Overview: securing level 8 / grade B
Unit / Hours / Beyond the ClassroomIntegers, powers and roots / 4
Geometrical reasoning: proof / 9
Proportion / 4
Sequences, functions and graphs / 6
Place value, calculations and checking / 6
Equations, formulae, identities and expressions / 5
Learning review 1
Transformations and coordinates / 4
Processing and representingdata; Interpretingand discussing results / 7
Equations, formulae, identities and expressions / 8
Probability / 6
Learning review 2
Note: This final year version of the Stage 6 scheme of work has the objectives which align with grade A descriptors removed from the core objectives section. They have been transferred to the ‘Next…’ section as a source of potential extension work, and are indicated by italics.
Integers, powers and roots
Autumn Term 4 hours / Previously…• Examine and extend the index laws to establish the meaning of negative, fractional and zero powers, including use of surd notation / Progression map
• Examine and extend the index laws to establish the meaning of inverse operations in relation to indices, i.e. the inverse operation of raising a positive number to power n is raising the result of this operation to power 1/n / Progression map
Next…
• Solve a problem using rational and irrational numbers, including surds
• Appreciate when results of calculations can be more elegantly and exactly communicated using surds and π, rationalising a denominator where appropriate, e.g. a trigonometrical solution / Progression map
Suggested Activities / Criteria for Success
NCETM Departmental Workshops
- Index Numbers
Geometrical reasoning: proof
Autumn Term 9 hours / Previously…• Examine and refine arguments in solutions to geometrical problems, distinguishing between practical demonstration and proof; produce simple proofs
• Use dynamic images to demonstrate invariant relationships between radii, chords and tangents in circles; develop arguments to explain and justify simple circle properties and theorems
• Solve geometrical problems using properties of lines, angles, polygons and circles; justify arguments and solutions using deductive reasoning
• Draw inferences about properties of similar 2-D shapes and use proportional reasoning to solve geometrical problems
• Use properties of 2-D and 3-D shapes to make accurate constructions on paper and using ICT; including constructing triangles from combinations of side and angle facts, reviewing and generalising findings to identify which of these conditions define unique constructions / Progression map
• Examine and create chains of deductive reasoning in solutions to more complex geometrical problems
• Examine and refine algebraic arguments presented to explain geometrical and numerical properties; choose and combine representations to present a convincing proof
• Examine and create proofs of the circle theorems; use circle theorems to solve problems
• Formalise existing knowledge of lines, angles and polygons by:
(i)using the congruence conditions (SSS, SAS, RHS, ASA) to deduce familiar properties of triangles and quadrilaterals, e.g. an isosceles triangle has two equal angles
(ii)explaining why standard constructions work, e.g. observing that lines joining points where compass arcs meet are sides of a rhombus / Progression map
Next…
• Engage with and explain the stages of a variety of proofs of Pythagoras’ theorem; use Pythagoras’ theorem to solve more complex 3-D problems
• Create a chain of reasoning to deduce the equation of a circle by applying Pythagoras’ theorem to the locus of a point / Progression map
Suggested Activities / Criteria for Success
Maths Apprentice
- Discover Thales’ Theorem by challenging pupils to join three points on the circumference of a circle to make a right angled triangle (see resources below and PowerPoint)
- Thales’ Theorem (includes proof by card sort)
- Geometrical reasoning – card sort proof (and PowerPoint version)
- GSP Resources: Circle Theorems
- Cabri Resources: Tangent and radius, Tangents to a circle, Angle at centre (move this to demonstrate angle in semi-circle too), Angles in same segment, Cyclic quadrilateral
- Investigating Angles in Circles: v2, v3, v4, v5
- Visualising: v7
- Manipulating Shapes II: v9
- Six point circles
- Eight point circles
- Twelve point circles
- Circle Theorems
- Squirty
- Partly Circles
- Triangles in Circles
- Subtended Angles
- Right Angles
- Lens Angle
Proportion
Autumn Term 4 hours / Previously…• Identify when a problem in number, algebra, geometry or statistics involves proportionality; use multiplicative methods fluently in solutions, including inverse calculations, e.g. with percentages
• Explore the behaviour of the reciprocal function (y = 1/x) for large and small values of x / Progression map
• Model real contexts where quantities vary in direct proportion, including repeated proportional change, e.g. growth/decay; use algebraic methods where appropriate and consider limitations of the model / Progression map
Next…
• Explore the historical and cultural roots of the number system and use algebra to justify and prove some of its features, e.g. that all recurring decimals can be expressed as a fraction / Progression map
Suggested Activities / Criteria for Success
NCETM Departmental Workshops
- Proportional Reasoning
Sequences, functions and graphs
Spring Term 6 hours / Previously…•Explore graphs of functions of the form y = xn(n an integer) and recognise their characteristic shapes; vary the values of a, b and c in functions such as y = ax2+ c, y = ax3+ c, y = (x + b)2using a graph plotter to explain how this transforms the graph
