# Mathematics

It is often said that math contains the answers to most of universe’s questions. Math manifests itself everywhere in the world that God created. One such example is the Golden Ratio. This famous Fibonacci sequence has fascinated mathematicians, scientist and artists for many hundreds of years. The Golden Ratio manifests itself in many places across the universe, including right here on Earth, it is part of Earth’s nature and it is part of us.

The Fibonacci sequence shows up in the most unexpected places. Here are some of them:

Flower petals
Number of petals in a flower is often one of the following numbers: 3, 5, 8, 13, 21, 34 or 55. For example, the lily has three petals, buttercups have five of them, the daisy has often 34 or 55 petals, etc.
Faces
Faces, both human and non-human, abound with examples of the Golden Ratio. The mouth and nose are each positioned at golden sections of the distance between the eyes and the bottom of the chin. Similar proportions can be seen from the side, and even the eye and ear itself.
Fruits, Vegetables and Trees
Spiralling patterns can be found on pineapples and cauliflower. Fibonacci numbers are seen in the branching of trees or the number of leaves on a floral stem; numbers like 4 are not. 3’s and 5’s, however, are abundant in nature.
Shells
Snail shells and nautilus shells follow the logarithmic spiral, as does the cochlea of the inner ear. It can also be seen in the horns of certain goats, and the shape of certain spider’s webs.

## Curriculum Intent

It is the intention of the St John Bosco Mathematics department to deliver a curriculum that is knowledge based and ambitious, designed to meet the needs of all our students. We aim to create an aspirational high-achieving culture while also considering our students individual needs and learning styles thus enabling all students to experience success.

We aim to develop the skills set out in the National Curriculum to promote an appreciation of Mathematics as a creative and highly transferable discipline. This will aid students in their further learning within mathematics at KS4 and KS5 as well as cross curricular subjects, apprenticeships, and employment, taking into account the improving local economy and labour market. We do not limit the life chances of any student as we offer a linear scheme in KS3 with all students having the full National Curriculum delivered to them.

We aim to provide students with a sense of enjoyment and curiosity about the subject together with an appreciation of the beauty and power of Maths in different cultures. We endeavour to provide support across a range of topics with an emphasis on problem-solving and developing Mathematical fluency through a mastery approach, making sure that all learning is embedded in long-term memory. This will build resilience and enable students to recall information to use in a variety of new learning opportunities as well as real life situations.

## Curriculum Content

### Year 7

Analysing and Displaying Data
Mode, Median and Range, Displaying data, Grouping data, Averages and comparing data and Line graphs and more bar charts.

Number Skills
Mental Maths, Addition and Subtraction, Multiplication, Division, Money and Time, Negative Numbers and Factors, multiples and primes.

Expressions, functions and formulae
Functions, Simplifying expressions, Writing Expressions, Substituting into formulae and Writing formulae

Decimals and Measures
Decimals and rounding, Length, mass and capacity, Scales and Measures, Working with decimals, Perimeter, Area and More units of measure

Fractions and Percentages
Comparing fractions, Simplifying fractions, Working with fractions, Fractions and decimals, Understanding percentages and Percentages of amounts

Probability
The language of probability, Calculating probability, Probability calculations, Experimental probability and Expected outcomes

Ratio and Proportion
Direct Proportion, Writing Ratios, Using Ratios, Proportions and Fractions and Proportions and Percentages

Lines and Angles
Measuring and drawing angles, Lines, angles and triangles, Drawing triangles accurately, Calculating angles, Angles in triangle and Quadrilaterals

Sequences and Graphs
Sequences, Pattern sequences, Coordinates and midpoints, Extending Sequences, Straight-Line Graphs and Position-to-term rules

Lines and Angles
Measuring and drawing angles, Lines, angles and triangles, Drawing triangles accurately, Calculating angles, Angles in triangle and Quadrilaterals

Sequences and Graphs
Sequences, Pattern sequences, Coordinates and midpoints, Extending Sequences, Straight-Line Graphs and Position-to-term rules

### Year 8

Number
Calculations, Divisibility and division, Calculating with negative integers, Powers and Roots, Powers, Roots and Brackets, Multiples and Factors

Area and Volume
Area of a Triangles, Area of a parallelogram and Trapezium, Volume of cubes and cuboids, 2D representations of 3D solids, Surface Area of Cubes and Cuboids and Measures

Statistics, Graphs and Charts
Pie Charts, Using Tables, Stem and Leaf Diagrams, Comparing Data, Scatter Graphs and Misleading Graphs

