# Mathematics

## Learning Journey & Sequencing Rationale

Mathematics can be applied in practical tasks, real life problems and within mathematics itself. The aim of the course is to develop mathematical vocabulary, improve mental calculation and use a range of methods of computation and apply these to 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

## Autumn 1

 Reasoning in Angles (3 weeks) and Probability (3 weeks) Skills Review of angles including angle-chasing accompanied with chains-of-reasoning (finding quantities by deductive reasoning about properties) and is a useful introduction to proof. Derive and utilise circle theorems for increased complexity angle-chasing accompanied with chains-of-reasoning is a useful introduction to proof: deductive reasoning about properties Use experimentation, observation, frequency tables, relative frequency as well as theoretical probability to make predictions about expectations of future events. Use sample spaces, two-way tables, tree diagrams and Venn diagrams to visualize combinations of independent or dependent events and to calculate conditional probability, probabilities of repeated events and repeated events without replacement. Knowledge The central idea of mathematics in general and geometry in particular, is the study of invariance. Most theorems in geometry can be seen as resulting from the study of what change is permitted that leaves some relationships or properties invariant.  For example, knowing that the angle in a semicircle, is an invariant 90 degrees provided that the point being moved remains on the circle; if it moves inside the circle, the angle is greater than 90 degrees, if it is outside it is less, which could mean that a valid definition of a circle would be the locus of points that subtend an angle of 90 degrees to the ends of a line segment. Geometry consists of three kinds of cognitive process: visualisation processes (for example, the visual representation of a geometrical statement), construction processes (when using tools, be these ruler and compass, or computer-based), and reasoning processes. Knowing that circle theorems require spatial thinking and visualisation and as well as deductive reasoning involving transformation or congruency arguments. These aspects are not separate; each gives rise to the other and each only exists in relation to the other. Rationale Geometry is one of the longest established branches of mathematics and remains one of the most important. Its development can be traced back through a wide range of cultures and civilisations with its origins in the surveying of land and in the design of religious and cultural artefacts.  If anything, geometry is becoming more important across many fields. This is not only because of the wide-ranging applications of geometry in everything from robotics to CGI (computer generated imagery) movies, from crystallography to architecture, from neuroscience to the very nature of our universe, but also because new geometrical ideas are being generated within such diverse fields as these. The challenge of making sense of situations that are unpredictable pervades everyday living.  Situations that lack information create conditions for uncertainty. Probability is one approach to reason about such uncertain situations. Uncertainty in some games and sports contests can be compelling and fun.  A player who can reason about uncertainty will be able to adopt strategic approaches, which will improve the player’s success rate in the long term.  Uncertainty is also apparent in everyday decision making.

## autumn 2

 Developing algebra thinking and argumentation (6 weeks) Skills Understand the difference between expressions, identities, equations Transform expressions including quadratics accompanied robust argumentation in the form of chains-of-reasoning Solve various equations including linear, quadratic, simultaneous linear and simultaneous non-linear and solve them by referring to real-life contexts. Solve quadratic equations other forms including find approximate solutions to quadratic equations graphically as well as by an iterative process. Solve inequalities and show the solution on a number line and using set notation; represent inequalities on graphs and interpret graphs of inequalities Sketch graphs of cubic functions, find the roots of cubic equations and solve cubic equations using an iterative process Knowledge Algebraic manipulation without any meaning or purpose is a source of mystery and confusion. Thus attributing ‘Meaning’ in algebra is crucial. ‘Meaning’ in school algebra comes from the way relations between quantities and variables are expressed. ‘Relations between quantities’ and ‘algebraic reasoning’ pervade mathematics. Know that solving simultaneous equations can be seen as the intersection of two functions and can be represented by a pair of values that ‘satisfy’ both equations, Know that a quadratic expression is a way of describing the area of a rectangular shape. Rationale Old Babylonian texts (1792-1750 b.c.e.) show a knowledge of how to solve quadratic equations, but it appears that ancient Egyptian mathematicians did not know how to solve them. Since the time of Galileo (1564 – 1642), they have been important in the physics of accelerated motion, such as free fall in a vacuum. The study of quadratics provides an arena for students to learn to go beyond visual assumptions and to use parameters as clues or tools to interrogate and display the characteristics of a function.

