Additional VET booklets and sets of virtual activities were developed for teachers and students as part of an Australian Studies in Science, Technology and Mathematics (ASISTM) project, Enhancing mathematics for Indigenous Vocational Education and Training students, conducted in 2008 at high schools and TAFEs in rural and remote Queensland communities. Further details are provided under each heading below.
Prevocational mathematics (12 PDF files)
The ARC Linkage VET project's activity with Bundamba State Secondary College focused on the Year 11 and 12 Prevocational Mathematics subjects taught at that college to less able mathematics students. The project used a series of intervention case studies to research learning. As part of this, 12 resource booklets were produced (six for Year 11 and six for Year 12). The booklets are based upon the notion of Renzulli (1977) that mathematics ideas should be developed through three stages:
- Stage 1: Motivate the students – pick an idea that will interest the students and will assist them to engage with mathematics.
- Stage 2: Provide prerequisite skills – list and then teach all necessary mathematics ideas that need to be used to undertake the motivating idea.
- Stage 3: Culminating task – end the teaching sequence by setting students an open-ended investigation to explore.
These booklets use Stage 3 as an assessment item and so we have added an assessment rubric whereby the culminating task can be used to check the knowledge held by the student. The booklets combine two approaches to teaching:
- structural activities that lead to the discovery and abstraction of mathematical concepts, processes, strategies and procedures; and
- rich-style tasks which allow students an opportunity to solve problems and build their own personal solution.
The booklets are numbered 11.1 to 11.6 (Year 11) and 12.1 to 12.6 (Year 12) .
Year 11 Prevocational Maths Booklet 11.1: "Using Numbers of Numbers"
This first learning unit focuses on students completing a variety of numerical activities that involve numerical procedures and common and decimal fractions. The prerequisite activities therefore include: (a) identifying and following numerical procedures to meet numerical outcomes; (b) understanding common fractions and the five meanings for these; (c) understanding decimal numbers and their relation to common fractions; (d) understanding how to interpret word problems; and (e) understanding how to tackle one-step and multi-step problems.
Year 11 Prevocational Maths Booklet 11.2: "The Big Day Out"
The motivating idea behind this booklet is that of setting up and budgeting for a trip to the Big Day Out, an Australian and New Zealand music festival. This task should have resonance with the students in that it is something they may have to do for themselves in their future. It is a task with many things to consider, so as well as being relevant, it has many options. The culminating task is thus motivating and open. The prerequisite skills for budgets are not so obvious. However, for this booklet, they consist of the following: (a) identifying and computing operations; (b) using tables; (c) budgeting and simple algebra; and (d) solving multi-step problems.
Year 11 Prevocational Maths Booklet 11.3: "Rating Our World"
This booklet focuses on how to use rate to determine measure. The culminating task requires students to engage with different types of rates in problems involving the costs associated with the home. The prerequisite skills needed include: (a) developing the language and symbols required for rate; (b) understanding the differences between rate and ratio; (c) recognising the three different rate problem types; and (d) understanding different methods for solving rate problems.
Year 11 Prevocational Maths Booklet 11.4: "Exchange Student"
The motivating idea behind this booklet is that of setting up and budgeting for a 26 week stay in a flat by two exchange students. This task should have resonance with the students in that it is something they may have to do for themselves in their future. It is a task with many things to consider, so as well as being relevant, it has many options. The culminating task is thus motivating and open. The prerequisite skills consist of the following: (a) identifying operations; (b) determining discounts; (c) using tables; (d) understanding metric conversions; (e) determining best buys; and (f) understanding the "exhausting all possibilities" and "breaking into parts" strategies.
Year 11 Prevocational Maths Booklet 11.5: "Planning a Roster"
The culminating task in this booklet involves preparing a staff roster for a new restaurant for a period of one week, trading 24 hours a day, and calculating the cost of wages for the same period. In preparing for this task, the following prerequisite skills appear necessary: (a) reading/constructing tables; (b) understanding 24 hour time; (c) understanding and calculating percentages; and (d) understanding addition/subtraction computation strategies.
Year 11 Prevocational Maths Booklet 11.6: "The Man from Hungary"
The culminating task for this unit is to use railway timetables to determine an appropriate schedule for a visitor, Mr Schien, to get to his appointments by railway. The task requires the following prerequisite skills: (a) understanding time relationships and representations; (b) calculating with time; (c) reading timetables; (d) understanding other travel-time factors; and (e) understanding fast and efficient time allocation.
