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

Arrangements and selections

Expected value

Probability of simple events

Probability of compound events

Independent events

Probability using tree diagrams

Probability trees with and without replacement

Reverse probability  Bayes' theorem

Bayes' theorem and medical testing

Probability patterns of discrete variables

Comparing distributions

The binomial distribution

A problem using binomial distribution

Bernoulli trials and the binomial formula

The normal distribution

Using Excel to calculate binomial probabilities

Using Excel to find the mean of a binomial distribution
Module 2: Statistics

Cumulative frequency and the Ogive

Cumulative frequency, quartiles and the interquartile range

Mean of grouped frequency tables

Constructing a stem and leaf diagram

Comparative stem and leaf diagram

Histograms  Understanding that area gives frequency

Introduction to sampling techniques

Random stratified sampling

Statistical correlation

Measuring correlation  Pearson's Correlation Coefficient

Using Excel to calculate Pearson's Correlation Coefficient

Using Excel to find Pearson's Correlation Coefficient, using the inbuilt function

Regression lines and their equation

The meaning of the coefficients in the equation of the regression line

Using Excel to calculate the equation of the regression line

Using Excel to find the equation of the regression line, using inbuilt function
Module 3: Synthetic Geometry

Properties of Shapes

Geometric reasoning  Part 1

Geometric reasoning  Part 2

Geometric reasoning  Part 3

Geometric reasoning  Part 4

Proof of Theorem 11

Proof of Theorem 12

Proof of Theorem 13
Module 4: Coordinate Geometry

Geometry with coordinates

Equation of a straight line

Parallel lines

Perpendicular lines

The graphs of y=kx^n

Translation of functions

Reflection of functions

Stretching functions

General transformation of functions

Equation of a circle 1  Centre (0,0)

Equation of a circle 2  Centre not (0,0)

Equation of a circle 3

Diameter of a circle

Tangent and normal to a circle

Touching circles

Prove that a line is a tangent to a circle

Distance from a point to a line

Angle between two lines

Dividing a line segment in a given ratio
Module 5: Trigonometry

Finding angles with sine ratio

Finding sides with the sine ratio

Finding angles with the cosine ratio

Finding sides with the cosine ratio

Finding angles with the tangent ratio

Finding sides with the tangent ratio

Finding a missing side with the Sine Rule

Finding a missing angle with the Sine Rule

Problem solving with the Cosine Rule

Introducing Radians

Using radians to find area of sector

Sketching and matching trigonometric functions

Graphing trigonometric functions. y = sinx  calculator work – in degrees

Graphing trigonometric functions. y = 3sin2x  calculator work – in radians

Introducing the Unit Circle

The Unit Circle – drawing sine and cosine

Comparing degrees to radians on the Unit Circle

Solving trigonometric equations

Proving the Sine Rule

Proving the Cosine Rule

Proving Trigonometric Identity – (Sin squared + cos squared = 1)

Finding areas with trigonometry

3D trigonometry – The cuboid

3D trigonometry – Rectangular based pyramid

Area of a segment
Module 6: Number Systems

Large numbers in standard form

Changing decimals to standard fom

Changing large numbers from standard form

Changing small numbers from standard form

Adding and multiplying simple powers

Working with indices

Negative indices

Fractional indices, numerator of 1

Index power equal to 0

Positive fractional indices all types

Negative fractional indices

Writing index numbers as a power of 2

Quadratic number patterns

Manipulating complex numbers and the complex conjugate

The Argand Diagram and Modulus

The meaning of i

Patterns with imaginary numbers

Rational or irrational

Root 2 is Irrational – Proof by contradiction

Finding the cube roots of 8

Changing the base of logarithms

Logarithmic equations

Limits of sequences

Arithmetic series

Geometric series

Infinite geometric series  Part 1

Infinite geometric series  Part 2

Deriving Amortisation formula from geometric series

The sum of the first n natural numbers – Proof by induction

Proof by induction applied to a geometric series

Further proof by induction – Multiples of 3

Further proof by induction – Factorials and powers
Module 7: Arithmetic  Financial Maths

Profit, markup and margin

Calculating profit with special offers

Percentage loss  Part 1

Percentage loss  Part 2

Simple interest – Calculating the interest

Simple interest – Calculating the rate

Simple interest – Calculating the principal

Simple interest – Calculating the period

Percentage changes using multipliers

Reverse percentages and VAT

Introducing compound interest

Compound interest and Annual Equivalent Rate

Compound interest APR with credit cards

Depreciation

Calculating APR

Calculating monthly interest from APR – 2 methods

Income tax, USC and PRSI

Net pay/Take home pay

Present value – Working out future value

Present value – Working out present value

Present value – Harder example

Present value and amortisation problem

Completing an amortization schedule

Savings and amortization
Module 8: Length, Area and Volume

Circumference of a circle

Area of a circle

Finding radius and diameter of a circle from its perimeter

Finding radius and diameter from area of circle

Area of trapezium

Area of acute angled triangles

Area of obtuse angled triangles

Area of parallelogram

Area of rectangles

Area Compound Shapes (rectangles)

