fluent in the fundamentals of mathematics. Reason ... · Mathematics Department – Medium Term...

Mathematics Department – Medium Term Plan

End Game The aims for our curriculum is to ensure that all pupils:

o Become fluent in the fundamentals of mathematics. o Reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language. o Can solve problems by applying their mathematics to a variety of routine and non-routine problems with increasing sophistication

1 2 3 4

Year 8 Fertile Question

Content Indices

Prime numbers

Fractions

Percentages

Algebra

2D shapes

Statistics

Ratio

Proportion

3D shapes

Concepts Number Number, Algebra Geometry

Statistics

Number

Number, Geometry

Knowledge Weeks 1 to 3

Review of Y7

Four operations

Order of operations

Negative numbers

Fractions 1

Percentages 1

Algebra 1

Weeks 4 to 7

Number: Indices and Primes Use integer powers and associated real roots

(square, cube and higher), recognise powers of

2, 3, 4, 5 and distinguish between exact

representations of roots and their decimal

approximations.

Find the prime factor decomposition of a

number.

Use prime factor decomposition to find LCM,

HCF, squares, and cubes.

Enumerate sets and unions/intersections of sets

systematically using Venn diagrams.

Weeks 8 to 10

Number: Fractions 2 Find a fraction of an amount.

Multiply and divide proper and improper

fractions and mixed numbers both positive

and negative.

Fraction x Integer

Fraction x Fraction

Fraction ÷ Integer

Weeks 11 to 13 Number: Percentages 2

Express one quantity as a percentage of another

Compare two quantities using percentages, and

work with percentages greater than 100%

Solve problems involving percentage

change, including:

Percentage increase, decrease

Original value problems

Simple interest in financial

mathematics

Weeks 14 to 20

Algebra 2

Substitute numerical values into formulae

and expressions, including scientific

formulae

Include all prior learning specifically fractions, decimals and negatives

Simplify and manipulate algebraic

expressions to maintain equivalence by:

multiplying a single term over a bracket

taking out common factors expanding products of two or more

binomials simplifying expressions involving

sums, products and powers

Rearrange formulae to change the subject,

where the subject appears once.

Use algebraic methods to solve linear

equations in one variable (including all

forms that require rearrangement)

Include equations with brackets Include fractional equations

Understand and use the concepts

Weeks 20 to 23

Geometry: 2D Shapes

Convert between cm2 and m2

Derive and apply formulae to calculate and

solve problems involving area of circles,

composite shapes and trapeziums.

Calculate and solve problems involving

perimeters of 2-D shapes (including

circles).

Include examples using algebra, fractions,

decimals, etc.

Weeks 24 to 26

Statistics

Construct and analyse stem and leaf diagrams,

including back to back.

For non-grouped data given in the form of a

table, find the mean, median, mode and range

Weeks 27 to 29

Number: Ratio

Change freely between related standard units

[for example time, length, area,

volume/capacity, mass]

Use ratio notation, including reduction

simplest form.

Divide a given quantity into two or more

parts.

Given information about one part, find the

whole or other part(s).

Understand that a multiplicative relationship

between two quantities can be expressed as a

ratio or a fraction.

Use compound units such as speed, unit

pricing and density to solve problems.

Weeks 30 to 32

Number: Proportion Solve problems involving direct and

inverse proportion, including graphical

and algebraic representations.

Examples may include:

Recipe problems

Best buy problems

Exchange rates

Weeks 33 to 35

Geometry: 3D Shapes

Use the properties of faces, surfaces, edges

and vertices of cubes, cuboids, prisms,

cylinders, pyramids, cones and spheres to

solve problems in 3D.

Convert between cm3 and m3

Know and use the fact that 1 litre =

1000cm3

Derive and apply formulae to calculate and

solve problems involving volume and

surface area of cuboids (including cubes)

and other prisms (including cylinders).

Construct and interpret plans and elevations of

3D shapes.

Week 36

Time at the beginning or end of the term for

consolidation gap filling, seasonal activities,

end of year assessments, etc.


Integer ÷ Fraction

Fraction ÷ Fraction

All of the above proper, improper,

mixed, positive and negative.

Find the whole amount, given a fraction of

the amount.

Find a fractional increase and decrease

and vocabulary of inequalities.

Represent the solution set to an

inequality on a number line and vice versa

Find the integer solutions of an inequality

Solve linear inequalities in one variable



o Become fluent in the fundamentals of mathematics. o Reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language. o Can solve problems by applying their mathematics to a variety of routine and non-routine problems with increasing sophistication.

1 2 3 4


Content Place value

Addition and

Subtraction

Multiplication

and division

Negative Numbers

Fractions

Percentages

Data Handling

Notation,

Substitution and Basic Manipulation

(Angles and

Lines)

Concepts Number Number

Number

Statistics

Algebra

Geometry

Knowledge Week 1 to 3

Number: Place Value Understand and use place value for decimals,

measures and integers of any size.

Order positive and negative integers, use the

number line as a model for ordering of the real

numbers; use the symbols =, =, <, >, ≤, ≥

Round numbers and measures to an appropriate

degree of accuracy [for example, to a number of

decimal places or significant figures].

Number: Addition and

Subtraction Use formal written

methods for addition and

subtraction of integers and

decimals.

Recognise and use

relationships between

addition and subtraction

including inverse

operations.

Calculate and solve

problems involving

perimeter.

(Week 7 -9)

Number: Multiplication

and Division

Week 10 -12

Number: Multiplication and Division

Calculate and solve problems involving area of

rectangles, triangles and parallelograms.

Calculate the mean average.

Use approximation through rounding to

estimate answers and calculate possible

resulting errors expressed using inequality

notation a < x ≤ b

Number: Negative Numbers

Calculate and solve problems involving area of

rectangles, triangles and parallelograms.

Calculate the mean average.

Use approximation through rounding to

estimate answers and calculate possible

resulting errors expressed using inequality

notation a < x ≤ b

Week 13 - 15

Number: Fractions 1 Represent fractions using diagrams and on a

number line.

