The School of Mathematics and Statistics at UNSW is pleased to offer a professional development day on **Thursday 8th June 2017** for high school mathematics teachers covering a selection of new topics in the current draft of the proposed HSC syllabuses for HSC Standard Mathematics, HSC Mathematics and HSC Mathematics Extensions 1 and 2.

The new Standard Mathematics Stage 6 (MS) will begin with year 11 in 2018. The NSW Educations Standards Authority (NESA) has recently announced a delay in the implementation of the new Mathematics Advanced Stage 6 (MA), Mathematics Extension 1 Stage 6 (ME) and Mathematics Extension 2 Stage 6 (MEX) which are now scheduled to begin with year 11 in 2019.

The day will begin with an overview of the proposed changes and include three 90 minute workshops on topics selected from the following list:

- Vectors and geometry (ME-V1)
- Vectors, lines and projectile motion (MEX-V1, ME-V1)
- Differential equations, direction fields and modelling (ME-C3)
- Teaching proof (ME-P1, MEX-P1, MEX-P2)
- Discovering proofs (ME-P1, MEX-P1, MEX-P2)
- Networks and paths (MS-N1, MS-N2)
- Critical path analysis plus the max-flow/min-cut theorem (MS-N3)
- Discrete random variables including the binomial (MA-S2, MA-S3, ME-S1)
- Simulations and digital technology (MA-S2, ME-S1)

The actual topics offered will depend on demand. You can indicate your preferred topics when signing up for the day (see below). Note that these workshops are intended to provide an introduction or refresher on topics that are likely to be in the final syllabuses so that teachers have a head start on preparation. They are not intended to be comprehensive and the details of the final syllabuses may vary from what is presented, however, he material should be interesting and useful to teachers regardless of the final form of the syllabuses.

Registration on the day will begin at 8:30am with the main program running from 9am to 3:00pm. The early bird price for this event is $50 to cover lunch and morning tea which will be provided. NESA accreditation is planned for this event. The day covers the Australian Professional Standards for Teachers 2.2.2, 2.3.2, 2.6.2, 3.4.2 and 6.2.2.

For further information, contact Dr Jonathan Kress (j.kress@unsw.edu.au).

#### Signing up for the day

To register for the day, please use the link below. The early bird price per participant is $50 (unti 1/6/2017) and $70 after that. Payment can be made online by credit card or by cheque.

The registration system allows multiple participants to register as a group. The participant details entered on the first page will be used for the invoice. Other participant details are collected on the following page. The registration system asks for the top 4 preferred topics by number (see above) for each participant. These are not binding but will help with planning the precise schedule on the day.

#### Information for the day

Below you will find the schedule for the day, a map showing the location of the registration desk where you will sign-in and some links for transport information.

**Schedule**

8:30am -- 9:00am | Registration |

9:00am -- 9:30am | Welcome and overview |

9:30am -- 11:00am | Session 1 |

11:00am -- 11:20am | Morning Tea |

11:20am -- 12:50pm | Session 2 |

12:50pm -- 1:30pm | Lunch |

1:30pm -- 3:00pm | Session 3 |

**Location of Registration desk**

Registration will be in the School of Mathematics and Statistics Foyer in the Central Wing of the Red Centre, UNSW Kensington Campus which can be found on the University Mall just below the large steps heading up to the Scientia Building. You will find the foyer at the end of the Red Centre that looks like the image below - the large red dot indicates the entrance point to the foyer.

The location is also indicated on the Google map below. Alternatively see H15 on the UNSW Campus map.

**Public Transport**

For Public Transport information, please see the UNSW Public Transport Page.

For the latest updates on public transport diversions and road closures due to the light rail construction see the UNSWLightRail twitter feed.

**Parking**

Parking is restricted on campus and in many nearby streets. If you plan to find a park on a nearby street you should arrive early and expect a long walk. Paid parking can be found close to the Red-Centre on the top floor of the Barker St carpark accessed via Gate 14 on Barker St. A pay and display parking permit obtained from a parking permit machine must be displayed and costs $6 for the first 2 hours and $3 per hour thereafter.

