Online Videos

Below are links to instructional videos that are freely available to those students who sign up with Daniel’s Maths Tuition NZ for at least a term. They range from basic NCEA Levels 1-3 material up to the final year of university, and cover the fundamentals of mathematics in a systematic way. They are not a substitute for your school or university course but complement the material you are learning, in tandem with our private tuition.

Back to Basics

  1. What are These Videos About?
  2. BEDMAS and Column Arithmetic
  3. Long Division and Continued Fractions
  4. Solving Quadratic Equations
  5. Exact Solutions vs Numerical Solutions
  6. How to Use a Calculator Properly

Solving Algebraic Problems

  1. Triangle Relations
  2. Rubber Duckies
  3. Substitution and Powers
  4. Areas and Quadratics
  5. Understanding Rational Functions
  6. Basketball and Cylinders

Tables, Equations and Graphs

  1. Plotting Data
  2. Courier Costings
  3. Parabolas and Exponentials
  4. Ball Chucking
  5. Rates of Cooling

Geometric Reasoning

  1. Isosceles Triangles
  2. Ferris Wheel
  3. Laws of the Circle
  4. Triangles within Triangles
  5. Joined Up Thinking
  6. Getting Your Bearings

Understanding Logic and Proofs

  1. Sets and Logic
  2. What are Functions?
  3. Where’s the Proof?
  4. Speaking Logically
  5. Mathematical Induction
  6. Strong Induction
  7. Recurrence Relations

A Gentle Introduction to Algebra

  1. Trigonometry
  2. Radians vs Degrees
  3. Complex Numbers
  4. Operations and the Polar Form
  5. Solving Complex Equations
  6. A Couple of Problems
  7. A Couple More Problems
  8. Vectors
  9. The Scalar Product
  10. Lines and Planes
  11. Systems of Equations
  12. Echelon Form
  13. Gaussian Elimination
  14. Gauss-Jordan Elimination

Using Algebraic Methods

  1. Simplifying And Factorising
  2. Mark it in a Calendar
  3. Discriminating between Solutions
  4. Reciprocal Polynomials
  5. Logarithm Laws
  6. Manipulating Logarithmic Equations
  7. Intersections of Curves
  8. Compound Interest
  9. Cubes vs Cuboids

Using Calculus Methods

  1. The Newton-Raphson Method
  2. Gradients of Functions
  3. Maximising Logos
  4. Recovering the Function
  5. It’s Just Not Cricket
  6. Slopes and Tangents
  7. Platonic Solids
  8. Taking the Plunge
  9. Building Houses

A User’s Guide to Basic Calculus

Portrait of Sir Issac Newton by Jean-Leon Huens.
  1. Types of Numbers
  2. Polynomials
  3. Rational Functions
  4. Trigonometric Functions
  5. Trigonometric Identities
  6. Sine and Cosine Rules
  7. Calculus at a Glance
  8. Rules of the Game
  9. Exponentials and Logarithms
  10. Derivatives of Inverse Functions
  11. Hyperbolic Functions
  12. Related Rates
  13. What is Integration?
  14. Basic Integration Techniques
  15. Integration-by-Parts
  16. Volumes and Surface Areas

Complex Number Problems

  1. Frankenstein’s Number “i”
  2. Absolute Values
  3. Finding Complex Roots
  4. Arguments and Loci
  5. Taking n-th Roots
  6. Manipulating Complex Expressions

Solving Differentiation Problems

  1. First Derivatives
  2. Parametric Form
  3. Gradients and Displacement
  4. Areas and Lengths
  5. Comparing Rates of Change
  6. Optimisation

Solving Integration Problems

  1. No sec^2(x), we’re British!
  2. Limits of Integration
  3. Separable D.E.s
  4. There’s a Hole in my Oil Tank
  5. The Trapezium Rule
  6. Areas between Curves

