# Mathematics

## Learning Journey & Sequencing Rationale

Mathematics can be applied in practical tasks, real-life problems and within mathematics itself. The course aims to develop deep learning through the mastery of mathematical vocabulary, conceptual understanding and mathematical reasoning used to solve a variety of problems.

The course of study should help you whether working individually or collaboratively to reason logically, plan strategies and improve your confidence in solving complex problems.

During Maths lessons you will learn how to:-

• Use and apply maths in practical tasks, real life problems and within mathematics itself.
• Develop and use a range of methods of computation and apply these to a variety of problems.
• Develop mathematical vocabulary and improve mental calculation.
• Consider how algebra can be used to model real life situations and solve problems.
• Explore shape and space through drawing and practical work using a range of materials and a variety of different representations.
• Use statistical methods to formulate questions about data, represent data and draw conclusions.

Engage in practical and experimental activities in order to appreciate principles of probability.

Key stage 3 overview

The HHS Mathematics department follows a tailored version of a mastery scheme of work. The curriculum is a spiral curriculum, which means prior learning is revisited and extended year upon year.

• Year 7 begins with an examination of the number system, followed by preparing learners for generalising with algebra. The themes in the autumn are built upon later in the spring, and the year ends with an in depth study of fractions.

•  In Year 8, learners begin by studying percentages before revisiting and extending the algebra learned 6 months prior. Learners then study a carefully sequenced set of topics to understand how algebra relates to the coordinate system. The end of the year applies some of this knowledge to geometry and statistics.

• In Year 9, learners build on their learning in Year 8 by studying probability, more challenging algebra, and trigonometry. The topics in the summer, quadratic equations and exponential growth, are designed to draw on themes from the whole of KS3.

Year 7

Year 7 begins by asking students to think more deeply about the number system, and the structures of numbers in general. Learners experience a range of number and numeral systems to develop their understanding of the base 10 place value system.Learners also explore the axioms of number and which operations they can be applied to. Generalisations such as algebra are introduced to communicate and explain mathematical ideas. Learners will then 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. After this has been secured, learners develop their algebraic reasoning, focusing on directed numbers and using algebraic notation. Finally, learners study prime factorisation and use is it to determine the highest common factor and lowest common multiple.

Later in the spring, learners are exposed to a range of new knowledge and skills in 2-D geometry, which will provide the foundation for further work in Years 8 and 9. This work also provides more contexts for students to practice the algebraic reasoning developed in Autumn term. Learners work with angles and revise facts involving angles. Learners explore and clarify definitions of parallel lines, perpendicular lines, and polygons. Learners are encouraged to conjecture and prove. Generalisations are expressed algebraically, and learners set up and solve equations involving special types of triangle and quadrilaterals.

In the summer learners will construct/draw triangles using initial information and conversely will deduce properties of triangles from completed constructions. The concept of area as a measurable quantity is introduced, and every opportunity is used to explore prior learning about arrays and formulae.

Lastly, at the end of the year, learners formally explore fractions and their relation to decimals. The extension of the number system to rational numbers is a highly important stage in learners’ mathematical development. 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. Students find fractions of amounts by considering the multiplication of an amount by a fraction. Bar models, line models and rectangular representations are used heavily throughout to represent the fractional amounts.

Year 8

Learners begin Year 8 by considering percentages as another representation of fractions. Bar models provide an excellent representation of percentage change and equivalence between amounts, hence are used throughout the unit to deepen understanding. As in Year 7, algebra is used to generalise and extend mathematical thinking. Next, sequences are derived from the same geometric patterns and other contexts. Different types of sequences are explored including linear, non-linear, arithmetic and geometric. In year 7 students explored the nature of equality and solved equations with one unknown where the unknown appeared on one side. In Unit 3 learners formalise methods for solving equations. Learners use inverse operations to transform equations with one and two steps and encounter equations involving a single bracket. Inequalities are then derived from the same contexts that were met in the previous unit.

In the second half term learners prepare to study linear graphs. This begins by learning about ratios. Bar models, double number lines and graphs are used to connect ratio notation with prior learning. Learners will be familiar with coordinates from work at primary school and in other subjects. The tasks in this unit will give learners opportunities to apply their understanding from previous units including negative numbers and geometric properties of triangles and quadrilaterals. Lastly, learners are expected to consider how transformations acting on an object produce different images. Reflection and rotation are introduced through previous experience of line and rotational symmetry. This is the foundation for exploring similarity and trigonometry later.

Throughout year 7, learners’ proportional reasoning was developed through experiences in multiplication, division and fractions. Learners study linear graphs and real life graphs and consider the functional relationships between different variables, including piecewise functions. Next, learners examine proportional relationships in familiar contexts before looking at the meaning of direct proportion in abstract. Students are encouraged to compare different approaches to solving problems involving direct proportion.

