Mathematics
Introduction
- Name: Raymund O’Connor
- Email: raymund.oconnor@csn.ie
Required for Class and Assessments
- Scientific Calculator (eg. as used for Leaving Certificate)
- Geometry Set (pencil, ruler, eraser, compass, protractor, etc)
IT Rules & Regulations
- From Student Handbook on www.csn.ie
Login / Log out of Office365/Moodle
- Username: csn20.firstname.lastname@csn.ie
- Password: !CSN.welcome
(Default username: student, !CSN.letmein)
File Management
- IT Induction (log into OneDrive – demo create, print, save files, etc)
- Create a folder for each component (files and folders)
- Backup to OneDrive (similar to Google Drive, iCloud, Dropbox, etc)
- Use OneDrive app on your smartphone to photograph your written work and save as a PDF.
- Submitting assessment work using Moodle
- Microsoft Excel – useful for calculating Mean, Standard Deviation, drawing graphs, etc)
Assessment (Deadlines must be adhered to)
- Assignment 1 – 30%
- Assignment 2 – 30%
- Exam – 40%
Sample Assessment Resources
Sample Word files (Download the relevant file below and save with your own name. Once you type your solutions save the Word file in PDF format and upload to Moodle)
- Assignment File: Maths Typed Solutions file – Ray O Connor
- Open Book End of Year Assignment: Maths End of Year Assessment – Typed Solutions Ray O Connor
- Equation Feature in Word https://support.office.com/en-us/article/can-i-write-an-equation-689ac0e5-bc51-4ecf-9427-fff8ed150858
- How to upload to Moodle Moodle Uploads
Examination – 40% Date available from Exams Department (End of April/Start of May) – Due to Covid19 the usual exam will be replaced by an open book assignment which must be typed, saved as a PDF file format and uploaded to Moodle.
Maths Study Support (Note: Room and time may change from year to year)
- Day: Details available from your course coordinator
Component
Resources
- Maths Tables – Maths_Formulae_Tables
- Log Tables Sin_Cos_Tan_Log_Tables –
- Useful videos re all Maths topics https://www.khanacademy.org/
- https://mathtv.com/
Self Directed Learning(Revision) – Maths Basics
- 0 Numbers (terminology)
- 1 Maths Rules – BIRDMAS – Add-Sub-Mult-Div
- Video – https://youtu.be/VwCLitK69PU
- 2 Simplifying Expressions
- 3 Simplifying Equations
- Solving Quad Equations using -b formula
- Factors/Factorising Factors and Factorising or Factors and Factorising
- 4 Factorising cubes
- Factorising – Exercises and Answers
- 5 Factorising and Adding Fractions
- 5 Working with fractions
- 6 Solving Equations with fractions
- Making a variable the subject of an equation – Make variable the subject of equation
- 7 Revision Exercises
- Test: https://forms.gle/XpCVQvY4rsXvvNaH6
Course Topics
1 MODELLING USING MATHEMATICS
1.1 Explain the concept of a mathematical model to include the difference between mathematical models and physical models
1.2 Explain the modeling process in diagrammatic form
1.3 Solve simple mathematical models to include identifying situations requiring mathematical modeling, and using appropriate mathematical skills and processes
1.4 Apply simple mathematical models to explain and predict behavior
- What is Maths Modelling? – https://www.youtube.com/watch?v=zbwyMigMFtQ
- Mathematical Modelling 1 Excel: Mathematical Modelling – Coordinate Geometry
- Examples/Exercises Mathematical Modelling Exercises – Area – Volume
- Mathematical Modelling – Definition
- Maths-Modelling-Diagrammatic-Form
- Solutions: Solutions to Exercises at end of Maths Modelling
- Real World Scenarios – Word Exercises and Solutions
- Real World Challenge – Snow Word: Real World Challenge – Snow
- Mathematical Model and Physical Models
- Making Predictions using Models
- Simple Mathematical Models to Explain and Predict Behavior
