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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.

Sequence

Mathematics is an interconnected subject in which pupils need to move fluently between representations of mathematical ideas. The programme of study for key stage 4 is organised into apparently distinct domains, but learners develop and consolidate connections across mathematical concepts. They should build on learning from key stage 3 to develop further fluency, mathematical reasoning and competence in solving increasingly complex problems. They also apply their mathematical knowledge wherever relevant in other subjects and financial contexts.  The expectation is that the majority of learners will move through the programme of study at broadly the same pace. However, decisions about when to progress will be based on the security of learners’ understanding and readiness to progress. Learners  who grasp concepts rapidly are challenged by being offered rich and sophisticated problems before moving on to new content. Those who are not sufficiently fluent with earlier material consolidate their understanding, including through additional practice, before moving on.

Year 10

At the beginning of year 10, learners review angle-chasing accompanied with chains-of-reasoning (finding quantities by deductive reasoning about properties) and is a valuable introduction to proof, leading to deriving and utilising circle theorems for increased complexity.

After this review, learners then move on to Probability, where deductive reasoning can be used in experimentation, observation, frequency tables, relative frequency, and theoretical probability to predict future events’ expectations.

As a continuation from KS3, learners review the use of sample spaces, two-way tables, tree diagrams and Venn diagrams to visualise combinations of independent or dependent events and to calculate conditional probability, probabilities of repeated events and repeated events without replacement.

The second half of the Autumn term will see learners develop algebraic thinking and argumentation. This will involve solving various equations, including linear, quadratic, simultaneous linear and simultaneous non-linear and solving them by referring to real-life contexts. They will begin solving quadratic equations, including finding approximate solutions to quadratic equations graphically and by an iterative process. They will solve inequalities and show the solutions on a number line and use set notation; represent inequalities on graphs and interpret graphs of inequalities. Learners of high ability will be expected to sketch graphs of cubic functions, find the roots of cubic equations and solve cubic equations using an iterative process.

In the Spring term, learners will focus on proportional reasoning and draw and interpret graphs of variables, namely, of direct and indirect proportion. Additionally, they will draw and interpret trigonometric graphs and functions using Pythagoras’ theorem and trigonometric ratios in 3D.

Vectors help develop ratio and proportional reasoning and offer the vital geometrical ideas of symmetry, invariance, transformation, similarity, and congruence. These ideas enable learners to build sound spatial and geometrical reasoning skills encompassed in engineering and physics. 

Learners will become fluent with compound measures’ underlying structure, including covariation and how one variable changes in relation to another. As learners connect compound measurements to their formulae, they will make sense of them and not use them as shortcuts. This phase of learning will encourage familiarity and fluency with ‘per’ as a keyword 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 those quantities compared do not have to be measured in the same units.

In the summer term, learners will be introduced to algebraic functions. They will begin with a 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, again strengthening ratio and proportional reasoning. Learners of high ability will create circle equations and find the tangent equation and perpendicular to a circle. The first half of the summer term will conclude with function notations including notation for inverse, composite functions and transforming functions including cubic graphs and reciprocal graphs.

The second half of the summer term will see learners visualise nets of 3D shapes and calculate surface area and volumes of prisms, pyramids, cones, and spheres. They will 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.

The year 10 POS will conclude with the 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 comparison, to two or more data sets.

Year 11

In year 11, learners will, in the Autumn term, revisit numbers & geometric reasoning. This will begin with a review of the rules on indices with numbers and with algebraic expressions. They will review converting ordinary numbers and numbers in standard form and calculate, compare, and order them in standard form.

To enhance geometric reasoning, learners will practise using a ruler and compass to construct angles, triangles, other regular polygons, the perpendicular bisector of a line, the shortest distance from a point to a line, and the bisector of an angle. They will use congruency arguments as a form of deductive reasoning. Draw a locus and use loci to solve problems use chains-of-reasoning to support explanations in an algebraic and geometric proof.

The second half of the autumn term will begin with Statistics and conclude with Algebra. Learners will learn to find the quarter and the interquartile range from stem-and-leaf diagrams.

Draw and interpret box plots, cumulative frequency tables, stem-and-leaf diagrams and histograms; find and interpret the median, quartiles and interquartile range, frequency density Understand frequency density in them. Furthermore, they will experience the Capture/recapture method and know why it is used. Learners will conclude the term by learning that iterative means carrying out a repeated action. They will use an iterative process to calculate the solutions to a quadratic equation accurately. The more able learners will understand the effect of transformations on quadratic equations in its function form relationship between translating a graph and the change in its function notation.

Our Year 11 SoW includes plenty of opportunities for scaffolding, and stretch and challenge. Regular assessment points with high-level QLA further inform teaching and learning. Our holistic and inclusive approach consists of a purpose-built SOW for at-risk learners of not achieve a grade at KS4. (Access to foundation SOW). The SoW reaches maturity at the end of the Autumn term, at which stage teachers use a portal called Pinpoint Learning, a focused solution for targeted needs. Pinpoint learning materials help teachers target a specific area of need in their classroom with clarity and flexibility. The Teacher Combined QLA Matrix on Pinpoint Learning contains the class average for each assessment taken at HHS, ordered from the earliest to the latest test. As topics repeat over multiple tests, you can see class progress, filtered to consistent weaknesses, strengths, and improved areas. Additionally, every student has a matrix for their independent revision! As we upload more tests after mocks, this becomes incredibly powerful as the exams approach at the end of year 11.

An example of the Teacher Matrix (filtered for consistent weaknesses)

 

 

 

 

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.