Course Planner: General Mathematics Units 1&2
Course Planner: General Mathematics Units 12
The following course documents are provided to assist teachers to plan their teaching sequence for VCE General Mathematics Units 1&2 using the student and teacher editions of MathsWorld General Mathematics Units 1&2.
Notes about this course planning document
· As prescribed in the study design, the material for each unit (i.e. in each semester), “covers four or more topics from at least three different areas of study”.
· In the tasks column, suitable student exercises, analysis and application tasks can be entered at the discretion of the mathematics faculty/teacher.
· More detailed teaching notes are provided in the teacher book.
· For each chapter, there is a Test A and Test B (parallel test). Each 40 mark test contains 5 multiple choice questions, 3 or 4 short answer and 1extended response question.
· For each chapter, there is a 15 item multiple choice quiz, available in Microsoft Word and Microsoft Excel (self-marking) formats At the discretion of the faculty/teacher, this could be placed on the school intranet for student use. Alternatively, it could be used as an additional test.
· Solutions to all exercises are available on the Solutions CD (available separately), and solutions to all analysis tasks are on the Teacher CD.
· Where possible, the topic order is based on chapter order for simplicity, and to reflect decisions made by the author team.
· In this document, there are two versions of this course planner
o Course A
§ Course A is based on Sample Course 2, as listed on page 51 of the VCE Mathematics Study Design (2006-2009).
§ It is designed to prepare students for Further Mathematics Units 3 and 4.
§ Where necessary because of intended Further Mathematics modules, material taken from Chapter 9 Non-linear relations and equations, Chapter 12 Sequences and Series, and/or Chapter 13 Variation could be used to supplement/replace the material indicated in the grids below.
§ The Bivariate data topic is placed in Unit 2 to assist any students making a transition from MM12 to GM12 midway through year 11.
o Course B
§ Course B is based on Sample Course 3, as listed on page 52 of the VCE Mathematics Study Design (2006-2009).
§ It is designed to prepare students for both Mathematical Methods Units 3 or 4 (or Mathematical Methods (CAS) Units 3 and 4) and Further Mathematics Units 3 and 4.
§ The Matrices topic is included because of its potential usefulness to both the new Matrices module in Further Mathematics and to Mathematical Methods (CAS).
General Mathematics (Course A) – Unit 1
Week(s) / Topic / Study design dot points / Chapter reference / Tasks/NotesWeek 1–4 / Univariate data / • categorical and numerical data
• data displays and their interpretation: frequency tables and bar charts for categorical data; dot plots, stem plots, frequency tables and histograms (including relative frequency, percentage frequency and cumulative frequency) for numerical data
• a summary of numerical data using measures of central tendency and spread: mean, median and mode; range, interquartile range (IQR); variance and standard deviation
• five-number summary for a set of data {minimum, Q1, Q2 = median, Q3, maximum}, and its graphical representation by box plot. / 1.1 Types of data
1.2 Displaying distributions: simple plots
1.3 Displaying distributions: tables, bar charts and histograms
1.4 Displaying distributions: frequency tables and polygons
1.5 Summarising distributions: centre
1.6 Summarising distributions: spread
1.7 Summarising and comparing distributions
Week 5–7 / Financial arithmetic / • cash flow in common savings and credit accounts, including interest calculations
• applications of simple interest and compound interest formulas
• comparison of purchase options, including cash, credit card, bank loans, time payments (hire purchase) and store cards
• appreciation and depreciation of assets, including investment of money, capital gains of physical assets and depreciation of assets by inflation. / 2.1 Simple interest and hire purchase
2.2 Compound interest
2.3 Appreciation and depreciation
Week 8–10 / Linear graphs and modelling / • determining gradients and intercepts and the equations of straight lines from graphs
• plotting and sketching straight lines given an equation
• determining points of intersection of straight line graphs by graphical and algebraic methods
• simple applications of linear modelling, such as fitting a line of best fit by eye to data values, identifying the equation of best fit, use of this line for prediction, informal discussion of closeness of fit
• constructing and interpreting line segment graphs. / 3.1 Straight line graphs
3.2 Modelling with linear functions
3.3 Fitting straight lines to data
Week 11–14 / Linear relations and equations / • substitution and transposition in linear relations, such as temperature conversion
• solving linear equations, including literal equations
• developing formulas from word descriptions; substitution of values in formulas
• the construction of tables of values from a given formula using technology
• linear relations defined recursively and simple applications
• the algebraic and graphical solution of simultaneous linear equations in two variables