- Predicting using probability – https://www.khanacademy.org/math/cc-seventh-grade-math/cc-7th-probability-statistics/cc-7th-theoretical-and-experimental-probability/v/making-predictions-with-probability
- Predicting using Linear Models – https://www.youtube.com/watch?v=aVDiAGZmcPo
- The Representation of Data – Rhinos Example
- Physical and maths model video: https://www.youtube.com/watch?v=alFPhyrfmC4
2 STATISTICS AND PROBABILITY
2.1 Discuss statistical concepts to include discrete and continuous variables, sampling, variance, skewness
2.2 Present information in a range of graphical and tabular forms, using pie charts, trend graphs, correlation diagrams (+/-), cumulative frequency curves, histograms and frequency tables with both discrete and continuous variables
2.3 Calculate the statistics for measuring and contrasting averages and dispersion of grouped data by calculating the mean, mode, median, weighted average, range, inter-quartile range and standard deviation
2.4 Calculate the number of possible outcomes on tests with no repetitions by using the Fundamental Principle of Counting, and Permutations and Combinations
2.5 Demonstrate an understanding of relative frequency and probability by using Information Technology simulations
2.6 Solve simple probability problems of one or two events including where two events are mutually exclusive and where two events are independent
2.7 Discuss findings, to include interpretation of results and distortions which may arise, and reasons for findings
- Statistics – Sampling
- Statistics Basics – Standard Deviation Exercise using Excel
- Mean Exercises
- MeanModeMedianExercises
- Weighted_Mean_Exercises
- Assorted Mean Exercises
- Grouped Mean Exercises
- Assorted Grouped Mean Exercises Q11 Solution Video – Q12 Solution Video
- Histogram_Pie_Trend_Bar_Charts – Exercise Histogram_Exercise
- Cumulative Frequency Curve – Inter Quartile Range
- Ogive_Exercises Solutions: Q3 and Q4 Solutions
- Data_Continuous_Discrete_Variables – Discrete and Continous Variables
- Standard_Deviation – Frequency Distribution – Standard Deviations
- Standard Deviation Worksheet – Standard Deviation Worksheet 2
- Exercise – Student Ages Data
- Excel file with formula to calculate Standard Deviation
- Video: Mean – Variance – Standard Deviation – Range – Dispersion –Khan Academy
- Excel files with solutions: Standard Deviation
- Microsoft Excel – Formula re Calculating Mean-Variance-StandardDeviation
- Relative frequency and probability: 2-5 Relative frequency and probability
Permutations/Combinations/Principle of Counting
- n! = n(n-1)(n-2) .. 3.2.1 Example: 4 paintings to hang on a wall. Then you have 4 x 3 x 2 x 1 = 24 ways of hanging paintings.
- Fundamental-Principle-of-Counting
- Definition of Permutation
- Definition of Combination
- Permutations(Arrangements & Combinations(Selections) – Permutations_Arrangement_Exercises_Solutions
- Permutations with blocks(units) – Arrangements with blocks
- Arrangements of blocks exercise 4 boys and 3 girls
- Permutations-and-Arrangements-Exercises-and-Solutions – Video: Solution to Q2 Dictionary Question
Probability
- Probability and Chance
- Sample Spaces
- Probability_Exercises – Probability_Exercises_Solutions
- Probability_Not_Or_Exercise_Solution
- Relative Frequency
- Dependent Probability: https://www.khanacademy.org/math/ap-statistics/probability-ap/probability-multiplication-rule/v/introduction-to-dependent-probability
- Independent Probability: https://www.khanacademy.org/math/ap-statistics/probability-ap/probability-multiplication-rule/v/independent-events-2
- 13 Statistics and Chance
3 FUNCTIONS AND GRAPHS
3.1 Describe the properties of basic mathematical functions to include linear, quadratic, exponential, log and trigonometric functions
3.2 Define the inverse of a function
3.3 Graph linear and quadratic functions showing the relationship between the domain and range