Express one quantity as a fraction of another.

Identify and use equivalent fractions.

Simplify fractions.

Compare and order fractions; use the symbols

=, =, <, >, ≤, ≥

Week 19 - 21

Number: Percentages 1 Define percentage as ‘number of parts per

hundred’

Interpret diagrams as percentages and vice

versa

Interpret fractions and percentages as operators,

with and without a calculator.

Interpret percentages as a fraction or as a

decimal

Week 22 -24

Statistics: Data Handling Understand the data handling cycle.

Understand the different types of data.

Collect, organise and interpret data.

Draw and interpret bar charts, pictograms, line

graphs and pie charts.

Week 25 - 27

Algebra 1 Introduction to algebra

variable.

expression, equation, formula, term, function

and identity.

Week 28 - 30

Algebra 1

Substitute numerical values into formulae and

expressions including scientific formulae

(including examples with negatives)

Simplify and manipulate algebraic expressions

to maintain equivalence by:

Use algebraic methods to solve simple linear

equations in one variable where the unknown

appears on one side of the equation.

Generate terms of a sequence from either a

term-to-term or a position-to-term rule.

Week 31 -34

Geometry: Angles and Lines Describe, sketch and draw using conventional

terms and notations: points, lines, parallel lines,

perpendicular lines, right angles, regular

polygons, and other polygons that are

reflectively and rotationally symmetric.

Derive and illustrate properties of triangles,

quadrilaterals, circles, and other plane figures

[for example, equal lengths and angles] using

appropriate language and technologies

Use a protractor to measure and draw angles.

Apply the properties of angles at a point, angles

at a point on a straight line, vertically opposite

angles.

Understand and use alternate and corresponding

angles on parallel lines.


Multiply and divide by 10,

100 and 1000

Use formal written

methods for multiplication

and division of integers

and decimals.

Recognise and use

relationships between

operations including

inverse operations.

Understand the order of

operations.

Use the concepts and

vocabulary of factors (or

divisors), common factors,

and highest common factor

(HCF).

Week 16 -18

Convert between mixed numbers and improper

fractions.

Convert between fractions and decimals

convert any fraction to a decimal.

Add and subtract any fraction.

that is a

multiple of the other.

in words.

Pupils should be taught to use and interpret

algebraic notation, including:

ab in place of a × b

a2 in place of a × a; a3 in place of a × a ×

a; a2b in place of a × a × b

b/a b ÷ a

as decimals

Derive and use the sum of angles in a triangle

and a quadrilateral.

Derive and use the sum of angles in a triangle

and use it to deduce the angle sum in any

polygon, and to derive properties of regular

polygons.

Week 35-36






o Become fluent in the fundamentals of mathematics. o Reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language. o Can solve problems by applying their mathematics to a variety of routine and non-routine problems with increasing sophistication.

1 2 3 4


Content Review of Year 7

Number: Operations, BIDMAS, Factors,

Powers, FDP, Rounding

Algebra: Expressions, Solving

Equations.

Ratio and Proportion.

Linear Graphs

Direct and Inverse Proportion

Sequences

Inequalities

Geometry (Transformations)

Geometry (Angles & Constructions)


Statistics and Probability

Geometry

(Pythagoras and

Trigonometry)

Standard Form

Concepts Number

Algebra

Algebra

Geometry

Geometry Algebra

Geometry

Number

Knowledge Week 1-5

Review of content taught in Year 7 & Year 8.

Four operations (N1, N2)

Order of operations (N3)

Negative numbers (N1, N2)

Multiples, factors, primes, powers, roots and

reciprocals (N4, N6)

Fractions, decimals and percentages (N1, N2)

Rounding and estimation (N14, N15)

Algebraic expressions, substitution, solving

simple linear equations and expanding brackets

(A1, A3, A4, A5, A6, A7, A21)

Ratio and proportion (R4, R5)

Week 6-9

Linear Graphs

Work with coordinates in all four quadrants.

Recognise, sketch and produce graphs of:

• Linear functions of one variable

• Quadratic functions of one variable

Algebraic manipulation including:

• Changing the subject of a formula

• Expansion and factorisation

• Understand and use standard mathematical

formulae

• Rearranging formulae to change the subject,

including where the subject appears more than

Week 10-11

Direct and Inverse Proportion

Interpret mathematical relationships both

algebraically and graphically e.g. direct and

inverse proportion and real life graphs.

Find approximate solutions to contextual

problems from given graphs of a variety of

functions, including piece-wise linear,

exponential and reciprocal graphs.

Week 12

Sequences

Recognise and generate geometric and

arithmetic sequences.

Week 13

Inequalities

Construct and solve equations and inequalities.

Week 14-17

Geometry (Transformations)

Identify properties of, and describe the results

of, translations, rotations, reflections and

enlargement applied to given figures.

Week 18-19


Week 20-21


Use scale factors, scale diagrams and maps.

Identify and construct congruent triangles, and

construct similar shapes by enlargement, with

and without coordinate grids. Derive and use

the standard ruler and compass constructions

(perpendicular bisector of a line segment,

constructing a perpendicular to a given line

from/at a given point, bisecting a given angle);

recognise and use the perpendicular distance

from a point to a line as the shortest distance to

the line.

Loci.

Week 22-26

Statistics and Probability

Statistics Use and interpret scatter graphs of bivariate

data and recognise correlation.

Draw and analyse frequency polygons.

For continuous data given in the form of a table

find an estimate of the mean, modal class

interval and class interval that contains the

median

Probability

Record, describe and analyse the frequency of

Week 27

Number (Standard Form) Interpret and compare numbers in standard

form A x 10n 1 ≤ A < 10, where n is a positive

or negative integer or zero. Add and subtract

two numbers in standard form.

Week 28-30

Number (Surds) Evaluate simple fractional and negative indices

in the form: a-n, a1/n, a-1/n where n is an

integer.

Understand what a surd is and simplify basic

surds.

