M1Maths

Modules - Algebra

Level 1       Level 2       Level 3       Level 4       Level 5       Level 6

Level 1 is Years 5-7, Level 6 Years 11-12.

The learning sequence for Algebra is somewhat unconventional. Algebra is treated as the study and use of relations, a relation being information which allows us to find the value of one quantity if we know the value of another quantity. As such, all new knowledge is presented as a technique for solving real-life problems: nothing is introduced as an abstract idea or as something that will be needed later. This avoids the common lament of 'When are we ever going to use this?'

By the time a student reaches the end of Level 4 or 5, s/he will know everything a student taught traditionally will know, but the knowledge will seem more coherent, meaningful and useful.

Modules:   A1:   Algebra - Level 1

Click the link in the second column for the required format. Word documents need to be downloaded and opened in Word to display properly. The second last column is the estimated time required to work through the module; the last column is the pre-requisite Level 1 modules.

Relations 1

• using relations given as tables, as graphs and as sets of ordered pairs

Relations 2

• the terms 'variable', 'independent' and 'dependent'

• converting between different forms of a relation

Patterns

• recognising patterns in sequences of numbers

• determining whether a relation has a pattern (from both tables and graphs)

• knowing that relations with patterns can be expressed as formulae

Discrete vs Continuous Relations

• distinguishing discrete and continuous relations

• presenting them appropriately as tables and graphs

Substitution

• substituting for the independent variable in a formula and doing the arithmetic to find the value of the dependent variable

• converting from formulae to other forms of a relation

• substituting into formulae with more than one independent variable

Equations

• substituting for the dependent variable in a formula and solving the resulting equation to find the value of the independent variable

• algebraic shorthand

Modules:   A2:   Algebra - Level 2

Click the link in the second column for the required format. Word documents need to be downloaded and opened in Word to display properly. The second last column is the estimated time required to work through the module; the last column is the pre-requisite Level 2 modules.

Writing Equations

• solving problems by modelling them with equations

Collecting Terms

• collecting terms to solve equations

Expanding

• expanding brackets to solve equations

Modules:   A3:   Algebra - Level 3

Click the link in the second column for the required format. Word documents need to be downloaded and opened in Word to display properly. The second last column is the estimated time required to work through the module; the last column is the pre-requisite Level 3 modules.

Undoing the Unknown

• undoing the unknown to solve equations

Choosing the Unknown

• writing and solving equations where the choice of unknown is not obvious

Squares and Fractions

• solving equations with squares and fractions

Rearranging Formulae

• changing the subject of a formula

Solving by Graphing

• using a graphics calculator or computer graphing program

• graphing relations and solving equations using a graphics calculator

Domain and Range

• domain and range

Functions

• functions and the language of functions

Linear Functions

• general form and graph shape

• using y=mx+c to convert between formulas, tables and graphs for linear functions

• finding the formula for a linear function from the gradient and points on the line

• equation solution methods

• applications

• finding equations of lines on the Cartesian plane

• parallel and perpendicular lines

• mid-point of a line segment

Reciprocal Functions

• general form and graph shape

• equation solution methods

• applications

Index Laws 1-5

• using the index laws for natural number powers

Index Laws 1-5

Alternative 'Learn' Section: Learning by Thinking

Modules:   A4:   Algebra - Level 4

Click the link in the second column for the required format. Word documents need to be downloaded and opened in Word to display properly. The second last column is the estimated time required to work through the module; the last column is the pre-requisite Level 4 modules.

Factorising

• factorising with circles

• expanding and factorising quadratics

Quadratic Functions

• general form and graph shape

• equation solution methods

• applications

• writing and solving quadratic equations

Simultaneous Equations - Linear

• solving pairs of linear simultaneous equations by equating, graphing, substitution, elimination and calculator

Inequalities

• solving single-unknown inequalities and graphing the solutions on a number line

Algebraic Fractions

• manipulating algebraic fractions

Algebraic Proofs

• algebraic proofs

Modules:   A5:   Algebra - Level 5

Click the link in the second column for the required format. Word documents need to be downloaded and opened in Word to display properly. The second last column is the estimated time required to work through the module; the last column is the pre-requisite Level 5 modules.

Polynomial Functions

• general form and graph shape

  equation solution methods

  applications

Index Laws 6-10

• zero, negative and fractional indices

• converting between fractional indices and surds

Index Laws 6-10

Alternative 'Learn' Section: Learning by Thinking

Power Functions

• general form and graph shape

• equation solution methods

• applications

Exponential Functions and Logs

• general form and graph shape

• equation solution methods and logs

• applications

Simultaneous Equations - General

• solving pairs of general simultaneous equations by equating, graphing and substitution

• solving larger sets of simultaneous equations

Finding Formulae for Functions

• finding formulae for functions from known value pairs

Model Functions

• finding model functions for data using technology

Calculator Equation Solving

• the equation solving function of a graphics calculator

Algebraic Transformations

• how changes to formulae affect their graphs and vice versa

Further Relations

• composition of functions

• inverse functions and y^2 = x

• discontinuities

• rational functions

• piecewise and step functions

• absolute value functions

• circles

Trigonometric Functions

• general form and graph shape of sinusoidal functions

• applications of sinusoidal functions

• drawing graphs of sinusoidal functions from the formulas and vice versa

• non-sinusoidal trigonometric functions

Trigonometric Equations

• solving equations derived from trigonometric functions

Logs

• the meaning of logs

• logarithmic functions

• equations from logarithmic functions

• the log laws

• logarithmic scales

Modules:   A6:   Algebra - Level 6

Click the link in the second column for the required format. Word documents need to be downloaded and opened in Word to display properly. The second last column is the estimated time required to work through the module; the last column is the pre-requisite Level 6 modules.

Combinations and the Binomial Expansion

• multiplication principle

• permutations and combinations

• expansion of (x+y)^n

• Pascal’s triangle

Arithmetic Sequences

• recursive and explicit formulae

• sum to n terms

Geometric Sequences

• recursive and explicit formulae

• sum to n terms

• sum to infinity

Linear Programming

• graphing formula inequalities on the Cartesian plane

• solving linear programming problems

Further Methods of Proof

• proof by deduction

• proof by exhaustion

• proof by contradiction

• proof by induction

Loci and Conic Sections

• loci

• conic sections

Background image: https://pixabay.com/illustrations/beach-background-sea-ocean-3892386/ (cropped) (Creative Commons licence)