Maths Program of Study KS3,4 & 5 by The Crypt School

Programme of Study 2022-2023 Mathematics - Our Vision Mathematics is essential for everyday life and understanding the world around us. It is not just about equations, numbers and calculations it is about deepening our understanding. At The Crypt we aim to empower our students by looking beyond the curriculum, to link mathematical concepts to practical and real-world examples. As a department we create a generation of problem solvers and logical thinkers that can think creatively to tackle problems. In Key Stage 3 we follow a scheme of work which focuses on a strong mastery approach, and we aim to develop both the mental and written mathematical skills. We aim to give a strong basis to move onto GCSE with topics on number, algebra, ratio and proportion, geometry and statistics. During year 11, some students have the opportunity to study AQA Certificate level 2 Further Maths GCSE and enhance their knowledge even further. Advanced mathematics is growing in popularity and is relevant to many careers. This is reflected by our popularity at KS5 with students taking Mathematics and Further Mathematics with students studying pure maths, statistics and mechanics.

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INTENT: In Year 7 we give students the opportunity to become more fluent and confident in the basic concepts studied at Key Stage 2, while developing their depth of understanding and their reasoning skills. Students apply fundamental concepts to new and challenging contexts such as algebraic manipulation, prime factorisation, Pythagoras, and probability. Students also start to learn basic calculator skills, practicing these skills in units on averages, area, and angles. Year

Term 1 Number

Baseline Assessment Numbers & Calculation Fractions Factors, Multiples & primes Decimals & Estimation

Term 2 Proportion

Term 3

Probability Venn Diagrams Ratio, Decimal & % percentage increase & decrease Ratio and Proportion Problem Solving Investigation Value for Money Non-Calculator assessment

Calculator assessment Angles and Parallel Lines Congruent Construction & Loci Area & Perimeter Circles Cuboids

Shape

Term 4 Sequences and Graphs

Term 5 Finding the Unknown

Term 6 Statistics & Transformations

Simplifying expressions Single Brackets Factorising Sequences Coordinates Plotting graphs

Solving Equations & Inequalities Pythagoras Problem Solving Investigation

Averages Statistical Diagrams Transformations Note these topics will be assessed in year 8

End of Year Non-Calculator assessment Calculator assessment

Year 7 End Points ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓

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Use the four operations with decimals, fractions and mixed numbers Round a number to a significant figure or decimal place and use it to estimate Use a sample space diagram Draw a Venn diagram and know what is meant by union and intersect Use non-calculator methods to find percentage of a number and increase and decrease by a percentage Use angle rules for around a point, straight line, in a triangle, in a quadrilateral, corresponding, alternate, co-interior and vertically opposite angles Find areas and perimeters of rectangles, triangles, parallelograms, and trapeziums. Find the circumference and area of a circle Find the nth term of a linear sequence Understand and use lines parallel to the axes, y=x and y=-x Simplify expressions by collecting like terms and expanding a single bracket Solve linear equation with unknowns on both sides Use Pythagoras to find a long or short side of a triangle Know and calculate mean, median, mode and range

INTENT: In Year 8 students will continue to deepen their understanding of topics taught in Year 7 and at primary school. For example, their work on algebraic expressions will be extended to include index laws, expanding double brackets and further their understanding of percentages by learning methods involving decimal multipliers and continue to work with the fundamental concepts of ratio and proportion that were first introduced at primary school. They will be introduced to new and exciting mathematical concepts which draw on previously taught skills and knowledge. For example, trigonometry, bounds, solving equations involving fractions and negative enlargement. These topics give students the opportunity to develop their critical thinking skills and their ability to answer more complex problem-solving questions. Students will also extend their knowledge of number systems and effective calculator use will feature heavily throughout the year. Year

Term 1 Number Prime factors, HCF & LCM Estimation and errors Recurring decimals Rules of indices Standard form Problem Solving Investigation

Term 2 Algebraic Thinking

Term 3

Expanding single and double brackets Identities Solving equations with unknowns on both sides Solving inequalities Changing the subject

Parallel Lines Angles in Polygons Congruency

Probability Non-Calculator assessment (will contain year 7 content)

Sample space diagrams Venn Diagrams and Tree Diagrams

Shape

Rotation Reflection Negative Enlargement Describing Transformations

Term 4 Proportion

Term 5 Investigating Data

Term 6 Mensuration

Introduction to Trigonometry (SOHCAHTOA) Percentage Multipliers

Simplifying Ratio

EOY Exams Non-Calculator & Calculator assessment

Non-Calc reverse percentages Compound Interest

Pythagoras

Calculator assessment

Year 8 End Points ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ 3 | Page

Find the HCF and LCM using product of primes and a Venn diagram Use the laws of indices for multiplication, division, and power to a power Understand and convert standard form Expand two brackets using FOIL Create a solve equations and inequalities which includes division Change the subject of a formula when two steps are required Draw and use a simple probability tree diagram Find a probability from a Venn diagram Describe and draw the four transformations Use Pythagoras with multistep problems Use SOHCAHTOA to find a length or angle Calculate mean from a frequency table Draw and use a line of best fit on a scatter graph and describe correlation Use and understand y=mx+c