Expressions and Equations
Algebraic Powers, Expressions and Brackets, Factorising Expressions, One-Step Equations, Two-Step Equations and The Balancing Method

Real-Life Graphs
Conversion Graphs, Distance-Time Graphs, Line Graphs, More Line Graphs, Real-Life Graphs and Curved Graphs

Decimals and Ratio
Ordering Decimals and Rounding, Place-Value Calculations, Calculations with Decimals, Ratio and Proportion with Decimals

Lines and Angles
Quadrilaterals, Alternate Angles and Proof, Angles in Parallel Lines, Exterior and Interior Angles and Solving Geometric Problems

Calculating with Fractions
Ordering Fractions, Adding and Subtracting Fractions, Multiplying Fractions, Diving Fractions and Calculating with Mixed Numbers

Straight-Line Graphs
Direct proportion Graphs, Gradients and Equations of Straight Lines

Percentages, Decimals and Fractions
Fractions and Decimals, Equivalent Proportions, Writing Percentages and Percentages of Amounts

### Year 9

Indices and Standard Form
Indices, Calculations and Estimates, More Indices and Standard Form

Equations and Formulae
Solving Equations, Substituting into Expressions, Writing and using Formulae, Using and Rearranging Formulae, Index Laws and Brackets and Expanding Double Brackets

Dealing with Data
Planning a Survey, Collecting Data, Calculating Averages, Displaying and Analysing Data and Presenting and Comparing Data

Multiplicative Reasoning
Enlargement, Negative and Fractional Scale Factors, Percentage Change, Compound Measures, Direct and Inverse Proportion

Constructions
Using Scales, Basic Constructions, Constructing Triangles and Using Accurate Scale Diagrams

Sequences, Inequalities, Equations and Proportion
Nth term of arithmetic sequences , Non-Linear Sequences, Inequalities, Solving Equations and Proportion

Circles, Pythagoras and Prisms
Circumference of a Circle, Area of a Circle, Pythagoras' Theorem, Prisms and Cylinders and Errors and Bounds

Graphs
Using y = mx + c, More straight-line graphs, Simultaneous equations, Graphs of quadratic Functions and More non-linear graphs

Probability
Mutually Exclusive Events, Experimental and Theoretical Probability, Sample Space Diagrams, Two-Way Tables and Venn diagrams

Comparing Shapes
Congruent and Similar Shapes, Ratios in Triangles, The tangent ratio, The sine ratio, The cosine ratio and Using Trigonometry to find angles.

### Year 10 and 11

Students will be assessed on the Edexcel exam board at the end of Y11 at either Higher or Foundation Tier. Topics are covered matching to the tier of entry and are carefully sequenced to enable students to build upon prior knowledge at KS3. All topics are taught at a mastery level and interleaved to other areas of math’s at all times. Staff are aware what has been delivered in previous year groups and their planning is adapted accordingly. Retrieval is used to reinforce prior knowledge and move onto new learning quickly. There is a huge emphasis on using problem solving and reasoning questioning in all year groups but especially in Y10 and Y11 where some topics may have been seen before.

##### Y10 and Y11 Foundation:

Unit 1 - Number Work

• We begin KS4 recapping some of the major number topics that will underpin most of the upcoming units. Negative numbers and Orders of operations are recapped and developed upon, as is estimating and rounding. Rounding is necessary in almost all upcoming units.

• Although HCF and LCM have been covered there is new learning in the form of using PPF and Venn Diagrams to answer more difficult questions.

Unit 2 – Algebra Simplification

• Similar to unit 1 we recap the major algebraic topics that will underpin most of the upcoming units. Expanding and factorising all brackets will be needed for unit 6 equations.

• Substitution will be taken to a much higher level than KS3 but will need the work covered in unit 1, including laws of indices as well as negative numbers.

Unit 3 - Graphs, tables and charts

• Some elements of this unit are to be quickly revisited and time spent emphasising on the problem solve and analysis side of these topics that will not have been covered at KS3. These topics are pie charts and scatter graphs.

• New content will be covered with students being asked for frequency polygons and also predicting the future with time series graphs.

Unit 4 – Fraction Work

• Another early topic that makes sure that basic fraction work both calculator and non-calc skills are secure from KS3. This mainly focusses on the 4 main operations with fractions but also extends to reverse fractions.

• These will also be needed in multiple upcoming topics such as gradients (Unit 10), scale factors in enlargements (Unit 11) and many of the formula with area and volume.

