# Prevocational mathematics

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:

1. structural activities that lead to the discovery and abstraction of mathematical concepts, processes, strategies and procedures; and
2. 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).

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).

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.