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. There is no coursework.

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

 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   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   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 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  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 All the content is from Year 7. These units were part of Year 7 Summer 1 and are to be retaught. 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, the uniqueness of the prime number decomposition is explored. This is used to show properties of particular numbers. Prime factorisation is used to determine the highest common factor and lowest common multiple.  The next two units are dedicated to fractions, building on knowledge of fractions from KS2. 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. Learners compare two or more fractions in a set and order them by their size. Learners extend their understanding of applying the four operations to non-integer values.Bar models, line models and rectangular representations are used heavily throughout to represent the fractional amounts.

autumn 2 - algebra

 Skills Generate terms of a linear sequences  Generate terms of a non-linear sequences  Identify different types of linear and non-linear sequences  Find a given term in a linear sequence Generalise the position to term rule for a linear sequence (nth term)  Derive equations and inequalities from contexts  Solve linear equations with an unknown on one side (revise from Year 7)  Solve linear equations with an unknown on both sides  Solve equations involving fractional terms and brackets  Forming and solving inequalities with unknown on one side  Forming and solving inequalities with an unknown on both sides  Represent the solution to an equation or inequality on a number line Knowledge Recall the key features of a linear sequence Understand the differences between linear and non-linear sequences Appreciate how the nth term rule relates to the nth position of a sequence Understand the meaning and conventions of the equals sign and inequality signs Classify expressions, equations, inequalities and identities  Use the language of solve, solution and unknown Interpret the solution to an equation based on the context from which it is derived  Interpret relationships expressed as inequalities Rationale This module gives learners the opportunity to develop and formalise the algebra they have become familiar with in Year 7. It has an increased level of challenge and complexity. This module begins by studying sequences. 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. In year 7 students explored the nature of equality and solved equations with one unknown where the unknown appeared on one side. In Unit 2 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. Equations are derived from familiar contexts and the solutions to these equations are interpreted within that context.  In unit 3, 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.

spring 1 - graphical representations

 Skills 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  Identify the equations of horizontal and vertical lines Identify parallel lines from algebraic representations  Plot coordinates from a rule to generate a straight line  Find the equation of a line from its graph Rearrange an equation in two variables into different forms Knowledge 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  Understand how the coordinate system is an infinite set of points Know what it means for a value to satisfy a rule Recognise a graph as representation of points that satisfy a rule Develop a rule into an algebraic representation  Develop concept of gradient using graphs of the form 𝑦 = 𝑎𝑥 before moving to equations of the form 𝑦 = 𝑎𝑥 + 𝑏  Identify key features of a linear graph including the y-intercept and the gradient  Make links between the graphical and the algebraic representation of a linear graph  Recognise different algebraic representations of a linear graph Rationale All the content in italics is from the Year 7 curriculum. The transformations unit was disrupted in the Spring of 2020. In this module, learners apply what they have learned about algebra and geometry in the Cartesian grid.  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.  THe next unit is students’ first formal introduction to straight line graphs. This begins with the plotting of discrete points beginning with 𝑛 = 1. The 𝑛-axis is replaced by the 𝑥-axis and discrete points are replaced with a continuous line to represent all coordinate pairs. 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 𝑦 coordinates. Learners are also able to explore the contexts of parallel lines and similar triangles. 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.

spring 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  Calculate with speed, distance and time Calculate with other compound measures such as density Check an equation to ensure the units are consistent on both sides Solve proportion problems, involving direct and indirect proportion Represent proportional relationships using tables and graphs  Represent proportional relationships algebraically Knowledge Understand the concept of ratio and use ratio language and notation  Connect ratio with understanding of fractions  Explore ratios in different contexts including speed and other rates of change Contrast ratio relationships involving discrete and continuous measures  Use speed and other rates of change to draw and interpret graphical representations  Explore density and concentration as other contexts for proportional relationships  Explore contexts involving proportional relationships  Understanding about graphs of proportional relationships  Meaning and properties of inverse proportional relationships  Investigate constant area as a context for indirect proportion  Represent inverse proportion relationships algebraically Rationale All the content in italics is from the Year 7 curriculum. In this module we combine the ratio unit in year 7 with the units on ratio and rate from year 8. This is a natural combination since the year 8 module involved a thorough recap of the year 7 content originally. Time is spent in the first unit reinforcing the notion of a ratio as an expression of a constant multiplicative relationship which can be between 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 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.  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. In the second week of this unit, learners will meet the concept of inverse proportion. Learners will encounter this in different contexts, notably by studying perimeter for a constant area.

Summer 1 - represent and reason with data

 Skills Find the mean, median mode and range from raw datasets  Use the mean, median and mode to compare data sets  Use an average plus the range to compare datasets  Find the mode, median and mean from tables and graphical representations (not grouped) Classify and tabulate data   Conduct statistical investigations using collected data   Construct scatter graphs  Examine clusters and outliers  Use a scatter graph to plot a line of best fit  Use a line of best fit to interpolate and extrapolate inferences Knowledge Understand the difference between an average and a measure of spread Have an understanding of when one average may be more appropriate than another Begin to understand the data handling cycle informally Explore methods of data collection including surveys, questionnaires and the use of secondary data  Appreciate the difference between discrete and continuous data Recognise the differences between univariate and bivariate data, and how they may be represented Analyse the shape, strength and direction to make conjectures for possible bivariate relationships Rationale This module explores a variety of methods of presenting data, with an emphasis on interpretation as well as production.  In the first unit, 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. 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. 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. In the second unit of the half term, students extend their understanding of statistical diagrams and measures to bivariate data. Students present the data in tables and in a scatter graph. They examine relationships between point to make simple inferences about association and covariation. The difference between correlation and causation are introduced and the idea of an explanatory variable.

summer 2 - area, volume and surface area

 Skills Round numbers to a required number of decimal places  Round numbers to a required number of significant figures  Identify rounding errors  Estimate quantities in a variety of contexts including area and perimeter  Identify and reason if an estimate is an over- or under-estimate  Use and apply formulae, including those with powers and roots Calculate with the area and circumference of a circle Area and circumference of a semi-circle and other sectors  Area and perimeter of composite shapes involving sectors of circles  Finding the volume and surface area of cuboids   Finding the volume and surface area of other prisms including cylinders  Finding the volume and surface area of composite solids  Solving equations  Convert between different units of area and volume Knowledge Appreciate the uses of rounding in different contexts Understand the concepts of length, area, volume and surface area Recall the anatomy of a circle Explore relationship between circumference and diameter/radius  Formula for circumference  Explore relationship between area and radius  Know the formula for area of a circle  Naming prisms, nets of prisms and using language associated with 3-D shapes Rationale In this module we have rearranged the unit on accuracy and rounding to go before work on measures of area and volume. This enabled time to be spent on year 7 units earlier in the year, and the accuracy unit complements the other units on measure. The units on angles originally in Year 8 will be taught in Year 9. ‘Accuracy and estimation’ provides an opportunity to consolidate understanding of rounding to a given decimal place which most learners will have met at primary school. 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. Estimation is encountered in a variety of contexts and is an opportunity to practice rounding and unit conversions.  Next, learners study circles. Learners explore the connection between the circumference of a circle and its diameter and through this are introduced to pi. Software and other visuals are used to give students the opportunity to see how formulae are derived. The unit ends with opportunities for students to apply their understanding to geometric problems involving the area and circumference of a circle.  Finally, learners formally meet volume as a measure of the space inside a 3D object. Students may need to revisit the names and properties of 2-D shapes before moving onto 3-D. Time is spent building and breaking down 3-D shapes, both with blocks and as nets. 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.

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.