Week 31-34

Geometry (Pythagoras and Trigonometry) Apply angle facts, triangle congruence,

similarity and properties of quadrilaterals to

derive results about angles and sides, including

angles in polygons, Pythagoras’ Theorem, and

use known results to obtain simple proofs.

Use Pythagoras’ Theorem and trigonometric

ratios in similar triangles to solve problems

involving right - angled triangles.

Know the formulae for: Pythagoras’ theorem

and the trigonometric ratios:

Apply them to find angles and lengths in right -

angled triangles and, where possible, general


once

Reduce a given linear equation in two variables

to the standard form y = mx + c

Calculate and interpret gradients and intercepts

of graphs of such linear equations numerically,

graphically and algebraically.

Use linear and quadratic graphs to estimate

values of y for given values of x and vice versa

and to find approximate solutions of

simultaneous linear equations.

Draw and measure line segments and angles in

geometric figures, including interpreting scale

drawings and use of bearings.

outcomes of simple probability experiments

involving randomness, fairness, equally and

unequally likely outcomes, using appropriate

language and the 0 - 1 probability scale.

Understand that the probabilities of all possible

outcomes sum to 1

Enumerate sets and unions/intersections of sets

systematically, using tables, grids and Venn

diagrams.

Generate theoretical sample spaces for single

and combined events with equally likely,

mutually exclusive outcomes and use these to

calculate theoretical probabilities.

triangles in two and three dimensional figures.

Week 35-36





End Game Foundation: Higher: ] For all students to achieve a minimum of 3 LOP in final examination, regardless of their starting point.

1 2 3 4

Year 10 Foundation

Fertile Question Does data always tell you the real Story? the whole Story? or the Truth?

Content Review of KS3- Numeracy, algebra, sequences

Averages and Range

Graphs, tables and charts

Fractions and percentages

Ratio and proportion

Parallel lines and interior and exterior angles in polygons.

Multiplicative reasoning

Graphs-linear graphs

Perimeter, area and volume

Quadratic equations and graphs

Right angles triangles: trigonometry and Pythagoras.

Probability

Concepts Algebra, Number and Data and Probability Number, Geometry, Ratio and proportion Ratio and proportion, Algebra, Geometry Algebra, Geometry, Data and probability

Knowledge Review of KS3: Integer/Decimal- 4 operations Basic indices, powers and roots Factors, multiples and primes Basic algebra: Simplifying, expanding and factorising singles brackets, substitution, solving linear equation and inequalities. Sequences. Averages and range Calculating median, mode, mean and range from a data set and from a table. Explain and understand Sampling. Carry out a statistical investigation and explain how biased data would result and how to eliminate it. Graphs, tables and charts Construct a frequency table for a continuous data set, deciding on appropriate intervals using inequalities Plan a journey using timetables. Produce and interpret: pictograms; composite bar charts; dual/comparative bar charts for categorical and ungrouped discrete data; bar-line charts; vertical line charts; line graphs; line graphs for time–series data; histograms with equal class intervals; stem and leaf (including back-to-back); Pie Charts Given two sets of data in a table, model the relationship and make predictions using scatter

Fractions and percentages Apply four operations to fractions (proper and improper, mixed number) Use fractions to solve problems. Convert between percentages, fractions and decimals. Find a percentage of a quantity, Calculate simple interest and percentage increases and decreases. Calculate a percentage profit or loss and find the original amount given the final amount after a percentage increase or decrease Ratio and proportion Use ratio notation, divide a given quantity into two parts, expression the division of a quantity into parts of a ratio, apply ratio to real contexts. Solve problem involving direct proportion; scales, recipes, currencies. Recognise and use direct proportion on a graph. Parallel lines and interior and exterior angles in polygons. Recap: apply the properties of angles at a point, angles at a point on a straight line, vertically opposite angles understand and use alternate and corresponding angles on parallel lines Derive and use the sum of angles in a triangle (e.g. to deduce and use the angle sum in any polygon, and to derive properties of regular polygons)

Multiplicative reasoning Convert between metric speed measures. Calculate average speed, distance and time. Graphs-linear graphs Draw graphs to represent relationships. Find the equations of straight-line graphs and from parallel and perpendicular lines. Draw and interpret graphs from real data. Use and draw distance–time graphs to solve problems and interpret a range of graphs. Perimeter, area and volume. calculate perimeters of 2D shapes, including circles calculate areas of circles and composite shapes know and apply formulae to calculate volume of right prisms (including cylinders)

Quadratic equations and graphs Recognise, simplify and factorise quadratic equations and function. Solve quadratic equations using a graph and algebraically. Right angles triangles: trigonometry and Pythagoras. Understand, recall and use Pythagoras theorem in 2D shapes. Understand, use and recall the trigonometric ratios sine, cosine and tan and apply them to find angles and lengths. Know the exact values of Sin, Cos and tan 0, 30, 45, 60 and 90. Probability Calculate simple probabilities and Understand mutually exclusive and exhaustive outcomes. Use two-way tables and sample space diagrams to record the outcomes from two events. Find and interpret probabilities based on experimental data. Use Venn diagrams, frequency tables and tree diagrams to work out probabilities. .


graphs and lines of best fit.

ADAPTED PEARSON ASSESSMENT



FULL GCSE EDEXCEL PAPER

Year 11 foundation.

Fertile Question

Content Transformations

Indices and standard form

Congruence, similarity and vectors

Algebra Extension

Revision Grid. Revision Grid.

