Maths
Subject/Curriculum Leader: Mr G Atkinson
The Mathematics Department at Norton College is made up of enthusiastic mathematicians, eager to spread both their passion for and understanding of the discipline of mathematics. As a team we are one of the largest departments in the school, with 4 full time teaching staff and two part-time staff. The department has five Maths classrooms that are fully equipped with a large Interactive displays and Wi-Fi connection.
The Maths department aims to equip all students with transferable numeracy skills; skills that will be used both within and beyond school. The department believes strongly in fostering an ethos and appreciation for a lifelong use of Maths as well as for meeting the demands of the curriculum. We invite all of our students to embrace the same level of resilience and understand that, although it is satisfying to solve a problem, the journey and mistakes made along the way are just as (if not more!) important.
Engaging, interactive and rigorous lessons encourage our students to discuss concepts, experiment with different methods, and compare solutions. We employ a variety of teaching techniques in our lessons to achieve the above such as jigsaws, loop card, and treasure hunts, while ensuring that in every lesson students have the quiet time necessary to answer questions independently as well.
The department has a range of extracurricular activities on offer. Every year a large number of students across all years are given the opportunity to take part in the UKMT Maths Challenges and we also take part in the Team Challenge events. As well as being fun these Maths Challenges undoubtedly have a hugely positive impact on students’ achievements at both GCSE and A Level. There is also a Chess club, where all abilities are welcome, and students enjoy learning a new game and, at times, beating their teachers.
We encourage students to have a Casio fx-83GTX calculator in every lesson from Years 7-11. Students will also be expected to arrive to all lessons with a full set of equipment: pen, pencil, ruler, protractor, a pair of compasses and a scientific calculator.
Key Stage 3
All pupils are provided with a student friendly scheme of learning, which is referred to as a learning journey. This is attached to the inside of their exercise books and allows them to see how the lessons within a topic support each other.
The Key Stage 3 course is covered over a period of 3 years and covers the main content domains;
o Number
o Algebra
o Ratio and Proportion
o Geometry and Measures
o Probability
o Statistics
We follow a Teaching for Mastery scheme of learning in Years 7, 8 and 9. There is a termly plan for both year groups. Every half term is split into blocks that ensure students spend enough time mastering a deep understanding of the topic being covered.
Our scheme of learning is designed with interleaving (revisiting topics within new contexts) as a key element. For example, Year 7 starts with developing algebraic thinking and further development of algebraic skills is then woven throughout the year so students reinforce and extend their knowledge and understanding.
We firmly believe that students who are successful with number are much more confident mathematicians, so we have continued to emphasise number work throughout. We also recognise, however, that arithmetic can be barrier to some students accessing other areas of the curriculum, so we have also incorporated the teaching and learning of calculator skills throughout the curriculum.
Year 7 - Terms 1 & 2:
Sequences – sequences are explored in detail using both diagrams and lists of numbers. Technology is used to produce graphs so students can appreciate and use the words “linear” and “non-linear” linking to the patterns they have spotted. Calculators are used throughout so number skills are not a barrier to finding the changes between terms or subsequent terms.
Algebraic Notation – The focus of this topic is developing a deep understanding of the basic algebraic forms, with more complex expressions being dealt with later. Function machines are used alongside bar models and letter notation, with time invested in single function machines and the links to inverse operations before moving on to series of two-step function machines and substitution into short abstract expressions.
Equality and Equivalence – In this section students are introduced to forming and solving one-step linear equations, building on their study of inverse operations. The equations met will mainly require the use of a calculator, both to develop their skills and to ensure understanding of how to solve equations rather than spotting solutions. This work will be developed when two-step equations are met in the next place value unit and throughout the course. The unit finishes with consideration of equivalence and the difference between this and equality, illustrated through collecting like terms.
Place Value and Ordering – In this unit, students will explore integers up to one billion and decimals to hundredths. Using and understanding number lines is a key strategy explored in depth and will be useful for later work on scales for axes. After being taught how to put numbers in order, this is a suitable point to introduce both the median and the range, separating them from other measures to avoid getting them mixed up. Rounding to the nearest given positive power of ten is developed, alongside rounding to one significant figure. Decimal places will come later, again to avoid too similar concepts being covered at the same time.
Fractions, Decimals and Percentages – Building on the recent work on decimals, the key focus for this topic is for students to gain a deep understanding of the links between fractions, decimals and percentages so that they can convert fluently between those most commonly seen in real-life. Whilst looking at percentages, pie charts will be introduced. In addition, various forms of representation of any fraction will be studied, focusing on equivalence. The focus is very much on a secure understanding of the most common fractions under one, but fractions above one will be touched upon.