Unit 5 – Percentage

• Percentage change, Percentage of an amount, Expressing as a Percentage are quickly recapped before huge emphasis is placed on single multipliers that then support the introduction of compound interest. This will be new for these students and focus will be placed upon working with all areas of compound interest and growth and decay. Also these questions will be in reverse.

Unit 6 - Equations, Inequalities and Sequences

• Equations both with 1 and 2 unknowns are recapped but more emphasis is placed on creating your own equations and then solving them. KS3 solving will be quickly recapped before exploring these create and solve questions. Many solutions will have non integer answers which will be supported by fraction work in Unit 4.

• These create and solve will also extend into inequality style questions which will further stretch the student.

• Sequences are revisited but again extended with all elements on nth term.

• Rearranging of formula will also be covered and taken further with some deep conversations of why some formula need to be rearranged.

Unit 7 – Angles

• Having not completed any geometry topics so far at KS4 we begin with quickly recapping the angle work up to parallel lines that has been covered at KS3. More emphasis is placed on explaining each step rather than just getting the answer.

• Regular polygons will be covered in depth with questions being given in reverse also. This tends to really stretch the understanding of angles in all polygons.

• Finally, having strengthened equations in the previous unit the link between angles and creating own equations are covered here for the first time.

Unit 8 - Averages and Range

• MMMR quickly recapped before exploring problem solve with averages, requiring students to confidently work backwards through all the operations with number that have been covered previously.

• Exploring which average to use will be introduced with much debate as to why some are more suitable than others.

• MMMR and frequency tables and grouped frequency tables will further develop students’ knowledge within this topic.

Unit 9 - Perimeter, Area and Volume 1

• Many areas and volumes are quickly recapped with problem solve being interleaved within. • What differs this unity from KS3 is its link between volume and capacity and as well as the links made between volume and density/pressure.

Unit 10 – Straight Line Graphs

• These students will revisit drawing some easier graphs from their KS3 which will then progress into y = mx + c which will be further developed from some earlier Y9 work.

• Substitution and rearranging formula from earlier units will help support this new learning as well as work with negatives.

Unit 11 - Transformations

• This topic is extended from KS3 where students will be asked to combine all for transformations.

Unit 12 - Right-Angled Triangles

• Indices and Pythagoras will be revisited from KS3 as Pythagoras will need these. Both Pythagoras and Trig will then be covered in depth and with problem solve elements added.

Unit 13 - Ratio and Proportion

• Simple ratios will be revised from KS3 such as simplifying and sharing ratios however these will extend further by introducing fractions with ratio as well as when there are 3 ratios within one question.

• Direct proportion will also be revised from KS3 but extend to inverse proportion as well as linking proportion to graphs.

• Speed will be covered here too as it is sensible to complete this with Time distance graphs that is here also

• Speed, Density and Pressure all to be covered here.

Unit 14 – Probability

• Basic probability will have already been covered in KS3 and revised here before extending to frequency trees, Venn diagrams and tree diagrams.

• This topic needs to follow work on operations with fractions and decimals as this will be required to fully access tree diagrams.

Unit 15 - Constructions, Loci and Bearings

• Constructions and Loci will be a new topic for these students that needs to be cover here before the end of Y11.

• Bearings is placed within the same unit and requires all knowledge of angles from earlier in Y10.

Unit 16 - Quadratic Equations and Graphs

• Multiplying and factorising single brackets will already have occurred in previous years. They will be revisited here briefly before extending to double brackets and factorising quadratics.

• We draw the quadratic graph here as the links can be made between expanding, factorising and drawing.

Unit 17 - Perimeter, Area and Volume 2

• Area, perimeter and volume 1, concentrated on basic areas and perimeters as well as volume of non-circular prisms. That knowledge is still required but expanded upon to further develop all circles and prism involving circles and sectors. Cones and spheres are also covered here. Substituting into formula from a previous year will be essential for students to be successful on these topics.

Unit 18 - Fractions, Indices and Standard Form

• Standard Form with basic index laws will be revisited from KS3 and Y10 enabling us to move onto calculations with standard form. Previous decimal work will also have to be revisited here to aide student learning.

Unit 19 - Congruence, Similarity and Vectors

• Prior knowledge form both the KS3 and KS4 course is used here, multiplying and dividing both fractions and decimals.

• Vector Arithmetic s closely linked to similarity so makes sense that it follows directly after. Prior knowledge on directed number will be revisited as is essential for students to access vector arithmetic.

Unit 20 - More Algebra

• Substitution will allow students to draw cubic and reciprocal graphs.

• Substitution and solving linear equations in Y9 will be further stretched with work on Simultaneous Equations.

