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# 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 evrything a student taught traditionally will know, but the knowledge will seem more coherent, meaningful and useful.

 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.
A1-1 pdf     docx

## Relations 1

• using relations given as tables, as graphs and as sets of ordered pairs
3 h
A1-2 pdf     docx

## Relations 2

• the terms 'variable', 'independent' and 'dependent'
• converting between different forms of a relation
3 h A1-1
A1-3 pdf     docx

## 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
3 h A1-2
A1-4 pdf     docx

## Discrete vs Continuous Relations

• distinguishing discrete and continuous relations
• presenting them appropriately as tables and graphs
2 h A1-3
A1-5 pdf     docx

## 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
3 h A1-4
A1-6 pdf     docx

## Equations

• substituting for the dependent variable in a formula and solving the resulting equation to find the value of the independent variable
• algebraic shorthand
7 h A1-5

 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.
A2-1 pdf     docx

## Writing Equations

• solving problems by modelling them with equations
3 h
A2-2 pdf     docx

## Collecting Terms

• collecting terms to solve equations
2 h A2-1
A2-3 pdf     docx

## Expanding

• expanding brackets to solve equations
3 h A2-2

 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.
A3-1 pdf     docx

## Undoing the Unknown

• undoing the unknown to solve equations
3 h
A3-2 pdf     docx

## Choosing the Unknown

• writing and solving equations where the choice of unknown is not obvious
2 h A3-1
A3-3 pdf     docx

## Squares and Fractions

• solving equations with squares and fractions
4 h A3-2
A3-4 pdf     docx

## Rearranging Formulae

• changing the subject of a formula
2 h A3-3
A3-5 pdf     docx

## Solving by Graphing

• using a graphics calculator or computer graphing program
• graphing relations and solving equations using a graphics calculator
2 h
A3-6 pdf     docx

## Domain and Range

• domain and range
2 h
A3-7 pdf     docx

## Functions

• functions and the language of functions
3 h A3-6
A3-8 pdf     docx

## 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
5 h A3-7 M3-3
A3-9 pdf     docx

## Reciprocal Functions

• general form and graph shape
• equation solution methods<
• applications
2 h A3-8
A3-10 pdf     docx

## Index Laws 1-5

• using the index laws for natural number powers
4 h
A3-10 pdf     docx

## Index Laws 1-5

Alternative 'Learn' Section: Learning by Thinking

5 h

 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.
A4-1 pdf     docx

## Factorising

• factorising with circles
5 h
A4-2 pdf     docx

• general form and graph shape
• equation solution methods
• applications
• writing and solving quadratic equations
6 h A4-1
A4-3 pdf     docx

## Simultaneous Equations - Linear

• solving pairs of linear simultaneous equations by equating, graphing, substitution, elimination and calculator
6 h
A4-4 pdf     docx

## Inequalities

• solving single-unknown inequalities and graphing the solutions on a number line
1 h
A4-5 pdf     docx

## Algebraic Fractions

• manipulating algebraic fractions
3 h A4-1
A4-6 pdf     docx

## Algebraic Proofs

• algebraic proofs
2 h

 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.
A5-1 pdf     docx

## Polynomial Functions

• general form and graph shape
• equation solution methods
• applications
3 h <
A5-2 pdf     docx

## Index Laws 6-10

• zero, negative and fractional indices
• converting between fractional indices and surds
3 h
A5-2 pdf     docx

## Index Laws 6-10

Alternative 'Learn' Section: Learning by Thinking

4 h
A5-3 pdf     docx

## Power Functions

• general form and graph shape
• equation solution methods
• applications
2 h A5-2
A5-4 pdf     docx

## Exponential Functions and Logs

• general form and graph shape
• equation solution methods and logs
• applications
5 h A5-2
A5-5 pdf     docx

## Simultaneous Equations - General

• solving pairs of general simultaneous equations by equating, graphing and substitution
• solving larger sets of simultaneous equations
6 h A5-3 A5-4
A5-6 pdf     docx

## Finding Formulae for Functions

• finding formulae for functions from known value pairs
3 h A5-5
A5-7 pdf     docx

## Model Functions

• finding model functions for data using technology
3 h A5-6
A5-8 pdf     docx

## Calculator Equation Solving

• the equation solving function of a graphics calculator
1 h A5-1
A5-9 pdf     docx

## Algebraic Transformations

• how changes to formulae affect their graphs and vice versa
4 h
A5-10 pdf     docx

## Further Relations

• composition of functions
• inverse functions and y^2 = x
• discontinuities
• rational functions
• piecewise and step functions
• absolute value functions
• circles
5 h A5-1
A5-11 pdf     docx

## 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
5 h M5-1
A5-12 pdf     docx

## Trigonometric Equations

• solving equations derived from trigonometric functions
5 h A5-11 M5-1
A5-13 pdf     docx

## Logs

• the meaning of logs
• logarithmic functions
• equations from logarithmic functions
• the log laws
• logarithmic scales
7 h A5-4

 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.
A6-1 pdf     docx

## Combinations and the Binomial Expansion

• multiplication principle
• permutations and combinations
• expansion of (x+y)^n
• Pascal’s triangle
6 h
A6-2 pdf     docx

## Arithmetic Sequences

• recursive and explicit formulae
• sum to n terms
4 h
A6-3 pdf     docx

## Geometric Sequences

• recursive and explicit formulae
• sum to n terms
• sum to infinity
3 h A6-2
A6-4 pdf     docx

## Linear Programming

• graphing formula inequalities on the Cartesian plane
• solving linear programming problems
3 h
A6-5 pdf     docx

## Further Methods of Proof

• proof by deduction
• proof by exhaustion