Concepts Geometry, Number, Algebra Geometry , Algebra Number, Algebra, Geometry, Ratio and Proportion, Data and Probability

Number, Algebra, Geometry, Ratio and Proportion, Data and Probability

Knowledge Transformations Describe and transform a given shape by reflection, rotation, translation and enlargement. Describe and transform 2D shapes using combined rotations, reflections, translations, or enlargements. Indices and standard form To know and use the laws of indices. Convert numbers from standard form into ordinary numbers and vice versa and calculate with standard form. FULL GCSE EDEXCEL PAPER

Congruence, similarity and vectors Use the basic congruence criteria for triangles (SSS, SAS, ASA and RHS); Identify shapes which are similar; including all circles or all regular polygons with equal number of sides; Calculate using column vectors, and represent graphically, the sum of two vectors, the difference of two vectors and a scalar multiple of a vector. Extension Algebra Change the subject of a formula. Identify expressions, equations, formulae and identities and Prove results using algebra Draw and interpret Draw and interpret graphs of cubic functions and reciprocals. Solve and write simultaneous equations algebraically and graphically. FULL GCSE EDEXCEL PAPER

Personalised Revision Grid prepared by class teachers with the intension of filling in GAPs in Exam paper. FULL GCSE EDEXCEL PAPER

Personalised Revision Grid prepared by class teachers with the intension of filling in GAPs in Exam paper. FULL GCSE EDEXCEL PAPER

Year 10 Higher

Fertile Question How do I make a reasonable Estimation?

Content Accuracy, estimating and bounds.

Algebra Review and changing the subject.

Standard form, indices and surds.

Fractions

Sequences

Ratio and proportion

Multiplicative reasoning.

Linear graphs , coordinate geometry and real life graphs

Area, perimeter and volume.

Similarity/ congruence

Further Trigonometry:

Probability

Averages and graphs

Complex quadratic equations.

Graphs of cubic and quadratic.

Concepts Number, Algebra, Algebra, Ratio and proportion Geometry, Algebra, Data and probability Data and probability, Algebra

Knowledge Accuracy, estimating and bounds Round numbers and measures to and appropriate degree of accuracy (d.p or s.f). Use inequality notation to specify error intervals due to truncation or rounding. Check answers using approximation

Sequences Generate terms of a sequence from either term-to –term or a position-to-term rule Recognise and use sequences of arithmetic progressions, Fibonacci, quadratic and geometric

Area, perimeter and volume. Know and apply formulae to calculate area of shapes and composite shapes surface area and volume of spheres, pyramids, cones, hemispheres, frustum and composite solids.

Averages and graphs Understand limitations of sampling. Interpret and construct table, charts and diagrams, including frequency tables, bar charts, pie charts, tables interpret histograms with equal and unequal


and estimation. Apply and interpret limits of accuracy, including upper and lower bounds and calculate the bounds of an expressions involving the four operations. Basic Algebra Review basics, solving equations, rearranging and solving equations. Linear simultaneous equations, factoring and expanding quadratic equations- simple. Simplifying and expanding quadratic brackets. More complex changing the subject Standard form, indices and surds Calculate with roots and with integers and fractional indices. Calculate with and interpret standard form A × 10n simplify surd expressions involving squares including expanding brackets and rationalise denominators Fractions. Apply four operations to fractions (proper and improper, mixed number) Change recurring decimals into fractions and vice versa. simplify and manipulate algebraic expressions involving algebraic fractions


sequences. Ratio and proportion. Use ratio notation, divide a given quantity into two parts, expression the division of a quantity into parts of a ratio, apply ratio to real contexts. Solve problem involving direct proportion; scales, recipes, currencies. Multiplicative reasoning: Solve problems involving direct and inverse proportion including graphical and algebraic representations Use and convert compound units such as speed, rates of pay, unit pricing, density and pressure. Use kinematics formulae to calculate speed and acceleration. Linear graphs ,coordinate geometry and real life graphs Find the gradient and y-intercept from a linear equation and calculate the equation of a line through 2 points, parallel and perpendicular to a given line. Draw and interpret distance–time graphs and real life linear graphs and understand velocity–time graphs.


Calculate the area of circles, areas of sectors of circles and arc lengths and sectors of circles. Similarity/ congruent Shapes. Use the basic congruence criteria for triangles. (SSS, SAS, RHS, ASA) Apply the concepts of congruence and similarity, including the relationships between lengths, area and volumes in similar figure. Trigonometry Understand, recall and use Pythagoras theorem in 2D and 3D shapes. Understand, use and recall the trigonometric ratios sine, cosine and tan and apply them to find angles and lengths. Know the exact values of Sin, Cos and tan 0, 30, 45, 60 and 90. Know and apply the sine rule and cosine rule to find unknown lengths and angles and trigonometric area to calculate the area, sides and angles of any triangle. Probability Review of frequency trees, theoretical probability, scales, expected frequency and independent and dependent events. Calculate and interpret conditional probabilities through representation using expected frequencies with two-way tables, tree diagrams and Venn diagrams.


class interval and line graphs for time series data and know their appropriate use. Calculate the mean, mode, median and range from a frequency table and from appropriate diagrams. Interpret and produce box plots to find quartiles, interquartile ranges and identify any outliers. Given two sets of data in a table, model the relationship and make predictions using scatter graphs and lines of best fit. Complex quadratic equations. simplify and manipulate algebraic expressions (including those involving surds and algebraic fractions) by expanding products of two or more binomials solve quadratic equations by completing the square and by using the quadratic formula solve quadratic inequalities in one variable solve two simultaneous equations in two variables where one is quadratic algebraically find approximate solutions to equations numerically using iteration Graphs of cubic, quadratic- identify turning points. Sketch quadratic and cubic functions. Know where a graph will cross the x-axis Understand maximum and minimum points. Find roots of an equation by completing the square and using the quadratic formula. FULL GCSE EDEXCEL PAPER

Year 11 Higher

Fertile Question

Content Transformations

Functions

Circle theorem

Vectors and geometric proof

Proportion and graphs

Revision Grid Revision Grid

Concepts Geometry, Algebra Geometry, Algebra, Ratio and Proportion Number, Algebra, Geometry, Ratio and Proportion, Data and Probability

Number, Algebra, Geometry, Ratio and Proportion, Data and Probability

Knowledge Functions and transformation of functions. Use function notation and find composite function and inverses. Interpret and analyse transformations of graphs of cubic, quadratic and trigonometric functions and write the functions algebraically Transformations. Describe and transform a given shape by reflection, rotation, translation and enlargement (fractional and negative scale) Draw and use scales on maps and scale drawings. Solve problems involving bearings. Construct triangles, bisect angles and construct the perpendicular bisector of a line.