Matrices and Linear Algebra

  1. What are Matrices?
  2. Matrix Multiplication
  3. Matrix Inversion
  4. Homogeneous Equations
  5. 2×2 Matrices
  6. Linear Transformations
  7. Eigenvalues
  8. What is the Cross Product?
  9. Applications of the Cross Product
  10. Linear Equations in 3-D
  11. Shortest Distances
  12. Computing NxN-Determinants
  13. 3×3-Eigenvalues and Eigenvectors
  14. Cramer’s Rule

Mechanics

  1. Newtonian Physics
  2. Vector Displacement
  3. Velocity-Time Graphs
  4. Vertical Projection of a Particle
  5. Catapulting Food and Medicine
  6. Collision of the Spheres
  7. Exploding Pies
  8. Driving Miss Daisy
  9. Inclined Planes
  10. Apr├Ęs-Ski
  11. Ropes and Pulleys
  12. Give us a Tow
  13. Walking a Tightrope
  14. Hanging by a Thread
  15. Bascule-Suspension Bridges
  16. The Great Glass Elevator

Scholarship Calculus

  1. The Plot Thickens
  2. A Complex Question
  3. Simultaneous Tan Functions
  4. Symmetries in Polynomials
  5. Roots of the Same Sign
  6. Double Angled Triangles
  7. GP Side Lengths
  8. Logarithmic Differentiation
  9. Drippers and Flasks
  10. Sectors and Chords
  11. De Moivre on Steroids
  12. Area of an Astroid
  13. Triangles inside Ellipses
  14. A Cheap Parlour Trick
  15. A Nasty Integral
  16. Predator-Prey Models

Differential Equations

  1. Introduction to D.E.s
  2. Solving Separable D.E.s
  3. First Order Homogeneous D.E.s
  4. Linear First Order D.E.s
  5. Euler’s Method and Runge-Kutta
  6. Second Order D.E.s
  7. More Second Order D.E.s
  8. Power Series Methods

Multi-Variable Functions

  1. Partial Derivatives
  2. Second Order Derivatives
  3. The Total Differential
  4. More Rates of Change
  5. Locating Stationary Points
  6. Application to Optimisation
  7. Partial Integration
  8. The Wave Equation
  9. The Heat Equation
  10. Further Topics in PDEs

Exploring Further Calculus

  1. The Completeness Axiom
  2. Limits of Sequences
  3. Ratio, Comparison and Alternating Series Tests
  4. The Real Topology
  5. The Algebra of Limits
  6. Left-hand and Right-hand Limits
  7. Intermediate Value Theorem and l’Hopital’s Rule
  8. Rolle’s Theorem and the Mean Value Theorem
  9. Taylor series and Stationary Points
  10. Big O(-) Notation
  11. Partitions
  12. Defining the Integral
  13. Uniform Continuity
  14. Constructing the Anti-Derivative
  15. Fundamental Theorem of Calculus

Fun with Number Theory

  1. Fermat’s Last Theorem
  2. Sums of Squares
  3. Polynomials and Irreducibility
  4. Algebraic vs Transcendental Numbers
  5. Classical Formulae for Roots
  6. Diophantine Approximation
  7. Rational Points on Conics
  8. Arithmetic Functions
  9. Euler’s Phi-function
  10. Properties of Congruences
  11. Solving Congruences
  12. Another Example
  13. Euclid’s Algorithm
  14. More Examples
  15. Legendre Symbols
  16. Residues and Non-Residues
  17. Gauss’ Lemma
  18. Law of Quadratic Reciprocity

Elements of Modern Cryptography

  1. What is Cryptography?
  2. Some Basic Cyphers
  3. Kerckhoffs’ Principle
  4. Index of Coincidence
  5. Estimating the I.C.
  6. Polyalphabetic Substitutions
  7. Combining Cryptosystems
  8. Feistel Cyphers
  9. Data Encryption Standard
  10. One-way Functions and Passwords
  11. Diffie-Hellman Key Exchange
  12. The Massey-Omura Cryptosystem
  13. The R.S.A. Cryptosystem
  14. The ElGamal Cryptosystem
  15. What is Information Theory?
  16. Standard Entropy
  17. Sources and Codes
  18. Digital Compression
  19. Key Equivocation
  20. The Unicity Distance
  21. Estimating the U.D. in Practice