The last professional development day for high school mathematics teachers was held on Thursday 9th May 2019.

A Randomised Assessments Workshop (RAW) was held after the PD day on Tuesday 21st May. For details see the Randomised Assignments Workshop section below.

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The day covered a selection of new topics in the new HSC syllabuses for HSC Standard Mathematics, HSC Mathematics and HSC Mathematics Extensions 1 and 2.

The new Standard Mathematics Stage 6 (MS) began with year 11 in 2018. The NSW Educations Standards Authority (NESA) has 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) began with year 11 in 2019. For all syllabus documents, see the stage 6 syllabuses from the NESA website.

The program included the following sessions:

- Vectors and geometry (ME-V1)
- Vectors, lines, forces and projectile motion (ME-V1)
- Vectors, lines and projections (MEX-V1, ME-V1)
- Differential equations, direction fields and modelling (ME-C3)
- Networks and paths (MS-N1, MS-N2)
- Critical path analysis (MS-N3)
- The max-flow/min-cut theorem (MS-N3)
- Discrete random variables including the binomial (MA-S2, ME-S1)
- Continuous random variables (MA-S2, MA-S3)
- Statistics in daily life (MS-S1, MS-S3, MS-S4, MA-S2, MA-S3)
- Randomised assessments
- The nature of proof (MEX-P1, MEX-P2)

Completing all 4 sessions at the UNSW Maths Teachers PD Day will contribute 5 hours of NESA Registered PD addressing 2.2.2, 2.3.2, 2.6.2, 3.4.2 from the Australian Professional Standards for Teachers towards maintaining Proficient Teacher Accreditation in NSW. |

For further information, contact Assoc Prof 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 $60 (until 02/05/2019) and $90 after that. Payment can be made immediately by credit card or an invoice can be generated for later payment.

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.

#### Information for the day

A map showing the location of the Central Lecture Block where the registration desk can be found is below along with some transport information.

**Schedule**

The schedule is subject to change but is likely to be as below.

Venue: Central Lecture Block (CLB) from 8:45am until 3pm. A detailed schedule is below.

CLB 2 | CLB 4 | Morven-Brown G6 | |

8:00am -- 8:45am | Registration (CLB Foyer) | ||

8:45am -- 9:00am | Welcome (CLB 2) | ||

9:00am -- 10:15am | Networks and paths | Vectors and geometry | Discrete random variables |

10:15am -- 10:35am | Morning Tea (CLB Foyer) | ||

10:35am -- 11:50pm | Critical Path Analysis* | The Nature of proof | Continuous random variables** |

11:50am -- 1:05pm | Differential equations | Vectors, lines, projections*** | Statistics in daily life |

1:05pm -- 1:45pm | Lunch (CLB Foyer) | ||

1:45pm -- 3:00pm | Max-flow/min-cut theorem* |
Vectors, forces, projectiles*** |
Randomised assessments |

* These sessions will asssume knowledge from the "Networks and paths" session.

** This session will assume knowledge from the "Discrete random variables" session.

*** This session will assume knowledge from the "Vectors and geometry" session.

**Location**

Central Lecture Block, UNSW Kensington Campus. See the Google map below for the location or find E19 on the UNSW Campus map.

**Public Transport**

For Public Transport information, please see the UNSW Public Transport Page. Coming from Central Station it's best to take the 891 Express from Eddy Avenue and get off at the High St Gate 8 stop. The current route map is here: 891 during light rail construction.

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 Central Lecture Block on the top floor of the Botany St carpark accessed via Gate 11 on Botany St. A pay and display parking permit obtained from a parking permit machine must be displayed. The cost can be found on the parking rates page.

The UNSW campus maps page has parking map to help you find a place to park.

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

#### Workshop descriptions

The descriptions below may be updated but we don't expect major changes before the day.

**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 how it can be used to find the angle between two vectors. This presentation will start from the very basics. The initial fundamental material has been written to be of use in lesson planning. The later content will be new to most teachers.

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

After a brief review of vectors through a game (please install the Kahoot app on your phone or bring your laptop), we will see how a straight line can be very naturally described using a point and a vector. The main focus of the session will be the applications of vectors to Mechanics. In particular, we will see how projectile motion can be restated in the language of vectors, and we will explore the benefits of this approach.

**3. Vectors, lines and projections** (MEX-V1, ME-V1)

Vector methods provide the simple and intuitive description of straight lines. This will be introduced in 2D and related to the Cartesian equation of a line but the real power of this approach becomes apparent in 3D where a single Cartesian equation for a line is no longer possible. This session will also introduce the idea of the projection of one vector onto another using the dot product.

