Higher Mathematics Home Page
The course is based on :
Algebra , Calculus , Geometry and TrigonometryAn external SQA exam is sat at the end of the course, with calculator and non-calculator papers.
Scholar
Formulae
Higher Maths Past Papers SQA Higher Maths Check List
National 5 Maths
Advanced Higher Maths
Prerequisite Knowledge
You must have a good grasp of the following :
- Completing the Square.
- Changing the subject of the formula.
- Manipulating Algebraic fractions.
- Factorising.
- Manipulating Surds and Indices.
- Manipulating quadratics.
- Basic Trigonometry.
Algebra
Algebraic Manipulation
Key Subskills
- Factorising polynomial expressions
- Solving polynomial equations
- Using the discriminant
- Solving quadratics
- Intersections of curves and straight lines
- Recurrence relations
- Limits of sequences
Exponentials and Logarithms
Key subskills
Simplifying numerical expressions using exponent and logarithm laws.
Solving logarithmic and exponential equations.
Applying exponent and logarithm laws in algebraic contexts.
Exponentials
Graphs of Logs and Exponentials
Functions and Graphs
Key subskills
Identifying and sketching related functions.
Determining composite and inverse functions.
Graphs
- Simple graphs of functions
- Reflections, translations, etc.
- y = −f(x)
- y = f(−x)
- y = f(x ± c)
- y = f(x) ± d
- y = f(bx)
- y = f(x)/b
- y = a f(x)
- y = f(x)/a
- y = a f(bx ± c) ± d
- y = x
- y = |f(x)|
- Sketching polynomial equations
- Graphing exponential & logarithmic functions
- Logarithmic–linear scales
- Log–log graphs
- Excel: exponential/log graphs
-
Interactive - Reflection
y = −f(x)
Calculus
Differentiation
Key subskills
Differentiating functions.
Tangents to curves at a point.
Increasing/decreasing functions.
Nature tables.
Optimisation problems.
Rate of change applications.
Background & First Principles
Main Differentiation
Tangents
Stationary Points
Curve Sketching
Applications of Differentiation
Integration
Key subskills
Integrating functions.
Definite integrals.
Area under curves.
Area between curves.
Geometry
The Circle
Key subskills
Determining and using the equation of a circle.
Using properties of tangency.
Determining intersections.
The Straight Line
Key subskills
Parallel & perpendicular lines.
Using m = tanθ.
Medians, altitudes, perpendicular bisectors.
Vectors
Key subskills
Working with 2D vectors.
Working with 3D coordinates.
Using vector components.
Trigonometry
Key subskills
Application of addition & double‑angle formulae.
Application of trigonometric identities.
Converting acosx ± bsinx to kcos(x ± α) or ksin(x ± α), k > 0.
General
Radians & Exact Values
Learning Outcomes
Algebra
Key Skills
- Manipulating algebraic expressions
- Identifying and sketching related functions
- Composite and inverse functions
- Solving algebraic equations
Learning Outcomes
- A1 Composite functions
- A2 Completing the square
- A3 Sketching after transformations
- A4 Finding f-1(x)
- A5 Sketching f′(x) from f(x)
- A6 Recurrence relations
- A7 Limits of sequences
- A8 Quadratic inequalities
- A9 Discriminant
- A10 Factorising cubic/quartic expressions
- A11 Solving cubic/quartic equations
- A12 Intersections of lines & curves
- A13 Simplifying log/exp expressions
- A14 Laws of logs & exponents
- A15 Solving log/exp equations
- A16 y = axb and y = abx
- A17 Straight‑line confirmation
- A18 Modelling with logs/exponentials
- A19 Sketching inverses of log/exp functions
Calculus
Key Skills
- Differentiating functions
- Investigating properties of functions
- Integrating functions
- Calculating definite integrals
- Applying differential calculus
- Applying integral calculus
Learning Outcomes
- C1 Differentiating algebraic functions
- C2 Tangents to curves
- C3 Increasing/decreasing functions
- C4 Stationary points & sketching
- C5 Differentiating ksin(x), kcos(x)
- C6 Chain rule
- C7 Optimisation
- C8 Rate of change
- C9 Integrating algebraic functions
- C10 ∫(x + q)ⁿ dx
- C11 ∫ksin(x), ∫kcos(x)
- C12 ∫(px + q)ⁿ dx
- C13 ∫psin(qx+r), ∫pcos(qx+r)
- C14 Solving dy/dx = f(x)
- C15 Definite integrals
- C16 Area under curves
- C17 Area between curves/lines
- C18 Recovering functions from rate of change
Geometry
Key Skills
- Rectilinear shapes
- Circles & graphs
- Modelling with sequences
- Vector connections
- Working with vectors
Learning Outcomes
- G1 Parallel & perpendicular lines
- G2 m = tanθ
- G3 Medians, altitudes, bisectors
- G4 Perpendicularity
- G5 Equation of a circle
- G6 Tangency
- G7 Line–circle & circle–circle intersections
- G8 Vector pathways in 3D
- G9 Collinearity
- G10 Internal division points
- G11 Scalar product & angle
- G12 Using the scalar product
- G13 Unit vectors i, j, k
Trigonometry
Key Skills
- Solving trigonometric equations
- Addition & double‑angle formulae
- Trigonometric identities
- Manipulating trig expressions
Learning Outcomes
- T1 Solving trig equations (degrees & radians)
- T2 Addition & double‑angle formulae
- T3 Trigonometric identities
- T4 Converting acosx ± bsinx to kcos(x ± α) or ksin(x ± α)
Books
A selection of resources available at Amazon
© Alexander Forrest