VCE Mathematical Methods Implementing the 2023-2027 study design

VCE Mathematical Methods Implementing the 2023-2027 study design

Asking Questions

General Outline 1. Brief background and new structure of VCE Mathematics 2023-2027 2. Outline of VCE Mathematical Methods Units 1 – 4 revisions 3. Discuss Investigations for Unit 1&2 leading to SACs in Unit 3&4 4. Address some FAQs (If time runs out, my contact details will be supplied)

The Study Design 2023-2027 Result of a thorough consultation and review process Published in February 2022 VRQA accredited from 2023 until 2027 VCAA undertake ongoing monitoring of study design implementation over the life of the study design VCAA support the implementation of the study design through an implementation program that includes webinars, on-demand videos, support materials and resources that provide guidance around study design awareness and clarification, sample course structures, suggested learning activities and sample investigations

VCE Mathematics studies 2023 - 2027 From 2023 there are four completely prescribed Units 1 – 4 VCE Mathematics studies: Foundation Mathematics General Mathematics Mathematical Methods Specialist Mathematics In each sequence, Units 1 and 2 contain assumed knowledge and skills for Units 3 and 4. Mathematical Methods Units 1 and 2 also contain assumed knowledge and skills for Specialist Mathematics Units 3 and 4. Specialist Mathematics Units 3 and 4 assumes concurrent study, or previous completion, of Mathematical Methods Units 3 and 4.

Mathematical Methods Units 1 & 2 The 2023 – 2027 Mathematical Methods study design prescribes the content for each of Unit 1 and Unit 2. The content for Unit 3&4 is combined. Unit 1 and 2 Areas of Study ‒ ‒ ‒ ‒ Functions, relations and graphs Algebra, number and structure Calculus Data analysis, probability and statistics

Mathematical Methods Unit 1 REVISED Areas of Study: U1 AOS1: Topic: Functions relations and graphs (dot-point 2) Points of inflection (including non-stationary) U1 AOS2: Topic: Algebra, number and structure (dot-point 5) Matrix representation no longer necessary for transformations U1 AOS4: Topic: Data analysis, probability and statistics Random variables and the distribution of results of experiments

Mathematical Methods Unit 1 Revised Key Knowledge and Key skills: U1 OC1 Key knowledge Explicit mention of vertical and horizontal asymptotes as key features of graphs Clarification of properties for probability U1 OC2 Key knowledge Key elements of algorithm design: sequencing, decision making, repetition including pseudocode U1 OC2 Key skills identify important information, variables, constraints and other key features to the investigation of various questions in a given context U1 O3 Key knowledge The role of computational thinking in problem solving and mathematical investigations The purpose and effect of sequencing, decision-making and repetition statements

Mathematical Methods Units 1 Assumed Student facility: U1 AOS1: Topic: Functions, relations and graphs Review of coordinate geometry Use of vertical line test to determine a function (as a sub-set of relations) U1 AOS2: Algebra, number and structure Inverse functions no longer appear in the AoS description, however, they remain in the key skills as a form of transformation

Mathematical Methods Unit 1 Removed/deleted: U1 AOS1: Topic: Functions, relations and graphs Graphs of polynomials of degree greater than 4, however, still implicitly present U1 AOS2: Algebra n Expansion of x a , however, still implicitly present (power forms defined in AoS1) U1 O1 Key skills Equations and by hand sketching of graphs of circles in the Cartesian plane (beyond operational knowledge of the unit circle in Unit 2)

Mathematical Methods Unit 2 REVISED Areas of Study: U2 AOS: Topic: Algebra, number and structure Clarification of the use of Newton’s method as an algorithm U2 AOS3: Topic: Calculus Informal treatment of the gradient to a curve at a point as a limit The central difference approximation and its graphical interpretation U2 AOS4: Topic: Data analysis, probability and statistics Simulation to estimate probabilities involving selection with and without replacement

Mathematical Methods Unit 2 REVISED Key Knowledge and Key skills: U2 OC1 Key knowledge Emphasis on the unit circle and explicit mention of asymptotes as key features of graphs The central difference approximation U2 O2 Key knowledge Key elements of algorithm design: sequencing, decision making, repetition, including use of pseudocode. U2 O3 Key knowledge The role of computational thinking in problem solving and mathematical investigations The purpose and effect of sequencing, decision-making and repetition statements

Mathematical Methods Unit 2 Assumed Student facility: U2 AOS1: Topic: Functions, relations and graphs Review of basic trigonometric ratios (SOHCAHTOA and applications) Measures of turn or rotation, degree to radian conversion Removed/deleted: U2 AOS3: Topic: Calculus Graphical and numerical approaches to approximating the gradient function Notations of derivative and first principles approach to derivatives Solving simple problems involving straight line motion Karnaugh maps? (Still a required technique)

Mathematical Methods Units 3 & 4 Areas of Study: Functions, relations and graphs Algebra, number and structure Calculus Data analysis, probability and statistics

Mathematical Methods Units 3 & 4 REVISED Areas of Study: AOS1: Topic: Functions and graphs Closer highlighting of ‘key features’ of the graphs of the prescribed functions AOS2: Topic: Algebra, number and structure Numerical solutions of polynomial equations (Newton’s method) Broader options of composite function nomenclature: (fog)(x) f(g(x)) AOS3: Topic: Calculus Key features of graphs now includes points of inflection Area approximations using the trapezium rule AOS4: Topic: Data Analysis, Probability and Statistics Expanding the use of simulations to illustrate variations in CI dependant upon the point sample