**4. 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 new Extension 1 syllabus introduces direction fields along with an expanded range of differential equations used for mathematical modelling in chemistry, biology and economics. In this presentation direction fields will be explained and used to explore the behaviour of important mathematical models of real world phenomena.

**5. 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.

**6. Critical Path Analysis ** (MS-N3)

Critical path analysis is a tool for analysing a multistage process by modelling it as a network and finding the bottlenecks. This networks related topic from the new Stage 6 Mathematics Standard syllabus will be explained through the use of examples.

**7. The max-flow/min-cut theorem** (MS-N3)

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. This networks related topic 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, ME-S1)

A new word has appeared in the new syllabus in the probability section: Random Variable. We will explain and use this new terminology to revisit well known concepts, in particular the Binomial distribution. Random variables can be continuous or discrete -we will explain that too-, and our talk will focus on the discrete ones (There is another session about the continuous ones). We will discuss expected value and variance of random variables, what they mean, and what their properties are.

**9. Continuous random variables** (MA-S2, MA-S5, MS-S5)

In this session, we introduce the basics of continuous random variables through discrete random variables. We will generalise the concepts of the expected value (mean), variance and quantiles to continuous random variable. To illustrate the theory, we will study examples including the continuous uniform random variable, the normal random variable and the Galton machine (Normal approximation to Binomial random variables).

**10. Statistics in daily life** (MS-S1, MS-S3, MS-S4, MA-S2, MA-S3)

In this session we will introduce some case studies and examples and show how introducing the language and concepts of statistics can help us understand what is happening. We will also discuss the Statistical Society of Australia Poster Competition as a possible project for year 11 and 12 students.

**11. Randomised assessments**

Assessment in mathematics is very often in the form of a supervised test with a relatively short time limit. While this is easy to control, it is not an environment that encourages mathematical thinking and communication. Assignments are set to become a bigger feature of HSC Mathematics with the introduction of the new syllabuses. They offer the chance for students to take their time to craft good answers and explore more complex problems and can better assess skills in these areas. However, traditional assignment answers can be copied, reducing the integrity of this form of assessment. Computer generated randomised assignments can help to reduce the incidence of copying and restore confidence in their use. They can also be used to encourage student perseverance by offering options for students to verify their answers. While a randomised assignment is not an investigative task, it can form a scaffold for such a task. Some examples of randomised assignments used in first year at UNSW and high schools will be presented and their advantages and limitations discussed.

**12. The nature of proof **(MEX-P1, MEX-P2)

This talk will look at the new topic to be introduced to the Extension 2 HSC course. As well as outlining the topic, we will look at many examples of the type mentioned in the syllabus as well as considering some harder examples of proof by induction.

#### Randomised assignments workshop using NUMBAS

A separate day devoted to a hands on workshop on setting randomised assignments using NUMBAS will be held on Tuesday 21st May (in the Red Centre, Room M020). The workshop will run from 9am to 3pm. Topic 10 in the PD Day described above is a showcase of some randomised assessments and discusses their advantages. This day is a hands on workshop where you will learn to create your own randomised assignments and quizzes. It is a repeat of the previous day that was run earlier this year.

NUMBAS is a free open source web based system for creating and the automatic grading of randomised assessments. These assessments can provide customised feedback and worked solutions. For on overview of NUMBAS see their website.

NUMBAS assessments run in a student's web browser, so no special software is required. They an be either stand alone webpages or integrated with LMSs such as Moodle, Canvas, etc. NUMBAS assessments and solutions can also be provided on paper. Teachers can draw from a bank of publically available questions but, more importantly, can easily construct their own, which can be shared within private groups of contributed to the public bank. While simple questions can be easily constructed with little expert knowledge, the system also allows more advanced users almost limitless freedom.

The day will start with an overview of NUMBAS but most of the day will be spent working with the system to construct questions and assessments. Participants should bring along examples of assessments that they would like to convert to this format. The questions and assessments produced on the day will be shared with the group and everyone should leave with a collection of assessments they can use in their teaching and the skills to create more.

Completing the Randomised assignments workshop will contribute 5 hours of NESA Registered PD addressing 3.4.1, 5.1.2 and 5.2.2 from the Australian Professional Standards for Teachers towards maintaining Proficient Teacher Accreditation in NSW. |

For further information, contact Assoc Prof Jonathan Kress (j.kress@unsw.edu.au). Please also get in touch if you came to the previous workshop and want more help to go further.

#### Signing up for the Randomised Assignments Day

To register for the day, please use the link below. The price per participant is $60 to cover catering. Morning and tea and lunch will be provided. Payment can be made online by credit card. Payment can be made immediately by credit card or an invoice can be generated for later payment.

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.