Vectors and geometric proof Add and subtract vectors algebraically and use column vectors. Solve geometric problems and produce proofs. Proportion and graphs plot and interpret graphs (including exponential graphs and trigonometric function) and graphs of nonstandard functions in real contexts, to find approximate solutions to problems such as simple kinematic problems involving distance, speed and acceleration interpret the gradient at a point on a curve as theo0zs instantaneous rate of change calculate or estimate gradients of graphs

Personalised Revision Grid prepared by class teachers with the intension of filling in GAPs in Exam paper.

Personalised Revision Grid prepared by class teachers with the intension of filling in GAPs in Exam paper.


Use loci to solve problems. Circle theorem recognise and use the equation of a circle with centre at the origin; find the equation of a tangent to a circle at a given point identify and apply circle definitions and properties, including: centre, radius, chord, diameter, circumference, tangent, arc, sector and segment apply and prove the standard circle theorems concerning angles, radii, tangents and chords, and use them to prove related results FULL GCSE EDEXCEL PAPER

and areas under graphs (including quadratic and other non-linear graphs), and interpret results in cases such as distance-time graphs, velocity-time graphs and graphs in financial contexts FULL GCSE EDEXCEL PAPER




End Game

1 2 3 4

Year 13

Fertile Question

Content GCSE A Level Transition Core Maths

Core Maths Statistics



Concepts • Proof • Algebra and Functions • Coordinate Geometry in the (x,y) Plane • Sequences and Series

Core Maths • Trigonometry • Exponentials and Logarithms Statistics • Statistical Modelling • Data Presentation and Interpretation

Core Maths • Differentiation • Integration Statistics • Probability • Statistical Distributions

Core Maths • Numerical Methods • Vectors Statistics • Statistical Hypothesis testing

Knowledge GCSE A Level Transition Pure Mathematics 1 Proof

1.1 Understand and use the structure of mathematical proof, proceeding from given assumptions through a series of logical steps to a conclusion; use methods of proof, including: Examples of proofs: Proof by deduction Proof by exhaustion Disproof by Counter Example

2 Algebra and functions 2.1 Understand and use the laws of indices for all rational exponents. 2.2 Use and manipulate surds, including rationalising the denominator. Students should be able to simplify algebraic 2.3 Work with quadratic functions and their graphs. The notation f(x) may be used The discriminant of a quadratic function, including the conditions for real and repeated roots.

Pure Mathematics 5 Trigonometry

5.1 Understand and use the definitions of sine, cosine and tangent for all arguments; Use of x and y coordinates of points on the unit circle to give cosine and sine respectively, Work with radian measure, including use for arc length and area of sector. 5.2 Understand and use the standard small angle approximations of sine, cosine and tangent 5.3 Understand and use the sine, cosine and tangent functions; their graphs, symmetries and periodicity. Knowledge of graphs of curves with equations such as y = sinx, y = cos(x + 30°), y = tan2x is expected. Know and use exact values of 5.4 Understand and use the definitions of secant, cosecant and cotangent and of arcsin, arccos and arctan; their relationships to sine, cosine and tangent; understanding of their graphs; their ranges and

Pure Mathematics 7 Differentiation

7.1 Understand and use the derivative of f ( x) as the gradient of the tangent to the graph of y = f ( x) at a general point (x, y); the gradient of the tangent as a limit; interpretation as a rate of change Understand and use the second derivative as the rate of change of gradient 7.2 Differentiate xn , for rational values of n, and related constant multiples, sums and differences. Understand and use the derivative of ln x 7.3 Apply differentiation to find gradients, tangents and normal. Use of differentiation to find equations of tangents and normals at specific points on a curve. maxima and minima and stationary points. points on inflection To include applications to curve sketching. Identify where functions are increasing or decreasing. To include applications to curve sketching. 7.4 Differentiate using the product rule, the quotient rule

Pure Mathematics 9 Numerical methods

9.1 Locate roots of f ( x) = 0 by considering changes of sign of f ( x) in an interval of x on which f (x) is sufficiently well behaved. Students should know that sign change is appropriate for continuous functions in a small interval. Understand how change of sign methods can fail. When the interval is too large sign may not change as there may be an even number of roots. If the function is not continuous, sign may change but there may be an asymptote (not a root). 9.2 Solve equations approximately using simple iterative methods; be able to draw associated cobweb and staircase diagrams. Understand that many mathematical problems cannot be solved analytically, but numerical methods permit solution to a required level of accuracy. Use an iteration of the form xn + 1 = f (xn) to find a root of the equation x = f (x) and show understanding of the convergence in geometrical terms by drawing cobweb and


2.4 Solve simultaneous equations in two variables by elimination and by substitution, including one linear and one quadratic equation. 2.5 Solve linear and quadratic inequalities in a single variable and interpret such inequalities graphically,. 2.6 Manipulate polynomials algebraically, including expanding brackets and collecting like terms, factorisation and simple algebraic division; use of the factor theorem. 2.7 Understand and use graphs of functions; sketch curves defined by simple equations including polynomials Graph to include simple cubic and quartic functions, The modulus of a linear function. Interpret algebraic solution of equations graphically; use intersection points of graphs to solve equations. Understand and use proportional relationships and their graphs. 2.8 Understand and use composite functions; inverse functions and their graphs. 2.9 Understand the effect of simple transformations on the graph of y = f(x), including sketching associated graphs: quartics, reciprocal, 2 2.10 Decompose rational functions into partial fractions (denominators not more complicated than squared linear terms and with no more than 3 terms, numerators constant or linear). 2.11 Use of functions in modelling, including consideration of

domains. Angles measured in both degrees and radians. 5.5 Understand and use sin, tan. cos=θ Understand and use sin2θ + cos2θ = 1 sec2θ = 1 + tan2 θ and cosec2θ = 1+ cot2θ These identities may be used to solve trigonometric equations and angles may be in degrees or radians. They may also be used to prove further identities. 5.6 Understand and use double angle formulae; use of formulae for sin (A ± B), cos (A ± B), and tan (A ± B), understand geometrical proofs of these formulae. To include application to half angles. Understand and use expressions for a cos θ + b sinθ in the equivalent forms of r cos (θ ±α ) or r sin (θ ±α ) Students should be able to solve equations such as a cos θ + b sin θ = c in a given interval. 5.7 Solve simple trigonometric equations in a given interval, including quadratic equations in sin, cos and tan and equations involving multiples of the unknown angle. These may be in degrees or radians 5.8 Construct proofs involving trigonometric functions and identities. 5.9 Use trigonometric functions to solve problems in context, including problems involving vectors, kinematics and forces.