Collecting Data Averages from frequency tables Pie Charts Scatter Graphs Cumulative Frequency

Area and perimeter Circles Volume & Surface Area of Prisms

Graphical Algebra Nth Term rule Geometric Sequences

Gradients Y=mx+c

INTENT: In year 9 we start the GCSE course, in addition to revisiting knowledge learnt throughout Years 7 and 8. We can further deepen understanding and give students the opportunity to develop their reasoning and problem-solving skills. We continue to develop trigonometry, probability and number skills we also introduce quadratics, bearings and surds. These topics will challenge students and develop resilience and confidence before moving into Year 10. In the units on angles and bearings, quadratics, volume and surface area students start to develop chains of reasoning, preparing them for the complex geometrical proofs they will construct at Key Stage 4 and beyond. Students will also be formally introduced to GCSE exam style questions and have the opportunity to develop exam technique. Year

Term 1 Number Product of prime factors, HCF & LCM Estimation Error Intervals Bounds Rules of Indices including fractional and negative Standard form Surds

Term 2 Manipulating Algebra Single and double bracket factorising Simplifying algebraic fractions Simultaneous equations Arithmetic Nth term rule Quadratic nth term rule

Term 3 Shape Circles, arcs and sectors Cones and Spheres Compound Units Bearings Calc Exam

Term 4 Proportion Problem Solving with ratio Ratio with algebra Direct & Inverse Proportion Pythagoras & SOHCAHTOA problems

Non-Calc Exam (contains year 8 content)

Year 9 End Points ✓ Understand and simplify surds ✓ Write and error interval and calculate with upper and lower bounds ✓ Calculate in standard form ✓ Factorise a quadratic with a=1 including a difference of two squares ✓ Solve a simultaneous equation algebraically ✓ Problem solve with volume and surface areas. ✓ Understand bearings and use it to calculate a return journey ✓ Use Pythagoras and SOHCAHTOA in multistep problems and real-life circumstances ✓ Draw, use and interpret cumulative frequency diagrams and box plots ✓ Find conditional probabilities and use set notation with Venn diagrams ✓ Use and interpret tree diagrams with multiple routes

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Term 5 Investigating Data Compound interest Frequency tables Cumulative Frequency Quartiles Box Plots Scatter Graphs

Term 6 Probability Sample space diagrams Two-way tables Set notation Venn diagrams Tree diagrams EOY Exams Non Calc and Calc Exams

INTENT: In year 10, students will get the opportunity to really see the interconnected nature of Maths and they will learn to apply the subject knowledge they have already acquired in more complex and sophisticated ways. Students will experience questions that go beyond routine and repetition so they will be required to think about what skills or concepts need to be applied in different contexts. Our aim in year 10 is to help students become more confident and resilient problem solvers by encouraging them to try out different methods to see what works and what does not. We aim for students to start to appreciate the actual process behind reaching an answer. Students will be exposed to more GCSE exam style questions and will have the opportunity to develop their exam techniques including common errors and misconceptions, layout, and workings, checking answers, mastering using a calculator, command words and how to tackle wordy questions. Some students will be aiming to study AQA Level 2 Further Maths, those students will be following a revised programme of study. Year 10 GCSE Programme of Study

10 GCSE Mathematics

Year

Term 1 Number and Equations

Term 2 Equations and Graphs

Term 3 Statistics and Trigonometry

Term 4 Proportion & Transformations

Term 5 Angles & Shapes

Term 6 Probability

Indices Surds Recurring decimals

Quadratic formula Creating and solving equations Y=mx+c Parallel lines Perpendicular lines Recognising graphs Real life graphs Velocity time graphs

Averages Cumulative Frequency Box Plots Histograms SOHCAHTOA Pythagoras Exact trigonometric values 3D Trigonometry Sine Rule Cosine Rule

Direct and inverse proportion Construction & Loci Transformations Rotation Reflection Translation Enlargement including negative scale factors Function notation Graph Transformations

Angles in polygons Congruency and Similarity

Product rule for counting Set notation Venn diagrams Tree diagrams Conditional probability

Solving linear equations Solving inequalities Simultaneous equations Solving quadratics by factorising

Non-calc and Calc assessment

Year 10 GCSE End Points ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ 5 | Page

EOY Exams Non-Calc and Calc

Vectors Drawing and understanding vectors Adding, subtracting & scalar multiples Vector Geometry Parallel vectors Co-linear

✓ ✓ ✓ ✓ ✓

Appy graph transformations using function notation Prove two triangles are congruent and apply similarity to length, area and volume Use product rule for counting “And” and “Or” rules for probability Understand vectors and use them for geometric proof

10 GCSE Mathematics & AQA Level 2 Further Maths

Year 10 GCSE and Level 2 Further Maths Programme of Study Year Term 1 Term 2 Number and Equations Equations and Graphs Indices Surds Recurring decimals Upper and lower bounds Solving linear equations Solving inequalities Simultaneous equations Solving quadratics by factorising Quadratic formula Creating and solving equations