Year 7 - Terms 3 & 4
Addition and subtraction – The focus for this topic is building on the formal methods of addition and subtraction students have developed at Key Stage 2. All students will look at this in the context of interpreting and solving problems. For those for whom these skills are secure, there will be even more emphasis on this. Problems will be drawn from the contexts of perimeter, money, interpreting bar charts and tables and looking at frequency trees. We believe all these are better studied alongside addition and subtraction rather than separately. Calculators should be used to check and/or support calculations, with significant figures and equations explicitly revisited.
Multiplication and division – This topic is dedicated to the study of multiplication and division, so allowing for the study of forming and solving two-step equations both with and without a calculator. Unit conversions will be the main context as multiplication by 10, 100 and 1000 are explored. As well as distinguishing between multiples and factors, substitution and simplification are also revisited and extended. Again, the emphasis will be on solving problems, particularly involving area of common shapes and the mean. Choosing the correct operation to solve a problem will also be a focus. There will also be some exploration of the order of operations, which will be reinforced alongside much of the content next term when studying directed number.
Fractions and percentages of amounts – This topic focuses on the key concept of working out fractions and percentages of quantities and the links between the two. This is studied in depth in Year 8.
Directed numbers – This block is designed to extend and deepen their understanding of directed number. Multiple representations and contexts will be used to enable students to appreciate the meaning behind operations with negative integers rather than relying on a series of potentially confusing “rules”. As well as exploring directed number in its own right, this block provides valuable opportunities for revising and extending earlier topics, notably algebraic areas such as substitution and the solution of two-step equations.
Arithmetic with fractions – This block builds on the Autumn term study of “key” fractions, decimals and percentages. It will provide more experience of equivalence of fractions with any denominators, and to introduce the addition and subtraction of fractions. Bar models and concrete representations will be used extensively to support this. Adding fractions with the same denominators will lead to further exploration of fractions greater than one.
Year 7 - Terms 5 & 6
– Constructing, measuring and using geometric notation Students will build on their skills using rulers, protractors and other measuring equipment to construct and measure increasingly complex diagrams using correct mathematical notation. This will include three letter notation for angles, the use of hatch marks to indicate equality and the use of arrows to indicate parallel lines. Pie charts will be studied here to gain further practice at drawing and measuring angles.
Developing geometric reasoning – This block covers basic geometric language, names and properties of types of triangles and quadrilaterals, and the names of other polygons. Angles rules will be introduced and used to form short chains of reasoning.
Developing number sense – Students will review and extend their mental strategies with a focus on using a known fact to find other facts. Strategies for simplifying complex calculations will also be explored. The skills gained in working with number facts will be extended to known algebraic facts.
Sets and probability – Fraction, decimal and percentage equivalence will be revisited in the study of probability, where students will also learn about sets, set notation and systematic listing strategies.
Prime numbers and proof – Factors and multiples will be revisited to introduce the concept of prime numbers. Venn diagrams will be used to solve more complex HCF and LCM problems. Odd, even, prime, square and triangular numbers will be used as the basis of forming and testing conjectures. The use of counterexamples will be addressed.
Year 8 - Terms 1 & 2
Ratio and scale – This unit focuses initially on the meaning of ratio and the various models that can be used to represent ratios. Based on this understanding, it moves on to sharing in a ratio given the whole or one of the parts, and how to use e.g. bar models to ensure the correct approach to solving a problem. After this we look at simplifying ratios, using previous answers to deepen the understanding of equivalent ratio rather than ‘cancelling’ purely as a procedure. We also explore the links between ratio and fractions and understand and use pi as the ratio of the circumference of a circle to its diameter.
Multiplicative change – Students now work with the link between ratio and scaling, including the idea of direct proportion, linking various forms including graphs and using context such as conversion of currencies which provides rich opportunities for problem solving. Conversion graphs will be looked at in this block and could be revisited in the more formal graphical work later in the term. Links are also made with maps and scales, and with the use of scale factors to find missing lengths in pairs of similar shapes.
Multiplying and dividing fractions – Students will have had a little experience of multiplying and dividing fractions in Year 6; here we seek to deepen understanding by looking at multiple representations to see what underpins the (often confusing) algorithms. Multiplication and division by both integers and fractions are covered, with an emphasis on the understanding of the reciprocal and its uses. Links between fractions and decimals are also revisited.