## spring 1

 Proportional reasoning (6 weeks) Skills Express variables, draw and interpret graphs of variables in direct and in indirect proportion Draw and interpret Trigonometric graphs and functions Using Pythagoras’ theorem and trigonometric ratios in 3D Knowledge Know that the fundamental concept behind ratio and proportional reasoning is the multiplicative relationship in which quantities, whether discrete or continuous, are compared using scalar multipliers. Know that the expert sense of ratio and proportional reasoning accumulate through use, over time, in many different mathematical, everyday, and scientific contexts. Learners need to know that they require repeated and varied experiences of ratio and proportional reasoning, over time, so that multiple occurrences of the words and the associated ideas and methods can be met, used, and connected whether consciously or tacitly. Understanding the constant relation between independent and dependent variables in direct and indirect proportional relations, indicated by k in y=kx, y=nkx and in y=1kx, y=1kn . Rationale Narrow ideas of ratio and proportion persist. These include ratio as the quotient of two numbers or the comparison as part to part and proportion as a comparison of part to whole or a comparison of ratios, with ‘proportional to’ indicating equality of ratios. These definitions complicate learners’ understanding of the constant relation between independent and dependent variables in proportional relations. The ratio between quantities is a comparison using the multiplicative relation, similar to the use of ‘difference’ to compare quantities in the additive relation. However, there is a crucial difference between ‘difference’ and ‘ratio’ which makes ratio significantly harder – its numerical form does not represent a magnitude in the same units as those it is comparing. When ratio can be counted the abstract nature of ratio is hidden in the concrete models and actions, but when it cannot be counted the ratio has significant imaginary qualities. Trigonometry (from the Greek for triangle measure) began as a branch of geometry, both for the practical task of surveying land and for astronomical calculations. Trigonometric functions are pervasive in many parts of pure and applied mathematics.  Working on trigonometry is an important theme because it provides a context for developing a relational (as compared to instrumental) understanding of the component concepts. For example, understandings of ratio can develop through work on similar right-angled triangles and the notion of sine, cosine, and tangent as multipliers, understanding of angle as a measure of turn develops through extension beyond 90 degrees. Trigonometry offers an opportunity to work on the algebra of functions in a meaningful context instead of being tempted to provide mnemonics and transformation aids, or memorised formulae.

## spring 2

 Vectors and compound measures (strengthening ratio & proportional reasoning) Skills Use vectors to strengthen ratio and proportion reasoning. Use symbolism associated with column vector notation including vector arithmetic, resultant vectors, parallel and co-linear vectors. Extend ideas of measurement to consider compound measures both continuous to strengthen ratio and proportion reasoning. Become fluent with the underlying structure of compound measures including covariation and how one variable changes in relation to another. Connect some compound measure to their formulae as a way to make sense of the formulae and not as a short-cut. Become familiar and fluent with ‘per’ is a key word in compound measures (e.g. miles per hour, or speed per second or grams per cubic centimetre, litres per kilometre, gallons per mile) or number of eggs per person or crisps per packet) and see that quantities to be compared do not have to be measured in the same units. Knowledge Know that ratios expressed as numbers can be scalar relations between quantities of the same stuff or a compound measure relating two kinds of stuff: either two types of continuous measures (speed, density, acceleration) or two different discrete items (eggs and people or crisps and packets). Appreciate that compound measures are also called ‘rates’ and are used interchangeably and are facilitated by fluency with ratio and proportion, fractions as division and gradients. Know that when fractions arise as a way to express rates, division is implied such as pounds per kilo, or miles per litre, or mass per cubic centimetre. Fluency with units of measure expressed as rates leads to less relying on remembering formulae and their short-cuts. Rationale Vectors are a useful topic to develop ratio and proportional reasoning and other the key geometrical ideas of symmetry, invariance, transformation, similarity, and congruence, and use these ideas to enable learners to build sound spatial and geometrical reasoning skills which are encompassed in engineering and physics.

