Mathematics

 

Curriculum intent

Our Mathematicians pursue an understanding of the modern world and how mathematics shapes it. They will develop fluency in key mathematical skills and through this the ability to be inquisitive about connections between areas of mathematics. They are encouraged to be enthusiastic individuals who use logical reasoning to analyse real-life problems to propose solutions. We want them to be resilient in their work, willing to learn from their mistakes and feeling safe to take risks to find solutions to problems. Our learners are curious and confident to question both methods and solutions, reflecting on their interpretations of problems. Students are able to articulate their mathematical understanding through the use of key vocabulary, clearly demonstrating mastery. Once fluency and mastery are achieved, students look to make generalisations and conjectures that they are prepared to test through experimentation and reasoning. Students are aware of how mathematics is interleaved into various subjects and career opportunities and supported to reach their goals.

Purpose

In Hayes School our aim is to get all of our pupils to think like mathematicians. Mathematics is vital to the modern world and pupils need to understand its structure and its beauty. Our curriculum and teaching is delivered by passionate mathematicians who revel in the opportunity to deliver their subject and just what inspired them when they were starting on their mathematical journey. We want our pupils to enjoy their mathematics and believe in their abilities.

Key features of learning


Mathematics is a cyclic subject with layers that build on the previous ones to increase the depth and understanding of our pupils. We build on the foundations from primary school with a curriculum which reinforces them in year 7 but also starts to open out to show more of the potential of the subject. Whilst the school is split into three Key Stages, mathematics is a subject which isn’t necessarily bounded by these conventions. All years are vital and contain content which the pupils will ultimately be examined on at the end of year 11. We recognise the importance of revisiting topics throughout the years, both to support and extend them. Interleaving is a key aspect of our curriculum to ensure that students know and remember more.
The Mathematics department places great value in considering current research into how pupils’ best learn and thrive in the subject. We try to use our meetings to do mathematics and discuss approaches to teaching topics which we know that the pupils find challenging. Recently this has involved a big focus on ratio which we have identified as a topic which has changed a lot over the years as to how it is examined.
Pupils can expect to be using the following techniques in lessons:
Goal Free Problems – Questions where the ultimate goal of the question has been removed with the pupils being encouraged to write any relevant mathematical facts that they can about the situation given. This encourages pupils to try something when faced with longer questions on exams which can look quite daunting.
Same Surface, Different Deep – A set of four questions based around a single piece of information. This could, for example, be a picture of a shape where the pupils could be asked to find area, perimeter, angles etc. This is training the pupils to recognise the need for a wide variety of skills and how to choose the correct one given the specific question.
Low Stakes Quizzes – Pupils are sitting regular quizzes based on topics which they have done over the last few months. The spacing out of these topics in the quiz is helping the pupils with their retention of knowledge. It is also supporting the pupils to prepare for being tested in more formal assessments by giving them practice in less stressful situations.
Confidence Scores – Pupils are regularly asked to provide confidence scores on questions which they have answered on how confident that they feel an answer is correct. This is partly to identify situations where the pupils feel confident with their answer but it is actually wrong. This is tapping into the ‘hyper-correction effect’ where we are more likely to remember something when we have been surprised by our initial answer being wrong.
Mathematics at A Level continues to build on the content which the pupils studied throughout the previous five years. It is one of the most popular A Levels in the school. We are also able to offer Further Maths as an A Level option.

How does our curriculum shape learners?


The subject of mathematics is one where people are quick to dismiss their abilities and almost wear their struggles as a badge of pride in later years. Pupils will leave us at a variety of ability levels, but they should all recognise that mathematics is a subject which they can do. We see it as vital that we aren’t just teaching the pupils so that they can gain a qualification but to develop key life skills. This is not just in their overall numeracy but also how they can use the problem-solving skills that they are taught in a variety of contexts.

Key concepts that underpin the curriculum
1 Recognise patterns and form generalisations
2 Proficiency in number
3 Fluency – Factual and Contextual
4 Application of algorithms
5 Problem solving
6 Analysis
7 Reasoning and Justification
8 Conjecture

 

End points for Academic Years


Year 7 By the end of Year 7 students can:

Number sense: Recognise patterns in linear sequences and generalise to the process for finding nth term. Understand meaning of equality in maths and generalise using expressions. Compare decimals and negatives and multiply by powers of 10, FDP equivalence, fraction of an amount, develop mental methods for arithmetic, fraction arithmetic, make and test conjectures. Concepts covered: 1, 2, 4
Algebra: Introduce conventions and apply algorithms for collecting like terms and forming expressions. Use function machines and recognise link to nth term and sequences. Use factual fluency to deduce value of unknowns (solve). Concepts covered: 1,3,4
Statistics: Analysis of pie charts brought up in context – understand proportional relationship between frequency and sector size. Concepts covered: 6 ,
Geometry: Have factual fluency for angle facts, and application of algorithms for constructing triangles from as few as three facts (eg SSS). Make conjectures about the sum of interior angles of polygons using prior knowledge of triangles and quadrilaterals. Concepts: 3,5,8.
Probability: Reasoning and justification by interpreting Venn diagrams, probability scale, sample spaces Concepts covered: 7

Year 8 By the end of year 8, students...
Number: Students will apply algorithms for manipulating ratios (eg to 1:n), percentage change using decimals multipliers, converting numbers between standard and ordinary forms. Concepts covered 4. Students will have developed proficiency in number sense by visiting multiplicative change and scale (proportional reasoning). Concepts covered 2. They have been encouraged to analyse solutions by estimation and error intervals Concepts covered 6
Algebra: Students will recognise patterns and form generalisation through learning about sequences and plotting them on the Cartesian plane (recognise pairs of x,y can be plotted and derived from formulae and explore concept of gradient). Concepts covered 1. They can apply algorithms for expanding brackets, factorising expressions, solve equations involving brackets etc, solve with unknowns on both sides and index rules. Concepts covered 4 and 7
Geometry: Students are more confident in applying algorithms for calculating areas of polygons and circles and solving problems in context, and have gained fluency in how to reflect shapes in lines. Concepts covered 4 and 5
Statistics: Students can analyse and interpret scatter graphs (correlation, line of best fit etc), interpret grouped frequency tables, two-way tables, measures of location. Concepts covered 6 and 8
Probability: Students can calculate probabilities from Venn diagrams, two-way tables and arriving at the product rule through reasoning and interpreting sample spaces. Concepts covered 7
Year 9 By the end of year 9, students...
Algebra: Students are able to expand double and triple brackets. They can factorise monic quadratic expressions including the difference of two squares, and draw straight line graphs from the equation including identifying gradient and y intercept and explaining the effect these have on the line. Concepts covered: 1, 2, 4, 6
Number: Students can work with percentages including repeat percentage, reverse percentage and compound interest. Students have investigated both direct and inverse proportion including algebraically and graphically. Concepts involving proportion have been introduced including, speed-distance-time, mass-density-volume and pressure-force-area. Standard form can be explained and students should be able to complete calculations using it. Students have built on the concept of rounding and can use upper and lower bounds including calculations with them and find error intervals. Concepts covered: 2, 3, 4, 5
Shape: Students have discovered Pythagoras’ theorem and can apply it to many situations. Students also encountered right-angled trigonometry and can use it to find both missing sides and angles. Concepts covered: 1, 2, 3, 4, 5
Probability: Students have built on their knowledge of the product rule when creating tree diagrams to calculate multi event probabilities including independent and conditional situations. Concepts covered: 2, 3, 5, 7
Year 10 By the end of year 10, students...
Algebra: Pupils have extended their knowledge of sequences by considering quadratic examples. The pupils who have considered quadratic equations in previous years and will now consider how to solve harder examples using the quadratic formula or completing the square method. Students can solve linear, quadratic and graphical inequalities. Students can identify and draw circles with an equation centred at the origin. Concepts covered: 1, 2, 4, 5
Number: Students have built on their knowledge of ratio and solve more complex problems. Students have investigated the properties of surds and can complete calculations with them. Concepts covered: 2, 4, 8
Data: Students can accurately draw various new charts such as histograms, cumulative frequency graphs, frequency polygons and boxplots and can read and calculate information from them. Concepts covered: 1, 3, 4, 5, 6, 7, 8
Shape: Students have built on their knowledge of reflections and are able to translate, rotate and enlarge shapes as well as describe all four transformations. Knowledge of constructions now includes bisecting lines and angles and completing questions on loci. Students can find volume of more complex solids such as pyramids, spheres and cones. Concepts covered: 2, 4, 5, 7
Probability: Pupils extend their knowledge of venn diagrams to include 3 categories and set notation. Concepts covered: 2, 3, 5, 6, 7
Year 11 By the end of year 11, students...
Students have expanded their algebraic knowledge, building on the representation of expressions in previous years to consider function notation. This also supports an extension to their knowledge of algebraic graphs when we consider transformations of graphs and the use of function notation to describe this. Students have extended their knowledge of solving equations to consider simultaneous equations, with some considering mixtures of quadratic and linear equations. Students can now consider proportional relationships, including direct and inverse examples. Concepts covered: 1, 2, 3, 4, 5, 7, 8
Velocity/Time graphs are introduced with basic links established to SUVAT equations and substitution into them. Some pupils will build on this if they take the subject at A Level. Concepts covered: 2, 3, 5
Angles is a recurring topic throughout the years in mathematics and the pupils build on their prior knowledge to look at a new set of relationships involving circles. Concepts covered: 1, 2, 3, 4, 5, 7, 8
Year 12  By the end of year 12, students...