Working in the Cartesian plane – Building on their knowledge of coordinates from KS2, students will look formally at algebraic rules for straight lines, starting with lines parallel to the axes and moving on to the more general form. They can explore the notions of gradient and intercepts, but the focus at this stage is using the equations to produce lines rather than interpretation of m and c from a given equation; this will be covered in Year 9. Use of technology to illustrate graphs should be embedded. Appreciating the similarities and differences between sequences, lists of coordinates and lines is another key point.
Representing data – Students are introduced formally to bivariate data and the idea of linear correlation. They extend their knowledge of graphs and charts from KS2 to deal with both discrete and continuous data.
Tables and probability – Building from the Year 7 unit, this short block reminds students of the ideas of probability, in particular looking at sample spaces and the use of tables to represent these.
Terms 3 & 4
Brackets, equations and inequalities – Building on their understanding of equivalence from Year 7, students will explore expanding over a single bracket and factorising by taking out common factors. The higher strand will also explore expanding two binomials. All students will revisit and extend their knowledge of solving equations, now to include those with brackets and with unknowns on both sides. Bar models will be recommended as a tool to help students make sense of the maths. Students will also learn to solve formal inequalities for the first time, learning the meaning of a solution set and exploring the similarities and differences compared to solving equations. Emphasis is placed on both forming and solving equations rather than just looking a procedural methods of finding solutions.
Sequences – This short block reinforces students’ learning from the start of Year 7, extending this to look at sequences with more complex algebraic rules now that students are more familiar with a wider range of notation.
Indices – Before exploring the ideas behind the addition and subtraction laws of indices (which will be revisited when standard form is studied next term), the groundwork is laid by making sure students are comfortable with expressions involving powers, simplifying e.g. 3x2y x 5xy3.
Fractions and percentages – This block focuses on the relationships between fractions and percentages, including decimal equivalents, and using these to work out percentage increase and decrease. Students also explore expressing one number as a fraction and percentage of another. Both calculator and non-calculator methods are developed throughout to support students to choose efficient methods. Financial maths is developed through the contexts of e.g. profit, loss and interest.
Standard index form – Standard index form is introduced to all students building from their earlier work on indices last term. The use of context is important to help students make sense of the need for the notation and its uses.
Number sense – This block provides a timely opportunity to revisit a lot of basic skills in a wide variety of contexts. Estimation is a key focus and the use of mental strategies will therefore be embedded throughout. We will also use conversion of metric units to revisit multiplying and dividing by 10, 100 and 1000 in context. We also look explicitly at solving problems using the time and calendar as this area is sometimes neglected leaving gaps in student knowledge.
Terms 5 & 6
Angles in parallel lines – This block builds on KS2 and Year 7 understanding of angle notation and relationships, extending all students to explore angles in parallel lines and thus solve increasingly complex missing angle problems. Links are then made to the closely connected properties of polygons and quadrilaterals. Students will also start to explore constructions with rulers and pairs of compasses.
Area of trapezia and circles – The formulae for the area of a trapezium and for the area of a circle is taught to all students. A key aspect of the unit is choosing and using the correct formula for the correct shape, reinforcing recognising the shapes, their properties and names, and looking explicitly at compound shapes.
Line symmetry and reflection – The teaching of reflection is split from that of rotation and translation to try and ensure students attain a deeper understanding and avoid mixing up the different concepts. Students will revisit and enhance their knowledge of special triangles and quadrilaterals and focus on key vocabulary such as object, image, congruent etc. Rotation and translations will be explored in Year 9.
The data handling cycle – Much of the statistics content in Key Stage 3 is a continuation of that studied at primary school, and many of the charts and graphs in this block have been used in Year 7 and earlier in Year 8. A particular focus is using charts to compare different distributions. We also explore when graphs may be misleading, an important real-life consideration.
Measures of location – Students have already met the median and the mean earlier in KS3. This block introduces the mode and also looks at when and why each average should be used. The previous block is built on as students have the opportunity to compare distributions, use these averages and the range. We also consider outliers, considering what effect these have on all the measures studied, and whether they should be included or excluded in our calculations.
Year 9 - Terms 1 & 2
Straight line graphs – Straight line graphs are explored and investigated ensuring pupils are able to understand the generic structure of a graph and how the variables will impact it (such as gradient and the y intercept). This includes working to interpret and represent straight line graphs using algebra.
