# Patterns and Relationships: Level 8

The key idea of patterns and relationships at level 8 is that results can be derived which can be used to solve problems.

At level 8, instead of using physical situations as a model for understanding the mathematics, the emphasis is on using mathematical models to predict properties of the situations.  This also links to the key idea of Equations and Expressions at level 8.

Once a problem solving method has been found, it can be used to solve whole classes of problems.  This means that problems can be solved without having to go back to the derivation of the method.  At this level it is sometimes the case that students are unable to derive a result by themselves and so need to rely on the mathematics that has gone before them.  However this does not mean that it is unimportant for students to understand what they are learning at this Level.

Relations of the form give rise to interesting shapes that occur in nature.  Conic sections can be used to solve many practical problems, especially when links are made to transformations and loci.

There are relationships between functions and their inverse and reciprocal functions, which may be shown graphically and algebraically.  The restrictions on the domains or ranges of inverse and reciprocal functions often have implications for practical problems.

Analyzing many cases of sets leads to ways of counting sets of objects.  Formulae for permutations and combinations are powerful tools for solving many problems, particularly in combinatorics and level 8 probability.

By considering a range of situations, methods can be found to facilitate the determining of a function of a given graph and optimizing given linear situations.  Many of the functions that have been learnt about can be used to model practical situations.  Once a situation has been modeled mathematically graphical interpretations or analytical techniques can be used to solve equations and inequations and hence solve practical problems.

This key idea develops from the key idea of patterns and relationships at level 7, where students are able to apply formulae to solve real world problems.