MATHEMATICS PATHWAYS Subject Guide
Mathematics allows for the exploration and growth in understanding the natural world and the interactions and relationships within. Our subjects provide the foundation for different pathways beyond secondary school. The knowledge and skills developed in Mathematical studies are interdisciplinary, as students will draw on these in their future academic pathways, personal and work lives. Mathematics provides both a framework for thinking and a means of symbolic communication that is powerful and concise. This allows students access to important ideas such as pattern recognition, algebra, functions and relations, logic, mathematical structure and working mathematically. These areas are essential concepts developed in the study of number, algebra measurement and geometry, statistics and probability.
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MATHEMATICS PATHWAYS Pathways Summary
FOUNDATION MATHEMATICS GENERAL MATHEMATICS
GENERAL MATHEMATICS MATHEMATICAL METHODS SPECIALIST MATHEMATICS
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MATHEMATICS PATHWAYS The diagram below shows the possible pathways and prerequisites for each subject from Year 10 to Year 12.
YEAR 10
Foundation Mathematics
YEAR 11
VCE Foundation Mathematics Unit 1 & 2
VCE General Mathematics Unit 1 & 2
VCE Mathematical Methods Unit 1 & 2
If doing Specialist Maths 1 & 2, students must also do Maths Methods 1 & 2
VCE Specialist Mathematics Unit 1 & 2
YEAR 12
VCE Foundation Mathematics Unit 3 & 4
VCE General Mathematics Unit 3 & 4
VCE Mathematical Methods Unit 3 & 4
If doing Specialist Maths 3 & 4, students must also do Maths Methods 3 & 4
VCE Specialist Mathematics Unit 3 & 4
Advanced Mathematics
Mathematics
Acceleration options are available for students should they meet the criteria.
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FOUNDATION MATHEMATICS Foundation Mathematics provides for the continuing mathematical development of students entering VCE and who do not intend to undertake further study in Mathematics beyond VCE. Foundation Mathematics focus on providing students with the mathematical knowledge, skills, understanding and dispositions to solve problems in real contexts for a range of workplace, personal, further learning, and community settings relevant to contemporary society. They are also designed as preparation for Foundation Mathematics Units 3 and 4 and contain assumed knowledge and skills for these units. The areas of study for Foundation Mathematics Units 1 &2 and Units 3&4 include ‘Algebra, number and structure’, ‘Data analysis, probability and statistics’, ‘Discrete mathematics’, and ‘Space and measurement’.
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FOUNDATION MATHEMATICS FOCUS AREAS
Unit 1 & 2
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Consists of 4 Areas of Study 1. Algebra, number and structure 2. Data analysis, probability and statistics
Content covered in Unit 2 includes: •
Construction, use and interpretation of formulas and symbolic expressions to describe relationships between variables and to model and represent generalisations and patterns
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Manipulation of symbolic expressions and solution of equations.
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Creation of a range of charts, tables and graphs to represent and compare data
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Explore personal financial services, income calculations, personal taxation and superannuation taxation as a community and contribution to government.
Measures of central tendency and simple measures of spread to summarise and interpret data and compare sets of related data.
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Explore consumer mathematics
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Interpreting, using and converting standard metric units and measures, including reading and interpretation of scales.
Description, representation and properties of simple and composite shapes and objects
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Explain and solve problems relating to location, maps, directories including birds-eye and street views, routes and itineraries.
3. Discrete mathematics 4. Space and measurement Content covered in Unit 1 includes: •
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Interpreting, using and converting time and duration of time including time and date specifications, conventions, schedules and timetables.
Number applications and their properties including the use of ratios, proportions, percentages and rates to solve problems. Collection and representation and interpretation of data to summarise and communicate findings and make possible conclusions.
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FOUNDATION MATHEMATICS OUTCOMES
Unit 1 & 2 Outcome 1: On completion of this unit the student should be able to use and apply a range of mathematical concepts, skills and procedures from selected areas of study to solve problems based on a range of everyday and real-life contexts. Outcome 2: On completion of this unit the student should be able to apply mathematical processes in non-routine practical 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: On completion of this unit the student should be able to apply computational thinking and use numerical, graphical, symbolic and statistical functionalities of technology to develop mathematical ideas, produce results and carry out analysis in practical situations requiring investigative, modelling or problemsolving techniques or approaches.
ASSESSMENT
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For each Unit 1 & 2 assessment consists of: •
Investigations and projects
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Assignments
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Tests of mathematical skills
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End-of semester examination.
