Higher Mathematics Home Page

The course is based on :

Algebra , Calculus , Geometry and Trigonometry

An 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


Exponentials and Logarithms


Key subskills
Simplifying a numerical expression, using the laws of logarithms and exponents.
Solving logarithmic and exponential equations.
Using the laws of logarithms and exponents.

Exponentials

Logarithms

Graphs



Calculus

Differentiation


Key subskills
Differentiating functions.
Determining the equation of a tangent to a curve at a given point by differentiation.
Determining where a function is strictly increasing/decreasing.
Using nature tables.
Using differentiation to determine the optimal solution for a given problem.
Solve problems using rate of change.

Tangents

Stationary Points

Curve Sketching

Applications of differentiation


Integration


Key subskills
Integrating functions.
Calculating definite integrals of polynomial functions.
Finding the area between a curve and the x axis.
Finding area under and between curves.

Basic Integration


Geometry




The Straight Line


Key subskills
Find the equation of parallel and perpendicular lines.
Using m = tan θ to calculate a gradient or angle.
Using properties of medians, altitudes and perpendicular bisectors.

General

Straight lines and circles


Vectors


Key subskills
Working with 2D vectors.
Working with 3D coordinates.
Using vector components

Basic Vectors



Course Overview and Learning Outcomes


Algebra

Manipulating algebraic expressions
Identifying and sketching related functions
Determining composite and inverse functions
Solving algebraic equations

Learning Outcomes

  • A1 Determining a composite function
  • A2 Completing the square in a quadratic expression
  • A3 Identifying or sketching a function after a transformation
  • A4 Determining f-1 (x) of functions
  • A5 Sketch f'(x) given the graph of y = f(x)
  • A6 Determining and using a recurrence relation
  • A7 Finding and interpreting the limit of a sequence, where it exists
  • A8 Solve quadratic inequalities
  • A9 Use the discriminant
  • A10 Factorising a cubic or quartic polynomial expression
  • A11 Solving a cubic or quartic polynomial equation
  • A12 Finding the coordinates of the point(s) of the intersection of a straight line and a curve or of two curves
  • A13 Simplifying a numerical expression using the laws of logarithms and exponents
  • A14 Using the laws of logarithms and exponents
  • A15 Solving logarithmic and exponential equations
  • A16 Solve equations involving y = axb and y = abx
  • A17 Use a straight line graph to confirm relationships of the form y = axb and y = abx
  • A18 Model mathematically situations involving the logarithmic or exponential function
  • A19 Sketching the inverse of a logarithmic or an exponential function

Older Learning Outcomes

E&F 1.1 Applying algebraic skills to logarithms and exponentials.
E&F 1.3 Applying algebraic and trigonometric skills to functions.

R&C 1.1 Applying algebraic skills to solve equations.
Apps 1.3 Applying algebraic skills to sequences.

Old Higher Course
Unit 2 LO1 Use the factor/remainder theorem and apply quadratic theory.
Unit 3 LO3 Logarithmic and exponential functions.


Calculus

Differentiating functions
Using differentiation to investigate the nature and properties of functions
Integrating functions
Using integration to calculate definite integrals
Applying differential calculus
Applying integral calculus

Learning Outcomes

  • C1 Differentiating an algebraic function
  • C2 Use differentiation to find the equation of a tangent to a curve at a given point
  • C3 Determining where a function is strictly increasing/decreasing
  • C4 Use stationary points to sketch graphs
  • C5 Differentiate ksin(x) and kcos(x)
  • C6 Differentiating a composite function using the chain rule
  • C7 Determining the optimal solution for a given problem
  • C8 Solving problems using rate of change
  • C9 Integrating an algebraic function
  • C10 Integrating functions of form f(x)=(x + q)n n ≠-1
  • C11 Integrating ksin(x) and kcos(x)
  • C12 Integrating functions of form f(x)=(px + q)n n ≠-1
  • C13 Integrate psin(qx+r) and pcos(qx+r)
  • C14 Solve differential eqns of form dy/dx=f(x)
  • C15 Calculating definite integrals of functions
  • C16 Finding the area between a curve and the x-axis
  • C17 Finding the area between a straight line and a curve or two curves
  • C18 Determine and use a function from a given rate of change and initial conditions

Older Learning Outcomes


Apps 1.4 Applying calculus skills to optimisation and area.


R&C 1.3 Applying calculus skills of differentiation.
R&C 1.4 Applying calculus skills of integration.

Old Higher Course
Unit 1 LO3 Use basic differentiation .
Unit 2 LO2 Use basic integration.
Unit 3 LO2 Use further differentiation and integration .

Geometry

Applying algebraic skills to rectilinear shapes
Applying algebraic skills to circles and graphs
Modelling situations using sequences
Determining vector connections
Working with vectors

Learning Outcomes

  • G1 Finding equations of parallel and perpendicular ines
  • G2 Use m = tanθ
  • G3 Use properties of medians, altitudes and perpendicular bisectors
  • G4 Determine whether or not two lines are perpendicular
  • G5 Determining and using the equation of a circle
  • G6 Using properties of tangency
  • G7 Determining the intersection of circles or a line and a circle
  • G8 Determining the resultant of vector pathways in three dimensions
  • G9 Working with collinearity
  • G10 Determining the coordinates of an internal division point of a line
  • G11 Evaluate a scalar product and determine the angle between two vectors
  • G12 Use the scalar product
  • G13 Use unit vectors i, j, k

Older Learning Outcomes

Apps 1.1 Applying algebraic skills to rectilinear shapes.

Apps 1.2    Applying algebraic skills to circles.
E&F 1.4 Applying geometric skills to vectors

Old Higher Course
Unit 1 - LO1-Use the properties of the Straight Line
Unit 2 LO4 Use the equation of the circle.
Unit 3 LO1 Use vectors in three dimensions

Trigonometry

Solving trigonometric equations
Application of the addition or double angle formulae
Application of trigonometric identities
Manipulating trigonometric expressions

Learning Outcomes

  • T1 Solving trigonometric equations in degrees or radians
  • T2 Application of the addition or double angle formulae
  • T3 Application of trigonometric identities
  • T4 Converting acos x± bsin x to k cos(x ±α) or k sin(x ±α, k > 0

Older Learning Outcomes

E&F 1.2 Applying trigonometric skills to manipulating expressions.
E&F 1.3 Applying algebraic and trigonometric skills to functions.
R&C 1.2 Applying trigonometric skills to solve equations.

Old Higher Course
>Unit 1 LO2 Associate functions and graphs and solve related equations. Unit 2 LO3 Apply trigonometric formulae.
Unit 3 LO4 Apply further trigonometric relationships .



© Alexander Forrest