With a secure foundation in basic algebra, learners will be ready to tackle problems in geometry and statistics. In the late spring learners return to angles by considering the angle sum theorem in polygons, and bearings. Learners learn about the properties of circles and mensuration of 3D shapes. Learners will be challenged to think carefully about proof and develop their spatial reasoning. In the first unit of the last half term, learners study univariate data. 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 some given hypotheses. In the second unit of the half term, students extend their understanding of statistical diagrams and measures to bivariate data. The difference between correlation and causation are introduced and the idea of an explanatory variable.

Year 9

Learners in Year 9 continue to build on the key themes in algebra, geometry and statistics. Their learning is extended to intermediate topics such as conditional probability, trigonometry and quadratic equations.

Learners begin by considering uncertainty and the language of probability in the context of idealised experiments and real contexts before considering combined events. Learners encounter a variety of tools and representations, including bar models and tree diagrams. This learning is formalised and extended in unit 3, where students encounter set theory for the first time, and use Venn diagrams to represent them.

Linear simultaneous equations are an important part of mathematical reasoning. The skills and knowledge required to solve equations like these are invaluable for technical and everyday contexts. The first unit focuses on the algebraic representation of linear equations and extends prior knowledge about equations into expressions and equations with two variables. In the second part of the module, learners revise linear graphs and represent systems  of linear equations graphically, linking the work earlier in the unit to their work on linear graphs in Year 8.

In the spring, learners will consolidate and extend their knowledge of the geometry of triangles. Learners explore construction and consider its relationship to congruence. A key part of this unit is determining the relationship between the minimum conditions for congruence in and the construction of triangles. Learners then study Pythagoras’ theorem, including some proofs and demonstrations, before moving onto similarity and trigonometry. These units are rich with opportunities to improve visual-spatial reasoning, problem solving and reasoning skills. There are also plenty of opportunities to link to careers and realistic situations.

Learners dedicate one half term to studying the features of quadratic expressions, equations and functions. This topic is a foundational topic in KS4 and KS5 mathematics. The module begins by examining quadratic expressions and how they are represented on a graph. The graph is then used to aid learners reason about the behaviour of a quadratic function. The graph is used to help learners interpret the meaning of a solution and link it to prior learning about equations. The module ends with factorising, solving and sketching quadratics.

In this module, learners extend their understanding of the number system into the realm of the logarithmic scale. Gaining an appreciation of exponential relationships and their differences to linear relationships is vital for further scientific study, as well as understanding exponential relationships in everyday life. In the first unit learners explore indices further and extend their understanding to include indices other than positive whole numbers. This is so learners are able to appreciate standard form and use it to describe very large and small numbers. Next, learners explore growth and decay in the context of repeated percentage change. This is linked to exponential relationships more broadly.

## autumn 1 - the number system

 Skills Recognise and name nine- and ten-digit numbers in base 10 Develop a sense of flexible number composition by solving problems involving time of day and quantities of time Use commutativity, associativity and distributivity to solve calculations efficiently Use the definitions of factors and multiples to find common factors and multiples  Express an integer as a product of its factors  Conjecture and make generalised statements e.g.:  Square numbers cannot be prime  The common multiples of 5 and 4 are always multiples of 20  Solve problems involving factors and multiples in unfamiliar contexts  Form written calculations, function machines and worded descriptions correctly embedding the order of operations  Form and identify equivalent calculations based on distributivity, commutativity and the order of operations  Form and interpret expressions involving variables Knowledge Understand the values of and relationship between of different place value columns in base 10 Understand a range of notation for quantities of time and time of day Have an awareness of different numerical systems and their representations Make use of and generalise the commutative, associative and distributive properties Compare and contrast scaling, area, repeated addition and grouping/sharing models for multiplication and division Develop number sense and efficient calculation strategies  Make links between efficient calculation strategies and the axioms  Understand the terms factor and multiple; recognise and define prime, square and cube numbers  Interpret and create representations of integers that reveal their structure  Understand the equal priority of addition with subtraction and multiplication with division in written calculations  Understand that operations of equal priority can be evaluated in any order  Understand that written calculations follow rules of ‘syntax’ determining the order of operations  Understand the higher priority of multiplication with division over addition with subtraction in written calculations Interpret the order of operations from written calculations, function machines and worded descriptions Rationale This module is fundamentally about asking students to think more deeply about the number system, and the structures of numbers in general.  The module begins with the ‘Numbers and Numerals’ unit, in which learners experience a range of number and numeral systems to develop their understanding of the base 10 place value system from primary years. In ‘Axioms and Arrays,’ learners develop their understanding of different models for multiplication and division. Learners also explore the axioms of number and which operations they can be applied to. Generalisations such as algebra are introduced to communicate and explain mathematical ideas. Learners will then 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. Lastly, learners 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.