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FOUNDATION MATHEMATICS FOCUS AREAS
Unit 3 & 4 Consists of 4 Areas of Study 1. Algebra, number and structure 2. Data analysis, probability and statistics 3. Discrete mathematics 4. Space and measurement
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Spatial and geometric constructions including transformations, similarity, symmetry and projections
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Measurements of perimeter, area, surface area and volume of compound shapes and objects
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Calibration and error in measurement, including tolerance, accuracy and precision.
Content covered in Unit 3 & 4 includes:
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Number and number operations.
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Direct and indirect variation explained using symbolic expressions, equations, formulas and graphs.
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Data collection requirements, modelling, predictions and comparisons.
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Money management and comparison of financial products and services
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FOUNDATION MATHEMATICS ASSESSMENT
In all assessment student may have access to a scientific calculator and one https://www.vcaa.vic.edu.au/assessment/vce-assessment/materials/Pages/index.aspx bound reference SAC for Unit 3 will contribute 40 per cent to the study score SAC for Unit 4 will contribute 20 per cent to the study score Units 3 and 4 is also assessed by a 2 hour end-of-year examination 2 contributing to 40% of the study score.
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GENERAL MATHEMATICS General Mathematics Units 1 and 2 offers a general course in mathematics covering a broad range of concepts to interest students and provide general preparation for employment and study of Further Mathematics Units 3 and 4. General Mathematics has a strong emphasis on preparation for Further Mathematics Unit 3 & 4 and mathematic demands for everyday life in the community, at work and at study. The areas of study are:
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Algebra and structure
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Arithmetic and number
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Discrete mathematics
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Geometry, measurement and trigonometry
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Graphs of linear and non-linear relations
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Statistics.
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GENERAL MATHEMATICS FOCUS AREAS
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Unit 1
Unit 2
Content covered in Unit 1 includes:
Content covered in Unit 2 includes:
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Display and describe categorical data distributions using frequency tables and bar charts
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Measurement of length, angle, area, volume and capacity
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Display and describe numerical data using histograms and stem plots
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Pythagoras’ theorem in two dimensions and application to practical problems
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Median, mean, range, interquartile range, standard deviation
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Perimeter and areas of triangles, quadrilaterals, circles and composite shapes
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Linear functions and graphs
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Fitting a linear model to data
Volume and surface area of solids and practical applications
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Linear programming problems, feasible areas and optimal solution/s.
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Using trigonometry to find the length of an unknown side or the size of an unknown angle in a right-angled triangle
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The sine and cosine rules to solve practical problems involving non-right angled triangles
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Sequences and series
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Linear programming problems, feasible areas and optimal solution/s.
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GENERAL MATHEMATICS OUTCOMES
Unit 1 & 2 Outcome 1: On completion of this unit the student should be able to define and explain key concepts as specified in the selected content from the area of study, and apply a range of related mathematical routines and procedures. Outcome 2: On completion of each unit the student should be able to select and apply mathematical facts, concepts models and techniques from the topics covered in the unit to investigate and analyse extended application in a range of contexts. Outcome 3: On completion of this unit the student should be able to select and use numerical, graphical, symbolic and statistical functionalities of technology to develop mathematical ideas, produce results and carry out analysis in situations requiring problem-solving, modelling or investigative techniques or approaches.
ASSESSMENT
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For each Unit 1 & 2 assessment consists of: •
Investigations and projects
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Assignments
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Tests of mathematical skills
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End-of semester examination.
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GENERAL MATHEMATICS General Mathematics Unit 3 and 4 provides for the study of non-calculus and discrete mathematics topics. They are designed to be widely accessible and provide preparation for general employment, business or further study, in particular where data analysis, recursion and financial modelling, networks and matrices are important. Students who have done only Mathematical Methods Units 1 and 2 will have had access to assumed key knowledge and key skills for General Mathematics Units 3 and 4 but may also need to undertake some supplementary study. In undertaking these units, students are expected to be able to apply techniques, routines and processes involving rational and real arithmetic, sets, lists, tables and matrices, diagrams, networks, algorithms, algebraic manipulation, recurrence relations, equations and graphs. Students would need to demonstrate relevant mental and by-hand approaches to estimation and computation.
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GENERAL MATHEMATICS FOCUS AREAS
Unit 3 In Unit 3 Further Mathematics, a compulsory Core area of study will be completed. The Core comprises ‘Data Analysis’ and ‘Recursion and financial modelling’. Content covered in Core - ‘Data analysis’ includes:
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representation of data – frequency tables, stem plots, box plots, histograms
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five-number summary and box plots
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mean and standard deviation
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correlation coefficient; least squares line; time series data and its analysis
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calculate the coefficient of determination, r2, and interpret in the context of the association
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construct a residual analysis to test the assumption of linearity
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identify key qualitative features of a time series plot.
