## Mathematics

## Key Stage 3

**Year Seven**

### Autumn

**Numbers and Numerals **In this unit students experience a range of number and numeral systems to develop their understanding of the base 10 place value system from primary years. First, students explore time to see that numeral systems can be used to group number in ways other than base 10 (while time still uses base 10 within groups). This provides students with a context in which to explore decomposition and regrouping of number. Students then meet the Indian number system. They convert between this and standard short form naming convention to strengthen recognition and naming of large numbers, increasing an understanding of why we cluster digits in groups of three. Finally, the Mayan number system is introduced as an informal way of meeting base 20 (and base 5) number composition. By seeing the pattern of ‘place value rows’ in the Mayan numeral system students can become more aware of the pattern that exists in base 10.

**Axioms and Arrays**In this two-week unit, students develop their understanding of different models for multiplication and division. Students also explore the axioms of number and which operations they can be applied to. The unit begins by looking at structures which lead to multiplication and the different models that can be associated with these structures. The array model is developed to explore the commutativity of multiplication before looking at this axiom with other models. The array model is again used to introduce the associativity and distributivity axioms. Students’ understanding of these axioms is developed by showing how they can be utilised to calculate efficiently.

Further use of these axioms takes place where students are asked the question: “What is the minimum number of multiplication tables they need to memorise?” It is shown that by utilising the axioms of number, and doubling and halving strategies, students can quickly generate other multiplication tables without the need for memorisation.

**Factors and Multiples**In this unit students are given the opportunity to explore the ‘structure’ of the natural numbers. Students will be introduced to factors, multiples and important sets of numbers such as prime numbers, square numbers and cube numbers. Once the fundamental concepts have been introduced students are given the opportunity to develop their understanding, conjecture, problem solve and generalise in a series of structured tasks. Throughout the unit, students consider the structure of number by considering some interesting and important properties e.g. a six-column number grid is used to show that any prime number greater than 3 can be written as 6n+1 or 6n-1. Students are exposed to a variety of representations and these are used to promote dialogue and develop understanding.

**Order of Operations **In this unit students will develop an explicit understanding of the order of operations based on the equal priority of addition with subtraction and multiplication with division. They will understand the higher priority of multiplication with division in written calculations and understand the use of brackets and vincula to manipulate this priority order. Throughout the unit, students will work with calculations represented via function machines, worded descriptions, area calculations of rectilinear shapes, and formal written calculations. Careful attention must be paid to identifying and representing equivalent calculations across these formats simultaneously to experience how the order of operations is embedded syntactically within each Students will also practise representing variables with letters within expressions accurately using the order of operations.

**Positive and Negative numbers**This unit provides opportunities to interpret negative quantities in practical situations allowing students experiences to shift their thinking from the idea that “measures” must be positive. Students will have experienced negative numbers in context from primary education and know that the number line extends beyond zero. Understanding is developed by placing number on a number line and using the number line as a representation for addition and subtraction of negative numbers as well as a scaling model for multiplying and dividing. Carefully structured tasks allow students to extend existing understanding of calculations with integer values to include negative numbers. By extending the axioms of positive numbers to negative numbers we will conclude that (−1) × (−1) = 1 and 𝑎 − (−𝑏) = 𝑎 + 𝑏. Teachers should be aware of language, using phrases such as ‘subtract negative three’ rather than ‘minus, minus three’. Laminated sheets with a selection of number lines (labelled and unlabelled) could prove a useful resource especially for demonstrating multiplication and division with negative numbers.

**Expressions, equations and inequalities **In this unit, students use algebraic notation to express the multiples of integers and see collecting like terms as a development of the distributive property. Students first use a range of representations and contexts including Cuisenaire rods, bar models, number pyramids to manipulate expressions into their simplest form as well as representing equality statements. Seen alongside ‘tracking calculations’, students will develop flexibility in expressing relationships algebraically as well as being comfortable substituting constant values for variables.

Once familiarity with this notation system has been developed, students will explore representations of equality, and understand the preservation of equality in written equations. This will then lead to exploration of inequalities utilising bar models and algebraic coding of perimeter as a problem-solving context. Finally, students will explore tracking calculations for counting strategies in growing dot patterns as another context for generalisation, introducing the foundation for later study of sequences.

### Spring

**Angles**

This unit covers estimating, measuring, drawing and calculating angles. Students describe, classify and identify types of angles using clear vocabulary, and measure and draw angles accurately. Students revise facts involving angles around a point, angles at a point on a straight line and vertically opposite angles from experiences in primary school. Students will begin to generalise these angle facts. Students explore and clarify definitions of parallel lines and perpendicular lines, and use rules around corresponding, alternate and co-interior angles. This will be built upon and applied in the following unit when exploring properties of quadrilaterals. Students being to formulate equations to show relationships between angles using the angle facts that are introduced in this unit. This is first done numerically before introducing unknown angles. Through this, students begin to solve equations with one unknown.