You can request to join the facebook group where links to materials will be posted.

#### Workshop descriptions

**1. Vectors and geometry** (ME-V1)

Vectors provide a powerful and elegant to way to describe and solve geometric problems and are essential in physics and engineering. Vectors will be introduced as both algebraic and geometric objects and used to prove results such as "the midpoints of the sides of a quadrilateral join to form a parallelogram". The dot product will be discussed and used to find the projection of one vector onto another.

**2. Vectors, lines and projectile motion** (MEX-V1, ME-V1)

Straight lines in two or three dimensions are most naturally described using vectors. After a brief introduction to vectors, the parametric description of straight lines will be explained and projectile motion explored using displacement, velocity and acceleration vectors.

**3. Differential equations, direction fields and modelling** (ME-C3)

A direction field is a tool for understanding the behaviour of the solutions to a differential equation even when explicit forms of those solutions are not known. The latest draft of the Extension 1 syllabus introduces direction fields along with an expanded range of differential equations used for mathematical modelling in chemisty, biology and ecomonics. In this presentation direction fields will be explained and used to explore the behaviour of important mathematical models of real world phenomena.

**4. Teaching proof** (ME-P1, MEX-P1, MEX-P2)

Proof is an essential feature of modern mathematics. It presents very particular challenges to the teacher, many of which centre on students' misconceptions as to what mathematics "really is". This session will consist of discussion as to what these misconceptions are and how they may be overcome. Contributions - including objections! - from participants will be welcomed. Some time will also be spent on the very important issue of how to write a clear proof.

**5. Discovering proofs** (ME-P1, MEX-P1, MEX-P2)

Examples of "proof problems" from both the current and new draft HSC syllabus will be presented, discussed with input from participants until a satisfactory solution is reached, and written up as a careful and logical argument. Various suggestions for discovering proofs will be developed. Reference will also be made to the teaching issues which arise from these examples, which will have been discussed in the "teaching proof" session; however, it is hoped that the present session will also be of use to those who have not attended the earlier session.

**6. Networks and paths** (MS-N1, MS-N2)

The rise of online social networks has put networks in a bright spotlight, not only for the general public but also for researchers across a wide range of research fields, including biology, psychology, computer science, physics and beyond. This workshop will present a useful glimpse into the study of networks. After presenting the basic definitions and properties of networks, paths, cycles, and trees, we will discuss and practice algorithms for solving practical problems on networks, such as finding shortest paths (as in Google Maps, for instance) and minimal spanning trees.

**7. Critical Path Analysis and the max-flow/min-cut theorem** (MS-N3)

Critical path analysis is a tool for analysing a multistage process by modelling it as a network and finding the bottlenecks. The Max-flow/Min-cut Theorem relates the maximum through put of a network (eg water pipes, roads, the internet, etc) to the minimum cuts required to break the network. These two networks related topics from the new Stage 6 Mathematics Standard syllabus will be explained through the use of examples.

**8. Discrete random variables including the binomial** (MA-S2, MA-S3, ME-S1)

In this session we will discuss random variables and the difference between continuous and discrete random variables. We concentrate on discrete random variables, giving examples including detailed discussion of the binomial distribution. We will discuss expected value and variance and the relationship between the mean of a sample and the expected value (the mean) of a random variable.

**9. Simulations and digital technology** (MA-S2, ME-S1)

In this session we will recall the binomial distribution, and also the sample proportion. We will use simulations - some using digital technology and some not using technology - to understand the distribution of the binomial distribution and the sample proportion. The new syllabus encourages schools and teachers to explore the use of technology. In this session we will introduce and demonstrate the computer software package R which is a free package very useful for many applications including such simulations. Participants are welcome to bring their own laptops and try using RStudio themselves.