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Year 11 Foundation GCSE Mathematics

Curriculum Overview

Autumn 1

Formulae and Kinematics Indices and Standard Form

Graphical Functions

Autumn 2 Volume and Surface Area Simultaneous Equations

Trigonometry

Spring 1 Revision Cycle 1 Revision Cycle 2 Revision Cycle 3

Spring 2 Revision Cycle 4 Revision Cycle 5 Revision Cycle 6

Summer 1

Past Papers from Exam Board

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Contents

Formulae and Kinematics ...................................................................................................................................................................................................................................... 3

Indices and Standard Form ....................................................................................................................................................................................... 6

Graphical Functions ................................................................................................................................................................................................ 10

Volume and Surface Area ....................................................................................................................................................................................... 13

Simultaneous Equations ......................................................................................................................................................................................... 17

Trigonometry ........................................................................................................................................................................................................... 20

Foundation GCSE Mathematics Exam Revision ..................................................................................................................................................... 23

Cycle 1 Aiming for Grades 1 to 3 ............................................................................................................................................................................ 24

Cycle 1 Aiming for Grades 4 to 5 ............................................................................................................................................................................ 25

Cycle 2 Aiming for Grades 1 to 3 ............................................................................................................................................................................ 26

Cycle 2 Aiming for Grades 4 to 5 ............................................................................................................................................................................ 27

Cycle 3 Aiming for Grades 1 to 3 ............................................................................................................................................................................ 28

Cycle 3 Aiming for Grades 4 to 5 ............................................................................................................................................................................ 29

Cycle 4 Aiming for Grades 1 to 3 ............................................................................................................................................................................ 30

Cycle 4 Aiming for Grades 4 to 5 ............................................................................................................................................................................ 31

Cycle 5 Aiming for Grades 1 to 3 ............................................................................................................................................................................ 32

Cycle 5 Aiming for Grades 4 to 5 ............................................................................................................................................................................ 33

Cycle 6 Aiming for Grades 1 to 3 ............................................................................................................................................................................ 34

Cycle 6 Aiming for Grades 4 to 5 ............................................................................................................................................................................ 35

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Formulae and Kinematics Students learn how to write a formula from a written description and use this formula to model various scenarios. As learning progresses students work with the various kinematics formulae. Prerequisite Knowledge • solve linear equations in one unknown algebraically (including

those with the unknown on both sides of the equation • translate simple situations or procedures into algebraic expressions • deduce expressions to calculate the nth term of linear sequence

Success Criteria • substitute numerical values into formulae and expressions,

including scientific formulae • understand and use the concepts and vocabulary of expressions,

equations, formulae, identities inequalities, terms and factors • understand and use standard mathematical formulae; rearrange

formulae to change the subject • know the difference between an equation and an identity; argue

mathematically to show algebraic expressions are equivalent, and use algebra to support and construct arguments

Key Concepts • When substituting known values into formulae it is important to

follow the order of operations. • Students need to have a secure understanding of using the

balance method when rearranging formulae. Recap inverse operations, e.g., x2=> √x

• When generating formulae, it is important to associate mathematical operations and their algebraic notation with key words.

Common Misconceptions • Students often consider to be incorrectly calculated as 2a3 as (2a)3.

Recap the order of operations to avoid this. • Students often have difficult generating formulae from real life

contexts. Encourage them to carefully break down the written descriptions to identify key words.

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Lessons

Writing Formulae

Substitution into Formulae

Kinematics Formulae

Rearranging Formulae

Revision & Problem-Solving Lessons

Working with Function Machines

Kinematics Formulae

Rearranging Formulae

Substitution into Formulae

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Additional Departmental Resources

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Indices and Standard Form

In this unit of work students learn how to work with numbers and terms written using index notation. Learning progresses from understanding the

multiplication and division rules of indices to performing calculations with numbers written in standard form.

Prerequisite Knowledge

• Apply the four operations, including formal written methods, to integers

• use and interpret algebraic notation

• count backwards through zero to include negative numbers

• use negative numbers in context, and calculate intervals across zero

Success Criteria

• use the concepts and vocabulary of highest common factor, lowest common multiple, prime factorisation, including using product notation and the unique factorisation theorem

• calculate with roots, and with integer indices

• calculate with and interpret standard form A x 10n, where 1 ≤ A < 10 and n is an integer.