**Classifying 2D shapes**

This unit involves analysing the geometric properties of polygons before focusing more closely on triangles and quadrilaterals. Students begin by classifying polygons by their properties including number of sides, number of line and rotational symmetries, types of internal angles, number of equal sides, and number of parallel sides. Students are encouraged to conjecture and prove which properties are the result of other properties. For example, if a polygon has as many line symmetries as sides then it must be regular. Students move on to look at triangles and quadrilaterals more closely. Some properties are proved, some are identified inductively (by generalising phenomenon based on a number of concrete examples) and some deductively (using previous results). Special types of triangles and quadrilaterals are defined and the properties of each are explored. Generalisations are expressed algebraically, and students set up and solve equations involving special types of triangle and quadrilaterals.

**Constructing Triangles and Quadrilaterals**This unit involves students drawing and constructing triangles and quadrilaterals. In addition, there will be time allocated for a further analysis of their geometrical properties. Students will draw and measure angles within this context allowing them to practise the skills learned in the previous unit. As a way of setting the scene for construction, the unit begins by introducing the anatomy of a circle. The minimum conditions for constructing triangles are explored and thereby informally introducing congruence. Students will construct/draw triangles using initial information and conversely will deduce properties of triangles from completed constructions. Students will then construct/draw quadrilaterals, providing an opportunity to further develop an understanding of their properties.

**Coordinates**In this unit students develop their understanding of the cartesian coordinate grid and solve problems in all four quadrants. Students will be familiar with coordinates from work at primary school and in other subjects. The tasks in this unit will give students opportunities to apply their understanding from previous units including negative numbers and geometric properties of triangles and quadrilaterals. The unit begins by revisiting their understanding of working with a number line in one dimension to working in two dimensions. Students solve problems involving midpoints and develop strategies for solving related problems. Students will begin to label horizontal and vertical lines by recognising the constant 𝑥 or 𝑦 coordinates along them.

**Area and Perimeter of 2d shapes**In this unit, the concept of area as a measurable quantity is introduced. This begins by revisiting arrays introduced in previous units for multiplication. Starting with rectilinear shapes, this is built upon to explore the area of other shapes including, triangles and special quadrilaterals. Reasoning about calculating the area of shapes is built up by decomposing shapes and connecting to existing knowledge. The generalised expressions for finding the area of shapes are used to introduce rearranging formulae. Solving equations and substituting for an unknown is also practised in this unit. Students also engage with generalised statements about the relationship between area and perimeter.

**Transforming 2d figures**In this unit, students are expected to consider how different transformations acting on an object produce different images. Reflection, rotation, translation and enlargement are first applied to an object on a grid before moving onto a coordinate axis. Reflection and rotation are introduced through previous experience of line and rotational symmetry.

There is a focus on language and consideration of the amount of information required to perform each transformation. The transformations are sorted into those which result in a congruent image (rotation, reflection and translation) and those which give a similar image (enlargements). The ratios within are considered and between the similar shapes produced when an object is enlarged by a given scale factor. This is the foundation for exploring similarity and trigonometry later. Combinations of transformations are explored, including which combinations can or cannot be described as a single transformation.

### Summer

**Prime Factor Decompostion**In this unit, students revisit the composition of numbers developed in unit 3. Student use the “lots of” representation to find the prime factor decomposition of a positive integer. The uniqueness of the prime number decomposition, for which indices notation is used, is explored. This is used to show properties of particular numbers e.g. if a number is a square or cube number and therefore identify square and cube roots. Prime factorisation is used to determine the highest common factor and lowest common multiple of two or more numbers. Different methods for finding these are introduced and students are required to reason why different methods work. Students manipulate the prime factor decomposition to find how many factors a number has and also to problem solve. During this unit, indices are used to denote powers greater than 2 for the first time.

**Conceptualising and comparing fractions **This unit builds on knowledge of fractions from KS2. Students explore multiple interpretations of fractions and establish useful language and representations to use alongside these. Fractions are considered simultaneously as a number, as a way of expressing division, as a continuous part-whole model and as a discrete part of a larger set. The representations are built to consider the concepts of equivalent fractions, working with improper and mixed fractions. Students are required to find equivalent fractions including simplifying by finding common factors in the numerator and denominator. Students also convert between different abstract representations of fractions including decimal fractions and mixed numbers. Teachers should use fraction notation for division alongside the division sign in order to consolidate understanding that fractions can be viewed as divisions. When considering the equivalence of fractions and decimals in this unit, the notion of a percentage is dealt with briefly; this is revisited and developed. Using different representations, students compare two or more fractions in a set and order them by their size.

