WhizzNotes

Mathematics Advanced

Functions, geometry and measurement, trigonometry, exponentials and logarithms, calculus, sequences, financial maths, statistics and networks for Year 11 and 12. Every topic starts from Year 9 and 10 and the whole Year 10 Advanced course is covered, so it also works as a Year 10 exam refresher.

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Pick your year on any chapter.

Unit 1 Functions

  1. 1The Algebra ToolkitExpanding, factorising, index laws, surds and solving equations. The skills every later chapter uses.
  2. 2Surds and Irrational NumbersRational and irrational numbers, simplifying surds, adding, multiplying and dividing them, and rationalising denominators.
  3. 3Algebraic FractionsSimplifying, multiplying, dividing, adding and subtracting algebraic fractions, and solving equations that contain fractions.
  4. 4Straight Lines and Coordinate GeometryGradient, y = mx + c, distance, midpoint, parallel and perpendicular lines, and linear models.
  5. 5Simultaneous Equations, Inequalities and RegionsLinear inequalities, simultaneous equations by substitution and elimination, word problems, and regions on the number plane.
  6. 6Quadratics and ParabolasSolving quadratics three ways, sketching parabolas, the vertex, and the discriminant.
  7. 7Working with FunctionsFunction machines, function notation, domain and range, the vertical line test, composite, odd and even functions.
  8. 8A Library of GraphsRecognise and sketch cubics, hyperbolas, exponentials, circles, absolute value and polynomial graphs.
  9. 9Polynomials and the Factor TheoremPolynomial language, expanding, long division, the remainder and factor theorems, finding zeros and sketching cubics.
  10. 10Graph TransformationsMoving, flipping and stretching graphs. Translations, reflections and dilations of any function.
  11. 11Modelling with FunctionsDirect and inverse variation, solving equations with graphs, and choosing a function to model real data.

Unit 2 Geometry and Measurement

  1. 12Congruent and Similar FiguresAngle facts, congruence tests, proving properties of quadrilaterals, similar triangles and scale factors for length, area and volume.
  2. 13Circle Geometry: Chords and AnglesCircle words, chord properties, the angle at the centre, angles in the same segment and in a semicircle, and cyclic quadrilaterals.
  3. 14Circle Geometry: Tangents, Chords and SecantsTangent and radius, tangents from an external point, the alternate segment theorem, and intersecting chords, secants and tangents.
  4. 15Area, Perimeter and Accuracy of MeasurementLength and area units, area of triangles, quadrilaterals, circles and sectors, composite shapes, and limits of accuracy and error.
  5. 16Surface Area and Volume of SolidsSurface area and volume of prisms, cylinders, pyramids, cones and spheres, capacity, and composite solids.

Unit 3 Trigonometric Functions

  1. 17Right-Angled TrigonometryPythagoras, sine, cosine and tangent, angles of elevation, bearings and exact values.
  2. 18The Sine Rule, Cosine Rule and Triangle AreaSolving triangles without a right angle. Sine rule, cosine rule, area formula and the ambiguous case.
  3. 19Radians and the Unit CircleArcs and sectors, measuring angles in radians, and trig ratios for any angle using the unit circle.
  4. 20Graphs of Trigonometric FunctionsThe waves of sin, cos and tan. Amplitude, period, phase shift and modelling things that repeat.
  5. 21Trigonometric Identities and EquationsKey identities, reciprocal ratios, proving identities and solving trig equations over a domain.

Unit 4 Exponential and Logarithmic Functions

  1. 22Exponential Functions and the Number eGrowth and decay, graphs of exponentials, Euler's number e and the derivative of e to the x.
  2. 23LogarithmsWhat a log is, the log laws, natural logs, log graphs and solving exponential equations.
  3. 24Scientific Notation and Logarithmic ScalesVery large and very small numbers, significant figures, and the log scales behind decibels, earthquakes and pH.

Unit 5 Calculus: Differentiation

  1. 25Rates of Change and the DerivativeFrom average speed to the gradient of a curve. Limits, first principles and what differentiable means.
  2. 26The Rules of DifferentiationPower rule, sums, product, quotient and chain rules, and the derivatives of eˣ, ln x, sin x, cos x and tan x.
  3. 27Tangents, Stationary Points and Curve SketchingEquations of tangents and normals, increasing and decreasing, stationary points, concavity and sketching curves.
  4. 28Applications of DifferentiationRates of change, motion in a straight line, and optimisation problems that find the best possible answer.

Unit 6 Calculus: Integration

  1. 29Anti-DifferentiationWorking backwards from a gradient. Primitives, the constant C, and the standard integrals.
  2. 30The Definite Integral and Area Under a CurveArea from rectangles and trapeziums, the trapezoidal rule, the fundamental theorem of calculus and signed area.
  3. 31Areas Between Curves and Applications of IntegrationArea between two curves, total change from a rate, and motion problems solved by integrating.

Unit 7 Sequences and Series

  1. 32Arithmetic Sequences and SeriesNumber patterns that add the same amount each step. The nth term, the sum of n terms and sigma notation.
  2. 33Geometric Sequences and SeriesPatterns that multiply by the same number each step. The nth term, sums, and the limiting sum of an infinite series.

Unit 8 Financial Mathematics

  1. 34Simple Interest, Compound Interest and DepreciationHow money grows and how assets lose value. Simple and compound interest, compounding periods and depreciation.
  2. 35Annuities and LoansRegular deposits and regular repayments. Future value, present value, interest factor tables and reducing balance loans.

Unit 9 Statistical Analysis

  1. 36Probability, Sets, Venn Diagrams and Two-Way TablesSample spaces, complements, Venn diagrams and two-way tables, set notation, the addition rule and conditional probability.
  2. 37Multi-Stage Experiments and Tree DiagramsTwo-step experiments, grids and tree diagrams, with and without replacement, and the "at least one" trick.
  3. 38Counting Principles, Arrangements and SelectionsThe multiplication principle, factorials, arrangements where order matters, selections where it does not, and counting in probability.
  4. 39Collecting, Displaying and Misusing DataTypes of data, census and samples, sampling methods and bias, stem-and-leaf plots, histograms, misleading graphs and time-series data.
  5. 40Data, Summary Statistics and Bivariate DataMean, median, quartiles, box plots, standard deviation, distribution shape, correlation and lines of best fit.
  6. 41Discrete Random VariablesProbability distributions, expected value, fair games, variance and standard deviation of a random variable.
  7. 42Continuous Random VariablesMeasured data, histograms as areas, probability density functions, cumulative distribution, median, mean and variance.
  8. 43The Normal DistributionThe bell curve, z-scores, the 68-95-99.7 rule, standard normal tables and finding probabilities.

Unit 10 Networks

  1. 44Networks, Paths and CircuitsVertices, edges and degrees, planar and isomorphic graphs, Euler trails and circuits, Hamiltonian paths and shortest paths.

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