3.4 Derive the inverse of a function from its algebraic expression
3.5 Calculate the equation of a straight line using a range of formulae to include distance between two points, slope, parallel lines and perpendicular lines
3.6 Solve maximum and minimum problems with limitations given by linear inequalities from graphs of linear inequalities and half planes
3.7 Analyse graphs of linear and quadratic functions for important properties to include domain and range, maximum and minimum values, increasing and decreasing intervals, periodicity
- Functions – Inverse Functions – Domain – Range
Functions – Inverse Functions – Domain – Range - Functions Exercises 1
- Functions Exercises 2 – Inverse Functions
- Simple function exercises https://mathbitsnotebook.com/Algebra1/Functions/FNNotationEvaluation.html
- Graphing Linear Functions Video
- Excel worksheets with examples of Linear and Quadratic Functions and Graphs Linear_and_Quadratic_Functions_and_Graphs
- Graphs creator(linear/quadratic/trig/etc) https://www.mathsisfun.com/data/function-grapher.php
- Linear and Quadratic Functions and Graphs in a given domain
- Graphs of Quadratic Functions and Questions on Graphs
- Solutions: Q4 Q5 (Q5 Graph drawn in Excel) Q6
- Draw graph in given domain and solve equations
- Pre Ass1b – Pre Ass1b Solution in class using:
- Quadratics_Exercises (https://www.stleos.nsw.edu.au)
- Linear_Inequalities_1
- Linear_Inequalities_2
- Excel file for calculating f(x) and -b formula Minus b Formula Calculator
- Max/min points – increasing/decreasing
- Functions-and-Graphs-Exercises-Vertex-Max-Min-Increasing – Solutions Solutions_in_Excel
- Graphing Cubic functions – Cubic Functions and Graphs Examples – Cubic Functions and Graphs in Excel
- Real Life examples of Linear Functions and Graphs (raymundoconnor.com)
- A4 printable graph paper Graph_paper_mm_A4
Linear Programming
- Graphing horizontal and vertical lines (also shading half planes)
- Drawing Lines (eg y=2x+3)
- Simultaneous Equations Simultaneous Equations
- Formula – Slope – Distance – Midpoint – Equation of a line
- Video: Plotting points on a Cartesian Place and Calculating the length of a Line Segment
- Video: Midpoint – Slope – Parallel and Perpendicular Lines
- Video: Equation of a line Also: y=mx+c formula explained
- Inequalities & Half planes (eg. x>-1, x<5) Video: Graphing Inequalities
- Lines_Midpoint_Slope_Equation_Formula_Exercises Excel: Linear Functions and Lines – Slope_Midpoint_Equation_Length
- Slopes of Parallel(S1 = S2) and Perpendicular (S1 * S2 = -1) Lines
- Line Exercises
- Graphing Inequalities and Shaded Regions
- Linear Programming Examples
- Linear Programming Exercises
- Linear Programming Exercises – Several
- Linear Programming Exercises – Several – Solutions
- Video – https://www.khanacademy.org/math/algebra/two-variable-linear-inequalities/modeling-with-linear-inequalities/v/u06-l3-t1-we3-graphing-systems-of-inequalities
- Exercises: Linear-Programming-Worksheet
Solutions: Linear Programming Worksheet and Solutions - Linear Programming Worksheet and Solutions
- Linear Programming Exercises 1 with Solution
- Linear Programming Exercises 2 with Solutions
- Linear Programming Exercises 3 with Solutions
- Question 10
- Question 11 – Important Question
- Video: https://www.youtube.com/watch?v=cafGkgECWbE
- Linear_Programming_PDF
- Linear Programming-Powerpoint EXCELLENT
- Linear Programming-1
- Resource: https://www.analyticsvidhya.com/blog/2017/02/lintroductory-guide-on-linear-programming-explained-in-simple-english/
- 11 Graphs, Functions and Rates
4 CALCULUS
4.1 Outline the key concepts of calculus to include limits, differentiation and integration
4.2 Explain the fundamental theorem of calculus
4.3 Calculate average rates of change for related variables x and y for a variety of standard functions y=f(x)