**Manipulating and Calculating Fractions**In this unit students extend their understanding of applying the four operations to non-integer values. This includes non-integers represented as fractions, decimal fractions and mixed numbers. Students find fractions of amounts by considering the multiplication of an amount by a fraction. The different conceptual meanings of × 𝑐 are explored as well as the equivalence of × 𝑐, 𝑎 × and × . Students build on this understanding and look at finding the whole given a fractional part before progressing onto the processes of fraction multiplication and division. After establishing methods and understanding for multiplying and dividing fractions, students move on to adding and subtracting fractions including those represented as decimals fractions. Bar models, line models and rectangular representations are used heavily throughout to represent the fractional amounts. Teachers should encourage students to use these to help them solve and demonstrate understanding of the problems even where not stated explicitly.

**Ratio (2 weeks)**

This unit explores ratio notation, language, representations and contexts. The ratio notation of 𝑎: 𝑏 is unpicked using bar models and this is used to show that in this notation it is possible to see the following fractional relationships: ା , ା , , , ା , ା . Students work on a series of problems that require them to use all of these relationships. Bar models, double number lines and line graphs are used throughout this unit to help develop and expand students’ conceptual understanding of ratio.

**Percentage **

In this unit, students work with percentages as another representation of ratios and fractions. The different ways of interpreting fractions are again referenced in this unit building on work from the previous unit in connecting ratios to fractions and decimals. In order to manipulate calculations involving percentages, students need to be able to use their knowledge of arithmetic with decimals and fractions. Therefore, it is important that students are also able to interpret percentages first as a number between 0 and 1 before moving onto work with percentages greater than 100. The unit progresses to the use of percentages to compare quantities and find a given percentage of a quantity. Students then increase and decrease quantities by a given percentage and find the original quantity given a percentage of the quantity. Bar models provide an excellent representation of percentage change and equivalence between amounts, hence should be used throughout the unit to deepen understanding.

**Year Eight**

### Autumn

**Sequences**

This unit on sequences builds on the work students have done in year 7. In the autumn term of year 7 students were introduced to algebraic notation and met sequences in the form of geometric patterns.

In this unit, sequences are derived from the same geometric patterns and other contexts. Students start with the term to term rules, before expressing the position to term rules algebraically. Different types of sequences are explored including linear, non-linear, arithmetic and geometric. Fibonacci sequences are also introduced as well as special sequences of numbers such as triangular and square numbers. Graphical representations of these sequences help draw out the difference in how they behave. Students use the position to term rule and other ways of reasoning to decide if a term is in a sequence or if two sequences share a term.

**Forming and Solving Equations **

In year 7 students explored the nature of equality by generating algebraic expressions and equations. Students also solved equations with one unknown where the unknown appeared on one side of the equation. In this unit students formalise methods for solving equations. Students use inverse operations to transform equations with one and two steps and encounter equations involving a single bracket. Equations are derived from familiar contexts and the solutions to these equations are interpreted within that context. Once secure with using inverse operations to solve equations, students should move on to working with equations where the unknown appears on both sides. Throughout the unit, students are encouraged to explore different transformations of an equation and reason which transformations are useful for solving different equation types.

**Forming and Solving Inequalities**

In this unit students develop their understanding of solving equations to solving an inequality with one unknown. In year 7 students developed their understanding of inequalities and the notation by expressing relationships between known and unknown values. In this unit inequalities are derived from the same contexts that were met in the previous unit. Solutions are built up by substituting numbers that satisfy the inequality. This develops an understanding that the solution to an inequality has a range of values. The unit continues with more formal strategies for solving inequalities. The same strategies for solving equations are developed in the context of inequalities. Students are required to reason why traditional “tricks” for solving inequalities work. For example, flipping the inequality sign when dividing or multiplying by a negative.

**Linear graphs**This unit is students’ first formal introduction to straight line graphs. In year 7 graphical representation were used to describe horizontal and vertical lines. They have also seen linear and non-linear sequences represented graphically using coordinates in the previous term. The graphical representation of a sequence is used as a starting point for defining linear and non-linear functions. This begins with the plotting of discrete points beginning with � = 1. The �-axis is replaced by the �-axis and discrete points are replaced with a continuous line to represent all coordinate pairs. This is emphasised by looking at a range of examples of coordinate pairs along the line including fractional and negative values. The relationship within and between (�, �) coordinate pairs is explored and linked to the equation of the line where the equation is presented in the forms � = �� + �, �� + �� = � and �� + �� + � = 0 Students should be encouraged to see the connections between the different mathematical representations of a relationship e.g. the equation, the coordinates and the graphical representation. Functions derived from real life contexts are used to help give meaning to the features of a linear graph. Students develop strategies for identifying and drawing graphs of linear functions.