• And finally, all our expanding single and double brackets as well as factorising will give students the opportunity to be successful with algebraic proof.

##### Y10 and Y11 Higher Sequence

Unit 1 - Number Work

• The unit begins with a quick recap of some number skills such as rounding and estimating that will be used in almost all upcoming units.

• We quickly move onto new content such as using PPF for HCF and LCM as well as some difficult Index laws.

• These will interleave nicely into an extension of the Standard from completed in KS3 before we introduce Bounds and calculations with bounds.

Unit 2 – Algebra Simplification

• We will build upon equations completed in KS3 to explore solving equations when we create our own or more obscure equations.

• These will extend into rearranging and also include rearranging where factorising is a key element. A recap of factorising will take place at the start of the unit as a result of this.

• Nth term is revisited from KS3 to quickly extend into Quadratic Sequences.

• Expanding Quadratics are then revisited to build towards factorising all quadratics including with a co-efficient and also difference of two squares.

Unit 3 – Interpreting and Representing Data

• Averages are quickly recapped before exploring problem solve questions that rely on this knowledge.

• Averages from tables (including groups) are then introduced.

• Pie Charts are covered with more of an emphasis on interpreting.

• Scattergraphs are revisited and built upon for work completed in KS3. Important to extend these style questions with some percentage questions that will now be added.

• Misleading graphs will also be introduced and students will be expected to be able to explain why it is misleading.

Unit 4 – Fractions, Ratios and Percentages

• This Unit recaps then extends on many topics introduced at KS3. Operations with Fractions and Ratio. It extends to difficult ratio sharing as well as when a third ratio is introduced.

• New learning takes the form of Compound Interest (Growth and Decay) as well as Terminating and Recurring Fractions.

Unit 5 – Angles and Trigonometry

• Angle rules up to parallel lines and Polygons are quickly recapped from KS3. More time is spent on problem solving questions relating to Regular Polygons.

• Pythagoras is recapped before extending to difficult problem solve as well as 3d Pythagoras.

• Right Angled Trig is introduced extending to 3d Pythagoras and Exact Values.

Unit 6 - Area and Volume

• This unit begins with the conversion of metric Areas and Volumes to help support this upcoming unit.

• Simple areas, perimeter and volumes are quickly recapped, concentrating on non-circular shapes. Links are made here to capacity.

• Circular Shapes are then recapped from KS3 however a much bigger emphasis will be on leaving answers in terms of pi and also dealing with difficult sectors.

• Cones spheres and frustums are then covered with many of these shapes interleaving. Substituting into formula from unit 2 will be essential for students to be successful in these topics.

Unit 7 – Graphs

• An extensive amount of time will be given to y = mx + c This will incorporate many things already covered such as substitution, rearranging and equations.

• Speed will be covered in preparation for Time Distance Graphs but we will extend to velocity graphs including curves. (Area covered in previous unit will help support trapezium needed here)

• All other higher graphs will be covered in depth including – Quadratic, Cubic and Reciprocal graphs.

Unit 8 – Transformations and Constructions

• Previous fraction work in KS3 will allow students to access all aspects of enlargement.

• Previous Graph work in Unit 7 will allow students to fully access reflection. • All combined trandsformations and negative enlargement to be covered here.

Unit 9 – Equations and Inequalities

• Multiplying and factorising single brackets will already have occurred in previous years. They will be revisited here briefly before extending to double brackets and factorising quadratics including with co-efficients and Difference of two squares.

• Solving Quadratics will now extend further to encompass the Quadratic Formula and Completing the Square.

• Prior knowledge on substitution and linear equations will allow students to access simultaneous equations.

Unit 10 - Probability

• Basic probability will have been already covered and revised here before extending to frequency tress and tree diagrams. Previous fraction and decimal work is revisited and embedded further here to allow students to fully access tree diagrams, including conditional as well as understanding the “And” and “OR” rules.

• Venn Diagrams are extended upon from KS3 to have the problem solve and equation element added to them.

Unit 11 – Similarity and Congruence

• A deep understanding on previous topics area and volume is needed as we now look at the relationships between similarity and volumes and areas.

• Squares, cubes and their roots (KS3) will also be revisited to deepen knowledge on this topic.

Unit 12 – More Trigonometry

• All higher level Trig is covered here. Sine and Cosine rules as well as area of a triangle. Prior knowledge of substitution and solving equations is needed here to use these formula.

Unit 13 Further Statistics

• Covered in this unit is Sampling and Stratified Sampling, UQ LQ and IQR from a list of numbers and why this is useful, Cumulative Frequency and boxplots. Drawing and interpreting histograms are also covered.

