# Patterns and Relationships: Level 7

The key idea of patterns and relationships at level 7 is that a number of important formulae exist that can be used to solve real problems.

Pythagoras’ theorem, concepts of points and lines on co-ordinate planes and the general form of linear equations allow formulae to be derived that can be used to describe the properties of pairs of points and lines on the Cartesian Plane. These formulae can then be used to solve problems in co-ordinate geometry.

Similarly Pythagoras’ Theorem and trigonometric ratios in right angled triangles can be extended to non-right angled triangles, giving rise to useful practical formulae involving angles and sides.

Useful formulae for solving practical problems in sequences and series can also be derived. In particular, any term of an arithmetic or geometric sequence or series can be determined in terms of an initial term, the common difference/ratio and the position of the term in the sequence.

Part of this thread which provides a direct continuation of the work in level 6 is that linear transformations preserve the essential shape of a graph. While the work at level 6 focused on linear and quadratic graphs, at level 7 these ideas are extended to all functions encountered. Graphs of functions can then be sketched using knowledge of key features such as roots and asymptotes. These techniques provide a way to sketch or visualize graphs from more ‘basic’ graphs. This work forms a foundation for the key idea of calculus at level 8.

This key idea develops from the key idea of patterns and relationships at level 6, where students explore the effects of transformations on functions.

This key idea is extended to the key idea of patterns and relationships at level 8 where derived results are used to solve problems.