• solving worded problems involving linear equations and simultaneous linear equations in two variables / 4.1 Linear equations
4.2 Applications of linear equations
4.3 Substitution and transposition in linear relations
4.4 Linear recursion
4.5 Simultaneous linear equations
Week 15–17 / Matrices / • a definition of a matrix
• matrix addition, subtraction, multiplication by a scalar and multiplication of matrices
• identifying inverse matrices and their properties
• applications of matrices in context, such as to stock inventories, solving simultaneous linear equations, transformations of the plane or networks
• calculator or computer applications for higher order matrices / 5.1 Matrices and simple definitions
5.2 Matrix multiplication
5.3 Inverse matrices
5.4 Solutions of simultaneous linear equations
General Mathematics (Course A) – Unit 2
Week 1–3 / Linear programming / • graphs of linear equations and inequalities
• solving linear simultaneous equations by algebraic, numerical and graphical methods
• graphical approaches to solving simple optimisation problems using linear programming. / 6.1 Graphs of linear inequations
6.2 Sketching systems of linear inequations
6.3 Formulation of linear programming problems
6.4 Solving linear programming problems
Week 4–7 / Bivariate data / • scatterplots
• informal interpretation of patterns and features of scatterplots
• correlation and regressions:
– using and interpreting the quadrant, q, correlation coefficient
– fitting a line with an appropriate linear association for a dependent variable with respect to a given independent variable by eye, using the two-mean method
– determining the equation of this line and using this equation for prediction (informally considering the closeness of fit—how close the data points are to the fitted line). / 7.1 Displaying and interpreting bivariate data
7.2 Correlation
7.3 Linear regression
Week 8–11 / Shape and measurement / • mensuration (angle, length, boundary, area, surface area and volume)
• Pythagoras’ theorem in two dimensions and simple examples in three dimensions
• similarity and symmetry in two dimensions and applications to maps, art, tessellations and plans
• similarity in three dimensions and application to scale models
• tests for similarity and symmetry. / 8.1 Mensuration in two dimensions
8.2 Mensuration in three dimensions
8.3 Pythagoras’ theorem in two and three dimensions
8.4 Similarity
8.5 Symmetry and tessellations
Week 12–15 / Trigonometric ratios and applications / • right-angled triangles and solutions to problems involving right-angled triangles using sine cosine and tangent
• two-dimensional applications, including angles of depression and elevation
• solution of triangles by the sine and cosine rules
• areas of triangles, including the formula
• circle mensuration: radian measure, arc length, area of sector and segments
• applications: for example, navigation and surveying in simple contexts. / 10.1 Right-angled triangles
10.2 Applications of trigonometry
10.3 Non-right angled triangles
10.4 Area of a triangle
10.5 Circle mensuration
Week 16–18 / Undirected graphs and networks / • description of networks in terms of faces (regions), vertices and edges
• faces (regions) and the application of Euler’s formula
• traversibility of a network:
– Eulerian paths and circuits, and applications
–Hamiltonian paths and circuits, and applications
• applications of networks to simple distance or time-minimisation problems
• trees and minimum spanning trees and applications. / 11.1 Vertices, edges, regions and Euler’s formula
11.2 Paths and circuits
11.3 Trees and subgraphs
11.4 Applications of networks
General Mathematics (Course B) – Unit 1
Week 1–4 / Univariate data / • categorical and numerical data
• data displays and their interpretation: frequency tables and bar charts for categorical data; dot plots, stem plots, frequency tables and histograms (including relative frequency, percentage frequency and cumulative frequency) for numerical data
• a summary of numerical data using measures of central tendency and spread: mean, median and mode; range, interquartile range (IQR); variance and standard deviation
• five-number summary for a set of data {minimum, Q1, Q2 = median, Q3, maximum}, and its graphical representation by box plot. / 1.1 Types of data
1.2 Displaying distributions: simple plots
1.3 Displaying distributions: tables, bar charts and histograms
1.4 Displaying distributions: frequency tables and polygons
1.5 Summarising distributions: centre
1.6 Summarising distributions: spread
1.7 Summarising and comparing distributions
Week 5–7 / Linear graphs and modelling / • determining gradients and intercepts and the equations of straight lines from graphs
• plotting and sketching straight lines given an equation
• determining points of intersection of straight line graphs by graphical and algebraic methods
• simple applications of linear modelling, such as fitting a line of best fit by eye to data values, identifying the equation of best fit, use of this line for prediction, informal discussion of closeness of fit
• constructing and interpreting line segment graphs. / 3.1 Straight line graphs
3.2 Modelling with linear functions
3.3 Fitting straight lines to data