The concept of gradient is introduced as the rate of change of the �- coordinates. Similar triangles are used to show the constant gradient of linear graphs and the gradient is linked the equation of the line in different forms. Parallel lines are identified from the equation where the equation is given in different forms and this is linked to the graphical representations. Students work on coordinate geometry problems by finding the equation of a line through two points and finding the equation of a line through a point with a given gradient.

**Accuracy and Estimation**

This unit provides an opportunity to consolidate their understanding of rounding to a given decimal place which most students will have met at primary school. This unit will use contexts involving measures and quantities that have been meet earlier such as area and perimeter, but the concepts from this unit can be revisited again in the summer term geometry units.

Significant figures are introduced through measuring contexts. Rather than meeting significant figures as a set of rules to follow, students are required to work out why the zero is 45.0 is significant, in terms of what it tells us about the accuracy of measure and contrast this with the 0 in 0.45. Estimation is encountered in a variety of contexts and is an opportunity to practice rounding and unit conversions.

### Spring

**Ratio, Real life graphs and rates of change **In this unit students build on the proportional reasoning taught in year 7 and allows students to experience the different ways of defining ratio and proportion. Throughout year 7, students’ proportional reasoning was developed through experiences in multiplication, division, fractions, decimals and percentages, as well as a unit introducing the formal notation for ratio. Time is spent in this unit reinforcing the notion of a ratio as an expression of a constant multiplicative relationship which can be between two quantities in the same unit e.g. fractions or between two quantities in different units e.g. speed measured in miles per hour. A variety of contexts, both previously experienced (groups of objects, unit conversion, scaling recipes, cost) and new (speed and average speed, density) are used to explore and clarify concepts. Having established ratio as an expression of a relationship between two quantities, this is applied to ratio problems where students are required to divide an amount into a given ratio and find different quantities given a ratio. The models of ratio tables, double number lines, bar models and linear graphs are used to support development of flexible strategies and understanding of procedures.

**Direct and Inverse Proportion**

This unit begins by looking at proportional relationships in familiar contexts building on their understanding of ratio from the previous unit. Students look at the meaning of direct proportion and make links to linear graphs from the last unit. Students will develop their proportional reasoning skills by solving a variety of problems. Students are encouraged to compare different approaches to solving problems involving direct proportion. In the second week of this unit, students will meet the concept of inverse proportion. Students will encounter different contexts where one variable increases, the other decreases, but their product remains constant. As a way into this, students are invited to make different rectangles with a constant area. The graphing of the side lengths will reveal an inversely proportional relationship. Students will use this understanding to solve problems involving inverse proportion.

**Univariate data **This unit explores a variety of methods of presenting data, with an emphasis on interpretation as well as production. Although students are not required to collect raw data, if they could collect their own data this would make the unit more meaningful. Some suggestions for how to do this as well as a large data set for students to work with for applying the statistical method taught in this unit. The unit presents a series of inquiry questions and students make hypotheses in relation to these. Each of the statistical methods taught in this unit are used to construct an argument for or against the hypotheses. Students begin by considering different ways of representing a data set such as in tables, bar charts, pictograms, line graphs and pie charts. Students organize data into different frequency distributions and look at the impact of different groupings. Misleading graphical representations of the data are presented and critiqued. In the second part of this unit, students begin to look at statistical measures and interpret these in terms of the data. Students calculate the mean, median, mode and range of ungrouped and grouped data. Time is spent discussing the different measures of centrality and the merits of each.

**Bivariate data**In this unit, students develop their understanding of statistical diagrams and measures to bivariate data. Students investigate ways of displaying a relationship between two dependent variables. Students present the data in tables and in a scatter graph. They examine relationships between point to make simple inferences about association and covariation. Types of correlation are discussed, and these are interpreted in the context of the data set. The difference between correlation and causation are introduced and the idea of an explanatory variable. As with the previous unit a large data set is provided for students to work with. Students could construct their own data set to contrast and compare with the one provided

### Summer

**Angles in Polygons**In this unit students review angle work in year 7 by revisiting angle facts including angles in parallel and perpendicular lines. Students work on more complex problems using these rules and these provide a context for setting up and solving equations. Students use their understanding of angles in triangles from year 7 to build up to a proof for the sum of the interior angles in polygons with 4 or more sides by dividing into triangles. The perspective is then shifted to investigate the sum of the exterior angles of a polygon and students develop an understanding of the relationship with between the exterior and interior angles. Students solve a variety of problems using the angle relationships built up in this unit and in previous units.

**Bearings **The introduction of bearings is another chance to apply and practice concepts involving angles that have been taught in year 7. The first part of this unit introduces the conventions for measuring, drawing and recording bearings. Once these conventions have been established, students then engage with a series of problems involving the angles rules that have been introduced up to this point. The problems that students encounter in these lessons also provide an opportunity to practice applying the conventions for rounding and applying scales when measuring lengths.

