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Project Maths - Higher Level
This course is for students interested in studying the Project Maths Higher Level Course in its entirety.
Description
This course is for students interested in studying the Project Maths Higher Level Course in its entirety. This course provides students with videos on all the Higher Level topics in one location listed by module and topic. In addition, a comprehensive assessment is provided which tests learners on the entire content of the Project Maths Higher Level Syllabus. These topics include Probability and Statistics, Geometry and Trigonometry, Numbers and Shapes, Algebra, Functions and Calculus.
Start Course NowModules
Module 1: Counting and Probability
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Arrangements and selections
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Expected value
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Probability of simple events
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Probability of compound events
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Independent events
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Probability using tree diagrams
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Probability trees with and without replacement
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Reverse probability - Bayes' theorem
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Bayes' theorem and medical testing
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Probability patterns of discrete variables
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Comparing distributions
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The binomial distribution
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A problem using binomial distribution
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Bernoulli trials and the binomial formula
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The normal distribution
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Using Excel to calculate binomial probabilities
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Using Excel to find the mean of a binomial distribution
Module 2: Statistics
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Cumulative frequency and the Ogive
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Cumulative frequency, quartiles and the inter-quartile range
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Mean of grouped frequency tables
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Constructing a stem and leaf diagram
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Comparative stem and leaf diagram
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Histograms - Understanding that area gives frequency
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Introduction to sampling techniques
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Random stratified sampling
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Statistical correlation
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Measuring correlation - Pearson's Correlation Coefficient
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Using Excel to calculate Pearson's Correlation Coefficient
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Using Excel to find Pearson's Correlation Coefficient, using the inbuilt function
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Regression lines and their equation
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The meaning of the coefficients in the equation of the regression line
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Using Excel to calculate the equation of the regression line
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Using Excel to find the equation of the regression line, using inbuilt function
Module 3: Synthetic Geometry
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Properties of Shapes
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Geometric reasoning - Part 1
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Geometric reasoning - Part 2
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Geometric reasoning - Part 3
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Geometric reasoning - Part 4
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Proof of Theorem 11
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Proof of Theorem 12
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Proof of Theorem 13
Module 4: Coordinate Geometry
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Geometry with coordinates
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Equation of a straight line
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Parallel lines
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Perpendicular lines
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The graphs of y=kx^n
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Translation of functions
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Reflection of functions
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Stretching functions
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General transformation of functions
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Equation of a circle 1 - Centre (0,0)
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Equation of a circle 2 - Centre not (0,0)
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Equation of a circle 3
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Diameter of a circle
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Tangent and normal to a circle
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Touching circles
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Prove that a line is a tangent to a circle
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Distance from a point to a line
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Angle between two lines
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Dividing a line segment in a given ratio
Module 5: Trigonometry
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Finding angles with sine ratio
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Finding sides with the sine ratio
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Finding angles with the cosine ratio
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Finding sides with the cosine ratio
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Finding angles with the tangent ratio
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Finding sides with the tangent ratio
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Finding a missing side with the Sine Rule
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Finding a missing angle with the Sine Rule
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Problem solving with the Cosine Rule
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Introducing Radians
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Using radians to find area of sector
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Sketching and matching trigonometric functions
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Graphing trigonometric functions. y = sinx - calculator work – in degrees
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Graphing trigonometric functions. y = 3sin2x - calculator work – in radians
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Introducing the Unit Circle
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The Unit Circle – drawing sine and cosine
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Comparing degrees to radians on the Unit Circle
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Solving trigonometric equations
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Proving the Sine Rule
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Proving the Cosine Rule
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Proving Trigonometric Identity – (Sin squared + cos squared = 1)
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Finding areas with trigonometry
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3-D trigonometry – The cuboid
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3-D trigonometry – Rectangular based pyramid
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Area of a segment
Module 6: Number Systems
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Large numbers in standard form
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Changing decimals to standard fom
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Changing large numbers from standard form
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Changing small numbers from standard form
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Adding and multiplying simple powers
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Working with indices
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Negative indices
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Fractional indices, numerator of 1