## summer 1

 Representing algebraic functions (6 weeks) Skills Review of linear graphs (gradients and y-intercepts); Investigate for parallel and perpendicular lines and make connections to (arithmetic) sequences and grapple with Geometric sequences and connections to growth and decay, strengthen ratio and proportional reasoning. Create equations of circles and find the equation of a tangent and perpendicular to a circle Work with function notation including notation for inverse and composite functions Transform functions including cubic graphs and reciprocal graphs Knowledge Knowing multiple meanings of ‘function’ provides better support to see functions as processes which could be combined as compound functions. Appreciate that graphs are sense-making objects, and that use of the function notation expresses the associated relations. Know how to devise algebraic expressions and then test them out graphically to see if they gave the expected values and behaviour. Know something about inverse functions from work in Trigonometry. Geometric sequences could be renamed multiplier sequences. Use of Geometric circa 16c. referred to ancient ways of multiplying by referring to methods associated with land measurements from Geo – earth/ land and metry – measure. Rationale The concept of algebraic functions is foundational to modern mathematics, and essential in related areas of the sciences.  Algebraic functions are used in finance, engineering, and design to name three. Learning about the concept of algebraic function is complex, with many high performing students possessing weak function understandings. Conceptions and reasoning patterns needed for a strong and flexible understanding of functions are more complex than what is typically assumed by designers of curriculum. Students who think about functions only in terms of symbolic manipulations and procedural techniques are unable to comprehend a more general mapping of a set of input values to a set of output values; they also lack the conceptual structures for modelling function relationships in which the function value (output variable) changes continuously in tandem with continuous changes in the input variable. These reasoning abilities have been shown to be essential for representing and interpreting the changing nature of a wide array of function situations they are also foundational for understanding major concepts in advanced mathematics

## summer 2

 Geometry and Locating ‘signals’ in ‘noisy’ data sets Skills Visualise nets of 3D shapes and calculate surface area and volumes of prisms, pyramids, cones, and spheres Envisage measuring volume as ‘packing’ a space with (cubic) units while capacity is ‘filling’ a container with iterations of units of a fluid (or small grains such as sand or rice) that take the shape of the container Experience statistics as a form of enquiry: compute estimates for each of the three averages of ungrouped and grouped data, distinguish between mean, mode and median and know when each might be useful (advantages and disadvantages) and their differing characteristics, analyse and interpret the results in context and in comparison, to two or more data sets. Knowledge Know the distinction between volume and capacity:  volume as the amount of 3D space something occupies (perhaps measured in cubic centimetres), and capacity as how much liquid a container can hold (measured, say, in millilitres). Professions that work with masses of data seek to make sense of massive variation in order to find results that will be useful to them or beneficial to others or both. Until a ‘result’ is deemed to be useful it is merely a ‘signal’. The key organising idea when working with massive data sets with massive variation is as an exercise in Exploratory Data Analysis where the answer is unknown. Exploratory Data Analysis utilizes ‘noisy’ processes within which can be detected a ‘signal’ from which corroborated claims can be made. Studying statistics is two-fold. Firstly, learning the computation of different ‘noisy’ processes and how to justify their relevance. Secondly, learning how to make inferences and predictions from data with massive variation but supporting these inferences with convincing reasoning. In this vein informal inferential reasoning is a key tool for this process. In search for meaning in simple statistics, a family of concepts called ‘averages’ and spread are the ‘noisy’ processes that we use to refine the data. Once a ‘signal’ is detected (the typical value for the mean, mode, median or spread), learning to make inferences about its usefulness within the context of the data and convincingly supported by reasoning is key. Conflicts between what students believe and what the data suggests drive the search for explanations. Rationale The scope of statistics spreads across the curriculum. Statistical reasoning draws on context and those contexts relate to many disciplines in the school curriculum. Making sense of mathematics involves describing, modelling, comprehending, and grappling with mathematical variation.  Reasoning with statistics is a variant of mathematical variation and at the same time serves as a tool for making sense of data in other subject areas or to be used by citizens seeking to be better informed. Insofar as statistics is a tool for rooting out possible causes or associations, it is invaluable to the practice of many disciplines, albeit with added complexity. Statistics is a form of enquiry, centred on inference making when working with data arising out of experimentation or observation rather than simply computing numerical and graphical representations of data The notion of informal inferential reasoning is an essential tool for the modern statistically literate citizen who needs to be able to reason with and about data.

## 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.