• simplify and manipulate algebraic expressions

• simplifying expressions involving sums, products and powers, including the laws of indices

Key Concepts

• To decompose integers into their prime factors’ students may need to review the definition of a prime.

• A base raised to a power of zero has a value of one.

• Students need to have a secure understanding in the difference between a highest common factor and lowest common multiple.

• Standard index form is a way of writing and calculating with very large and small numbers. Students need to have a secure understanding of place value to access this.

Common Misconceptions

• One is not a prime number since it only has one factor.

• x2 is often incorrectly taken as 2x.

• Students often have difficulty when dealing with negative powers. For instance, they assume, 1.2 × 10-2 to have a value of -120.

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Lessons

Prime Factor Decomposition

Highest Common Factor and Lowest Common Multiple

Rules of Indices

Standard Index Form

Adding and Subtracting in Standard Form

Calculations with Standard Form

Reciprocals and Negative Powers

Simplifying Surds

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Revision & Problem-Solving Lessons

Primes and Prime Factors

Prime Factors, HCF and LCM

Rules of Indices

Writing Numbers in Standard Form

Calculations with Standard Form

Surds

Problem Solving with Standard Form

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Additional Departmental Resources

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Graphical Functions

Students learn how to create and use a table of results to plot a quadratic, cubic and reciprocal graphs. As learning progresses, they use these

graphs to model a range of scenarios and estimate solutions to equations.

Prerequisite Knowledge

• plot graphs of equations that correspond to straight-line graphs in the coordinate plane

• recognise, sketch and interpret graphs of linear functions

Success Criteria

• Recognise, sketch and interpret graphs of quadratic functions, simple cubic functions, the reciprocal function y = 1/x with x ≠ 0.

• Solve quadratic equations by finding approximate solutions using a graph

Key Concepts

• To generate the co-ordinates students, need to have a secure understanding of applying the order of operations to substitute and evaluate known values into equations.

• Quadratic, Cubic and Reciprocal functions are non-linear which means they do not have straight lines. All graphs of this nature should be drawn with smooth curves.

• Students need to gain an understanding of the shape of each function in order to identify incorrectly plotted coordinates

Common Misconceptions

• Students often have difficulty substituting in negative values to complex equations. Encourage the use of mental arithmetic.

• By identifying lines of symmetry in each function students will have a greater understanding of the typical shapes for each function.

• By creating the table of results students will be more able to choose a suitable scale for their axes.

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Lessons

Plotting Quadratic Equations

Solving Quadratics Graphically

Plotting Cubic Equations

Plotting Reciprocal Functions

Revision & Problem-Solving Lessons

Plotting Curved Graphs

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Additional Departmental Resources

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Volume and Surface Area

Students learn how to calculate the total surface area and volume of cuboids, prisms and convergent shapes. Throughout the topic students

develop their algebraic notation, and application of Pythagoras’ Theorem to aid their problem-solving skills.

Prerequisite Knowledge

• use standard units of measure and related concepts (length, area, volume/capacity

• know and apply formulae to calculate: area of triangles, parallelograms, trapezia;

• know the formulae: circumference of a circle = 2πr = πd, area of a circle = πr2; calculate: perimeters of 2D shapes, including circles; areas of circles and composite shapes

Success Criteria

• know and apply formulae to calculate the volume of cuboids and other right prisms (including cylinders)

• know the formulae to calculate the surface area and volume of spheres, pyramids, cones and composite solids

Key Concepts

• To calculate the volume of a prism, identify the cross-section and calculate its area. The volume is a product this area and its depth.

• When calculating surface areas encourage students to illustrate their working by either writing the area on the faces of the 3D representation or create the net diagram so all individual faces can be seen.

• While students are not necessarily required to derive the formulae for the volume and surface area of complex shapes they do need to be proficient with substituting in known values.

Common Misconceptions

• Students often forget to include units when calculating volumes and areas.