Year 12 Prevocational Maths Booklet 12.1: "Beating the Drought" - available soon
This first learning unit in Year 12 focuses on weather activities and studies them in terms of probability (using a dice), and statistics (using graphs, tables and maps). The prerequisite skills therefore include: (a) understanding concepts of probability; (b) applying probability to real-world situations; (c) understanding and constructing bar graphs and histograms; (d) understanding and constructing tables of data; and (e) understanding and constructing pictograms.
Year 12 Prevocational Maths Booklet 12.2: "Monopoly" - available soon
This second learning unit in Year 12 focuses on Monopoly, a game of chance that involves buying properties and charging rent as a board is circled. This involves an understanding of outcomes and the various probabilities of these outcomes occurring, some understanding of games and the strategies for winning them, particularly games that involve probability, and an understanding of the particular focus of Monopoly, the various aspects of the finance of property. The prerequisite activities therefore include: (a) probability (outcomes and likelihood of outcomes); (b) games (types and strategies for winning and ways in which probability can help understand a game); (c) property finance (money, rent, savings, borrowing, and so on); and (d) statistics (e.g. methods of display such as tables and graphs).
Year 12 Prevocational Maths Booklet 12.3: "How Tall is the Criminal?" - available soon
The culminating task in this booklet engages students with different types of ratios in problems requiring calculations, including calculating the height and other body measurements of a criminal based only on a footprint found at a crime scene. The basis of the unit is the notion of ratio and equivalence of ratio (proportion). Ratio is a multiplicative way of identifying comparative relationships similar to fractions but relating part to part instead of part to whole. Ratio is normally for like attributes, for example, length to length, mass to mass or time to time. Multiplicative comparison across different attributes is usually called rate. The prerequisite activities focus on (a) introducing ratio and proportion; and (b) applying ratio and proportion to measurement problems.
Year 12 Prevocational Maths Booklet 12.4: "Design a Kitchen" - available soon
The culminating task here is to design a kitchen. This requires creating a picture of the kitchen with all the components well organised and determining the cost of building this kitchen and buying all appliances. Thus the prerequisite mathematics for this booklet is: (a) developing knowledge of space, shape and visual imagery; (b) calculating perimeter, area and volume; (c) understanding percent and discount; (d) understanding rate, ratio and scale; and (e) using tables to summarise data.
Year 12 Prevocational Maths Booklet 12.5: "Healthy Eating" - available soon
The culminating task "Healthy Eating" deals mainly with statistics, tables and graphs. Therefore, the prerequisite skills required are: (a) creating graphs (bar graphs, histograms and line graphs); (b) constructing and interpreting tables; and (c) calculating statistical averages (mean, mode, and median).
Year 12 Prevocational Maths Booklet 12.6: "Rocking Around the World" - available soon
The culminating task in this booklet requires students to organise an itinerary for a band who want to play in as many cities as they can in 10 days. The prerequisite mathematics that underlies this task involves: (a) working out time zones based on angle; (b) calculating with time; (c) using tables to organise data; and (d) solving complex problems by breaking into parts and exhausting all possibilities.
Basic mathematics (2 PDF and 54 PowerPoint (zipped) files)
The following two basic mathematics booklets and eight sets of virtual materials (54 PowerPoint slides) were produced as material to support Indigenous students completing a variety of vocational certificates at Shalom Christian College and Wadja Wadja High School.
Basic Maths Booklet VB1: Mathematics behind Whole Number Place Value and Operations (PDF, 1.2MB)
This VB1 booklet looks at two- and three-digit whole numbers in terms of numeration, addition and subtraction, and multiplication and division. It focuses on concepts and strategies associated with viewing number in separated or place value terms. Virtual materials for use with this booklet can be found in the following folders: Bundling Sticks, MAB and Virtual Finance.