Students will be able to recognise patterns and form generalisations in all areas of mathematics, and will be able to reason mathematically in order to test or prove the truth of various mathematical statements.
Algebra remains the cornerstone of a significant amount of the content covered. Students will have reinforced skills covered in previous years but take their learning forwards into binomial expansion and geometric problems involving circles. They have a deeper algebraic understanding of the concept of proof.
The topic of calculus (differentiation and integration) is introduced with students now using these skills to solve problems in a more logical and reasoned way. Students will also have a developed understanding of links between these concepts and problems in mechanics are made.
Students are also introduced to the concept of exponentials and logarithms, further expanding their skills of algebraic manipulation.
Students meet the ‘SUVAT’ equations in KS4 but their derivation and use are expanded in year 12 mechanics. They will have a greater confidence in how to solve problems involving forces, looking at Newton’s laws.
Statistics builds on knowledge from KS4 and introduces the concept of testing hypotheses, expanding the students’ ability to interpret the relevance of statistical tests. Through their analysis of large sets of weather data they better appreciate the practical applications of their skills.
Further mathematicians will have developed sophistication and fluency across the full spectrum of mathematics, from an understanding of the logical and axiomatic foundations of the subject to developing problem solving skills at an undergraduate level in both pure and applied mathematics, including engineering and science.

Complex numbers: Students have greater knowledge of number sets to include complex numbers, which are crucial to many applications of maths, including engineering and physics. They also have a greater knowledge and understanding of complex number arithmetic, complex solutions of polynomial equations, and geometric representations of complex numbers.

Matrices: Students will have expanded their knowledge of number representations to include matrices. They will have explored the array structure of matrices, matrix arithmetic and algebra, and how matrices are used to describe linear transformations that are crucial in many areas, including computer science and physics.
Mathematical proof: Students will be introduced to the concept of inductive proof, and will be able to use this logical “domino” concept in order to formally prove the truth various mathematical statements.
Decision mathematics: Decision mathematics is a branch of mathematics that is both new to pupils and relatively new in the development of mathematics as a subject. The overarching theme of decision mathematics is optimisation and it therefore has many real world applications such as in business, economics and computer science. Students understand the mathematics that underpins: how a sat nav determines the shortest route; how a business can maximise its profit and minimise its costs; how a rail network can inspect all of its tracks by covering the shortest possible distance; and how a construction company plans a skyscraper project to be completed in the quickest time whilst accounting for all of the interconnected activities.
Series: Students have built on their basic knowledge of sequences and series, with the key development for pupils now being a formalisation of the notation and arithmetic of series, which equips students with the mathematical language required to explore the more advanced mathematical analysis and proof concepts that arise throughout these A Levels.
Matrices, complex numbers, roots of polynomials, series, proof by induction, decision maths

