale drawings • use bearings to specify direction • find simple loci, both by reasoning and by using ICT, to produce shapes and paths. 13 | Exploring mathematics | Tier 4 (red) Statistics 1: Probability (6 hours) Before they start, pupils should be able to: In this unit, pupils learn to: • understand and use the probability scale from 0 to • identify the mathematical features of a context or problem 1 • find probabilities based on equally likely outcomes Probability (276–283) in simple contexts • estimate probabilities based on experimental data in a frequency table. • conjecture and generalise, identifying exceptional cases or counterexamples • select appropriate procedures and tools, including ICT • make accurate diagrams and graphs on paper and on screen • record methods, solutions and conclusions and to: • interpret results of an experiment using the language of probability and appreciate that random processes are unpredictable • know that, if the probability of an event occurring is p, then the probability of it not occurring is 1 – p • use diagrams and tables to record all possible mutually exclusive outcomes for single events and for two successive events • compare estimated experimental probabilities with theoretical probabilities, recognising that: – if an experiment is repeated, the outcome may and usually will be different – increasing the number of times an experiment is repeated generally leads to better estimates of probability. 14 | Exploring mathematics | Tier 4 (red) Statistics 2: Enquiry 1 (6 hours) Before they start, pupils should be able to: In this unit, pupils learn to: • design a data collection sheet or questionnaire to • identify the mathematical features of a context or problem Statistics • construct frequency tables for discrete data, (248–273) grouped where appropriate in equal class Solving problems intervals (28–29) use in a simple survey • interpret pie charts • understand and use the mean of discrete data • use the range and one of the mode, median or mean to describe a set of data. • conjecture and generalise • try out and compare mathematical representations • select appropriate procedures and tools, including ICT • make accurate diagrams and graphs on paper and on screen • refine own findings and approaches on the basis of discussion with others • record methods, solutions and conclusions, relating them to the context • evaluate alternative strategies and approaches and to: • discuss a problem that can be addressed by statistical methods and identify related questions to explore • decide which data to collect to answer a question the sample size and degree of accuracy needed, and identify possible sources • plan, construct and use two-way tables for recording discrete data • construct and interpret: – bar charts and frequency diagrams for grouped discrete data – pie charts for categorical data – simple scatter diagrams • calculate statistics for sets of discrete data, recognising when it is appropriate to use the range, mean, median and mode and, for grouped data, the modal class • construct and interpret stem-and-leaf diagrams, and compare two simple distributions using the range and one of the mode, median or mean • relate summary statistics and findings to the questions being explored. 15 | Exploring mathematics | Tier 4 (red) Statistics 3: Enquiry 2 (7 hours) Before they start, pupils should be able to: In this unit, pupils learn to: • design a data collection sheet or questionnaire to • identify the mathematical features of a context or problem use in a simple survey • conjecture and generalise • construct frequency tables for discrete data, Statistics grouped where appropriate in equal class (248–275) intervals Probability (284–285) • construct and interpret bar-line graphs and frequency diagrams for grouped discrete data • try out and compare mathematical representations • select appropriate procedures and tools, including ICT • make accurate diagrams and graphs on paper and on screen • refine own findings and approaches on the basis of discussion with others Solving problems • interpret pie charts • record methods, solutions and conclusions, relating them to the context (28–29) • calculate the mean, including from a simple • evaluate alternative strategies and approaches frequency table and to: • find the mode, median and range, and the modal class for grouped data, and use them to compare two simple distributions. • compare estimated experimental probabilities with theoretical probabilities • decide which data to collect to answer a question, the sample size and degree of accuracy needed, and identify possible sources • plan, construct and use frequency tables with equal class intervals for gathering continuous data • construct and interpret: – bar charts and frequency diagrams for continuous data – simple line graphs for time series • compare two simple distributions using the range and one of the mode, median or mean, relating summary statistics and findings to the questions being explored • write about and discuss the results of a statistical enquiry, justifying the methods used, and using ICT as appropriate Revision 1 and 2 - here are the process objectives Previous learning Objectives based on NC levels 5 and 6 (mainly level 5) Pupils should already be able to apply and use In this unit, pupils consolidate their ability to: many of the skills shown on the right. This unit • identify the the mathematical features of a context or problem offers is an opportunity to consolidate and refine these skills. • relate the current problem or structure to previous situations • select appropriate procedures and tools • calculate accurately, using mental methods or a calculator as appropriate • manipulate numbers, algebraic expressions and equations, and apply routine algorithms • make accurate mathematical diagrams and graphs • estimate, approximate and check working, giving accurate solutions appropriate to the context or problem • evaluate the efficiency of alternative strategies and approaches • record methods, solutions and conclusions 16 | Exploring mathematics | Tier 4 (red) and to: Number content objectives are as in the unit 17 | Exploring mathematics | Tier 4 (red)