# GCSE

## Year 10

##### Intent

In Year 10, pupils develop fluency in proportional reasoning and apply to varied problems. Pupils consider accuracy of calculations and use standard form. Pupils encounter a variety of different diagrams for solving probability problems. In algebra they study sequences and join up all their prior learning on linear algebra and quadratics as well as using function notation for the first time and formal numerical methods for solving equations. Geometric reasoning is extended to circles and trigonometry moves on to right-angled and non-right-angled triangles.

##### Learning Journey

##### Key Concepts and Themes

- Understand ratio notation and interpret ratio as a fraction, applying this to geometric problems.
- Use equivalent and unitary ratios to solve problems.
- Understand direct and inverse proportionality, and construct appropriate graphs.
- Calculate percentage change and reverse percentage.
- Calculate repeated percentage change including compound interest.
- Apply ratio and proportion to similar shapes by considering scale factors.
- Generate linear sequences from given rules.
- State nth term rule for linear sequences.
- Recognise and use sequences for triangular and cube numbers.
- Solve problems involving Fibonacci type sequences.
- Expanding and factorising algebraic expressions, including quadratics and the difference of two squares

##### Vocabulary

- Ratio
- Proportion
- Unitary
- Equivalent
- Direct proportion
- Inverse proportion
- Scale factor
- Enlargement
- Simple interest
- Compound interest
- Linear sequence
- Arithmetic progression
- Term
- Factorise
- Expand
- Quadratic

##### Key Concepts and Themes

- Solving linear equations, including those involving brackets, fractions and the unknown on both sides.
- Writing and rearranging formulae.
- Solving and forming linear simultaneous equations.
- Round numbers to appropriate degrees of accuracy.
- Write error intervals for rounded values and measurements.
- Calculate with roots and indices and use rounding with mental methods to estimate calculations.
- Use standard form and calculate with numbers in standard form.
- Calculate with fractions to solve problems.
- Simplify and calculate with surds.

##### Vocabulary

- Mutually exclusive
- Relative frequency
- Dependant
- Independent
- Conditional
- Sample space
- Error interval
- Limits of accuracy
- Lower and upper bounds
- Standard form
- Irrational number
- Surd
- Rationalise denominator

##### Key Concepts and Themes

- Finding the equation of straight lines.
- Equations of parallel and perpendicular lines.
- Solving linear simultaneous equations graphically.
- Understand mutually exclusive events and calculate theoretical probabilities.
- Use relative frequency diagrams.
- Apply the addition and multiplication rules for probability.
- Construct probability tree diagrams for dependant and independent events.
- Use Venn diagrams, two-way tables, listing methods and sample spaces to solve probability problems

##### Vocabulary

- Factorise
- Expand
- Binomial
- Quadratic
- Simultaneous equations
- Iteration

##### Key Concepts and Themes

- Solve quadratic equations algebraically and graphically.
- Use algebraic methods to sketch graphs of quadratic functions.
- Understand and apply circle theorems in geometric reasoning.
- State the equation of a circle and find the equation of a tangent.

##### Vocabulary

- Function
- Inverse
- Composite function
- Cubic
- Reciprocal
- Exponential
- Translation
- Reflection
- Chord
- Tangent
- Arc
- Sector
- Segment

##### Key Concepts and Themes

- Use trigonometric functions to solve problems in right-angled triangles.
- Use sine and cosine rules in non-right-angled triangles.
- Use trig to find the area of a triangle.
- Solve problems in similar shapes and use bearings.
- Convert measures including estimated conversions for common metric/imperial measures and apply in problems.
- Apply dimensional analysis.
- Derive and use formulae for compound measures.

##### Vocabulary

- Sine
- Cosine
- Tangent
- Pythagoras’ theorem
- Similar shapes
- Bearings

##### Key Concepts and Themes

- Represent inequalities on a number line.
- Solve linear and quadratic inequalities.
- Representing inequalities in graphically and solving inequalities in two variables.
- Understand and interpret scatter graphs, including lines of best fit.
- Intepreting and comparing distributions.
- Carrying out measure of central tendency and spread on ungrouped and grouped data.

##### Vocabulary

- Inequality
- Correlation
- Mean
- Median
- Mode
- Range
- Interquartile range

##### Skill Development

- Formalise and apply knowledge of ratio and proportion, including trigonometric ratios, in working with measures and geometry, and in working with proportional relations algebraically and graphically
- Make and test conjectures about the generalisations that underlie patterns and relationships; look for proofs or counter-examples; begin to use algebra to support and construct arguments
- Explore what can and cannot be inferred in statistical and probabilistic settings, and express arguments formally
- Assess the validity of an argument and the accuracy of a given way of presenting information
- Develop mathematical knowledge, in part through solving problems and evaluating the outcomes, including multi-step problems and financial contexts
- Make and use connections between different parts of mathematics to solve problems
- Select appropriate concepts, methods and techniques to apply to unfamiliar and non-routine problems; interpret solutions in the context of the given problem

## Year 11

##### Intent

In year 11, pupils deepen understanding working with Circles and Trigonometry in order to be able to solve more complex problems. Pupils consider different ways to solve quadratics both algebraically and graphically depending on whether they want to the roots or the vertex and intercept. Year 11 pupils will begin to explore the concepts behind formal proof and how to construct rigorous mathematical arguments across a range of topics.