**Circles**Within this unit, students explore the connection between the circumference of a circle and its diameter and through this are introduced to pi as the constant linking the relationship between the two measures. When introducing the area of a circle, dynamic geometry software can be used to show how the area formula for a circle can derived by slicing a circle into sectors. We can decrease the width of these sectors, so they become infinitely small, and the area can be shown to be equivalent to the area of a rectangle where one side length is the radius and the other is half the circumference. Exposure to this proof is important as often this type of formula can seem like a trick and applied with little understanding of the meaning behind it. However, it is not important for them to remember the exact steps of the proof. The unit ends with opportunities for students to apply their understanding to geometric problems involving the area and circumference of a circle.

**Volume and Surface area of prisms**In this unit, students formally meet volume as a measure of the space inside a 3- D object. Students may need to revisit the names and properties of 2-D shapes before moving onto 3-D. Time should be spent building and breaking down 3-D shapes, both with blocks and as nets, to fully understand the properties of these shapes and aid students as they progress onto finding volume and surface area. Students develop their own methods for finding the volume of prisms and before any exposure to the conventional formulae for cuboids, cylinders and other uniform prisms. Different units for volume are explored and converting between these units is an opportunity to practice fractional arithmetic and proportional reasoning. The primary focus for this unit is on developing an understanding of volume and use of formulae within this context. Students are expected to understand the concept of surface area as an application of the area formulae that they encountered in previous units earlier in the year and in year 7.

**Year Nine**

### Autumn

**Place value & Number Properties**In this topic pupils will deepen there understanding on place value by visiting elements such as ordering of decimals, related calculations and the use of the product rule for counting and calculations. This sets the foundations for the following topic of

**4 Rules – Decimals**

This is a development of the pupils understanding of adding, subtracting, multiplying and dividing decimals. This allows pupils to revisit basic skills as well as developing it further, particularly with the use of AO3 problems.

**Factors, Multiples & Primes**

Within this area, pupils will recap factors and multiples through their do now activities and use this as the foundations of working out HCF, LCM and prime factor decomposition.

**Indices Powers & Roots**

Pupils will gain an understanding of how powers and indices operate and how they can be used and manipulated in context of the problem they will be facing. They will look at multiplying and dividing, powers of zero, negative powers and how to estimate using roots.

**Rounding & estimation**

Pupils will visit fundamental elements of numeracy within this topic by looking at significant figures, estimation and error intervals. Pupils will differentiate between decimals places and significant figures.

**Ratio**

In this topic, pupils will have the chance to gain an understanding of ratios by simplifying ratios, writing them in the form of 1:n and n:1. Students will draw on similarities to fractions and ratios.

**FDP**

In this topic, pupils will start drawing on their understanding of fractions, decimals and percentages. They will learn how to convert from one to the other. They will then go on to compare and order FDP problems.

**Fractions**

This develops from the previous topic by delving deeper into fractions. Pupils will learn how to multiply, divide, add and subtract fractions within AO3 types of problems. They will also be introduced to reciprocals and expressing one number as a fraction of another.

**Percentages**

Pupils will be introduced on how to use the multiplier and how this can be used to work out increases and decreases. Students will also be given an insight on hhow to find the amount of something following a percentage change.

**Proportion**

Students will learn how to apply direct and inverse proportion in real life conntexts, they will also learn about best buys and value for money when purchasing a good or service. They will also look at exchange rates and how to adapt recipes.

### Spring

**Notation**Students will be introduced how to use fundamental notation when using algebra. They will look at key terminologies, symbols and expressions. This will ensure they have an understanding when looking at algebra more deeply.

**Simplifying Index Laws**

Students will learn how to multiply, divide, add and subtract algebraic fractions. This will enable them to apply their understanding of fractions in context of algebra. Students will also gain a deeper understanding of how to simplify expressions and algebraic fractions.

**Expanding and Factorising**

Students will continue their algebraic journey by exploring expanding and factorising algebraic expressions. They will look at linear and quadratics, as well as understanding the difference of two squares.

**Expression and substitution**

Students will delve into this topic by understanding how functions machines and algebra are synonymous. Students will explore the application of algebra across topics such as area/perimeter, number properties and formulae.

**Linear Equations**

Students continue their algebraic journey by exploring how to solve linear equations which contain single and multiple steps, subjects in the denominator, negatives, brackets/fractions and unknown on both sides. They will then go onto explore how to solve and substitute equations too.

**Linear inequalities**

Pupils will gain an understanding of inequality symbols and how they are used in number lines and equations. They will draw on their understanding of solving linear equations and apply it too solving equations with inequalities within it.