4.4 Differentiate simple standard functions using a table of derivatives
4.5 Use the Product Rule, Quotient Rule and Chain Rule to calculate the derivative of composite functions
4.6 Integrate standard integrals, polynomials, trigonometric and exponential functions
4.7 Calculate the area enclosed between a curve and the x-axis using integration
4.8 Apply differentiation to solve simple rates of change models to include maximum and minimum
4.9 Apply integration to solve simple practical real life problems
Videos
- Differentiation – Derivative – f'(x) – dy/dx – (see Maths tables Video: https://youtu.be/NNnvgpNEB_w)
- The following videos cover examples and exercises found in the notes below (4-5 Differentiation)
- Differentiating polynomial equations – https://youtu.be/ZAfKwWzSaOE
- Evaluating derivatives https://youtu.be/SvNH-BilB4U
- Evaluating derivatives 2 https://youtu.be/WOVhKY9hTYI
- Differentiation using Product Rule https://youtu.be/3fBqGnTaIig
- Differentiation using Quotient Rule https://youtu.be/rQw7Bg8E8WE
- Differentiation using Chain Rule https://youtu.be/BmjWL5LqLgM
- Second Derivative 1 https://youtu.be/olFoFnw3XRM
- Second Derivative 2 https://youtu.be/w0UT-HxVrmE
- Slope and Turning Points https://youtu.be/wXjrA3W8PF4
- Quadratic graph https://youtu.be/Fn73UiV6CIs
- Quadratic graph continued https://youtu.be/3pnGXz60DAE
- Graphs using differentiation https://youtu.be/UmqgoqZtf4c
- Graphs using differentiation continued https://youtu.be/KCuMQyC4twQ
- Cubic Graph & Turning Points https://youtu.be/I4svq9tc88M
- Find where Cubic curve cuts X-axis https://youtu.be/mth8lmlIY9M
- Speed and Acceleration https://youtu.be/1wFD7i6gKzQ
- Speed/Acceleration P539Q9 https://youtu.be/P_cJK5H_h5s
- Velocity (Rocket Example) https://youtu.be/Ft-P7mMcIaE
- Integration – Integrals – f – (see Maths tables https://youtu.be/NNnvgpNEB_w)
- Integration – Power Rule https://youtu.be/SlakHC5VjBs
- Integrals – Trigonometric https://youtu.be/WlOhmdx8GQA
- Definite Integrals 1 – https://youtu.be/OlaS_h5RLJA
- Definite Integrals 2 – https://youtu.be/YRCvp6STb_k
- Area between graph and X-axis https://youtu.be/2zQiqzNTFgs
- Area under graph y = x^2 +2 https://youtu.be/IsAjRDUszmo
- Volume generated by rotating area between curve and x-axis about the X-axis https://youtu.be/1sbFpuQx3cQ
Notes and Exercises
- Differentiation
- 4-0 Limits – 4-0 les10.1
- 4-1 Key concepts – Differentiation – Integration
- 4-3 Average Rates of Change of Related Variables
- 4-3 Average Rates of Change of Related Variables
- 4-5 Differentiation
- Differentiation – Exercises and Solutions
- Find the local maximum and minimum turning points for the following function
- Differentiation – Assorted Exercises
- Max Area – Differentiation Question – Derivatives – Slope – Find Max Area Question 10 – Q10 Max Min Points
- 4-8 Differentiation – Maximum and Minimum
- Applied Maximum and Minimum Problems
- Integration
- 4-6 Integration
- 4-7 Integration – Area under curve Solutions: Q5 Solution from 4-7 Area under curve notes
- Area under a Curve 3 examples – Integration
- Find the area between graph and x-axis exercises
- Area between curve and Y-axis – click here for Khanacademy.org
- Area between curve and the Y axis
- Area Under a Curve – Exercises with Answers
- 4-9 Integration – Volume
- 4-9-1 Integration – Area and Volume Exercises
- 4-9-1 Integration – Volume Exercises – Example 2 Math_Utah_Edu_Volume (complex)
- Rotating about y-axis video
- Exercise with Solutions – Find Volume rotated about x and y axis
- Q4a
- Q5a Q5c Q5e
- Q6a
- Volume when rotating about Y using Integration – Volumes_by_Integration_Examples
5 Complex Numbers
5.1 Explain what is meant by a complex number
5.2 Represent complex numbers on the Argand diagram to include distinguishing between the modulus and the argument