Forming and Solving equations – From initially looking at one and two step equations, this unit moves onto exploring and manipulating inequalities and formulae. This also includes using negative numbers and non-integers with pupils also working on these within context
Testing conjectures – Although this section is small it is important in the application of logical reasoning. Starting initially with exploring patterns that pupils will be familiar (e.g. number), the unit then moves onto to look at generalisation through and with algebra.
Three dimensional shapes – From knowing the names of 2-D to 3-D shapes, pupils will be expected to be able to sketch these shapes and recognise nets of more common 3-D Shapes. By visualising the nets pupils will gain a understanding for calculating the surface areas of these shapes. The unit also explores plans and elevations which are commonly used by designers such as architects and engineers.
Constructions & Congruency – Continuing on from three dimensional shapes pupils will explore the significance of the detail in scale drawings. From this pupils will proceed onto construction and use these skills to explore loci. Towards the end of the topic pupils will work with congruency and specifically triangles, reflecting and reinforcing the previously learnt facts of construction.
Terms 3 & 4
Numbers – Although pupils will have come across different types of number, this section looks to consolidate and explore the interconnecting properties of number. Starting with Integers, real and rational numbers pupils will also need to know what they are not; this then leads the onto looking at surds. The final section is about consolidating previous work on fractions and then using this knowledge in working with different types of number in context, including standard form.
Using percentages – Pupils will have worked with percentages before (starting in year 7) and so in year 9 they will work with the more complex applications such as reverse and compound percentages. By the end of this topic they should be able to confidently do these calculations with and without a calculator.
Maths & money – This topic follows immediately on from percentages as it explores the application of percentages on money and the different language used within a financial context, including statements and bank balances.
Deduction – This block is designed to extend and deepen their understanding of conjecture through looking specifically at geometric problems including angles. It aims to link the previous work on construction and geometrical reasoning.
Rotation & translation – These two elements are important within transformations, and therefore have a section devoted to just these two. In looking at rotation and translation pupils will also consolidate their prior learning on reflection, with exercises designed to allow pupils to work on a multiple transformations within the same question.
Pythagoras’ Theorem – Initially pupils will explore and understand the properties of a right angle triangle including where ratio and algebra is used before proceeding onto how to label a triangle using the correct notation. This provides the foundation for working with Pythagoras’ Theorem in a variety of contexts including 3 dimension.
Terms 5 & 6
Enlargement & Similarity – Continuing on from the previous terms work where pupils worked on congruency and transformations, they will now look at and work on enlarging shapes with both positive, negative and fractional scale factors. This provides them with an ideal starting point into working with similarity including calculations involving ratio and missing angles/ sides.
Solving ratio & proportion problems – Consolidating their previous work on ratio and proportion, pupils will begin working on inverse relationships. By being able to generalise these relationships in algebra this will deepen their own understanding and application of proportionality problems in context.
Rates – This unit focuses on compound measurements. These are; speed, distance and time, and density, mass and volume problems. Being able to represent these in a graph they will also be able to work through flow problems, which includes the rates of changes of the units.
Probability – Pupils will already have looked at and worked on probability problems. This is developed further in this block by examining the language used in probability such as relative frequency, before moving onto different types of representation including sample space diagrams and tree diagrams. Using these diagrams will help pupils in their understanding and the application of conditional probabilities.
Algebraic representation – The final section in Year 9 looks at graphs that have previously not been explored in detail, such as quadratic and reciprocal graphs. The unit then moves onto investigating and solving simultaneous equations through graphing. The last piece of work in this section allows pupils to work through how different inequalities can be represented by graphs.
Assessment at Key Stage 3
Students will be graded in line with the GCSE grades (Grades 9-1 with the addition of Entry Level and Novice Level) to allow students to see their progress across all five years of their secondary education.
During the first term of Year 7, all students will take a baseline assessment to assess their current strengths and weaknesses and determine which set we feel will suit each student best.
All students in Years 7, 8 and 9 throughout the year will take mini assessments during each topic, both self and peer marked, to help them identify their own progress against their targets. All pupils will be expected to take an assessment at the end of each term. This will help identify areas that they are struggling with and where additional support is needed.
In Key Stage 4 (Years 10 and 11) we follow a Mastery Scheme of Learning.
There are six content domains covered within the specification: number; algebra; ratio, proportion, and rates of change; geometry and measures; probability; and statistics. which is detailed below.
There are two overlapping tiers of entry, with students being entered for the most appropriate tier. Movement between tiers is possible and the final level of entry will be decided at the time of entry for GCSE.