**Perimeter, Area & Measures**

In this topic, students will learn and remember key formulas which will help them solve areas of particular shapes. They will then be exposed to how area and algebra can be used together to work out measures and area/perimeter.

**Pythagoras Theorem**

Students will be introduced to the magical world of right-angled triangles and what Pythagoras theorem actually is. They will be introduced to key terms such as the hypotenuse and how this theorem can help them solve other geometric problems too.

### Summer

**Properties of Shapes **This half term, students will begin looking at Geometry, therefore it is imperative that students have a deeper understanding of the properties of particular shapes and how they are used.

**Angle facts**

Students will start to explore the world of angles and the rules of angles in particular shapes. Students will differentiate between regular and irregular Polygons and understand how to work out exterior and interior angles.

**Angles in Parallel lines**

Developing on from angle facts, students will learn the significance of parallel lines and how key angles are worked out and explained.

**Circles**

Students will delve into the world of Circles by gaining a deeper understanding and development from previous learning surrounding circles. They explore how to calculate arc lengths, sectors of circles and learn how to work out area of compounded shapes which include circles.

**Volume**

Students will move from 2D shapes to 3D shapes by beginning to look at volume of key shapes. They will learn formulas and apply them to problem solving questions.

**Surface Area**

Students will continue their journey into exploring 3D shapes by looking at surface area. Students will understand that surface area is also a key component of 3D shapes. They will learn formulas which are a development of area of 2D shapes.

**Basic vectors**

Students will gain their first exposure to vectors and their special notations. Students will learn to add, subtract and multiply vectors use the resultant vectors for their problems. They will then go on to explore parallel lines and vectors too.

**Sequences**

Students will begin to delve deeper into sequences by learning how to calculate the nt term of an expression and how to use this to work out terms within this pattern.

**Plans & Elevations**

This topic allows pupils to look at the nets of 3D shapes (both regular and irregular) and look at the how to draw them too.

## Key Stage 4

**Year Ten**

### Autumn

**Foundation Students:**

**Rearranging the formulae**Students begin their key stage 4 journey by developing upon their algebraic knowledge attained in Key stage 3. Students will begin by rearranging formulae’s which require single, multiple and complex formulae.

**Linear graphs**

In this topic, students will aim to grasp the concept of real-life graphs and the application of them. Pupils will engage in tasks which require them to interpret and explain particular graphs.

**Y=mx+c**

The complexity of this topic requires pupils to build up their understanding of this topic to ensure they secure their knowledge of it for long term application. They will learn how to derive the gradients and intercepts of straight-line graphs. They will then move onto find the equations of perpendicular and parallel lines.

**Compound measures**

This topic will allow pupils to draw on similarities between Science and Maths more clearly. They will learn how to calculate, speed/distance time, pressure/force/area and density/mass/volume.

**Quadratic graphs, turning points and roots**

Students will delve further into graphs and algebra in this unit by learning how to draw quadratics graphs, find the turning points and use lines of symmetry.

**Linear simultaneous equations**

In this topic area, students will further their skills of solving equations by solving simultaneous equations. They will be explore how to solve them algebraically and graphically and draw on conclusions what they are actually doing when solving simultaneous equations.

**Further graphs**

This will enable pupils to investigate graphs even further, by being able to plot, identify, illustrate and interpret graphs such cubic and reciprocal graphs.

**Higher Students:**

**Rearranging the formulae**Students begin their key stage 4 journey by developing upon their algebraic knowledge attained in Key stage 3. Students will begin by rearranging formulae’s which require single, multiple and complex formulae.

**Linear graphs**

In this topic, students will aim to grasp the concept of real-life graphs and the application of them. Pupils will engage in tasks which require them to interpret and explain particular graphs.

**Y=mx+c**

The complexity of this topic requires pupils to build up their understanding of this topic to ensure they secure their knowledge of it for long term application. They will learn how to derive the gradients and intercepts of straight-line graphs. They will then move onto find the equations of perpendicular and parallel lines.

**Compound measures**

This topic will allow pupils to draw on similarities between Science and Maths more clearly. They will learn how to calculate, speed/distance time, pressure/force/area and density/mass/volume.

**Quadratic graphs, turning points and roots**

Students will delve further into graphs and algebra in this unit by learning how to draw quadratics graphs, find the turning points and use lines of symmetry.

**Further Expanding and Factorising**

Higher spec students will explore quadratics and cubics in more detail. They will learn to expand and simplify triple bracket, learn the quadratic formula and solve by comleting the square. In addition to this, they will learn to sketch quadratic graphs too.

**Linear simultaneous equations**

In this topic area, students will further their skills of solving equations by solving simultaneous equations. They will be explore how to solve them algebraically and graphically and draw on conclusions what they are actually doing when solving simultaneous equations.