Mathematical Methods Units 3 & 4 ASSUMED background knowledge and skills: AOS2: Topic: Algebra Review of algebra of polynomials and equating coefficients AOS3: Topic: Calculus Review of average and instantaneous rates of change, tangents and derived function

Mathematical Methods Unit 3&4 Revised Key Knowledge and Skills: U3/4 OC1 Key knowledge Introduction of the trapezium rule, explicit mention of vertical and horizontal (not oblique) asymptotes, Newton’s method algorithm U3/4 OC2 Key knowledge Specific and general formulations of concepts used to derive results Key elements of algorithm design: sequencing, decision making, repetition including pseudocode U3/4 OC2 Key skills Identify important information, variables, constraints and other key features of investigations Use algorithms, patterns, models and simulation to solve problems in a given context U3/4 OC3 Key knowledge/key skills The role of computational thinking in problem solving and mathematical investigations The purpose and effect of sequencing, decision-making and repetition statements

Mathematical Methods Units 3 & 4 No longer present: AOS1: Topic: Algebra Simple functional relations (Matrices for) transformations of functions AOS3: Topic: Calculus Identification of intervals over which a function is constant Identification of the maximum rate of increase or decrease Specified contexts for definite integrals (i.e. basic kinematics, cumulative growth AOS4: Topic: Data analysis, probability and statistics Median of a continuous random variable

Mathematical Methods Units 3 – 4 In Units 3&4, the assessment weighting used to calculate the final study score has been slightly re-adjusted (p. 12 in 2023-2027 study design)

Mathematical Investigation Units 1 & 2 For each unit: This comprises one to two weeks of investigation into one or two practical or theoretical contexts or scenarios based on content from areas of study and application of key knowledge and key skills for the outcomes. Investigation is to be incorporated in the development of concepts, skills and processes for the unit, and can be used to assess the outcomes.

Mathematical Investigation Units 1 & 2 There are three components to mathematical investigation: Formulation: overview of the context or scenario, and related background, including historical or contemporary background as applicable, and the mathematisation of questions, conjectures, hypotheses, issues or problems of interest. Exploration: investigation and analysis of the context or scenario with respect to the questions of interest, conjectures or hypotheses, using mathematical concepts, skills and processes, including the use of technology and application of computational thinking. Communication: summary, presentation and interpretation of the findings from the mathematical investigation and related applications.

Mathematical Investigation Units 1 & 2 Can be implemented in a variety of ways to meet requirements: as an investigative learning activity or activities as an assessment task or tasks a combination of the above Time range 1 to 2 weeks: Minimum one longer activity/task, or two shorter activities/tasks over the equivalent of 1 week duration Up to 2 weeks, a suitable combination, for example, two longer activities/tasks, one longer and two shorter activities tasks, three medium activities/tasks, four shorter activities/tasks

Mathematical Investigation Units 1 & 2 Provides a natural and effective context for addressing Outcome 2 and Outcome 3 Can be used to support computational thinking and experimentation Is one of the assessment task types listed for Units 1 and 2 Provides preparation for the application, problem solving and modelling tasks in Specialist Mathematics Units 3 and 4.

Sample Investigation topic Introduction This task involves students modelling an approximate tidal pattern for a chosen coastal town and to explore the effect rising sea levels might have on their chosen location, with consideration of the increased reach of high tide. Students are encouraged to use online elevation maps and predictive maps to help them assess the potential impact of rising sea levels; several relevant links are provided below. Australian tide information: https://tides.willyweather.com.au/ Elevation Finder Map: https://www.freemaptools.com/elevation-finder.htm Interactive maps indicating potential impact of rising sea levels:

SACs in Units 3 & 4 – Application Task Outcomes Outcome 1 Define and explain key concepts as specified in the content from the areas of study and apply a range of related mathematical routines and procedures. Outcome 2 Apply mathematical processes in non-routine contexts, including situations with some open-ended aspects requiring investigative, modelling or problem-solving techniques or approaches, and analyse and discuss these applications of mathematics. Outcome 3 Apply computational thinking and use numerical, graphical, symbolic and statistical functionalities of technology to develop mathematical ideas, produce results and carry out analysis in situations requiring investigative, modelling or problem-solving techniques or approaches. Marks allocated 15 20 15 Assessment task Application task A function and calculus-based mathematical investigation of a practical or theoretical context involving content from two or more areas of study, with the following three components of increasing complexity: introduction of the context through specific cases or examples consideration of general features of the context variation or further specification of assumption or conditions involved in the context to focus on a particular feature or aspect related to the context

SACs in Units 3 & 4 - Application

Pseudocode Introduction Pseudocode is a tool for representing algorithms without the use of a particular programming language and related syntax. It is written in a combination of plain English and common symbols, and describes, in a detailed step-bystep manner, the processes used in the algorithm. An example for finding the smaller of two numbers a and b Algorithm: minimum of two numbers input a, b if a b then print a else print b end if

Computational Thinking ‒ Decomposition ‒ Break down the problem into simpler, less complex components Pattern Recognition ‒ Classify patterns in data and organizing data logically ‒ Representation and interpretation Abstraction ‒ Removing non essential information and focusing on principal structure only Algorithms ‒ A sequence of instructions that can be performed Link is HERE.

Some FAQs Will Support material be available and when? Will sample SACs be published by VCAA? Will pseudocode be examinable? Has the SAC structure (effectively) changed? What does “algorithmic” thinking “look” like? What is the role of “computational” thinking?

Contact Michael MacNeill – Curriculum Manager, Mathematics Victorian Curriculum and Assessment Authority Level 7, 2 Lonsdale Street, Melbourne 3000, Victoria T 61 3 7022 5538 M 0438 602 765 Email: [email protected]