• It is important to differentiate between those which are prisms and those which are not. Encourage students to identify the cross-section whenever possible.

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Lessons

Volume of Prisms

Surface Area of Prisms

Volume and Surface Area Investigation

Metric Conversions of Volume and Area

Cylinders

Spheres

Surface Area of Square Based Pyramids

Pyramids

Volume of a Cone

Surface Area of a Cone

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Revision & Problem-Solving Lessons

Volume and Surface Area of Cuboids

Volume of Prisms

Spheres, Cones and Pyramids

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Additional Departmental Resources

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Simultaneous Equations

Students learn how to set up and solve a pair of simultaneous equations using the elimination method. Learning progresses from solving equations where the

coefficients are equal to setting up a pair of equations with different coefficients from known facts.

Prerequisite Knowledge

• Solve two simultaneous equations in two variables algebraically;

• Find approximate solutions to simultaneous equations in two variables using a graph

• Translate simple situations or procedures into algebraic expressions or formulae; derive an equation

Success Criteria • Solve two simultaneous equations in two variables

algebraically; • Find approximate solutions using a graph

Key Concepts • For every unknown an equation is needed. • Students need to have a secure understanding of adding

and subtracting with negatives when eliminating an unknown.

• Coefficients need to be equal in magnitude to eliminate an unknown.

• Students need to check their solutions by substituting the calculated values into the original equations.

Common Misconceptions • Students often struggle knowing when to add or subtract the

equations to eliminate the unknown. Review addition with negatives to address this.

• Equations need to be aligned so that unknowns can be easily added or subtracted. If equations are not aligned students may add or subtract with non like variables.

• Students often try to eliminate variables with their coefficients being equal

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Lessons

Graphical Methods

Elimination Method

Problems with Simultaneous Equations

Revision & Problem-Solving Lessons

Simultaneous Equations by Elimination

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Additional Departmental Resources

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Trigonometry

Students are guided to discover the Sine, Cosine and Tangent ratios of right-angled triangles. As learning progresses, they learn how to

calculate a missing angle and length in right-angled triangles.

Prerequisite Knowledge

• express a multiplicative relationship between two quantities

• understand and use proportion as equality of ratios

• know the formulae for: Pythagoras’ theorem, a2 + b2 = c2

• apply angle facts, triangle congruence, similarity and properties of quadrilaterals to conjecture and derive results about angles and sides, including Pythagoras’ Theorem and the fact that the

• base angles of an isosceles triangle are equal, and use known results to obtain simple proofs

Success Criteria

• know the trigonometric ratios, Sin ϑ = Opp/Hyp, Cos ϑ = Adj/Hyp, Tan ϑ = Opp/Adj.

• apply them to find angles and lengths in right-angled triangles and, where possible, general triangles in two dimensional figures.

• know the exact values of Sin ϑ and Cos Sin ϑ for ϑ = 0°, 30°, 45°, 60°, and 90°.; know the exact value of Tan ϑ for 0°, 30°, 45° and 60°.

Key Concepts

• Sin, Cos and Tan are trigonometric functions that are used to find lengths and angles in right-angled triangles.

• The ‘hypotenuse’ is opposite the right angle, the ‘opposite’ refers to the side that is opposite the angle in question and ‘adjacent’ side runs adjacent to the angle.

• The inverse operations of sin, cos and tan are pronounced arcos, arcsin and arctan.

Common Misconceptions

• Students often have difficulty knowing which trigonometric ratio to apply. Encourage them to clearly label the sides to identify the correct ratio.

• Use SOHCAHTOA as a memory aid as students often forget the trigonometric ratios.

• When using trigonometric ratios to calculate angles students often forget to use the inverse functions.

• Students often try to apply right-angled formulae to non-right-angled triangles.

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Lessons

Introducing Trigonometry with Right-Angled Triangles

Angles in a Right-Angled Triangle

Lengths in a Right-Angled Triangle

Problems with Right-Angled Triangles

Revision & Problem-Solving Lessons

Applying 2D Trigonometry

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Additional Departmental Resources

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Foundation GCSE Mathematics Exam Revision

The program of revision reminds students of the key concepts, at their potential target grade. It is based around a cycle of formal fortnightly, hour

long assessments which are provided by examination boards.