6 Exponentials and logarithms 6.1 Know and use the function ax and its graph, where a is positive. Understand the difference in shape between a < 1 and a > 1 Know and use the function

and the chain rule, including problems involving connected rates of change and inverse functions. Differentiation of cosec x, cot x and sec x and differentiation of arcsin x, arcos x, and arctan x are required. Skill will be expected in the differentiation of functions generated from standard forms using products, quotients and composition 7.5 Differentiate simple functions and relations defined implicitly or parametrically, for first derivative only. The finding of equations of tangents and normals to curves given parametrically or implicitly is required. 7.6 Construct simple differential equations in pure mathematics and in context, (contexts may include kinematics, population growth and modelling the relationship between price and demand).

8 Integration 8.1 Know and use the Fundamental Theorem of Calculus Integration as the reverse process of differentiation. Students should know that for indefinite integrals a constant of integration is required. 8.2 Integrate xn (excluding n = −1) and related sums, differences and constant multiples. Given f ′(x) and a point on the curve, Students should be able to find an equation of the curve in the form y = f (x).. 8.3 Evaluate definite integrals; use a definite integral to find the area under a curve and the area between two curves Students will be expected to be able to evaluate the area of a region bounded by a curve and given straight lines, or

staircase diagrams. 9.3 Solve equations using the Newton-Raphson method and other recurrence relations of the form xn+1= g(xn) Understand how such methods can fail. For the Newton-Raphson method, students should understand its working in geometrical terms, so that they understand its failure near to points where the gradient is small. 9.4 Understand and use numerical integration of functions, including the use of the trapezium rule and estimating the approximate area under a curve and limits that it must lie between. 9.5 Use numerical methods to solve problems in context.

10 Vectors 10.1 Use vectors in two dimensions and in three dimensions Students should be familiar with column vectors and with the use of i and j unit vectors in two dimensions and i, j and k unit vectors in three dimensions. 10.2 Calculate the magnitude and direction of a vector and convert between component form and magnitude/direction form. Students should be able to find a unit vector in the direction of a, 10.3 Add vectors diagrammatically and perform the algebraic operations of vector addition and multiplication by scalars, and understand their geometrical interpretations. The triangle and parallelogram laws of addition. Parallel vectors. 10.4 Understand and use position vectors; calculate the distance between two points represented by position vectors. 10.5 Use vectors to solve problems in pure


limitations and refinements of the models. 3

3 Coordinate geometry in the (x,y) plane 3.1 Understand and use the equation of a straight line, including the forms y – y1 = m(x – x1) and ax + by + c = 0; To include the equation of a line through two given points, and the equation of a line parallel (or perpendicular) to a given line through a given point. Gradient conditions for two straight lines to be parallel or perpendicular. m′ = m for parallel lines and m′ = m − 1 for perpendicular lines Be able to use straight line models in a variety of contexts.. 3.2 Understand and use the coordinate geometry of the circle including using the equation of a circle in the form (x – a)2 + (y – b)2 = r2 Students should be able to find the radius and the coordinates of the centre of the circle given the equation of the circle, and vice versa. point. 3.3 Understand and use the parametric equations of curves and conversion between Cartesian and parametric forms.

4 Sequences and series 4.1 Understand and use the binomial expansion of (a + bx)n for positive integer n. 4.2 Work with sequences including those given by a formula for the n th term and those generated by a simple relation of the form xn + 1 = f(xn); increasing sequences;

e x and its graph. 6.2 Know that the gradient of ekx is equal to kekx and hence understand why the exponential model is suitable in many applications. Realise that when the rate of change is proportional to the y value, an exponential model should be used. 6.3 Know and use the definition of loga x as the inverse of a x, where a is positive and x > 0. Know and use the function ln x and its graph. a ≠ 1 Know and use ln x as the inverse function of ex 6.4 Understand and use the laws of logarithms: 6.5 Solve equations of the form a x =b Students may use the change of base formula. Questions may be of the form, e.g. 23x – 1 = 3 6.6 Use logarithmic graphs to estimate parameters in relationships of the form y = axn and y = kbx, given data for x and y 6.7 Understand and use exponential growth and decay; use in modelling ; consideration of limitations and refinements of exponential models.

Statistics 1 Statistical sampling

1.1 Understand and use the terms ‘population’ and ‘sample’. Use samples to make informal inferences about the population. Students will be expected to comment on the advantages and disadvantages

between two curves. 8.4 Understand and use integration as the limit of a sum. 8.5 Carry out simple cases of integration by substitution and integration by parts; understand these methods as the inverse processes of the chain and product rules respectively (Integration by substitution includes finding a suitable substitution and is limited to cases where one substitution will lead to a function which can be integrated; integration by parts includes more than one application of the method but excludes reduction formulae.) 8.6 Integrate using partial fractions that are linear in the denominator. Integration of rational expressions such as those arising from partial fractions, 8.7 Evaluate the analytical solution of simple first order differential equations with separable variables, including finding particular solutions (Separation of variables may require factorisation involving a common factor.) 8.8 Interpret the solution of a differential equation in the context of solving a problem, including identifying limitations of the solution; includes links to kinematics.

Statistics 3 Probability

3.1 Understand and use mutually exclusive and independent events when calculating probabilities. 3.2 Understand and use conditional probability, including the use of tree diagrams, Venn diagrams, two-way tables. 3.3 Modelling with probability,

mathematics and in context, (including forces).