Area of compound shapes

Area of Compound shapes (Triangles)

Volume and surface area

Sectors and arcs

Area of a segment

Volume of compound solid
Module 9: Complex Numbers

Manipulating complex numbers and the complex conjugate

The Argand Diagram and Modulus

The meaning of i

Patterns with imaginary numbers

Writing complex numbers in polar form

Multiplying and dividing in polar form (proof)

Multiplying and dividing in polar form (example)

Proof of De Moivre’s Theorem

Complex numbers when solving quadratic equations

Cubic equations with complex roots

Finding the cube roots of 8
Module 10: Expressions and Formulae

Factorising quadratics

Factorising – Difference of two squares

Expanding brackets (grid method)

Expanding brackets (FOIL method)

Expanding any two brackets

Rearranging simple formulae

Rearranging simple formulae  2 steps

Rearranging formulae with squares and square roots

Rearranging formulae new subject appearing twice

Simplifying Surds

Further Calculations with Surds

Completing the Square

General Completing the Square

Binomial expansion
Module 11: Solving Equations

Equations with linear functions in the denominator

Quadratic equations using the formula

Quadratic equations non unitary x squared

Quadratic equations both brackets the same sign

Quadratic equations brackets with different signs

Quadratic equations that have to be rearranged

Solving simultaneous equations graphically

Simultaneous equations both negative signs

Simultaneous equations negative and positive signs

Simultaneous equations both positive

Hidden quadratics

Use of the discriminant

Equations with indices

Logarithmic equations

Solving exponential equations

From roots to functions

Nonlinear simultaneous equations

Factor Theorem  Part 1

Factor Theorem  Part 2

The Remainder Theorem

General Remainder and Factor Theorem

Finding roots of cubic equation

Modulus equations

Using graphs to solve modulus equations
Module 12: Inequalities

Finding inequalities from shaded regions

Solving linear inequalities with fractions

Solving quadratic inequalities – Method 1

Solving quadratic inequalities – Method 2

Solving quadratic inequalities – Method 3

Nonunitary x^2  Trial and error method

Nonunitary x^2  Algebra method

Nonunitary x^2  Graphical method

Special Cases (use SQI DoTS)

Rational functions

Modulus equations

Using graphs to solve modulus equations

Modulus inequalities

Modulus function on a graph

Graphical solution of modulus inequalities
Module 13: Functions

Plotting quadratic graphs from table of values

Plotting cubic graphs from table of values

Plotting reciprocal graphs from table of values

Plotting exponential graphs from table of values

Matching equations and sketches

Sketching parabola using completing the square

Completing the square and sketching the full method

Matching functions and graphs – quadratic / exponential

Translation of functions using f(xa)

Translating of functions using f(x)+a

Sketching functions in x direction

Sketching functions in the y direction

The Discriminant

Inverse and bijective functions
Module 14: Calculus  Differentiation

Basic differentiation of y = x^n

Differentiation of polynomials

Equation of a tangent

Equation of a normal

Harder questions on normals

General differentiation of x^n

Second derivative

Differentiation of tangents

Using different variables

Differentiating products and quotients

The Chain Rule

The Product Rule

The Quotient Rule

Differentiation of logarithms

Differentiation of y = sinx

Differentiation of y = cosx

Differentiation and rates of change

Rate of change  Example 1

Rate of change  Example 2

Differentiation and turning points

Differentiation and matching graphs

Applications of differentiation 1 (displacement / velocity / acceleration)

Applications of differentiation 2 (voltage / current)

Maximum volume of box

Exploring relationships between graphs of cubic functions and their differentials

Exploring relationships between graphs of trigonometric functions and their differentials

Exploring relationships between graphs of exponential functions and their differentials

Inverse functions differentiated
Module 15: Calculus  Integration

Basic integration

The integral sign

Harder integration

Definite integration

Area between lines

Basic differential equations

Integration of Trig functions (y=cos4x)

Integration of exponentials

Introducing the Trapezoidal Rule

Trapezoidal Rule into integration

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 problemsolving 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 DeMoivres 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 ‘antiderivative’ 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
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