**Further graphs**

This will enable pupils to investigate graphs even further, by being able to plot, identify, illustrate and interpret graphs such cubic and reciprocal graphs.

### Spring

**Foundation Students:**

**Probability**Students will begin a detailed study of probability this half term, they will be able to work out relative frequency, use tree and venn diagrams which in turn allows them to understand elements of conditional probability too. Students will spend time grasping the concepts and applying them to AO3 styled problems.

**Standard Index form**

In this topic area, students will begin to learn a skill which relaxes the tensions of working with microscopical or astronomical numbers. They will learn how to convert to and from standard form and to do calculations with it too.

**Simple Interest**

Students will explore the world of how money is created in our western world. They will learn how to calculate simple interest with and without a calculator, and how to work with reverse percentages.

**Ratio (Further)**

This topic will enable students to work with ratios and fractions, sharing in a given ratio, identify changes within ratios and deduct linear functions. Students will begin to draw on comparisons between ratios and fractions.

**Growth & Decay**Students will learn the difference between simple and compounded interest. They will learn the difference between the two and how they can be used in our real world. They will learn how to calculate compound interest and depreciation, along with repeated percentage change.

**Higher Students:**

**Probability**Students will begin a detailed study of probability this half term, they will be able to work out relative frequency, use tree and venn diagrams which in turn allows them to understand elements of conditional probability too. Students will spend time grasping the concepts and applying them to AO3 styled problems.

**Capture & Recapture**

This topic is short and brief as it allows pupils to expand their knowledge further from Probability. It allows pupils to grasp a deeper understanding in a particular setting which is similar to that of Probability styled problems.

**Standard Index form**

In this topic area, students will begin to learn a skill which relaxes the tensions of working with microscopical or astronomical numbers. They will learn how to convert to and from standard form and to do calculations with it too.

**Proportion (Furthher)**

Students will learn how to work with direct and inversely proportional problems. They will learn how to apply their algebraic skills to these problem by solving for k. They will learn how to work with linear and non-linear styled problems and learn to deduce what the corresponding graphs and tables would look like.

**Simple Interest**

Students will explore the world of how money is created in our western world. They will learn how to calculate simple interest with and without a calculator, and how to work with reverse percentages.

**Ratio (Further)**

This topic will enable students to work with ratios and fractions, sharing in a given ratio, identify changes within ratios and deduct linear functions. Students will begin to draw on comparisons between ratios and fractions.

**Growth & Decay**

Students will learn the difference between simple and compounded interest. They will learn the difference between the two and how they can be used in our real world. They will learn how to calculate compound interest and depreciation, along with repeated percentage change.

**Recurring decimals**

This topic stretches students understanding of converting between decimals and fractions. Students will draw upon their algebraic knowledge and converting between FDP to do calculations which involve recurring decimals.

### Summer

**Foundation Students:**

**Statistics**Students will learn most of their data and statistical skill in this topic area. They will learn key statistical language such as bias and sampling. They will then go onto learn averages and problem solving when applying the skills. Students will also learn how to interpret statistical analysis from frequency tables (discrete and continuous data). Students will then move onto time series, Pie charts, scatter graphs and frequency polygons.

**End of year revision programme**

**Higher Students:**

**Statistics**

Students will learn most of their data and statistical skill in this topic area. They will learn key statistical language such as bias and sampling. They will then go onto learn averages and problem solving when applying the skills. Students will also learn how to interpret statistical analysis from frequency tables (discrete and continuous data). Students will then move onto time series, Pie charts, scatter graphs and frequency polygons.**Surds**Students will begin to get comfortable with leaving their answer in surd format following this topic. They will learn to simplify, multiply, divide, add, subtract using surds. They will also use the technique of expanding and simplifying from algebra in this topic too. Students will also learn how to rationalise denominators and how to apply it to geometric problems.

**Bounds**

Students will learn how to apply the error intervals they learn in key stage to work out bounds of accuracy. They will also learn how to use bounds to leave their work to a suitable degree of accuracy.

**Trigonometry**

Students will gain a deeper understanding into right-angled triangles. Students will learn trigonometric functions and apply them. In this topic students will learn to find the angle and lengths of right angled triangles. They will then apply this skill to more complex problems. Students will learn to use trigonometry to non-calculator and calculator based problems.

**Similar shapes**

In this topic, pupils will learn about the properties of similar shapes and how lengths, areas and volumes are applicable to the concept of mathematically similar shapes.

**Quadratic sequences**

This is a development of linear sequences as students will not look to work out the nth term of quadratic sequences and other terms in the sequence.