5.3 Solve quadratic equations with complex roots
5.4 Perform mathematical functions on complex numbers including addition, subtraction, multiplication, division, conjugate, modulus, and plot on an Argand diagram
Videos
- Writing a complex number in Polar form https://youtu.be/ZH8dcgxr-Zc
- Complex number to a power and applying De Moivres Theorem https://youtu.be/gECgxudKcsM
- Solving equations with Complex roots https://youtu.be/k0uvv-iBqdI
- Solving equations with Complex roots continued https://youtu.be/7y4iAdCTnXU
Notes
- Argand_Diagram
- Addition_Subtraction
- Multiplication_Division
- Conjugate_Exercises
- Solving Equations for Re and Im
- Modulus
- Modulus and Argument of a Complex Number
- Solve Quadratic Equations with Complex Roots
- Various Exercises
- Complex-Numbers-in-Polar-Form
- Complex Numbers in Polar Form Video https://www.youtube.com/watch?v=UvXDRwD4XoQ
- Complex Numbers and Powers – De Moivre’s Theorem De Moivres Theorem Solutions: Solutions_to_De_Moivres_Theorem_Powers_Exercises
5.5 Apply de Moivre’s Theorem to finding powers of Z and the cube root of 1
- Cube_root_of_1
- 5 Complex Numbers
- Evaluating_cube_roots
- Evaluating_the_cube_roots_1
- Finding Powers and Roots of Complex Numbers in Polar Form
- Solving Equations using De Moivres Theorem Video: https://www.youtube.com/watch?v=6OvQwM7KAIU
- Solving_Equations_Using_De_Moivres_Theorem_Exercise_1
- Solving_Equations_Using_De_Moivres_Theorem_Exercise_2
6 TRIGONOMETRY
6.1 Explore the uses of trigonometry in everyday life
6.2 Define sine, cosine and tangent functions as related to the unit circle
6.3 Solve practical, simple problems using appropriate trigonometric formulae and terminology, including the sine, cosine and tangent ratios for right angled triangles, area of triangle=1/2absin C, Sine Rule, Cosine Rule
6.4 Analyse the functions y = sinx, y = cosx, y = tanx and y = asinbx from plotted graphs by determining period, and amplitude.
- Theorem Pythagaros
- Define cos sin tan Answer: In_Terms_of_K
- Unit Circle
- Quadrants – if sinA= -0.8090 find two possible angles of A
- Solving_for_Angles_In_Quadrants
- Right Angled Triangles Exercises
- Non-Right Angled Triangles Formula – Sine and Cosine Rule
- Video: https://youtu.be/9hAl0-_prww
- Sine Rule – Exercises – Solution to Q9 from Sine Rule – Exercises
- Cosine-Rule-Exercises – Solutions to Cosine Rule – Exercises
- Patterns_Period_Range – Find period and range for each of the following
- Trigonometry Graphs Solving Problems – Trigonometric Functions – Graphs – Period – Range Q and A
- UnitCircle
- Special_Triangles
- 12 Geometry and Trigonometry
- 15 Complex Numbers and Trigonometry
- 6-1 Trigonometry
- 6-2 Trigonometry Unit Circle
- 6-3 Solving Triangles using SOHCAHTOA
- 6-4-2_SineCosineRule
- 6-5-TrigonometricGraphs
Sample Exam Papers
End of Year Review
Moodle Quizes
- Text file: 1 Mathematical Modelling Quiz – Moodle Word file with Answers: Maths Modelling Quiz – ANSWERS
Additional Notes, Exercises and Resources
Frequency Distribution – Standard Deviations
0 Mathematics5N1833ResourceList 1 Numbers 1 Statistics Charts 1 Statistics Interpretation of results and distortions 1 Stats Weighted Average – Modelling 
2 Maths Modelling 
2-5 Relative frequency and probability 
3 Linear Programming Example 
4-Linear-Programming-Exercises-Solutions 
6-2-1_Using_Logs_re_Angles0 Mathematics5N1833ResourceList
__________________________________________________________________________________
Online Tests
- Algebra https://docs.google.com/forms/d/1IWdlNKSQZUuGNtsQNHbXdFMwKYW1g6ptknPda62z1G4/edit
- Probability & Stats https://docs.google.com/forms/d/1LdZdPAtBX5EuVyR4WnD4Iqejnm4GzmA2HkYwq1VJMiQ/editDe Moivres TheoremExercise with Solutions – Find Volume rotated about x and y axis
Sample Exam Paper
Teaching Maths Resources
- Video how to teach Maths using OneNote – click here