Virt1 Bundling Sticks (zipped, 2.9MB)
The virtual materials in Bundling Sticks help students relate visual materials to language and symbols. They are also designed to strengthen students' understanding of place value. As students progress through these activities the complexity increases from an understanding of language-symbols-materials to applying it in simple addition situations moving on to more complex division scenarios. This zipped folder includes the following PowerPoint files:
- BS1 Place Value 2-D Materials to Language
- BS2 Place Value 2-D Materials to Symbols
- BS3A Operations 2-D Addition
- BS3B Operations 2-D Subtraction
- BS4A Operations 2-D Multiplication
- BS4B Operations 2-D Division
- BS5 Operations 2-D Additive Subtraction
Virt2 MAB (zipped, 2.6MB)
MAB virtual materials are designed to help students relate real materials to language and to symbols. Once students have developed their understanding of this abstract idea, they move on to simple and complex mathematical questions to reinforce this understanding and see how maths is really evident in the real world. This zipped folder includes the following PowerPoint files:
- MAB1 Place Value 3,4-D Materials – Language
- MAB2 Place Value 3,4-D Materials – Symbols
- MAB3A Operations 3,4-D Addition
- MAB3B Operations 3,4-D Subtraction
- MAB4A Operations 3,4-D Multiplication
- MAB4B Operations 3,4-D Division
- MAB5 Operations 3,4-D Additive Subtraction
Virt3 Virtual Finance (zipped, 4.9MB)
These materials are designed to equip students with the ability to relate money (material) to the language and symbolic ($10, etc.) representation of it. The materials then continue on with practical applications of money that students would come across in the real world. This zipped folder includes the following PowerPoint files:
- VFA 1A Place Value Language to Materials
- VFA 1B Place Value Materials to Language
- VFA 2A Place Value Symbols to Materials
- VFA 2B Place Value Materials to Symbols
- VFA 3A Operations Addition
- VFA 3B Operations Subtraction
- VFA 4 Operations Shopkeeper Subtraction
- VFA 5A Operations Multiplication
- VFA 5B Operations Multiplication and Subtraction
- VFA 6A Operations Division
- VFA 6B Operations Division and Subtraction
Basic Maths Booklet VB2: Mathematics behind Whole-Number Numeration and Operations (PDF, 1.4MB)
This VB2 booklet looks at two- and three-digit whole numbers in terms of numeration, addition and subtraction, and multiplication and division. It focuses on concepts and strategies associated with a variety of models and materials (99 board, number line, arrays and digit cards on place value charts) that were not included in VB1 – seeing number in a number line/rank way (99 board, number line) as well as a place value/separated way (digit cards on place value charts). Number is also seen in a multiplicative way (arrays and digit cards). Virtual materials for use with this book can be found in the following folders: 99 Board, Number Line, Array, Multiplicative Structure and Decimal Money.
Virt4 99 Board (zipped, 807KB)
Using a 10×10 board with the numbers 0-99, the 99 Board virtual materials are designed to help students locate and understand different properties of numbers and begin to develop an understanding of how numbers are related to each other. This zipped folder includes the following PowerPoint files:
- 99 Board1 – Reading
- 99 Board2 – Patterns
- 99 Board3 – Place Value
- 99 Board4 – Seriation
- 99 Board5 – Comparing & Ordering
- 99 Board6 – Addition
- 99 Board7 – Subtraction
- 99 Board8 – Additive Subtraction
Virt5 Number Line (zipped, 1.1MB)
These materials use number lines to demonstrate to students the way numbers are connected, and how we can use number lines to reinforce understanding of addition, subtraction, multiplication and division. This zipped folder includes the following PowerPoint files:
- Blank Number Lines
- Number Line1 – Reading
- Number Line2 – Positioning
- Number Line3 – Rounding
- Number Line4 – Addition
- Number Line5 – Subtraction
- Number Line6 – Additive Subtraction
- Number Line7 – Multiplication
- Number Line8 – Division
Virt6 Array (zipped, 780KB)
The Array virtual materials are designed to help students relate different objects set in arrays (rows and columns) into mathematical concepts. The concepts that these focus on are multiplication and division. The materials are designed to help students deepen their understanding of the reversibility of numbers and the links between numbers with a similar factor, for example 5 × 4 + 4 × 2 = 4 (5 + 2). This zipped folder includes the following PowerPoint files:
- Array1 – Multiplication (reading)
- Array2 – Multiplication (matching)
- Array3 – Multiplication (creating)
- Array4 – Multiplication (computation)
- Array5 – Division (reading)
- Array6 – Division (matching)
- Array7 – Division (creating)
Virt7 Multiplicative Structure (zipped, 131KB)
This virtual material helps students understand the decimal system we use and focuses on multiplying and dividing by factors of 10.
Virt8 Decimal Money (zipped, 1.8MB)
This is a practical set of questions that are interactive and enable students to answer questions they would come across in the real world with money. The questions are specifically designed to bring in the concept of decimal places. This zipped folder includes the following PowerPoint files:
- Decimal Money 1– Numeration
- Decimal Money 2 – Addition and Subtraction
- Decimal Money 3 – Multiplication
- Decimal Money 4 – Division
Construction and building (3 PDF files)
The following three booklets were produced as material to support Indigenous students completing courses or certificates associated with construction at the Northern Peninsula Area College, Bamaga.