Year 13  By the end of Year 13, students...
Calculus: Students have built on the concepts covered in year 12 and can now apply a greater range of techniques to solve problems. Practical examples are covered to show how differential equations, for example, can be applied to ‘real life’ problems. Concepts covered: 1, 2, 4, 5
Trigonometry: Earlier in the curriculum, students have started solving ‘simple’ trigonometric equations and they build on this in year 13 with the introduction of new identities and ratios. By the end of the year they will use these skills to support their ability to solve problems involving calculus. Concepts covered: 2, 4, 5, 7
Algebraic manipulation skills continue to develop with a focus on fractions. Students also consider situations where a new parameter is introduced and how this expands their ability to solve problems. Students also consider situations where equations can’t be solved and more complex methods of estimating answers are introduced. Concepts covered: 1, 4, 8
Sequences and Series are built upon supporting pupils’ ability access problems involving compound interest. Concepts covered: 1, 5
Statistics: Pupils build on the distributions covered in the previous year by looking at the normal distribution. Recognition of tests for correlation are also considered to aid their appreciation of links between variables and thus better equipping them to better appreciate statistics from the world around them. Concepts covered: 2, 3, 4, 5, 6
Mechanics: Pupils continue their skills in Mechanics with the formal introduction of friction into problems, thus making them more relevant to the modern world. The concept of moments, turning forces, is also introduced, expanding the pupils’ skill set to access problems of a more complicated nature. Concepts covered: 3, 5, 7
Further Maths  Further Maths:
In addition, by the end of year 13, the students studying further maths at A level…
The students at A Level will be studying calculus with the further maths students having a more developed understanding of complicated differential equations and how they link to more complex ‘real world’ problems. This further demonstrates the links between what they study in class and how they might apply their knowledge to in later careers. They will also look at the further methods to solve problems involving calculus thus expanding the repertoire of skills that they are able to apply. Concepts covered: 3, 4, 5, 6
Students will all study Cartesian coordinates from primary school onwards but the further maths students explore polar coordinates and can use it as a new way to define positions in a 2D space. Concepts covered: 4, 5
Complex numbers are covered in year 12 with the year 13 students continuing this to look at De Moivre’s theorem and how this links to series. Concepts covered: 2, 5
Hyperbolic functions: Students have furthered their knowledge of different types of functions to include hyperbolic functions, which arise from the unit hyperbola in the same way that trigonometric functions arise from the unit circle. Hyperbolic functions help mathematicians understand and explore hyperbolic geometry (a non-Euclidean geometry) and help architects, engineers and scientists understand naturally occurring shapes, such as that of a freely hanging chain, suspension bridges and cobwebs. Concepts covered: 5

Students look at a unit of further mechanics which compliments work that they do in Physics, this being a popular choice of subject for our Further Mathematicians. This expands on the concepts covered in the normal A Level to look at momentum, collisions, Hooke’s law and the topics of Work/Energy/Power. The increased level of work continues to add a level of ‘realism’ to the problems that they can access. Concepts covered: 5, 6, 7

The second ‘applied’ unit that the pupils look at is about Decision Maths. Students will have explored the applications of algorithms; these may be to find minimum connectors between points, the shortest route from one place to another or how we can use algorithms to schedule tasks to finish in the most efficient way, for example. They can also employ the use of inequalities to calculate the maximum profit or minimum loss from a business model. There are real world applications here to planning tasks and some clear links to computer programming. Concepts covered: 4, 5, 6, 7



What will you see in Maths Lessons?

Intelligent practise;
Clear objectives;
Strategies encouraging meta-cognition
Supportive maths conversations;
An environment where mistakes are welcome as an opportunity to learn;
Whole class questioning and dialogue;
Differentiation that supports and challenges;
Integration for inclusion
AfL to assess for learning;
Use of AfL to support pupil progress and, if required, to adjust learning throughout the lesson;
Misconceptions are addressed;
Qualified mathematicians who plan appropriately;
Modelling through teacher demonstration

 

What will you see in Maths books?

Date and title
Clear understanding of learning objective
Keywords and definitions
Notes and exemplars
Self-assessment in red
Teacher feedback in purple
Mini-quizzes
Intelligent practice and minimally different progressions
Homework in the back of the book
Do Now tasks to engage students and link to prior knowledge as they enter the classroom

What formative assessment will you see in Maths?

Mini-white boards
Open questions that are carefully scaffolded and targeted to support pupil progress
Cold calling
Targeted and specific verbal feedback
Present misconceptions encouraging meta-cognition
Opportunities for discussion
Confidence scores
Differentiation by outcome allowing teachers to gauge progress
Mechanisms that allow teacher to gauge understanding (eg tallies, thumbs up/down etc)

What is the department currently reading and why?

Mr Barton – How I wish I taught Maths (following an inspirational visit from Mr Barton to learn more about his methods)
Jo Morgan – Compendium of Mathematical Methods (to discover new methods for topics that we already teach)
Daniel Sobell and Sara Alston – The inclusive classroom, a new approach to differentiation (to ensure that vulnerable pupils are well supported)
Becky Francis, Antonia Tereshchenko and Becky Taylor – Re-assessing ability grouping (to support our mixed attainment teaching at KS3)
Chris McGrain and Mark McCourt – Mathematical tasks, the bridge between teaching and learning (to support gaps and encourage learning after lockdown)
Tom Sherrington – The learning rainforest; great teaching in real classrooms
Mathematical Beauty (Daniel Pearcy) - To further explore the depth, connectedness and beauty of the maths that we teach to our pupils, so as to enhance pupil enjoyment and engagement in my lessons.

 

Click here to access the KS3 Maths curriculum map. 

Click here to access the KS4 Maths curriculum map. 

Click here to access the KS5 Maths curriculum map. 

Click here to access the KS5 Further Maths curriculum map.