Term 4 Proportion & Transformations

Term 5 Angles & Shapes

Term 6 Probability

SOHCAHTOA Pythagoras Exact trigonometric values 3D Trigonometry Sine Rule Cosine Rule Direct and inverse proportion

Construction & Loci Transformations Rotation Reflection Translation Enlargement including negative scale factors Function notation Graph Transformations Angles in polygons

Congruency and Similarity Circle Theorems

Product rule for counting Set notation Venn diagrams Tree diagrams Conditional probability

Non-calc and Calc assessment

Year 10 GCSE End Points ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓

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Y=mx+c Parallel lines Perpendicular lines Recognising graphs Real life graphs Velocity time graphs Averages Cumulative Frequency Box Plots Histograms

Term 3 Statistics and Trigonometry

Rationalise a denominator Negative and fractional indices Algebraically prove any recurring decimal is a fraction Solve quadratics through factorising and quadratic formula Linear graphs, y=mx+c, parallel and perpendicular lines Draw and interpret histograms Gradients and areas underneath velocity time graphs Sine and cosine rule to find a missing angle and length, use 1/2absinC for area of a triangle Appy graph transformations using function notation Know and use circle theorems Prove two triangles are congruent and apply similarity to length, area and volume Use product rule for counting “And” and “Or” rules for probability Understand vectors and use them for geometric proof

EOY Exams Non-Calc and Calc

Vectors Drawing and understanding vectors Adding, subtracting & scalar multiples Vector Geometry Parallel vectors Co-linear

INTENT: Year 11, students will learn the remaining content required to be successful in GCSE Mathematics as well have the opportunity to fully consolidate their learning throughout key stage 3 and year 10. Students will become confident in interpreting and communicating mathematical information in a variety of forms appropriate to the information and context. They will also concentrate on acquiring the mathematical skills required to select and apply techniques to solve problems. There will be an emphasis on revision and retrieval of content as well as exam question practice to further develop their exam techniques. From the student’s year 10 exams some students will also be studying the enrichment qualification Level 2 Further Mathematics

11 GCSE Mathematics

Year

Term 1 Further Algebra

Term 2 Circles and Proof

Term 3 Fractions, Graphs, Sequences and Surds

Term 4

Expanding 3 brackets Completing the Square Sketching Graphs Quadratic Inequalities Quadratic & Linear Simultaneous Functions

Circle Theorems Proof Iteration Cones and Spheres

Algebraic fractions

Assessment Week

Y=mx+c Equation of a circle Tangent of a circle Quadratic sequences

Mock Exams

Complex Surds

Year 11 GCSE End Points ✓ Use identities and comparing coefficients to find missing values ✓ Completing the square ✓ Quadratics ✓ Functions ✓ Know and use circle theorems ✓ Simplify and calculate with algebraic fractions ✓ Algebraic proof ✓ Understand and use iteration

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Term 5 Revision

Term 6 Revision

GCSE EXAMS

11 GCSE Mathematics & Level 2 Further Mathematics

Year

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Term 1 Further Algebra

Term 2 Proof and Geometry

Term 3 Further Maths

Term 4 Further Maths

Term 5

Expanding 3 brackets Completing the Square Sketching Graphs Quadratic Inequalities Quadratic & Linear Simultaneous Functions Proof Algebraic fractions

Equation of a circle Tangent of a circle

Factor theorem Factorising cubics Completing the square Simultaneous equations w/3 unknowns Functions Matrices

Sine & Cosine rule Trigonometry graphs Trigonometric identities CAST diagrams

Expanding 3 brackets Completing the Square Sketching Graphs Quadratic Inequalities Quadratic & Linear Simultaneous Functions Proof Algebraic fractions

Iteration

Further Maths Differentiation Differentiation Tangents/Normals Stationary Points Binomial Expansion Mock Exams

Year 11 GCSE & Level 2 Further Mathematics End Points ✓ Use identities and comparing coefficients to find missing values ✓ Completing the square ✓ Quadratics ✓ Functions ✓ Know and use circle theorems ✓ Simplify and calculate with algebraic fractions ✓ Algebraic proof ✓ Understand and use iteration

Sequences Equations of circles

✓ ✓ ✓ ✓ ✓

Revision

Term 6 GCSE EXAMS Equation of a circle Tangent of a circle Iteration

Find the equation of a tangent to a circle Differentiate and find stationary points Use a CAST diagram and solve trigonometric equations 𝑠𝑖𝑛 𝑥 Use the identities 𝑠𝑖𝑛2 𝑥 + 𝑐𝑜𝑠 2 𝑥 = 1 and 𝑡𝑎𝑛 𝑥 = 𝑐𝑜𝑠 𝑥 Use the factor theorem and algebraic long division to factorise cubics ✓ Binomial expand (𝑎𝑥 + 𝑏)𝑛 ✓ Solve simultaneous equations with 3 variables ✓ Calculate with matrices

A level Mathematics INTENT: One of the requirements of the new A-level specification is to test the content synoptically and for students to apply the knowledge they have in unfamiliar areas. Students will be aiming to ‘draw together information from different areas of the specification’ and ‘apply their knowledge and understanding in practical and theoretical contexts’. Use of calculators in exams is more important now and students will learn to use the modes and abilities of their calculator efficiently. The course consists of material that covers both Pure and Applied Mathematics. Two thirds of the course is focused on Pure Mathematics covering subjects such as advanced algebra and calculus. One third is split evenly between statistics and mechanics. There are two versions of the program of study depending on whether the lead teacher is a statistics or mechanics specialist.