##### Learning Journey

##### Key Concepts and Themes

- Understand and apply circle theorems in geometric reasoning.
- State the equation of a circle and find the equation of a tangent.
- Use trigonometric functions to solve problems in right-angled triangles.
- Use sine and cosine rules in non-right-angled triangles.
- Use trig to find the area of a triangle.
- Solve problems in similar shapes and use bearings.
- Solve quadratic equations by factorisation.
- Approximate solutions to quadratics by graphing.
- Solve quadratic equations using the quadratic formula or by completing the square.
- Identify the key points of quadratic graphs and use to sketch.
- Solve quadratic and linear simultaneous equations.

##### Vocabulary

- Chord
- Tangent
- Arc
- Sector
- Segment
- Theorem
- Angle
- Normal
- Sine
- Cosine
- Tangent
- Pythagoras’ theorem
- Similar shapes
- Bearings
- Root (of function)
- Parabola
- Vertex
- Turning point
- Solve
- Variable
- Equation

##### Key Concepts and Themes

- Draw a tangent to a curve.
- Estimate the gradient at a point on a curve.
- Find the area under a linear graph.
- Estimate the area under a non-linear graph.
- Solve worded problems involving gradients/areas.
- Repeated proportional (percentage) change.
- Compound interest and depreciation.
- Identify a quadratic sequence and find missing terms.
- Find terms in a quadratic, cubic or fractional sequence when given the nth term.
- Combine nth terms of linear sequences to make complex sequences in the form of fractions.
- Find the nth term of a quadratic sequence in the form an² +bn +c.
- Construct a cumulative frequency diagram.
- Find median and IQR
- Draw a box plot from a cumulative frequency diagram.
- Construct histograms with both equal and unequal class intervals.
- MMMR from grouped frequency tables.
- Draw frequency trees.

##### Vocabulary

- Gradient
- tangent
- area
- velocity
- speed
- time
- distance
- instantaneous
- average
- variable
- curve
- percentage
- proportional
- depreciation
- compound
- growth
- decay
- Sequence
- Linear
- Quadratic
- Relationship
- Arithmetic
- Geometric
- Progression
- nth term
- Difference
- Coefficient
- Representation
- data
- cumulative
- frequency
- mean
- median
- mode
- range
- interquartile range
- quartile

##### Key Concepts and Themes

- Plot and interpret data used in a scatter graph, describing the correlation and relationship between the two variables.
- Draw and use a line of best to make predictions on data.
- Describe trends in populations based on numerical results.
- Add and subtract vectors.
- Draw vectors.
- Multiply column vectors by a scalar and be able to represent this as a diagram.
- Combine sums and scalar multiplication of vectors.
- Solve geometrical problems involving vectors.
- Solve vector proof problems.
- Represent inequalities on a number line and using set notation.
- Express integer values of an inequality.
- Solve linear inequalities.
- Recognise and represent inequalities on a graph.
- Solve quadratic inequalities and be able to represent this graphically

##### Vocabulary

- Variable
- Data
- Scatter Graph
- Independent
- Dependant
- Correlation
- Line of Best Fit
- Trends
- Outlier
- Anomaly
- Sample
- Representative
- Stratified
- Random
- Predict
- Column
- vector
- distance
- direction
- resultant
- scalar multiple
- coefficient
- Inequality
- integer
- set notation
- solve
- equation
- coordinate
- linear
- graph
- variable
- quadratic.

##### Key Concepts and Themes

- Use identities to equate coefficients.
- Understand and construct mathematical arguments.
- Use algebra to write generalised expressions to prove results.
- Construct geometric proofs.
- Convert recurring decimals to fractions using mathematical proof.
- Calculate the arc length given the angle subtended at the centre and vice versa.
- Find the area of a sector given the radius and angle.
- Find the volume of composite shapes made from pyramids, cones, spheres and frustums.
- Find the surface area of complex composite solids.
- Enlarge a shape on a coordinate grid using a centre of enlargement.
- Enlarge and describe a shape using fractional and negative scale factors.
- Draw and accurately describe translations including the use of column vectors.
- Draw and accurately describe reflections and rotations.
- Be able to combine a variety of transformations to produce a final result on a grid.

##### Vocabulary

- Identity
- equivalent
- equation
- coefficient
- expression
- algebra
- reasoning
- prove
- geometrical
- angle
- infinity
- recurring
- decimal
- fraction
- irrational
- terminating
- Circle
- arc
- radius
- circumference
- sector
- segment,
- pi
- volume
- tetrahedron
- hemisphere
- frustum
- Enlargement
- similarity
- scale factor
- fractional
- quadrant
- invariance
- reflection
- vector
- cartesian

##### Skill Development

- Make and test conjectures about the generalisations that underlie patterns and relationships; look for proofs or counterexamples; develop further their use of algebra to support and construct arguments.
- Accurately interpret statistical and geometrical representations, and express arguments formally.
- Assess the validity of an argument and the accuracy of a given way of presenting information.
- Justify their reasoning in an articulate way using formal mathematical language.
- Develop mathematical knowledge, in part through solving problems and evaluating the outcomes, including multi-step problems, financial contexts and worded scenarios.
- Make and use connections between different parts of mathematics to solve problems.
- Select appropriate concepts, methods and techniques to apply to unfamiliar and nonroutine problems; interpret solutions in the context of the given problem.

## Mathematics News

- Return of UKMT Maths Challenge

Senior challenge on Wednesday 10th November 2021 (15/10/2021)