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Equations and Expressions: Level 7

The key idea of equations and expressions at levels 7 and 8 is that equations can be treated as objects, which can undergo a variety of transformations while preserving the same solutions.

At Level 7 there is an emphasis on exponents, surds and logarithms, not only as functions of numbers but also as functions of unknowns.  The relationships between equations in these different forms are important as they can be exploited to provide new methods of solving equations.  The equation ax = b can be transformed into log (ax) = log b and hence  .

Surds are also useful for solving quadratic equations by the method of completing the square, and hence deriving the quadratic formula.  This greatly expands the number of quadratic equations that may be solved from those with rational solutions to all quadratics with real solutions.

A greater variety of non-linear equations, including exponential, logarithmic and trigonometric equations, are encountered so that the links to Level 7 Patterns and Relationships become very important for providing graphical approaches to finding solutions.  It is therefore recommended that students have access to graphics calculators.  The trigonometric equations are just restricted to the domain [0, 2π], paving the way to more solutions within larger domains at Level 8 Equations and Expressions.  Graphical approaches are also important for providing a conceptual understanding of the solutions of simultaneous equations where one of the equations is non-linear.

Because at this level a greater variety of transformations of equations are carried out, it becomes even more important to relate mathematical solutions of equations to the problem context.  During the often large number of steps required to solve equations students are working purely abstractly.  Following this, as part of the problem solving process, the solutions should be interpreted and verified.

This key idea develops from the key idea of equations and expressions at level 6.

The detail of this key idea is extended in the key idea of equations and expressions at level 8.