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Year

Term 1

Term 3

PURE MATHS

TEACHER 1 Quadratics ● Quadratic Functions ● Simultaneous Equations ● Inequalities

TEACHER 1 Trigonometry ● Sine and Cosine Rule ● Trigonometric graphs ● Solving equations through CAST diagrams ● Trigonometric Identities

TEACHER 1 Calculus - Differentiation ● Maxima and minima ● Optimisation Problems

LEAD TEACHER 1 (6 Lessons per fortnight Pure & Statistics) TEACHER 2 (3 lesson per fortnight Pure & Mechanics)

Term 2

Calculus – Differentiation ● From first principles ● Differentiation ● Liebnitz notation ● Linear coordinate geometry ● Tangents and Normals

TEACHER 2 Factor Theorem and Cubics ● Factor theorem ● Algebraic Long division ● Solving Cubics

STATISTICS T1 Logarithms and Exponentials ● Laws of logarithms ● Solving logarithmic equations ● Solving exponential equations

TEACHER 2 Indices and Surds ● Index laws ● Manipulating surds ● Rationalising a denominator

Curve Sketching ● Sketching Cubic, quartic and reciprocal grams ● Curve sketching graph transformations

Sampling and Definitions ● Sampling Methods ● Key definitions ● Large Data Set

Term 4

PURE MATHS TEACHER 1 Proof ● By exhaustion ● Algebraic ● Disprove by counter example

Term 5

PURE MATHS

TEACHER 2 Algebraic Fractions (A level) ● Simplifying ● Four operations

TEACHER 1 Partial Fractions (A level) ● Standard partial fractions ● Repeated roots ● Improper fractions

Binomial Expansion ● With n as an integer

Calculus – Integration ● Introduction to integration ● Reverse differentiation ● Areas under graphs

Coordinate Geometry ● Equation of a circle ● Coordinate geometry problems

Describing Data ● Histogram ● Means and Standard deviations ● Comparing Distributions ● Bivariate Data

Term 6

Iteration ● Location of roots ● Spider and staircase diagrams

Logarithms and Exponentials 3 ● e*x and ln x ● Real life exponentials ● Logarithmic Data

TEACHER 2 Trapezium Rule (A level)

STATISTICS T1 Probability ● Notation ● Tree diagrams ● Venn diagrams ● Independence and Mutually exclusive

MECHANICS T2

Forces and Units ● Standard units and basic dimensions ● Force units and balanced forces ● Resultant forces

Vectors ● Notation ● Magnitude ● Unit vectors ● Angles with an axis

MECHANICS T2 Variable Acceleration ● Calculating displacement, velocity and acceleration using calculus

Discrete Probability ● Binomial Distribution

Hypothesis Testing ● Binomial hypothesis testing p-value and critical regions

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Midterm Exam

Comprehend and critique mathematical arguments, proofs and justifications of methods and formulae, including those relating to applications of mathematics. Recognise the underlying mathematical structure in a situation and simplify and abstract appropriately to enable problems to be solved.

Evaluate, including by making reasoned estimates, the accuracy or limitations of solutions. Understand the concept of a mathematical problem solving cycle, including specifying the problem, collecting information, processing and representing information and interpreting results, which may identify the need to repeat the cycle Understand, interpret and extract information from diagrams and construct mathematical diagrams to solve problems, including in mechanics.

MECHANICS T2 Connected Particles ● Lifts ● Pulleys

Connected Particles ● Cars pulling trailers

Understand that a mathematical model can be refined by considering its outputs and simplifying assumptions; evaluate whether the model is appropriate.

Constant Acceleration ● SUVAT Proof ● SUVAT equations

Initial Assessment

Understand and use language and symbols associated with set theory, as set out in the appendices. Apply to solutions of inequalities and probability.

Interpret and communicate solutions in the context of the original problem.

Dynamics ● Use of F=ma Motion in a straight line ● Displacement time graphs ● Velocity time graphs

Understand and use mathematical language and syntax as set out in the content.

Construct extended arguments to solve problems presented in an unstructured form, including problems in context.

