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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

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 includes a purpose-built SOW for at-risk learners of not achieving 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, which can be 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.

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

Working with number & Geometric Reasoning (6 weeks)

Skills

  • Review the rules on indices with numbers and with algebraic expressions.
  • Review converting between ordinary numbers and numbers in standard form and calculate, compare, and order numbers in standard form
  • Use 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
  • 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 algebraic and geometric proof

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. Indeed, it could be that as much new geometry is being devised by geometers working outside of established university mathematics departments as within.

autumn 2

Statistics and Algebra (6 weeks)

Skills

  • 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
  • Experience the Capture/recapture method and know why it is used
  • Know that iterative means carrying out a repeated action.
  • Experience an iterative process to accurately calculate the solutions to a quadratic equation.
  • Understand the effect of transformations on quadratic equation in its function form relationship between translating a graph and the change in its function notation

Knowledge

  • Know the relationships between box-plots, cumulative frequency diagrams and histograms.
  • Know the differences and similarities between ‘cumulative’ and ‘accumulative’.
  • Know that the area under a velocity-time graph is the distance travelled.
  • Know that the tangent to a curve graph is a straight line that touches the graph at a point. the gradient at a point on a curve is the gradient of the tangent at that point.
  • Know that the straight line that connects two points on a curve is called a chord. the gradient of the chord gives the average rate of change and can be used to find the average speed on a distance-time graph.

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.

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.