Week 8–10 / Linear relations and equations / • substitution and transposition in linear relations, such as temperature conversion
• solving linear equations, including literal equations
• developing formulas from word descriptions; substitution of values in formulas
• the construction of tables of values from a given formula using technology
• linear relations defined recursively and simple applications
• the algebraic and graphical solution of simultaneous linear equations in two variables
• solving worded problems involving linear equations and simultaneous linear equations in two variables / 4.1 Linear equations
4.2 Applications of linear equations
4.3 Substitution and transposition in linear relations
4.4 Linear recursion
4.5 Simultaneous linear equations
Week 11–14 / Matrices / • a definition of a matrix
• matrix addition, subtraction, multiplication by a scalar and multiplication of matrices
• identifying inverse matrices and their properties
• applications of matrices in context, such as to stock inventories, solving simultaneous linear equations, transformations of the plane or networks
• calculator or computer applications for higher order matrices / 5.1 Matrices and simple definitions
5.2 Matrix multiplication
5.3 Inverse matrices
5.4 Solutions of simultaneous linear equations
Week 15–17 / Linear programming / • graphs of linear equations and inequalities
• solving linear simultaneous equations by algebraic, numerical and graphical methods
• graphical approaches to solving simple optimisation problems using linear programming. / 6.1 Graphs of linear inequations
6.2 Sketching systems of linear inequations
6.3 Formulation of linear programming problems
6.4 Solving linear programming problems
General Mathematics (Course A) – Unit 2
Week 1–3 / Bivariate data / • scatterplots
• informal interpretation of patterns and features of scatterplots
• correlation and regressions:
– using and interpreting the quadrant, q, correlation coefficient
– fitting a line with an appropriate linear association for a dependent variable with respect to a given independent variable by eye, using the two-mean method
– determining the equation of this line and using this equation for prediction (informally considering the closeness of fit—how close the data points are to the fitted line). / 7.1 Displaying and interpreting bivariate data
7.2 Correlation
7.3 Linear regression
Week 4–7 / Non-linear relations and equations / • substitution and transposition in non-linear relations, such as volume mensuration formulas, radiation intensity, and distance and simple logarithmic scales
• developing formulas from word descriptions; substitution of values into formulas
• construction of tables of values from a given formula by using a calculator, computer algebra system or spreadsheet
• solution of non-linear equations using algebra, tables, graphs and simple numerical approaches such as bisection, secants or simple iteration (including the use of continued fractions to approximate the irrational roots of a quadratic equation) to determine a root for an equation over an interval in which it is known to exist
• solution of word problems using non-linear equations
• the solution of simultaneous equations arising from the intersection of a line with a parabola, circle or rectangular hyperbola using algebra, graphs, tables and simple iteration. / 9.1 Formulas, substitution and transposition
9.2 Solution of non-linear equations
9.3 Simultaneous non-linear equations
Week 8–11 / Trigonometric ratios and applications / • right-angled triangles and solutions to problems involving right-angled triangles using sine cosine and tangent
• two-dimensional applications, including angles of depression and elevation
• solution of triangles by the sine and cosine rules
• areas of triangles, including the formula
• circle mensuration: radian measure, arc length, area of sector and segments
• applications: for example, navigation and surveying in simple contexts. / 10.1 Right-angled triangles
10.2 Applications of trigonometry
10.3 Non-right angled triangles
10.4 Area of a triangle
10.5 Circle mensuration
Week 12–15 / Sequences and series / • sequences and series as maps between the natural numbers and the real numbers, and the use of technology to generate sequences and their graphs
• sequences generated by recursion: arithmetic , geometric and fixed-point iteration* (for example, )
• practical applications of sequences and series, such as financial arithmetic, population modelling and musical scales.
[* Note that fixed-point iteration is covered in chapter 9 (Non-linear equations and relations).] / 12.1 Arithmetic sequences and series
12.2 Geometric sequences and series
12.3 Applications of sequences and series
Week 16–18 / Variation / • numerical, graphical or algebraic approaches to direct, inverse and joint variation
• transformation of data to establish relationships between variables; for example, to linear
• modelling of certain non-linear data using the relationship, , where k is a positive real number
• modelling of data using the logarithmic function
y = alog10(x) + b, where a is a positive real number. / 13.1 Direct variation
13.2 Other types of variation
13.3 Non-linear data and transformations
MathsWorld General Mathematics Units 1&2 Teacher edition 10