This qualification consists of three equally weighted written examination papers at either Foundation tier or Higher tier. Paper 1 and 3 are calculator assessments and Paper 2 is non calculator assessment. Each paper is 1 hour and 30 minutes in duration and each paper contains a total of 100 marks. Each paper has a range of question types; some questions will be set in both mathematical and non-mathematical contexts.
The qualification will be graded and certificated on a nine-grade scale from 9 to 1 using the total mark across all three papers, where 9 is the highest grade.
The available grades are as follows:
• Foundation tier: grades 1 to 5.
• Higher tier: grades 4 to 9 (grade 3 allowed).
Year 10:
Autumn Term 1
Congruence, Similarity and Enlargement
Having already worked on Congruence and Enlargement from Year 9, pupils explore the conditions for congruency and similarity. They then move onto working on various connections and calculations that can be performed across all these areas.
Trigonometry
In this section they begin by revisiting Pythagoras Theorem, where they also explore exact solutions in conjunction with missing sides. This leads onto all aspects of applying and using the ratio of sides of a triangle in calculating missing sides and angles using sohcahtoa. Building upon this the Year 10s move onto the sine and cosine rules for non-right-angle triangles.
Autumn Term 2
Representing Solutions of equations and inequalities
In this block they begin by consolidating their prior understanding of equations and inequalities, before expanding onto representation which aids in deepening understanding. The unit ends with Year 10s exploring and working with quadratics.
Simultaneous Equations
Initially they will explore the context of a simultaneous equation and what a solution represents. This is followed on by working through and identifying the most appropriate method for solving where there are sometimes numerous methods; thus, ensuring fluidity in the approach and application of the maths.
Spring Term 1
Angles and Bearing
In this topic pupils work with calculating bearings and the link to angles. They are then presented with the opportunity of connecting these problems to previously learnt topics such as Pythagoras, trigonometry and sine/ cosine rules.
Working with circles
In this section the Year 10s will cover all the different elements that are connected with circles; from Circle Theorems, lengths or arcs, areas of sectors and volume and surface areas of spheres.
Vectors
Although pupils will have come across vectors previously, their understanding will be developed in this topic to enable them to explore shapes with vectors, before moving onto constructing geometric arguments and proof.
Spring Term 2
Ratios and Fractions
Initially the Year 10s will revisit ratio and fractions from previous years before quickly moving onto working with these in algebraic contexts. The pupils will link them in the context of graphs and 1:n before moving onto area and volume problems.
Percentages and interest
In this topic pupils will explore complex percentage problems before moving onto growth and decay and iterative processes.
Probability
Probability can be seen as slightly abstract by some pupils, but by looking at various ways of representing probability, from Venn Diagrams to Tree Diagrams we are able to not only cover a wide range of scenarios such as conditional probabilities but also ensure pupils have a secure understanding of how to apply the calculations effectively.
Summer Term 1
Collecting, representing and interpreting data
This unit focusses on statistical analysis and different ways of representing and interpreting data.
Non-calculator methods
In this section we revisit the four rules of fraction decimal arithmetic, before moving onto calculations with irrational numbers. We finish by working on financial mathematical problems.
Summer 2
Types of number and sequences
Students explore different types of sequences, from ones learnt earlier such as arithmetic before moving onto geometric and quadratic sequences. We also look at sequences involving fractions and irrational numbers.
Indices and roots
In this unit pupils work with all indices and identify the connection with roots.
Manipulating expressions
In this section pupils will work with manipulating various forms of algebraic fractions before working on calculations. This is then used to investigate and explore proof.
Year 11
Autumn Term 1 (Graphs)
The first topic Year 11’s work with is focussed on graphs; the three main areas are Gradients and lines, Non-linear and using graphs. This enables us to draw together their understanding and develop further into applying and combing graphs including circles, quadratics and cubic.
Autumn Term 2 (Algebra)
In the second half of the term the Year 11’s will move into the more abstract elements of algebra (quadratics and changing the subject) before working on all the various nuances of functions.
Spring Term 1 (Reasoning)
The first half of the spring term is focussed on reasoning and involves the main three strands: Multiplicative; Geometric and Algebraic. We will through the numerical elements, visual and then the abstract providing a secure understanding of reasoning within Mathematics.
Spring Term 2 (Revision and Communication)
In the first part of this half term pupils will work on developing their understanding of transforming and construction in Maths. They will then move onto some work with statistics including Venn diagrams. The final section of the half term the Year 11s will work on problems and problem solving. The aim is to provide pupils with as much opportunity to develop their fluency within the subject.
Summer Term (Revision and GCSEs)