**End of Year revision programme**

**Year Eleven**

### Autumn

**Foundation Students:**

**Pythagoras theorem **Students will begin the year with learning about right angled triangles, beginning with Pythagoras. Students will learn the properties of a right-angled triangle and about the hypotenuse. Students will learn how to find missing lengths and other geometric problem involving triangles.

**Trigonometry**

Students will gain a deeper understanding into right-angled triangles. Students will learn trigonometric functions and apply them. In this topic students will learn to find the angle and lengths of right angled triangles. They will then apply this skill to more complex problems. Students will learn to use trigonometry to non-calculator and calculator based problems.

**Bearings and scale drawings**

Students will learn about bearings and how to apply them to problem solving. They will also be exposed to how to scale their drawings and how to use angles and Pythagoras/Trigonometry within it.

**Transformations of shapes**

Students will learn how to rotate, reflect, translate and enlarge shapes in this topic. They will learn how to apply all transformations in a single form and understand how to interpret these transformations too.

**Congruence**

In this topic, students will learn and Understand that distances and angles are preserved under reflections, so that any figure is congruent under this transformation. Students will learn the rules of triangles using SSS, SAS, ASA and RHS.

**Higher Students:**

**Algebraic proof**Students will begin the academic year by continuing work on algebra. Students will learn terminologies and apply them to problems. Students will learn to work with equivalent statements and how to work out given terms.

**Solving quadratics and further simultaneous equations**

Students will continue their work in the algebraic world by exploring the algebraic formula, solving quadratics from algebraic fractions and non-linear simultaneous equations.

**Functions**

Students will begin to look at functions, composite functions and inverse functions. They will then go onto explore problems which require them to set up equations and use function notations.

**Iteration**

Students will use their understanding of rearranging the formulae to use the iterative formula to find roots which are suitable.

**Quadratic Inequalities**

In this topic students will learn how to graph inequalities by shading regions and drawing upon graphical solutions. Students will also learn how to use quadratic graphs to solve inequalities.

**Bearings**

In this topic students will learn to read, interpret and measure drawings which include angles, bearings, Pythagoras and trigonometry. Students will be expected to draw upon skills and knowledge previous learnt and build upon it.

**Circle theorems**

Students develop their understanding of circles further by exploring the seven theorems which will enable them to work out angles, give rational explanations and interpret the application of these theorems.

**Further Trigonometry and trigonometric graphs**

Students will build upon their understanding of trigonometry by learning about for triangles which are not right-angled. They will learn about the sine and cosine rules and how they translate onto trigonometric graphs. Students will also learn how to apply transformations to these trigonometric graphs.

### Spring

**Foundation Students:**

**Vectors**Students will gain a deeper understanding of vectors and have exposure to vectors and their special notations. Students will learn to add, subtract and multiply vectors use the resultant vectors for their problems. They will then go on to explore parallel lines and vectors too.

**Similar Shapes**

In this topic, pupils will learn about the properties of similar shapes and how lengths, areas and volumes are applicable to the concept of mathematically similar shapes.

**Construction & Loci**

Students will learn how to use their compasses effectively and use them to construct shapes and solve problems involving loci. Students will learn how to draw bisectors segments and intersecting lines.

**Higher Students:**

**Statistics further**In this topic students will develop their understanding further by delving into histograms, cumulative frequency, box plots and quartiles. Students will apply this deeper understanding upon the skills and knowledge previously learnt. This will enable pupils to tackle AO3 problem solving questions.

**Transformations**

Students will learn how to rotate, reflect, translate and enlarge shapes in this topic. They will learn how to apply all transformations in a single form and understand how to interpret these transformations too.

**Congruence**

In this topic, students will learn and Understand that distances and angles are preserved under reflections, so that any figure is congruent under this transformation. Students will learn the rules of triangles using SSS, SAS, ASA and RHS.

**Vectors**

Students will be exposed to the addition, multiplication and division of vectors in order to work out the magnitude. Students will learn how to work with parallel vectors and to solve geometric problems when given a ratio. They will also produce geometrical proofs to work with parallel vectors.

**Gradients and areas under a curve**

Students will explore a deeper understanding into graphs and be able to estimate areas under a curve between linear and non-linear equations. Students will also explore distance/time graphs and velocity/time graphs.

**Kinematics**

Students will once again draw upon comparisons between their physics lessons by applying the kinematic formulas to calculate speed, velocity and acceleration.

**Graphical Transformations**

In this topic students will learn how to transform quadratic, cubic and reciprocal graphs using function notations. They will also learn how to transform trigonometric graphs such as sine/cosine/tangent graphs.

**Construction & Loci**

Students will learn how to use their compasses effectively and use them to construct shapes and solve problems involving loci. Students will learn how to draw bisectors segments and intersecting lines.

**Revision programme**

### Summer

If you require any further information please email maths@steds.org.uk.