STATISTICS T1

Discrete Probability Discrete random variables

MECHANICS T2

Year End Points Construct and present mathematical arguments through appropriate use of diagrams; sketching graphs; logical deduction and precise statements

Understand and use modelling assumptions

Year 12 Pure and Applied

MOCK EXAMS

Statistics and Mechanics Exam

Year

Term 1

Term 3

PURE MATHS

TEACHER 1 Quadratics ● Quadratic Functions ● Simultaneous Equations ● Inequalities

TEACHER 1 Trigonometry ● Sine and Cosine Rule ● Trigonometric graphs ● Solving equations through CAST diagrams ● Trigonometric Identities

TEACHER 1 Calculus - Differentiation ● Maxima and minima ● Optimisation Problems

LEAD TEACHER 1 (6 Lessons per fortnight Pure & Mechanics) TEACHER 2 (3 lesson per fortnight Pure & Statistics)

Term 2

Calculus – Differentiation ● From first principles ● Differentiation ● Liebnitz notation ● Linear coordinate geometry ● Tangents and Normals

Logarithms and Exponentials ● Laws of logarithms ● Solving logarithmic equations ● Solving exponential equations

TEACHER 2 Indices and Surds ● Index laws ● Manipulating surds ● Rationalising a denominator

Curve Sketching ● Sketching Cubic, quartic and reciprocal grams ● Curve sketching graph

TEACHER 2 Factor Theorem and Cubics ● Factor theorem ● Algebraic Long division ● Solving Cubics

MECHANICS T1 Forces and Units ● Standard units and basic dimensions ● Force units and balanced forces ● Resultant forces

Term 4

PURE MATHS TEACHER 1 Proof ● By exhaustion ● Algebraic ● Disprove by counter example

Term 5

PURE MATHS TEACHER 1 Functions (A level) ● Function notation ● Range and domains ● Composite functions ● Inverse Functions

Binomial Expansion ● With n as an integer Logarithms and Exponentials 3 ● e*x and ln x ● Real life exponentials Calculus – Integration ● Introduction to integration ● Reverse differentiation ● Areas under graphs

Coordinate Geometry ● Equation of a circle ● Coordinate geometry problems

Logarithmic Data

TEACHER 2 Algebraic Fractions (A level) ● Simplifying ● Four operations

Term 6

PURE MATHS TEACHER 1 Binomial Expansion (A level) ● n is negative or fractional

Partial Fractions (A level) ● Standard partial fractions ● Repeated roots ● Improper fractions

TEACHER 2 ● Trapezium Rule (A level)

MECHANICS T1 Vectors ● Notation ● Magnitude ● Unit vectors ● Angles with an axis

Variable Acceleration ● Calculating displacement, velocity and acceleration using calculus

Sampling and Definitions ● Sampling Methods ● Key definitions ● Large Data Set

Probability ● Notation ● Tree diagrams ● Venn diagrams ● Independence and Mutually exclusive

STATISTICS T2

Discrete Probability ● Discrete random variables ● Binomial Distribution

Hypothesis Testing ● Binomial hypothesis testing p-value and critical regions

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Midterm Exam

Recognise the underlying mathematical structure in a situation and simplify and abstract appropriately to enable problems to be solved.

Understand the concept of a mathematical problem solving cycle, including specifying the problem, collecting information, processing and representing information and interpreting results, which may identify the need to repeat the cycle Understand, interpret and extract information from diagrams and construct mathematical diagrams to solve problems, including in mechanics. Translate a situation in context into a mathematical model, making simplifying assumptions. Use a mathematical model with suitable inputs to engage with and explore situations (for a given model or a model constructed or selected by the student). Interpret the outputs of a mathematical model in the context of the original situation (for a given model or a model constructed or selected by the student).

Describing Data ● Histogram ● Means and Standard deviations ● Comparing Distributions ● Bivariate Data

Initial Assessment

Comprehend and critique mathematical arguments, proofs and justifications of methods and formulae, including those relating to applications of mathematics.

Evaluate, including by making reasoned estimates, the accuracy or limitations of solutions.

Connected Particles ● Cars pulling trailers ● Lifts ● Pulleys

STATISTICS T2

Understand and use language and symbols associated with set theory, as set out in the appendices. Apply to solutions of inequalities and probability.

Interpret and communicate solutions in the context of the original problem.

Constant Acceleration ● SUVAT Proof ● SUVAT equations

STATISTICS T2

Understand and use mathematical language and syntax as set out in the content.

Construct extended arguments to solve problems presented in an unstructured form, including problems in context.

MECHANICS T1

Dynamics ● Use of F=ma Motion in a straight line ● Displacement time graphs ● Velocity time graphs

Year End Points Construct and present mathematical arguments through appropriate use of diagrams; sketching graphs; logical deduction and precise statements

Understand that a mathematical model can be refined by considering its outputs and simplifying assumptions; evaluate whether the model is appropriate. Understand and use modelling assumptions

Year 12 Pure and Applied

MOCK EXAMS

Statistics and Mechanics Exam

Year

LEAD TEACHER 1 (6 Lessons per fortnight Pure & Statistics) TEACHER 2 (3 lesson per fortnight Pure & Mechanics)

Term 1

PURE MATHS

Term 3

PURE MATHS

Term 4

PURE MATHS

TEACHER 1 Trigonometry

TEACHER 1 Calculus - Differentiation

TEACHER 1 Calculus - Integration

TEACHER 1 Parametric Equations 2

● Introduction to radians ● Arc length and area ● Reciprocal trigonometric functions ● Inverse trigonometric functions ● Identities ● R Formula ● Small angle approximations ● Differentiation of trig from first principles