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Index power equal to 0
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Positive fractional indices all types
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Negative fractional indices
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Writing index numbers as a power of 2
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Quadratic number patterns
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Manipulating complex numbers and the complex conjugate
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The Argand Diagram and Modulus
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The meaning of i
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Patterns with imaginary numbers
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Rational or irrational
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Root 2 is Irrational – Proof by contradiction
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Finding the cube roots of 8
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Changing the base of logarithms
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Logarithmic equations
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Limits of sequences
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Arithmetic series
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Geometric series
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Infinite geometric series - Part 1
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Infinite geometric series - Part 2
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Deriving Amortisation formula from geometric series
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The sum of the first n natural numbers – Proof by induction
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Proof by induction applied to a geometric series
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Further proof by induction – Multiples of 3
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Further proof by induction – Factorials and powers
Module 7: Arithmetic - Financial Maths
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Profit, markup and margin
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Calculating profit with special offers
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Percentage loss - Part 1
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Percentage loss - Part 2
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Simple interest – Calculating the interest
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Simple interest – Calculating the rate
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Simple interest – Calculating the principal
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Simple interest – Calculating the period
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Percentage changes using multipliers
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Reverse percentages and VAT
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Introducing compound interest
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Compound interest and Annual Equivalent Rate
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Compound interest APR with credit cards
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Depreciation
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Calculating APR
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Calculating monthly interest from APR – 2 methods
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Income tax, USC and PRSI
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Net pay/Take home pay
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Present value – Working out future value
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Present value – Working out present value
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Present value – Harder example
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Present value and amortisation problem
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Completing an amortization schedule
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Savings and amortization
Module 8: Length, Area and Volume
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Circumference of a circle
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Area of a circle
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Finding radius and diameter of a circle from its perimeter
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Finding radius and diameter from area of circle
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Area of trapezium
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Area of acute angled triangles
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Area of obtuse angled triangles
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Area of parallelogram
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Area of rectangles
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Area Compound Shapes (rectangles)
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Area of compound shapes
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Area of Compound shapes (Triangles)
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Volume and surface area
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Sectors and arcs
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Area of a segment
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Volume of compound solid
Module 9: Complex Numbers
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Manipulating complex numbers and the complex conjugate
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The Argand Diagram and Modulus
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The meaning of i
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Patterns with imaginary numbers
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Writing complex numbers in polar form
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Multiplying and dividing in polar form (proof)
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Multiplying and dividing in polar form (example)
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Proof of De Moivre’s Theorem
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Complex numbers when solving quadratic equations
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Cubic equations with complex roots
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Finding the cube roots of 8
Module 10: Expressions and Formulae
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Factorising quadratics
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Factorising – Difference of two squares
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Expanding brackets (grid method)
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Expanding brackets (FOIL method)
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Expanding any two brackets
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Rearranging simple formulae
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Rearranging simple formulae - 2 steps
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Rearranging formulae with squares and square roots
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Rearranging formulae new subject appearing twice
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Simplifying Surds
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Further Calculations with Surds
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Completing the Square
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General Completing the Square
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Binomial expansion
Module 11: Solving Equations
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Equations with linear functions in the denominator
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Quadratic equations using the formula
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Quadratic equations non unitary x squared
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Quadratic equations both brackets the same sign
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Quadratic equations brackets with different signs
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Quadratic equations that have to be rearranged
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Solving simultaneous equations graphically
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Simultaneous equations both negative signs
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Simultaneous equations negative and positive signs
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Simultaneous equations both positive
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Hidden quadratics
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Use of the discriminant
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Equations with indices
