The teaching and learning of algebraic linear graphs in the school have experience problems with regards to understanding this topic. The learner's inability to understand algebraic linear graphs as a result of scaling was evidence in this study. Learners experience difficulties in interpreting, constructing and predicting from linear graphs without the knowledge of scaling. Four tasks were considered necessary for constructing linear graphs. This research focused on scaling task. The objective of the research is to establish the fact that scaling has influence on the understanding of algebraic linear graphs. An empirical method of research was employed to carry out this research. The results proved that scaling has influence on the understanding of algebraic linear graphs at grade 9 (Form B). Learners will find it easy to construct, interpret and make prediction from a graph drawn by scaling. / Mathematical Sciences / M.A. (Mathematics Education)

The teaching and learning of algebraic linear graphs in the school have experience problems with regards to understanding this topic. The learner's inability to understand algebraic linear graphs as a result of scaling was evidence in this study. Learners experience difficulties in interpreting, constructing and predicting from linear graphs without the knowledge of scaling. Four tasks were considered necessary for constructing linear graphs. This research focused on scaling task. The objective of the research is to establish the fact that scaling has influence on the understanding of algebraic linear graphs. An empirical method of research was employed to carry out this research. The results proved that scaling has influence on the understanding of algebraic linear graphs at grade 9 (Form B). Learners will find it easy to construct, interpret and make prediction from a graph drawn by scaling. / Mathematical Sciences / M.A. (Mathematics Education)

We say a graph is locally P if the induced graph on the neighbourhood of every vertex has the property P. Specically, a graph is locally traceable (LT) or locally hamiltonian (LH) if the induced graph on the neighbourhood of every vertex is traceable or hamiltonian, respectively. A locally locally hamiltonian (L2H) graph is a graph in which the graph induced by the neighbourhood of each vertex is an LH graph. This concept is generalized to an arbitrary degree of nesting, to make it possible to work with LkH graphs. This thesis focuses on the global cycle properties of LT, LH and LkH graphs. Methods are developed to construct and combine such graphs to create others with desired properties. It is shown that with the exception of three graphs, LT graphs with maximum degree no greater than 5 are fully cycle extendable (and hence hamiltonian), but the Hamilton cycle problem for LT graphs with maximum degree 6 is NP-complete. Furthermore, the smallest nontraceable LT graph has order 10, and the smallest value of the maximum degree for which LT graphs can be nontraceable is 6. It is also shown that LH graphs with maximum degree 6 are fully cycle extendable, and that there exist nonhamiltonian LH graphs with maximum degree 9 or less for all orders greater than 10. The Hamilton cycle problem is shown to be NP-complete for LH graphs with maximum degree 9. The construction of r-regular nonhamiltonian graphs is demonstrated, and it is shown that the number of vertices in a longest path in an LH graph can contain a vanishing fraction of the vertices of the graph. NP-completeness of the Hamilton cycle problem for LkH graphs for higher values of k is also investigated. / Mathematical Sciences / D. Phil. (Mathematics)