Content covered in Core - ‘Recursion and financial modelling’ includes: •
concept of first order linear recurrence and its use in generating the terms in a sequence
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concepts of financial mathematics including simple and compound interest, nominal and effective interest rates
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concept of a recurrence relation to determine the depreciating value of an asset.
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GENERAL MATHEMATICS FOCUS AREAS
Unit 4 Two modules selected from the following four Matrices, Networks, Geometry and Measurement, or Graphs and relations. Models to be confirmed on commencement of Unit 3.
Content covered in ‘Geometry and measurement’ includes: •
Surface and volume of spheres, cylinders, cones, pyramids and prisms, and their composites
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Solving right and non-right-angled triangles, including the sine rule, and their applications to solving practical problems in two and three dimensions
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Arc length of a sector of a circle, and the areas of segments and sectors with practical applications
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Use of trigonometry and Pythagoras’ theorem in two and three dimensions to solve problems involving the solution of right-angled triangles within a sphere.
Content covered in ‘Matrices’ includes: •
represent information in a matrices and complete elementary matrix operations
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construct a transition matrix
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construct a transition matrix to model transitions in a population.
Content covered in Networks includes:
Content covered in ‘Graphs and relations’ includes:
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construct graphs, digraphs and networks and their matrix
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recognize and solve a variety of problems including minimum connector problem, flow problems and shortest path problem.
Straight-line graphs, line segment graphs and step graphs and their use to model and analyse practical situations
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recognize that a problem is an example of the matching problem and solve it by inspection or using the Hungarian algorithm
Simultaneous linear equations in two unknowns and their use to model and analyse practical situations including break-even analysis, where cost and revenue functions are linear
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Non-linear graphs, coordinates of points of intersection for applications such as break-even analysis with nonlinear cost and revenue.
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solve scheduling problems by using critical path analysis.
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GENERAL MATHEMATICS OUTCOMES
Outcome 1: On completion of this unit the student should be able to 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: On completion of this unit the student should be able to 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: On completion of this unit the student should be able to 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.
ASSESSMENT
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Unit 3 School-assessed Coursework: 21% (Combined total of application task and modelling task)
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Unit 4 School-assessed Coursework: 16% (Two tasks a Modelling and problem-solving task)
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Units 3 and 4 Examination 1: 30%
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Units 3 and 4 Examination 2: 30%
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MATHEMATICAL METHODS Mathematical Methods is completely prescribed and extend the study of simple elementary functions to include combinations of these functions, algebra, calculus, probability and statistics, and their applications in a variety of practical and theoretical contexts. They also provide background for further study in, for example, science, humanities, economics and medicine. Mathematical Methods extends the introductory study of simple elementary functions of a single real variable, to include combinations of these functions, algebra, calculus, probability and statistics, and their applications in a variety of practical and theoretical contexts. The areas of study consists of •
Functions and graphs
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Calculus
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Algebra
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Probability and statistics
Concepts are developed in complexity and students will encounter sophisticated problem types presented within a context.
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MATHEMATICAL METHODS FOCUS AREAS
Unit 1 & 2 Mathematical Methods Unit 1 & 2 extends on prior knowledge of algebra, and introduces students to calculus, further probability and statistics and their applications. The units are designed as preparation for Mathematical Methods Units 3 and 4 and contain assumed knowledge and skills for these units. Unit 1 and 2 Areas of Study 1. Functions, relations and graphs
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Power and inverse functions
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Approximate methods to solve problems including bisection method
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Average and instantaneous rate of change
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Probability concepts including random experiments, sample spaces, basic probabilities
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Counting methods and their applications to probability
2. Algebra, number and structure
Content covered in Unit 2 includes:
3. Calculus
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Functions, relations and graphs including exponentials, logs and circular functions
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Use of inverse functions and transformations to model, analyse and solve problems
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Tangents, normal to a graph
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Limits, derivatives and integration of functions
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Applications of differentiation and integration to solve problems
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Probability concepts including the addition rule, considering conditional and independence.
4. Data analysis, probability and statistics Content covered in Unit 1 includes:
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replace with and keep bullet points:
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Linear functions and coordinate geometry
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Algebraic manipulation including factorising and division of polynomials
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Functions and Relations notation, domain and range
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Polynomial functions
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MATHEMATICAL METHODS FOCUS AREAS
Unit 3 & 4
Content covered in Unit 3 & 4 includes:
The focus for Unit 3 Mathematical Methods is in the study areas of ‘Functions and graphs’, ‘Algebra’ and the application of derivatives and differentiation from the ‘Calculus’ area of study.