•Sketch and interpret graphs that model real-life situations, including those generated from other subjects such as science; use mathematical argument to justify features of their shapes
• Compare graphical, algebraic and geometrical representations, including mapping diagrams, to explain the effect of:
(i)rotating the line y = mx + c through 90° about any point,
(ii)reflecting the line y = mx + c in the line y = x;
derive properties of perpendicular lines and of the inverse function / Progression map
• Explore connections between the form of the equation and the resulting graphs of quadratic and cubic functions such as: y = (x + 2)(x – 5), y = (x – 2)(x2+ 7x + 12), y = x2– 2x + 1, y = x3+ 3;
include features such as roots of the equation, intercepts and turning points
• Apply knowledge of mathematical functions to problems involving:
(i)optimisation, using numerical, algebraic and graphical, techniques, including maxima and minima
(ii)using ICT to fit a curve to data from a real context such as a science experiment
(iii)repeated proportional change, e.g. compound interest / Progression map
Next…
•Explore graphs of exponential and trigonometrical functions and recognise their characteristic shapes; apply to the graph y = f(x) the transformations y = f(x) + a, y = af(x), y = f(x + a), y = f(ax) for linear, quadratic, sine and cosine functions; use a graph plotter to explain the effect of transformations on the graph and generalise to other functions
•Set up a mathematical model of a real-life context or problem, identifying the variables and their functional relationship; use graphs and sketches to explain the behaviour of the variables and to explain or justify the effect of assumptions in the model / Progression map
Suggested Activities / Criteria for Success
NRICH
- Doesn’t Add Up
Place value, calculations and checking
Spring Term 6 hours / Previously…• Convert between ordinary and standard index form representations, using significant figures as appropriate; justify the representation used and choice of accuracy in relation to the problem and audience for the solution
• Select mental or written strategies and calculating devices appropriate to the stage of the problem; calculate accurately with reciprocals, powers, trigonometrical functions and numbers in standard form
• Examine and refine estimates and approximations of calculations involving rounding
• Identify a range of checking strategies and appreciate that more than one way may be necessary in the context of the problem / Progression map
• Engage in mathematical tasks where using numbers in standard form is essential to the calculations involved; critically examine the effect of numerical representations on the accuracy of the solution, e.g. understand how errors can be compounded in calculations
• Select and justify an appropriate and efficient combination of methods of calculation, i.e. mental, written, ICT or calculator to solve problems
• Critically examine alternative methods, compare strategies for:
(i)calculating (including calculating devices)
(ii)checking;
recognise the limitations of some approaches / Progression map
Next…
• Communicate the solution to a problem, explaining the limitations of accuracy, using upper and lower bounds
• Reflect on a solution to a problem commenting constructively on the choice of calculating strategies
• Appreciate when results of calculations can be more elegantly and exactly communicated using surds and π, rationalising a denominator where appropriate, e.g. a trigonometrical solution / Progression map
Suggested Activities / Criteria for Success
Equations, formulae, identities and expressions
Autumn Term 5 hours / Previously…• Develop fluency in transforming linear expressions; expand the product of two linear expressions of the form x ± n and factorise simple quadratic expressions; establish identities such as the difference of two squares; compare and evaluate different representations of the same context; identify equivalent expressions and confirm by transformation
• Use algebraic representation to synthesise known rules of arithmetic, including the commutative and distributive laws; justify these generalisations, e.g. using spatial representations; use algebraic argument to generalise the index laws for multiplication and division to include zero, negative and fractional powers / Progression map
• Expand and factorise quadratic expressions; simplify or transform algebraic fractions, e.g. by factorising and cancelling common factors; compare and evaluate different representations of the same context; identify equivalent expressions and confirm by transformation
• Appreciate the generality of the forms a + b = c and ab = c, where each term can itself be an expression; use this insight into structure to develop fluency in transforming more complex equations / Progression map
Next…
• Use symbols and representations consistently to present a formal proof, e.g. deriving the formula for solving quadratic equations / Progression map
Suggested Activities / Criteria for Success
NRICH
- Plus Minus
- AlwaysPerfect
- What's Possible
- Odd Squares
- Perfectly Square
LEARNING REVIEW 1
Transformations and coordinates
Spring Term 4 hours / Previously…• Use precise language and notation to describe and generalise the results of combining transformations of 2-D shapes on paper and using ICT, including:
(i)rotations about any point,
(ii)reflections in any line,
(iii)translations using vector notation,