The revision cycle takes place over 7 lessons per fortnight with the assessment taking place in the 7th lesson. Lesson 1 in the next cycle is used

to go through the assessment with the students. This ensures every student has sufficient exam style practice applying the skills they revised in

the previous five lessons.

All revision lessons begin with a range of traditional questions that remind students of the skill they are revising. As learning progresses, they

apply this skill to solve wider ranging problems that connect to other areas of mathematics.

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Cycle 1 Aiming for Grades 1 to 3

Order of Operations

Calculations with Negative Numbers

Converting Between Metric and Imperial Units

Area and Perimeter of Rectangles

Collecting Like Terms

Problems with Function Machines Assessment Set 1 Paper 1

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Cycle 1 Aiming for Grades 4 to 5

Multiplication with Decimals

Division with Decimals

Written Methods to Solve Worded Problems

Plotting Straight Line Graphs

Rules of Indices

Problems with Ratios Assessment Set 1 Paper 1

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Cycle 2 Aiming for Grades 1 to 3

Model Solutions to Set 1 Paper 1

Algebraic Products and Brackets

Highest Common Factors

Angle Properties

Equivalent Fractions and Mixed Numbers

Powers and Roots

Assessment Set 1 Paper 2

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Cycle 2 Aiming for Grades 4 to 5

Model solutions to Set 1 Paper 1

Substitution into Formulae

Factorising Expressions

Pythagoras’ Theorem

Expectation and Mutually Exclusive Events

Nth Term of Arithmetic Sequences

Assessment Set 1 Paper 2

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Cycle 3 Aiming for Grades 1 to 3

Model Solutions Set 1 Paper 2

Writing Probabilities

Multiples and Lowest Common Multiple

Volume and Surface Area of Cuboids

Percentages of an Amount Balance Method to Solve Two-Step Equations

Assessment Set 1 Paper 3

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Cycle 3 Aiming for Grades 4 to 5

Model Solutions Set 1 Paper 2

Averages from a Grouped Frequency Table

Angles in Parallel Lines

Problems with Proportional Reasoning

Prime Factors, HCF and LCM Constructing Loci

Assessment Set 1 Paper 3

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Cycle 4 Aiming for Grades 1 to 3

Model solutions to Set 1 Paper 3

Constructing Triangles and Circles

Fractions of an Amount Area of 2D Shapes

Fractions, Decimals and Percentages

Best Value Ratio Problems Assessment Set 2 Paper 1

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Cycle 4 Aiming for Grades 4 to 5

Model solutions to Set 1 Paper 3

Density and Pressure Enlarging Shapes on a Grid Reverse Percentages

Writing Numbers in Standard Form

Problems with Simultaneous Equations

Assessment Set 2 Paper 1

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Cycle 5 Aiming for Grades 1 to 3

Model Solutions to Set 2 Paper 1

Isometric and Elevation Drawings Adding and Subtracting with Fractions

Area of Triangular Shapes

Primes and Prime Factors

Multiplying with Decimals

Assessment Set 2 Paper 2

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Cycle 5 Aiming for Grades 4 to 5

Model solutions to Set 2 Paper 1

Performing and Describing Transformations

Venn Diagrams and Set Notation Adding and Subtracting with Fractions

Calculations with Standard Form Probability Trees and Independent Events

Assessment Set 2 Paper 2

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Cycle 6 Aiming for Grades 1 to 3

Model Solutions Set 2 Paper 2 Averages and Range

Direct Proportion

Pie Charts

Percentage Changes Rounding and Estimates Assessment Set 2 Paper 3

Model Solutions to Set 2 Paper 3

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Cycle 6 Aiming for Grades 4 to 5

Model Solutions Set 2 Paper 2

Compound Percentage Changes

Equation of Straight-Line Graphs

Multiplying and Dividing with Fractions

Problems with Circular Shapes

Trigonometry

Assessment Set 2 Paper 3

Model Solutions to Set 2 Paper 3