## autumn 2 - expressionS and equations

 Skills Compare and order positive and negative numbers  Use positive and negative numbers to express change and difference  Calculate using all four operations with positive and negative values  Form and manipulate expressions involving negative numbers  Collect like terms to simplify expressions and understand that this is a result of the distributive property e.g. 3a +2a = (3 + 2)a = 5a Substitute numerical values into and evaluate expressions  Use the distributive property to identify equivalent expressions involving a single bracket and the expanded form e.g. 3(a + b) = 3a + 3b Use two equations to form another related equation or inequality e.g. if a = b and b = c then a = c, a + 1 > b, 2a + b = 3c etc.  Use different contexts, including sequences, to construct expressions, equations and inequalities Knowledge Interpret negative numbers in a variety of contexts  Understand the meaning of absolute value  Use number lines to model calculations with negative numbers Explore scaling with negative multipliers  Develop understanding of algebraic notation including:  a × b = ab   y + y + y = 3y a × a = a²  Use the vinculum, a/b = a ÷ b  Develop understanding of the equality and inequality signs  Represent algebraic expressions using a variety of models including arrays and bar models Rationale This module provides learners with the opportunity to develop their algebraic reasoning, while extending their knowledge of the number system. It provides a thorough introduction to the key concepts and procedures in algebra that underpin the rest of secondary mathematics. This first unit focuses on directed numbers. It provides opportunities to interpret negative quantities in practical situations.  Students will have experienced negative numbers in context from primary education. 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.  In the second unit, ‘expressions, equations and inequalities,’ students use algebraic notation and make links to work on the basic properties of arithmetic established in the previous module. Once familiarity with this notation system has been developed, students will explore representations of equality. Finally, learners will explore counting strategies in growing dot patterns, supporting the  later study of sequences

## spring 1 - 2-D Geometry

 Skills Draw and measure acute and obtuse angles reliable to the nearest degree  Estimate the size of a given angle  Use angle facts: angles at a point, angles at a point on a straight line, vertically opposite angles to find missing angles Generalisations and reason with unfamiliar angle situations Use angle facts around corresponding, alternate and co-interior angles to find missing angles  Find unknown angles. Form algebraic expressions. Solve for unknowns. Know and use the angle sum of triangles and quadrilaterals  Generalise results for properties of special types of triangles and quadrilaterals  Form and solve equations from contexts arising from properties of triangles and quadrilaterals  Construct triangles and quadrilaterals for given conditions using ruler, protractor and compasses  Recognise when two triangles are congruent Knowledge Understand an angle as a measure of turn, and identify the features that determine the size of an angle Know the classification of different types of angles Recall and derive basic angle facts Define parallel and perpendicular lines  Classifying polygons by symmetry, regularity, intersection of diagonals, number of parallel sides  Classify triangles and quadrilaterals according to properties (angles, regularity, symmetry)  Explore constructions through use of dynamic geometry software  Become familiar with the different cases of minimum conditions for the construction of triangles Rationale This module has two purposes. First is to expose learners to a range of new knowledge and skills in 2-D geometry, which will provide the foundation for further work in Years 8 and 9. Second, to provide more contexts for students to practice the algebraic reasoning developed in Autumn term. The first unit covers angles. Learners describe, classify and identify types of angles using clear vocabulary, and measure and draw angles accurately. Learners revise facts involving angles from experiences in primary school. Learners explore and clarify definitions of parallel lines and perpendicular lines, and use rules around corresponding, alternate and co-interior angles. Students being to formulate equations to show relationships between angles using the angle facts.  Next learners, study the geometric properties of polygons before focusing more closely on triangles and quadrilaterals. Learners are encouraged to conjecture and prove. Special types of triangles and quadrilaterals are defined. Generalisations are expressed algebraically, and learners set up and solve equations involving special types of triangle and quadrilaterals. Lastly, students construct triangles and quadrilaterals. 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.