● Chain, product and quotient ● Derivations of inverses ● Shapes of functions

● Area between two curves ● Integration by cover up ● Rational functions ● Partial fractions ● Trigonometric identities ● By parts ● Substitution ● Standard Results

● Differentiating parametrics ● Integrating parametrics

TEACHER 2 Sequences and Series ● Recurrance relationships ● Arithmetic Sequences ● Geometric Sequences

Functions ● Definition, domains and ranges ● Inverse function ● Compound functions

TEACHER 2 Binomial Expansion ● with fractional and negative powers

Modulus

Conditional Probability ● Applied to tree diagrams and Venn diagrams

Normal Distribution ● Finding probabilities ● Working backwards ● Z values and finding mean and standard deviation

Variable Acceleration ● Calculating displacement, velocity and acceleration using calculus

PURE MATHS

Understand and use language and symbols associated with set theory, as set out in the appendices. Apply to solutions of inequalities and probability

Solving Differential Equations ● Connected rates of change ● Separation of variables

Understand and use the definition of a function; domain and range of functions. Comprehend and critique mathematical arguments, proofs and justifications of methods and formulae, including those relating to applications of mathematics.

MECHANICS T2 Vectors in 3D

Recognise the underlying mathematical structure in a situation and simplify and abstract appropriately to enable problems to be solved. Construct extended arguments to solve problems presented in an unstructured form, including problems in context. Interpret and communicate solutions in the context of the original problem.

TEACHER 2 Calculus - Differentiation ● Implicit differentiation ● Normals, tangents and turning points

Proof ●

Proof by contradiction

Numerical Methods ● Iteration ● Newton Rapheson ● Trapezium Rule

Year End Points Construct and present mathematical arguments through appropriate use of diagrams; sketching graphs; logical deduction; precise statements involving correct use of symbols and connecting language Understand and use mathematical language and syntax as set out in the content.

● The graphs of parametric equations ● Parametric to cartesian

STATISTICS T1

Moments ● Multiple pivots and suspensions

Term 5

Parametric Equations 1

● Graphs ● Solving equations

MECHANICS T2

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Term 2

STATISTICS T1

Approximating Distributions ● Approximating Binomial with a normal distribution ● Normal distribution hypothesis testing

Hypothesis Testing ● Normal hypothesis testing p-value ● Correlation hypothesis testing p-value

Understand that many mathematical problems cannot be solved analytically, but numerical methods permit solution to a required level of accuracy. Evaluate, including by making reasoned estimates, the accuracy or limitations of solutions, including those obtained using numerical methods. Understand the concept of a mathematical problem solving cycle, including specifying the problem, collecting information, processing and representing information and interpreting results, which may identify the need to repeat the cycle Understand, interpret and extract information from diagrams and construct mathematical diagrams to solve problems, including in mechanics.

MECHANICS T2 Projectiles ● From ground level ● From a height ● At an angl

MECHANICS T2 Statics ● At an angle ● Coefficient of friction

Dynamic ● At an angle ● Coefficient of friction

Year 13 transitional Exam

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Year 13 Assessment Week

Year 13 Mock Exams

Understand that a mathematical model can be refined by considering its outputs and simplifying assumptions; evaluate whether the model is appropriate. Understand and use modelling assumptions

Year

Term 1

PURE MATHS

Term 2

Term 3

Term 4

PURE MATHS

TEACHER 1 Trigonometry

TEACHER 1 Calculus - Differentiation

TEACHER 1 Calculus - Integration

TEACHER 1 Parametric Equations 2

● Chain, product and quotient ● Derivations of inverses ● Shapes of functions

● Area between two curves ● Integration by cover up ● Rational functions ● Partial fractions ● Trigonometric identities ● By parts ● Substitution ● Standard Results

● Differentiating parametric ● Integrating parametric

Functions ● Definition, domains and ranges ● Inverse function ● Compound functions

Term 5

PURE MATHS Revision

Understand and use mathematical language and syntax as set out in the content. Understand and use language and symbols associated with set theory, as set out in the appendices. Apply to solutions of inequalities and probability

Solving Differential Equations ● Connected rates of change ● Separation of variables

LEAD TEACHER 1 (6 Lessons per fortnight Pure & Mechanics) TEACHER 2 (3 lesson per fortnight Pure & Statistics)

Parametric Equations 1 TEACHER 2 Sequences and Series

TEACHER 2 Binomial Expansion

● Recurrance relationships ● Arithmetic Sequences ● Geometric Sequences

● with fractional and negative powers

Modulus ● Graphs ● Solving equations

Moments ● Multiple pivots and suspensions

Variable Acceleration ● Calculating displacement, velocity and acceleration using calculus

TEACHER 2 Calculus Differentiation ● Implicit differentiation ● Normals, tangents and turning points

Mechanics T1 Proof ●

Proof by contradiction

Mechanics T1

Projectiles

Statics

● From ground level ● From a height ● At an angle

● At an angle ● Coefficient of friction

Numerical Methods

Dynamic

● Iteration ● Newton Rapheson ● Trapezium Rule

● At an angle ● Coefficient of friction

Statistics T2

Conditional Probability

Normal Distribution

● Applied to tree diagrams and Venn diagrams

● Finding probabilities ● Working backwards ● Z values and finding mean and standard deviation