(iv)a transformation and its inverse;
generate and analyse patterns, e.g. Islamic designs
• Enlarge 2-D shapes using positive, fractional and negative scale factors, on paper and using ICT; use reciprocals as a multiplicative inverse in the context of enlargement; recognise the similarity of resulting shapes and explain the effect of enlargement on perimeter
• Apply the properties of similar triangles and Pythagoras’ theorem to solving problems presented on a 2-D coordinate grid; use a 3-D coordinate grid to represent simple shapes
• Recognise and use reciprocals as a multiplicative inverse in contexts such as enlargement / Progression map
• Enlarge 3-D shapes; identify and explain the effects of enlargement on areas and volumes of similar shapes and solids; relate this understanding to practical contexts, e.g. in biology / Progression map
Next…
• Explain and demonstrate graphically the effects of combining translations, using vector notation, including:
(i)the rule for addition of vectors
(ii)scalar multiplication of a vector (repeated addition) / Progression map
Suggested Activities / Criteria for Success
NCETM Departmental Workshops
- Enlargement
Processing and representing data; Interpreting and discussing results
Spring Term 7 hours / Previously…
• Construct on paper and using ICT suitable graphical representations, including:
(i)histograms for grouped continuous data with equal class intervals,
(ii)cumulative frequency tables and diagrams,
(iii)box plots,
(iv)scatter graphs and lines of best fit (by eye);
justify their suitability with reference to the context of the problem and the audience
• Use an appropriate range of statistical methods to explore and summarise large data sets, justifying the choices made; include grouping data, estimating and finding the mean, median, quartiles and interquartile range
• Find patterns and exceptions and explain anomalies; including interpretation of social statistics and evaluation of the strength of association within bi-variate data (correlation, lines of best fit) / Progression map
• Process data drawn from problems involving seasonality and trends in a time series; choose and combine statistical methods to analyse the problem, including moving averages
• Interpret and compare distributions, including cumulative frequency diagrams; make and discuss inferences, using the shape of the distributions and measures of average and spread, including median and quartiles / Progression map
Next…
• Choose and combine suitable graphical representations to progress an unfamiliar or non-routine enquiry, including histograms with equal or unequal class intervals / Progression map
Suggested Activities / Criteria for Success
Equations, formulae, identities and expressions
Spring Term 8 hours / Previously…• Construct linear equations and simple linear inequalities (one variable) to represent real-life situations or mathematical problems; solve linear equations and inequalities, representing the solution in the context of the problem
• Construct a pair of simultaneous linear equations to represent real-life situations or mathematical problems; examine and compare algebraic methods of solution; use graphical representation to explain why the intersection of two lines gives the common solution and why some cases have no common solution and others have an infinite number / Progression map
• Select and justify optimum methods for solving a pair of simultaneous linear equations in a variety of contexts; construct several linear inequalities in one and two variables to represent real-life situations or mathematical problems; solve the inequalities graphically, identifying and interpreting the solution set in the context of the problem / Progression map
Next…
• Construct simple quadratic equations to represent real-life situations or mathematical problems and solve them using factorisation, graphical or trial and improvement methods; justify the number of solutions using algebraic or graphical arguments and select appropriate solutions, interpreting their accuracy / Progression map
Suggested Activities / Criteria for Success
NCETM Departmental Workshops
- Simultaneous Equations
- Inequalities
- CD Heaven
- Matchless
Probability
Autumn Term 6 hours / Previously…• Identify when the events in a problem are mutually exclusive or independent; use and interpret tree diagrams to represent outcomes of combined events and to inform the calculation of their probabilities; decide when to add and when to multiply probabilities
• Use relative frequency as an estimate of probability, including simulation using ICT to generate larger samples; discuss its reliability based on sample size and use to interpret and compare outcomes of experiments / Progression map
• Interpret the effect on probability of contexts involving selection with and without replacement; choose and combine representations to communicate probabilities as part of a solution to a problem
• Explore a relevant and purposeful problem involving uncertainty; estimate risk by modelling real events through simulation; justify decisions based on experimental probability and comment on the effect of assumptions and sample size on the reliability of conclusions / Progression map
Next…
• Recognise when and how to work with probabilities associated with independent and mutually exclusive events when interpreting data / Progression map
Suggested Activities / Criteria for Success
NRICH
- The Better Bet
LEARNING REVIEW 2
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