## spring 2 - the cartesian plane

 Skills Read and write coordinates in all four quadrants, including non-integer coordinates  Solving geometric problems involving missing coordinates Finding the mid-point of a line segment or two points  Recognise and plot horizontal and vertical lines on a coordinate axis Finding the area of rectilinear shapes   Finding the area of 2-D shapes including triangles, and special quadrilaterals   Rearrange formulae to make a different subject  Reflection of an object in a mirror line Identify horizontal and vertical mirror lines and their equations  Rotation of an object using the centre of rotation  Translating shapes by a given number of units (positive or negative) Combining transformations and which combinations can be expressed as a single transformation  Simple enlargements with positive scale factors Knowledge Understand the 2-D coordinate system and start to understand its continuous nature. Use the language of horizontal and vertical lines Know that area is the 2D space inside a shape Develop understanding of counting strategies in arrays to using similar strategies to calculate the area of shapes Generalise formulae for finding the area of 2-D shapes using the language of height, base, width, length etc.  Reason about generalised statements of the relationship between area and perimeter  Exploring the ratios of sides lengths within and between shapes produced by an object being enlarged by a given scale factor  Use the language of congruence and recognise which transformations produce congruent shapes Rationale In this module, learners start to apply what they have learned about algebra and geometry in the Cartesian grid.  First learners study coordinates in all four quadrants. Learners will be familiar with coordinates from work at primary school and in other subjects. The tasks in this unit will give learners opportunities to apply their understanding from previous units including negative numbers and geometric properties of triangles and quadrilaterals.  Next, the concept of area as a measurable quantity is introduced. This begins by revisiting arrays. 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. Lastly, learners are expected to consider how transformations acting on an object produce different images. 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 by whether they result in a congruent image. This is the foundation for exploring similarity and trigonometry later.

## summer 1 - fractions

 Skills Write a number as a product of primes   Find the highest common factor and lowest common multiple using the prime factorisation   Find squares, square roots, cubes and cube roots using prime factorisation   Calculate with the use of a calculator, including squares, cubes, square roots and cube roots    Recognise and name equivalent fractions  Convert fractions to decimals   Convert between mixed numbers and improper fractions   Compare and order numbers (including like and unlike fractions)   Convert simple fractions and decimals to percentages  Express one quantity as a fraction of another   Find a fraction of a set of objects or quantity  Find the whole given a fractional part  Multiply and divide fractions by a whole number or fraction  Solve word problems involving multiplication of a fraction by a whole number or fraction using models and equations to represent the problem  Add and subtract fractions and mixed numbers with like and unlike denominators   Calculate with decimals Knowledge Identify and use the language of factors and multiples, square numbers, cube numbers, prime number, triangular numbers Develop an appreciation for the fundamental theorem of arithmetic Use indices to record repeated multiplication    Interpret, use and explore multiple representations of fractions such as area diagrams, bar models and number lines Understand that a fraction is at once a proportion, a quotient and a number Understand the place of fractions on the number line, and appreciate the infinite number of rational numbers Know that a decimal is equivalent to a decimal fraction Rationale In this module learners formally explore fractions and their relation to decimals. The extension of the number system to rational numbers is a highly important stage in learners’ mathematical development. To begin, learners revisit the composition of numbers developed in unit 3. The uniqueness of the prime number decomposition 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. During this unit, indices are used for powers greater than 2 for the first time.  The next two units are dedicated to fractions, building on knowledge of fractions from KS2. Unit 14 has learnings exploring multiple interpretations of fractions. 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 notion of a percentage is dealt with briefly. Learners compare two or more fractions in a set and order them by their size. In Unit 15, learners extend their understanding of applying the four operations to non-integer values. Students find fractions of amounts by considering the multiplication of an amount by a fraction. 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.

## summer 2 - ratio and proportion

 Skills Compare two or more quantities in a ratio  Recognise and construct equivalent ratios  Express ratios involving rational numbers in their simplest form  Construct tables of values and use graphs as a representation for a ratio  Compare ratios by finding a common total value  Solve ratio and proportion problems in a variety of contexts  Express a quantity as a percentage of another  Convert between fractions, decimals and percentages Compare two quantities using percentages  Find a percentage of an amount with and without a calculator  Increase and decrease a quantity by a given percentage  Find a quantity given a percentage of it Knowledge Understand the concept of ratio and use ratio language and notation  Connect ratio with understanding of fractions  Understand percentages as a ratio of two quantities where one quantity is standardised to 100  Understand percentages as a fractional operator with a denominator of 100  Understand and interpret percentages over 100  Interpret a percentage as a fraction and decimal Rationale In this module, learners extend their understanding of fractions to percentages and ratio. This module forms an important bridge towards proportional reasoning. Unit 16 explores ratio notation, language, representations and contexts. The ratio notation of 𝑎: 𝑏 is unpicked using bar models. Learners 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 learners’ conceptual understanding of ratio.  In the final unit of Year 7, learners work with percentages as another representation of ratios and fractions. The different ways of interpreting fractions are again referenced. The unit progresses to the use of percentages to compare quantities and find a given percentage of a quantity. Learners 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 are used throughout the unit to deepen understanding.

## knowledge OrganiserS

A knowledge organiser is an important document that lists the important facts that learners should know by the end of a unit of work. It is important that learners can recall these facts easily, so that when they are answering challenging questions in their assessments and GCSE and A-Level exams, they are not wasting precious time in exams focusing on remembering simple facts, but making complex arguments, and calculations.

We encourage all pupils to use them by doing the following:

• Quiz themselves at home, using the read, write, cover, check method.
• Practise spelling key vocabulary
• Further researching people, events and processes most relevant to the unit.

## Maths mastery intro

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