# UNSW Mathematics Teachers Professional Development Day

The School of Mathematics and Statistics at UNSW is pleased to offer a professional development day on Thursday 10th November for high school mathematics teachers covering a selection of new topics in the current draft of the proposed HSC syllabi for HSC Mathematics and HSC Mathematics Extensions 1 and 2.

The day will begin with an overview of the proposed changes with an emphasis on the practicalities of teaching the new syllabus. The day will include three 90 minute workshops on topics selected
from the following list:

#### Workshop descriptions

Learning statistics by doing a study

An effective way to teach statistics is to contextualise it in a research study - start with a research question, work out how to collect data, actually collect it, analyse it, and interpret it.  Some examples and the impact they have had will be presented before participants do some research of their own!

Simulations of random variables

This workshop will involve hands on experiments and technology to explore the concept of randomness and to help develop an understanding of and familiarity with binomial and other random variables.  Participants will be introduced to RStudio, a free and open source software package that can be used in the classroom.  They will learn about its capabilities through a series of practical exercises including simulations of random processes with particular emphasis on the statistics topics that are new in the current draft of the stage 6 HSC syllabuses.

Vectors and geometry

This workshop will introduce vectors and the operations of addition and scalar multiplication both in abstract geometric terms and as coordinate vectors in both two and three dimensions.  The scalar product will be defined and its relationship to the angle between two vectors will be explored.  Participants will learn how to give a vector description of a straight line and use this to find the point of intersection, when it exists, and angle between two lines.  Some simple orthogonality proofs using abstract vectors will also be covered.

An introduction to matrices with applications

This workshop will introduce matrices and matrix multiplication as a way of encoding linear simultaneous equations and linear discrete evolution processes, eg Markov models, Leslie models, Leontief models.  Solving 2x2 linear systems using a matrix inverse will presented along with the use of the determinant to detect when the solution is not unique.  Finding successive states in the linear evolution processes by matrix multiplication will be explored.

Differential equations and modelling

Applications of separable differential equations in areas such as chemistry and population models will be presented.  Gradient fields will be introduced as a way of understanding the solutions of these models with different initial conditions and their long time behaviour.

Networks

The rise of online social networks has put networks and graph theory in a bright spotlight, not only for the general public but also for researchers across a wide range of research fields, from biology to psychology to computer science and beyond. This workshop will present a useful glimpse into graph theory, first providing the basic definitions and properties of networks, paths, cycles, and trees, and then presenting and practicing algorithms for finding shortest paths (as in Google Maps, for instance) and minimal spanning trees more generally.

Proof

This workshop will introduce “The Nature of Proof” topic from the draft extension 2 syllabus.  This is designed to provide students with a solid basis on which to build their understanding of what constitutes a proof in mathematics.  Precise notation will be established for implications and the quantifiers “for all” and “there exists”.  Then various forms of proof will be introduced by example, including direct proof, induction, proof by cases, proof by contradiction and the contrapositive.  The use of counterexamples will be emphasised for disproving a statement.