Construction and Building Booklet VC1: Mathematics behind Dome Construction using Earthbags (PDF, 1.0MB)
Booklet VC1 covers circles, circle formulae, circle construction, circle relationships, and domes. It is based on the idea that it is important to give students a wide understanding and feeling for circles (e.g., why are the earth, moon, and sun spheres; why does spinning clay give a circular result; how is the circle related to ecology, balance, recycling; why are circles good for wheels) as part of formal mathematics ideas such as the formulae for circumference, area, and volume. This means that the booklet has investigations that look at circle puzzles and designs, and at what can be called circle metaphors (that is, the role of circles in culture – e.g., circle of life, circle of time, circle of friends, etc.), and other uses of circles (e.g., Ying and Yang), including investigations of uses of circles in Indigenous cultures.
Construction and Building Booklet VC2: Mathematics behind Dome Construction using Earthbags (PDF, 760KB)
Booklet VC2 covers rate, ratio, speed and mixes and is based on the idea that it is important to give students a wide understanding and feeling for rate and ratio as well as how to apply rate and ratio to Earthbag buildings. It looks at multipliers and a technique called double number line to assist students to understand how to solve rate and ratio problems. Booklet VC2 also contains activities and investigations. However, as students progress the focus of the booklet tends to remain on straightforward activities with the occasional investigation.
Construction and Building Booklet VC3: Mathematics behind Construction in Horticulture (PDF, 1.5MB)
Booklet VC3 focuses on the mathematics behind construction in horticulture, introducing measurement, and focusing on angle, shape, area and volume, and optimisation.
Engineering and mechanics (3 PDF files)
The following three booklets were produced as material to support Indigenous students completing certificates associated with engineering and mechanics at the TAFE campuses on Thursday Island (run by Tropical North Queensland TAFE) and Palm Island (Barrier Reef Institute of TAFE).
Engineering and Mechanics Booklet VE1: Mathematics behind Small Engine Repair and Maintenance (PDF, 1.3MB)
Booklet VE1 focuses on the numeration basis of units; metric units and conversions; fractions and imperial units; and perimeter, area and volume formulae.
Engineering and Mechanics Booklet VE2: Mathematics behind Small Engine Repair and Maintenance (PDF, 1.0MB)
Booklet VE2 focuses on the nature of rate and ratio; the double number line method; rate, time, distance and fuel consumption; and ratio, fuel mixes, levers, gears and compression.
Engineering and Mechanics Booklet VE3: Mathematics behind Metal Fabrication (PDF, 1.7MB)
Booklet VE3 focuses on uprights, crosspieces and division; construction and angles; construction strength and balance; fabrication, nets and solid shapes; and the formulae.
Marine studies (3 PDF files)
The following three booklets were produced as material to support Indigenous students completing courses associated with the Marine Studies Coxswain Certificate at the Thursday Island campus of Tagai State Secondary College. It may also be applicable to the Coxswain Certificate at Queensland TAFEs, most notably at the Tropical North Queensland TAFE campus at Thursday Island.
Marine Studies: Coxswain Cert. Booklet VM1: Mathematics behind Handling Small Boats and Ships (PDF, 780KB)
Booklet VM1 focuses on angle and direction; using compasses on boats/ships; and distance and speed.
Marine Studies: Coxswain Cert. Booklet VM2: Mathematics behind Handling Small Boats and Ships (PDF, 805KB)
Booklet VM2 focuses on the nature of rate and ratio; using rate to calculate speed and fuel consumption; and using ratio to determine tides.
Marine Studies: Coxswain Cert. Booklet VM3: Mathematics behind Modelling Marine Environments (PDF, 1.4MB)
Booklet VM3 focuses on percentage; coverage and density; box model; and modelling environments.
Pest management (2 PDF files)
The following two Pest Management booklets were produced as material to trial with pesticide students completing Unit 6 as part of the Open Learning Institute of TAFE course Certificate III in Asset Management (Pest Management – Technical).
Cert III Pest Management Booklet VP1: Perimeter-Area-Volume (PDF, 870KB)
Booklet VP1 focuses on how to teach formulae for perimeter, area and volume as required for Unit 6. It leads onto Booklet VP2 which looks at ratio and proportion for mixing chemicals for spraying.
Cert III Pest Management Booklet VP2: Ratio in Mixing Chemicals (PDF, 870KB)
Booklet VP2 focuses on using ratio and proportion to correctly mix chemicals for pesticide spraying as required for Unit 6. It follows on from Booklet VP1 which looks at area and volume formulae for the same unit.
Retail (1 PDF file)
The following booklet was produced as material to support Indigenous students completing certificates associated with retail at the TAFE and post-senior campuses on Palm Island run by Barrier Reef Institute of TAFE in conjunction with Kirwan State High School.
Retail Booklet VR1: Mathematics behind Handling Money (PDF, 1.4MB)
Booklet VR1 focuses on meanings for number and operations; money as whole number; money as decimal numbers; addition and subtraction of money; and multiplication and division of money.