Year 13 transitional Exam

Year 13 Assessment Week

Vectors in 3D Statistics T2

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● The graphs of parametric equations ● Parametric to cartesian

Mechanics T1

Statistics T2 Approximating Distributions ● Approximating Binomial with a normal distribution ● Normal distribution hypothesis testing

Year 13 Mock Exams

Statistics T2 Hypothesis Testing ● Normal hypothesis testing p-value Correlation hypothesis testing p-value

Translate a situation in context into a mathematical model, making simplifying assumptions. Use a mathematical model with suitable inputs to engage with and explore situations (for a given model or a model constructed or selected by the student). Interpret the outputs of a mathematical model in the context of the original situation (for a given model or a model constructed or selected by the student). Understand that a mathematical model can be refined by considering its outputs and simplifying assumptions; evaluate whether the model is appropriate. Understand and use modelling assumptions

A level Further Mathematics INTENT: Inspiration and excellence are at the heart of everything we do. Our main aim is to inspire confidence in our students and stimulate their interest in Mathematics, both within the curriculum and the wider world. We hope that this in turn will spark a love of learning and an improved ability to model and solve problems, applying the Mathematical knowledge gained in lessons to a variety of different circumstances. We believe that this will enable our students, irrespective of background, to flourish. Students will be aiming to ‘draw together information from different areas of the specification’ and ‘apply their knowledge and understanding in practical and theoretical contexts’. Use of calculators in exams is more important now and students will learn to use the modes and abilities of their calculator efficiently. The course consists of material that covers both Pure and Applied Mathematics. Two thirds of the course is focused on Pure Mathematics covering subjects such as complex numbers and matrices. One third is split evenly between further statistics and further mechanics.

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Year

Term 1

CORE MATHS

Term 2

CORE MATHS

Term 3

CORE MATHS

Term 4

CORE MATHS

TEACHER 1 Complex Numbers 1

TEACHER 1 Matrices 2

TEACHER 1 Polar Coordinates

TEACHER 1 Vectors

● Arithmetic ● Conjugate ● Quadratics & Roots ● Argand diagram ● Modulus Argument form ● Loci ● Problem Solving

● Determinate ● Inverses ● Singular ● Transformations ● Invariant Points ● Matrices PPQ

● Sketching polar graphs ● Polar to cartesian ● Intersecting lines

● Vector equation of a line ● Cartesian equation ● Intersection and Matrices ● Dot Product ● Angle between vectors ● Distances

Complex Numbers 2 Maclaurin Expansion ● Expansions ● Combining standard results

Matrices 1 ● Introduction ● Arithmetic

TEACHER 2 Summations ● Standard results

● Complex Polynomials

FStats T1 Discrete Random Variables ● Expectation and

Roots of Polynomials ● Sum and product of roots of a quadratic ● Roots of a cubic ● Roots of a quartic ● Relationship between roots

Algebraic Inequalities ● Solving inequalities algebraically

variance ● Coding ● Sums and differences ● Functions

Chi Squared ● Hypothesis testing for

independence on a contingency table

Volume of Revolution ● Revolution around x

axis ● Revolution around y axis

Term 5

Term 6

Year End Points OT1: Mathematical argument, language and proof

CORE MATHS

TEACHER 1 Hyperbolic Functions • Definitions and graphs • Inverse • Identities and equations

TEACHER 1 Complex Numbers (A level) • Exponential form • Roots of complex

FSTAT T1

Proof of Differences (A level) • Proof of differences with

Poisson • Definitions • Sum of distributions • Hypothesis Testing • Type I and Type II errors

numbers

• Euler’s form

partial fractions

● Proof of differences ● Proof by induction summations, division and recurrance

TEACHER 2 Graphs ● Linear rational equation & Inequalities ● Quadratic rational equations and inequalities ● Parabolas, ellipses and hyperbola

Mean Value Theorem ● Finding mean value of

a function using integration

Continuous Random Variables ● Expectation and

● Proof by induction Matrices

Confidence Intervals ● Introduction to normal

distribution ● Confidence interval for a mean

Momentum ● Conservation ● Experimental law ● Collisions ● Impulse

Dimension Analysis ● Analysis ● Consistency

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OT1: Mathematical Modelling

● Mean and Quartiles

FMech T2 Proof

OT2: Mathematical problem solving OT2.1 Recognise the underlying mathematical structure in a situation and simplify and abstract appropriately to enable problems to be solved. OT2.2 Construct extended arguments to solve problems presented in an unstructured form, including problems in context OT2.3 Interpret and communicate solutions in the context of the original problem. OT2.6 Understand the concept of a mathematical problem solving cycle, including specifying the problem, collecting information, processing and representing information and interpreting results, which may identify the need to repeat the cycle. OT2.7 Understand, interpret and extract information from diagrams and construct mathematical diagrams to solve problems, including in mechanics.