Unit 14 Further Equations and Graphs

• Sim Equations with Lines meeting Quadratics and Circles and Graphing Inequalities relies on prior knowledge of y = mx + c

• Multiplying Cubic Brackets and Understanding Roots relies on students being able to expand a double bracket.

• Iteration relies on students being able to substitute.

Unit 15 – Circle Theorems

• Circle Theorems are introduced here but students must recall all their angle work from KS3 and Y10.

• Circle Problems with Tangents again require a deep understanding of y = mx + c and again is a grade 9 topic so to give students the best chance at being successful it is late on in the course.

Unit 16 – More Algebra

• Algebraic Fractions are covered in their entirety and require recall of operations with fractions as well as expanding and factorising brackets from previous units.

• Surds are covered in their entirety with previous knowledge of square numbers, fraction work and expanding brackets all coming together to make surds accessible to students.

• Functions are introduced and again are underpinned by students recalling information on substitution, and creating and solving equations.

• And finally, all our expanding single and double brackets as well as factorising will give students the opportunity to be successful with algebraic proof.

Unit 17 – Vectors and Geometric Proof

• Vector Arithmetic will also be seen by these students for the first time but having already covered directed number this will make this very accessible. An easy topic but is left until now as it closely relates to vector geometry which is a grade 9 topic.

• Vector Geometry will be new and will require students to recall operations with fraction sand ratios

Unit 18 – Proportion and Graphs

• Direct and Inverse Proportion - Formula follows on from work on direct and inverse proportion without the formula in KS3.

• Transforming Functions stand alone as a discreet topic and are grade 9. Thus we cover them here at the end of the course. We will also Transform Trig Graphs at the same time.

### Key Stage Five

At A-Level we follow the Edexcel 2- year scheme of work.
The curriculum at KS5 is balanced between Applied and Pure Mathematics.

Pure Maths - This is the basic maths that will be familiar to you from GCSE, particularly the study of number, algebra, graphs and trigonometry. These ideas are extended to form the basic ‘tool kit’ on which all Maths is based. Calculus, the mathematics of growth and change, on which so many applications depend is introduced and developed.​

Statistics -This is the mathematics of data or information processing and extends many ideas including averages, standard deviation and probability distributions.​

Mechanics -This is the study of forces and the movement of objects and has many applications in the sciences particularly physics and engineering.

### Year 12

Term One

• Pure - Algebraic expressions, Quadratics, Equations & inequalities, Graphs and Transformations, Straight line graphs, Circles, Algebraic methods, Binomial expansion.

• Statistics - Data collection, Measures of locations and spread, Representations of data

Term Two

• Pure - Trigonometric ratios, Trigonometric identities and equations, Vectors, Differentiations

• Statistics - Correlation, Probability, Statistical distributions, Regressions and correlation, Conditional probability

Term Three

• Pure - Differentiation, Integration, Exponentials and logs, Algebraic expressions, Quadratics, Equations & inequalities

• Statistics - Hypothesis testing, Conditional probability, Normal Distribution

### Year 13

Term One

• Pure - Graphs and Transformations, Straight line graphs, Circles, Algebraic methods, Binomial expansion, Parametric equations, Numerical Methods.

• Mechanics - Variable acceleration, Modelling, Constant acceleration, Moments, Forces and Friction, Projectiles.

Term Two

• Pure - Trigonometric ratios, Trigonometric identities and equations, Vectors, Differentiations.

• Mechanics - Forces and motion, Variable acceleration, Application of forces, Further Kinematics.

Term Three

• Pure - Integration, Vectors.

• Mechanics - Further Kinematics.

#### Revision support:

Students use a variety of resources in their learning:​

• Integral- Homeworks and topic assessment​

• Dr Frost Maths- Homeworks​

• Pearson Exercise Books- in class activities and preparation for class (flipped learning)

• Zig- Zag Maths- consolidation and retrieval practices

### Extra-curricular Enrichment and Visits

• Chess club – Tuesday lunchtime and Thursday after school

• Super Saturday revision sessions

• Half termly revision sessions

• Year 7 catch up tutoring

• Y11 tuition sessions

• UKMT Maths Challenges – Junior, Intermediate and Senior

• Reward trips – including visits to Breakout room Liverpool

• Partner school with Liverpool University STEM subjects – including taster days

• Enrichment days - financial Maths and careers in Maths

• Celebration of Special days – Pi Day, NSPCC number day, World Maths day, Maths Party Day Liverpool

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