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Functions relations and graphs including polynomial, powers, exponentials, logarithms, circular and inverse functions.
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Sum, difference, product and composite functions.
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Transformation and combinations of these functions, including simple piecewise (hybrid) functions
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Graphs and equations of the derivative function, tangents and normal to a function,
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Calculate probabilities, analyse and solve problems for both discrete and continuous random variables
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Statistical inference, including definition and distribution of sample proportions, simulations and confidence intervals
The focus for Unit 4 Mathematical Methods is on the applications of anti-differentiation and discrete and continuous random probability distributions. Unit 3 and 4 Areas of Study 1. Functions, relations and graphs 2. Algebra, number and structure 3. Calculus 4. Data analysis, probability and statistics
Note All Unit 1 & 2 Content is assumed knowledge and assessable in Unit 3 & 4 in addition to the Unit 3 and 4 content.
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MATHEMATICAL METHODS FOCUS AREAS
Mathematical Methods Unit 1 to 4: Outcome 1: On completion of this unit the student should be able to 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: On completion of this unit the student should be able to 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: On completion of this unit the student should be able to 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.
ASSESSMENT
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Mathematical Methods Unit 1 & 2
Mathematical Methods Unit 3 and 4
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Investigations and projects
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Assignments
Unit 3 School-assessed Coursework: 20% (Application Task)
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Tests of mathematical skills
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End-of semester examination.
Unit 4 School-assessed Coursework: 20% (total combined Modelling and Problem Solving task)
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Units 3 and 4 Examination 1: 20%
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Units 3 and 4 Examination 2: 40%
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SPECIALIST MATHEMATICS Specialist Mathematics course content highlights mathematical structure, reasoning and proof and applications across a range of modelling contexts with an appropriate selection of content for each of Unit. Specialist Mathematics provides a course of study for students who wish to undertake an in-depth study of mathematics or related domains such as engineering and science, with an emphasis on concepts, skills and processes related to mathematical structure, modelling, problem-solving, reasoning and proof. Mathematical Methods Units 1 and 2 and Specialist Mathematics Units 1 and 2 should be studied concurrently and is assumed preparation for Specialist Mathematics Units 3 and 4 Study of Specialist Mathematics Units 3 and 4 also assumes concurrent study or previous completion of Mathematical Methods Units 3 and 4.
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SPECIALIST MATHEMATICS FOCUS AREAS
Unit 1 & 2 Unit 1 and 2 Areas of Study 1. Algebra, number and structure 2. Discrete mathematics Content covered in Unit 1 includes:
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Number including definitions and properties of natural, rational and complex numbers.
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Principles of proofs
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Graph theory
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Logic propositions, connectives, truth values, truth tables and Karnaugh maps
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Boolean algebras and binary number systems
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Sequences and series, arithmetic and geometric sequences and their partial sums
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Combinatorics
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Matrices.
Content covered in Unit 2 includes: •
Functions, relations and their equations and graphs
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Probability and statistic concepts for discrete random variables
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Trigonometry including sine rule, cosine rule and compound and double angle formulas.
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Transformations including rotations of functions and coordinates
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Vectors and Vector algebra
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Complex numbers including conversion between cartesian and polar form of complex numbers.
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SPECIALIST MATHEMATICS FOCUS AREAS
Unit 3 & 4
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Logic and proof
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Functions, relations and graphs
2. Calculus
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Complex numbers
3. Data analysis, probability and statistics
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Calculus
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Differential calculus and integral calculus
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Differential equations
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Kinematics: rectilinear motion
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Vectors
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Vector and cartesian equations
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Vector calculus
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Distribution of linear combinations of random variables
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Distribution of the sample mean
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Confidence intervals for the population mean
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Hypothesis testing for a population mean with a sample drawn from a normal distribution.
Unit 3 and 4 Areas of Study 1. Algebra, number and structure
4. Discrete mathematics 5. Functions, relations and graphs and Space and measurement.
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Content covered in Unit 3 & 4 includes:
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SPECIALIST MATHEMATICS OUTCOMES
Outcome 1: On completion of this unit the student should be able to 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: On completion of this unit the student should be able to 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: On completion of this unit the student should be able to 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.
ASSESSMENT
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Specialist Mathematics Unit 1 & 2
Specialist Mathematics Unit 3 and 4
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Investigations and projects
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Assignments
Unit 3 School-assessed Coursework: 20% (Application Task)
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Tests of mathematical skills
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End-of semester examination.
Unit 4 School-assessed Coursework: 20% (Total combined Modelling and Problem Solving task)
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Units 3 and 4 Examination 1: 20%
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Units 3 and 4 Examination 2: 40%.
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