Statistics

5 Statistical hypothesis testing 5.1 Understand and apply the language of statistical hypothesis testing, developed through a binomial model: null hypothesis, alternative hypothesis, significance level, test statistic, 1-tail test, 2-tail test, critical value, critical region, acceptance region, p-value; 5.2 Conduct a statistical hypothesis test for the proportion in the binomial distribution and interpret the results in context. 5.3 Conduct a statistical hypothesis test for the mean of a Normal distribution with known, given or assumed variance and interpret the results in context.


decreasing sequences; periodic sequences. 4.3 Understand and use sigma notation for sums of series. 4.4 Understand and work with arithmetic sequences and series, including the formulae for nth term and the sum to n terms 4.5 Understand and work with geometric sequences and series, including the formulae for the nth term and the sum of a finite geometric series; the sum to infinity of a convergent geometric series, including the use of |r| < 1; modulus notation The proof of the sum formula should be known. 4.6 Use sequences and series in modelling.

associated with a census and a sample. Understand and use sampling techniques, including simple random sampling and opportunity sampling. Students will be expected to be familiar with: simple random sampling, stratified sampling, systematic sampling, quota sampling and opportunity (or convenience) sampling. Select or critique sampling techniques in the context of solving a statistical problem, including understanding that different samples can lead to different conclusions about the population.

2 Data presentation and interpretation

2.1 Interpret diagrams for single-variable data, including understanding that area in a histogram represents frequency. Students should be familiar with histograms, frequency polygons, box and whisker plots (including outliers) and cumulative frequency diagrams. Connect to probability distributions 2.2 Interpret scatter diagrams and regression lines for bivariate data, including recognition of scatter diagrams which include distinct sections of the population Use to make predictions within the range of values of the explanatory variable and the dangers of extrapolation. Derivations will not be required. Variables other than x and y may be used. 2.3 Interpret measures of central tendency and variation, extending to standard deviation.

including critiquing assumptions made and the likely effect of more realistic assumptions

4 Statistical distributions 4.1 Understand and use simple, discrete probability distributions including the binomial distribution, as a model; calculate probabilities using the binomial distribution. 4.2 Understand and use the Normal distribution as a model; find probabilities using the Normal distribution 4.3 Select an appropriate probability distribution for a context, with appropriate reasoning, including recognising when the binomial or Normal model may not be appropriate.

5 Statistical hypothesis testing 5.1 Understand and apply the language of statistical hypothesis testing, developed through a binomial model: null hypothesis, alternative hypothesis, significance level, test statistic, 1-tail test, 2-tail test, critical value, critical region, acceptance region, p-value; 5.2 Conduct a statistical hypothesis test for the proportion in the binomial distribution and interpret the results in context. 5.3 Conduct a statistical hypothesis test for the mean of a Normal distribution with known, given or assumed variance and interpret the


Data may be discrete, continuous, grouped or ungrouped. Understanding and use of coding. Measures of central tendency: mean, median, mode. Measures of variation: variance, standard deviation, range and interpercentile ranges. Use of linear interpolation to calculate percentiles from grouped data is expected. 2.4 Recognise and interpret possible outliers in data sets and statistical diagrams.

results in context.


End Game

1 2 3 4

Year 13

Fertile Question

Content AS-A2 Transition Core 3 Mechanics 1

Core 3 Mechanics 1 Core 4

Mechanics 1 Core 4

Concepts Core 3

Algebraic fractions

Functions

Numerical methods

The exponential and log functions

Transforming graphs of functions Mechanics 1

Kinematics

Moments

Core 3

Further Trigonometry

Trigonometry

Differentiation Core 4

Partial fractions

Vectors Mechanics 1

Dynamics

Core 4

The binomial expansion

Coordinate geometry

Differentiation

Integration Mechanics 1

Statics

Vectors

Knowledge C3

Chapter 1: Algebraic Fractions (completed at the end of Yr 12)

Add, subtract, multiply and divide algebraic fractions, simplifying the answer

Use algebraic long division to simplify improper algebraic fractions

Work with identities containing algebraic fractions - evaluate coefficients

Chapter 2: Functions (completed at the end of Yr12

Understand function notation and substitution into f(x)

Identify the domain and range for a

C3

Chapter 6/7: Trigonometry - Further trigonometric identities and their applications.

Know the definitions and graphs of sec x, cosec x and cot x

Know and prove the identities tan2x + 1 = sec2x and 1 + cot2x = cosec2x

Solve trig equations containing one or more of sec, cosec and cot

Know the definitions and graphs of arcsin x, arccos x and arctan x

Use compound angle formulae [e.g. sin (A + B)] to solve trig equations or to prove trig identities

Prove and use the double angle

M1

Chapter 4: Statics of a particle

Resolve forces into horizontal and vertical components and use with F = ma

Resolve forces into perpendicular and parallel components on a slope and use with F = ma

Use friction = µR in problems involving F = ma in any of the above contexts

Understand tension and thrust and how to represent them on a force diagram

Solve connected particle problems involving pulleys, including particles on slopes

C3

C4

M1


given function, possibly by using a graph sketch

Substitute into or find an expression for fg(x) or gf(x) - known as composite functions

Find an expression for the inverse function f-1(x) of a given function f(x)

Know and use the domain/range match ups between f(x) and f-1(x)

Solve equations involving f(x), g(x), fg(x), f-1(x) etc.

Chapter 3: The exponential and log function (completed at end of Yr 12)

Know that ex differentiated is ex (main definition of ex)

Know and use that if y = ex then x = ln y or if f(x) = ex then f-1(x) = ln x

Know the graphs of ex, e-x and ln x and be able to transform them e.g. y = e3x+2

Solve problems containing exponential or ln equations

Chapter 4: Numerical methods

Know how to show that the equation f(x) = 0 has a root in a given interval by sign change

Use graph sketching to demonstrate the number and location of roots of an equation

formulae - learn them (inc 3 versions for cos 2x)

Prove and use the half angle formulae

Know how to write given trig expressions in the form R sin (x +/- a) or R cos (x +/- a)

Solve trig equations by using any of the above formulae/expressions

Chapter 8: Differentiation.

