# 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.

## knowledge Organiser

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.