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Logarithmic equations
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Solving exponential equations
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From roots to functions
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Non-linear simultaneous equations
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Factor Theorem - Part 1
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Factor Theorem - Part 2
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The Remainder Theorem
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General Remainder and Factor Theorem
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Finding roots of cubic equation
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Modulus equations
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Using graphs to solve modulus equations
Module 12: Inequalities
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Finding inequalities from shaded regions
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Solving linear inequalities with fractions
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Solving quadratic inequalities – Method 1
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Solving quadratic inequalities – Method 2
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Solving quadratic inequalities – Method 3
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Non-unitary x^2 - Trial and error method
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Non-unitary x^2 - Algebra method
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Non-unitary x^2 - Graphical method
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Special Cases (use SQI DoTS)
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Rational functions
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Modulus equations
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Using graphs to solve modulus equations
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Modulus inequalities
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Modulus function on a graph
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Graphical solution of modulus inequalities
Module 13: Functions
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Plotting quadratic graphs from table of values
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Plotting cubic graphs from table of values
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Plotting reciprocal graphs from table of values
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Plotting exponential graphs from table of values
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Matching equations and sketches
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Sketching parabola using completing the square
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Completing the square and sketching the full method
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Matching functions and graphs – quadratic / exponential
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Translation of functions using f(x-a)
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Translating of functions using f(x)+a
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Sketching functions in x direction
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Sketching functions in the y direction
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The Discriminant
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Inverse and bijective functions
Module 14: Calculus - Differentiation
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Basic differentiation of y = x^n
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Differentiation of polynomials
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Equation of a tangent
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Equation of a normal
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Harder questions on normals
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General differentiation of x^n
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Second derivative
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Differentiation of tangents
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Using different variables
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Differentiating products and quotients
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The Chain Rule
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The Product Rule
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The Quotient Rule
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Differentiation of logarithms
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Differentiation of y = sinx
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Differentiation of y = cosx
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Differentiation and rates of change
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Rate of change - Example 1
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Rate of change - Example 2
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Differentiation and turning points
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Differentiation and matching graphs
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Applications of differentiation 1 (displacement / velocity / acceleration)
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Applications of differentiation 2 (voltage / current)
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Maximum volume of box
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Exploring relationships between graphs of cubic functions and their differentials
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Exploring relationships between graphs of trigonometric functions and their differentials
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Exploring relationships between graphs of exponential functions and their differentials
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Inverse functions differentiated
Module 15: Calculus - Integration
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Basic integration
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The integral sign
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Harder integration
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Definite integration
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Area between lines
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Basic differential equations
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Integration of Trig functions (y=cos4x)
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Integration of exponentials
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Introducing the Trapezoidal Rule
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Trapezoidal Rule into integration
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Finding the area between a quadratic function and a straight line
Module 16: Project Maths- Higher Level Assessment
Learning Outcomes
Having completed this course students will be able to:
Describe the concepts of probability
Understand outcomes of random processes
Describe statistical reasoning with an aim to becoming a statistically aware consumer
Find, collect and organise data
Represent data graphically and numerically
Analyse, interpret and draw inferences from data
Develop synthesis and problem-solving skills
Determine the slope of a line given it’s equation
Find the equation of a line perpendicular to a given line and passing through a given point
Compute the angle between two given lines
Determine the equation of a circle having a given centre and radius
Find the equation of a tangent to a given circle at a specified point
Find distance and angle using Sine and Cosine Rules
Find length of an arc and area of a sector using circular measure
Solve the trigonometrical equations sin(x) = r and cos(x) = r in general form
Determine the sum of an arithmetic series
Determine the sum of a geometric series
Work out the repayment on a loan
Work out the future value of an investment
Represent a complex number in polar form
Use De-Moivres Theorem to simplify an expansion
Solve linear simultaneous equations with 3 unknowns
Factorise expressions of order 2 and 3
Search for roots of cubic equations and solve them
Solve one linear equation and one equation of order 2 with two unknowns
Solve basic inequalities
Solve inequalities involving fractions
Basic Differentiation of functions (including trig, exp and log)
The rules of differentiation (product rule, quotient rule, chain rule)
Determine the local maxima and local minima turning points of a curve
Understand rate of change of distance, area and volume
Understand the meaning of ‘anti-derivative’ and Indefinite Integration
Basic Integration of algebraic functions
Basic Integration of Trigonometrical and Exponential functions
How to use integration to find an area and use of the Trapezoidal Rule
Certification
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