variance

Proof

OT1.1 Construct and present mathematical arguments through appropriate use of diagrams; sketching graphs; logical deduction; precise statements involving correct use of symbols and connecting language, including: constant, coefficient, expression, equation, function, identity, index, term, variable. OT1.2 Understand and use mathematical language and syntax as set out in the content OT1.3 Understand and use language and symbols associated with set theory, as set out in the content. OT1.4 Understand and use the definition of a function; domain and range of functions. OT1.5 Comprehend and critique mathematical arguments, proofs and justifications of methods and formulae, including those relating to applications of mathematics

FMech T2

Work, Energy & Power

Circular Motion

● Work, energy and power ● Gravitational potential energy ● Kinetic energy ● Tension & EPE

● Angular Motion ● Constant speed ● Connected particles

FMech T2 Circular Motion (A level) ● Tangential acceleration ● Conical pendulum

OT3.1 Translate a situation in context into a mathematical model, making simplifying assumptions. OT3.2 Use a mathematical model with suitable inputs to engage with and explore situations (for a given model or a model constructed or selected by the student). OT3.3 Interpret the outputs of a mathematical model in the context of the original situation (for a given model or a model constructed or selected by the student). OT3.4 Understand that a mathematical model can be refined by considering its outputs and simplifying assumptions; evaluate whether the model is appropriate OT3.5 Understand and use modelling assumptions.

Midterm Exam

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Year 12 Exam Week

Further Statistics Assessment

MOCK EXAMS

Further Statistics and Further Mechanics Exam

Year

Term 1

CORE MATHS

Term 2

CORE MATHS

Term 3

CORE MATHS

Term 4

CORE MATHS

TEACHER 1 Complex Numbers 1

TEACHER 1 Matrices 2

TEACHER 1 Integration 1

TEACHER 1 Integration 2

● Demoivres introduction ● Trig multiples to powers ● Powers to multiples ● Nth roots of unity

● Factorising ● Eigen vectors and Eigen values ● Diagonalisation

● Improper integrals

● Integrating Partial

Vectors ● Vector product ● Equation of line and area of triangle ● Planes, angles and intersections ● Distances

Matrices 1 ● Determinate of 3x3 ● Sim equations and geometric interpretation

2nd Order Differential Equations ● Solve homogenous ● Solve nonhomogenous ● Particular solutions

● Mid-ordinate rule ● Euler’s Method

TEACHER 2 Hyperbolic Functions 1 ● Reciprocal hyperbolic functions ● Differentiating and integrating hyperbolic functions

TEACHER 2 Hyperbolic Functions 1

Year End Points

CORE MATHS

OT1: Mathematical argument, language and proof

TEACHER 1 Differential equations ● 1st order differential equations ● Simple harmonic motion ● Damped and forced motion ● Coupled equations

● Osbourne’s rule ● Identities in proof

OT2.1 Recognise the underlying mathematical structure in a situation and simplify and abstract appropriately to enable problems to be solved. OT2.2 Construct extended arguments to solve problems presented in an unstructured form, including problems in context OT2.3 Interpret and communicate solutions in the context of the original problem. OT2.6 Understand the concept of a mathematical problem solving cycle, including specifying the problem, collecting information, processing and representing information and interpreting results, which may identify the need to repeat the cycle. OT2.7 Understand, interpret and extract information from diagrams and construct mathematical diagrams to solve problems, including in mechanics.

● Proving an expansion ● L’Hopitals rule

FMech T1

Circular Motion

● Inclined planes ● Conical pendulum

● Vertical motion

Graphs ● Modulus functions ● Rational with oblique asymptotes ● Composite transformations

Centre of Mass

OT1: Mathematical Modelling

● Particles and rods ● Composite ● Volume of revolution ● Toppling or sliding

Collisions ● Collisions with walls ● Impulse

FStat T2 Chi Squared ● Yates correction

FMech T2

Continuous random variables

Work, Energy & Power

Circular Motion

● Exponential distribution

● Work, energy and power ● Gravitational potential energy ● Kinetic energy ● Tension & EPE

Continuous random variables

T-Tests

● Cumulative distribution ● Rectangular distribution

● Hypothesis testing with t-tests

Confidence Intervals

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OT1.1 Construct and present mathematical arguments through appropriate use of diagrams; sketching graphs; logical deduction; precise statements involving correct use of symbols and connecting language, including constant, coefficient, expression, equation, function, identity, index, term, variable. OT1.2 Understand and use mathematical language and syntax as set out in the content OT1.3 Understand and use language and symbols associated with set theory, as set out in the content. OT1.4 Understand and use the definition of a function; domain and range of functions. OT1.5 Comprehend and critique mathematical arguments, proofs and justifications of methods and formulae, including those relating to applications of mathematics

OT2: Mathematical problem solving

Maclaurin & Limits

FMech T1 TEACHER 2 Numerical Methods

fractions ● Polar integration ● Reduction formula

Term 5

● Angular Motion ● Constant speed ● Connected particles

● Mean and variance unknown

Year 13 transitional Exam

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Year 13 Assessment Week

Year 13 Mock Exams