Know and use dy/dx = 1/ (dx/dy) and dx/dy = 1/(dy/dx)

Know the differentials of sin [f(x)], cos [f(x)] and tan [f(x)]

Know the differentials of sec [f(x)], cosec [f(x)] and cot [f(x)]

Know and use the differentials of ef(x) and ln [f(x)]

Know and use the chain rule (or quick methods) to differentiate a function of a function

Know and use the product rule

Know and use the quotient rule

M1

Chapter 3: Dynamics of a particle moving in a straight line.

Use conservation of momentum with colliding particles

Use conservation of momentum with exploding shells and

Solve connected particle problems involving a car and trailer including on a slope

Chapter 6: Vectors.

Understand that vectors can represent any quantity with magnitude and direction

Calculate the magnitude and direction of a given vector and interpret the magnitude

Understand how to calculate and use unit vectors

Use F = ma and v = u + at as vector equations for 2-D acceleration problems

Use r = r0 + tv to find the position of a particle moving in 2-D at time t

Use ArB = rA - rB to find the position vector of A relative to B

Calculate the closest distance between two moving objects using modulus of ArB

Calculate the time for which two objects are within a certain distance of each other

C4

Chapter 2: Coordinate geometry in the (x,y) plane.

Eliminate the parameter to find an


Rearrange f(x) = 0 into the form x = g(x) to obtain an iterative equation xn+1 = g(xn)

Substitute x1, x2, x3 etc. into an iterative equation to obtain successive approximations

Justify a root of f(x) = 0 to a given degree of accuracy by substituting upper and lower bounds for a sign change

Chapter 5: Transforming graphs of functions

Sketch the graph of y = |f(x)| for a given f(x)

Sketch the graph of y = |f(x)| or y = f(|x|) from the graph of y = f(x)

Know and apply the graph transformations covered in C2 to functions covered in C3

Sketch two graphs of the form y = |mx + c| and determine the points of intersection

M1

Chapter 1: Mathematical models in mechanics.

Chapter 2: Kinematics of a particle moving in a straight line.

Know the significance of modelling assumptions, and how they affect the calculations in a problem

bullets/guns

Use conservation of momentum with exploding shells and bullets/guns

Use conservation of momentum with jerk in a string on connected particles

Use IMPULSE = CHANGE IN MOMENTUM

C4

Chapter 5: Vectors

Find the vector for a line segment between two points and find its modulus

Find the vector equation for a straight line through two given points

Find the vector equation for a straight line through a given point and parallel to a given line

Determine whether two lines intersect and find the point if they do intersect

Use scalar product to find the angle between the directions of two vectors

Use scalar product to find the angle between two lines given in vector form

Calculate the angles, lengths and

equation between x and y

Use the chain rule to find dy/dx and then find a tangent, normal or stationary point

Use ∫ y dx = ∫ y (dx/dt) dt to find the area between a parametric curve and the x axis

Use π∫ y2 dx = π∫ y2 (dx/dt) dt to find the volume of revolution for a parametric curve

Chapter 3: The binomial expansion.

Understand how to use the expansion (1 + x)n for negative and rational values of n

Use |x| < 1 to identify range of validity for a given expansion e.g. (1 + 3/2x)-2 then |3/2x| < 1

Expand expressions such as (3-2x)1/2 using (1 + x)n appropriately

Use given information to find p and n for (1 + px)n or (a + px)n

Expand expressions such as (1 - 2x)(1 + 3x)-1/3

Use partial fractions and then expand appropriately

Identify and substitute a small value of x into an expansion to approximate a value

Chapter 4:Differentiation

Review differentiation rules and


Know and use v = u + at

Know and use v2 = u2 + 2as

Know and use s = ut + 1/2 at2

Know and use s = vt - 1/2 at2

Know and use s = (u + v)t/2

Apply the above equations to vertical motion under gravity using g = 9.8ms-2

Sketch velocity/time graphs from given information

Use the gradient of a section of a velocity/time graph to calculate acceleration

Use the area under a velocity/time graph to calculate/equate to total distance

Chapter 5: Moments

Use Fd and Fd sin θ to calculate the moment of a force about a point

Understand that clockwise = anticlockwise in equilibrium situations

Calculate with moments when forces are given as vectors and points as co-ordinates

Solve balance problems for uniform rods

Solve balance problems for non-uniform rods

Solve balance problems when on

area for a triangle made with three given points

Find the perpendicular distance from a point to a line

Chapter 1: Partial Fractions

Know how to split into partial fractions involving up to three linear denominators

Know how to split into partial fractions when one denominator has a repeated factor

Use algebraic long division with improper algebraic fractions and then find partial fractions

techniques from C1, C2 and C3

Differentiate a pair of parametric equations in order to find dy/dx

Obtain dy/dx for an implicit equation e.g. y3 - 3xy2 + 5xy - 2x2 = 50

Differentiate functions involving ax

Chapter 6: Integration

Review integration rules and techniques from C1 and C2

Use integrals for the 6 trig functions of the type ∫ sin (ax + b) dx

Use the integrals ∫ eax + b dx and ∫ (ax + b)n dx

Find integrals of the form ∫ f/(x) [f(x)]n dx

Find integrals of the form ∫ f/(x)/f(x) dx

Use partial fractions to set up integrals of the form ∫ f/(x)/f(x) dx

Use the six integrals of the squares of trig functions e.g. ∫ cos2x dx

Use integration by parts NOT involving ln

Integrate by parts involving ln

Use integration by substitution/change of variable

Find integrals of the form ∫sin 5x


the point of tilting about one pivot

cos 3x dx

Evaluate definite integrals for any of the above types

Use A = ∫ y dx to find the area between a curve and the x axis

Use V = π ∫ y2 dx to find the volume of revolution around the x axis

fluent in the fundamentals of mathematics. Reason ... · Mathematics Department – Medium Term...

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