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New Curriculum New Opportunities New Curriculum New Opportunities Editors Part time Michael Westbrook, Chief Editor Marty Ross David Treeby June Penney Catriona Sexton Patrick Walsh Ann Kilpatrick Jen Bowden Allason Mcnamara Jeanne Carroll Michelle Huggan Sue Ferguson Debora Lipson Published by: THE MATHEMATICAL ASSOCIATION OF VICTORIA FOR THE FORTY-SEVENTH ANNUAL CONFERENCE December 2010 i Published 2010 by The Mathematical Association of Victoria FOREWARD “Cliveden” 61 Blyth Street The theme for the Conference ‘New Curriculum – New Opportunities’ has encouraged Brunswick VIC 3053 Authors to contribute to new and different aspects of the curriculum taking the best of Designed by Idaho Design & Communication the past and adapting it for the needs of the future as well as taking advantage of the new Produced by Big Print technologies. Topics have ranged from pedagogy to practical, from modelling to maths in context, from perceptions to school performance and whilst many articles were written by academic researchers and teacher educators, it is gratifying to see how many classroom teachers have National Library of Australia Cataloguing-in-Publication Data: contributed. This year has unfortunately seen a reduction in the number of papers. But the year has also seen a different editorial panel mainly consisting of practicing teachers from Primary and Secondary schools, Government and Private , rather than academics. This has resulted in a few teething and communication problems as Reviewers struggled with teaching commitments and the demands of editing. The team has, however, enjoyed the experience of working with diverse Authors and has gained many insights which they will apply to their own teaching. ISBN: Enjoy the Conference © Copyright 2010 Permission to reproduce material from this publication must be sought from the owner of copyright, and due acknowledgement be given to Mathematical Association Editors Part time of Victoria as publisher. Michael Westbrook, Chief Editor Marty Ross While every care has been taken to trace and acknowledge copyright, the publishers David Treeby June Penney tender their apologies for any accidental infringement where copyright has proved Catriona Sexton Patrick Walsh untraceable. The MAV would be pleased to come to a suitable arrangement with the Ann Kilpatrick Jen Bowden rightful owner in each case. Allason Mcnamara Jeanne Carroll Michelle Huggan Sue Ferguson Should readers wish to contact any of the authors, full addresses are available from the Debora Lipson Professional Officer, the Mathematical Association of Victoria. The opinions expressed in this publication do not necessarily reflect the position or endorsement of the Editors or of the Mathematical Association of Victoria. ii iii DIDACTIC BY STEALTH .................................................................................................... 107 CONTENTS Terry Lockwood Peter Galbraith FOREWARD .................................................................................................................................. III PROBLEM SOLVING – A GRADE SIX PRIMARY SCHOOL EXPERIENCE YEARS: 5 TO 8 ........................................................................................................................... 117 AN INTRODUCTION TO THE ALGEBRA OF RANDOM VARIABLES ...........1 Ian Bull John Kermond TEACHING STUDENTS ‘AT RISK’: HUME NUMERACY INTERVENTION WORKING MATHEMATICALLY PROGRAM .................................................................................................................................. 126 IN A RURAL CONTEXT .........................................................................................................17 Mark Waters and Pam Montgomery Michael Mitchelmore & Heather McMaster QUADRILATERAL QUARRELS........................................................................................ 135 STUDENTS’ PERCEPTIONS OF MATHEMATICS TEACHING AND LEARN- Allan Turton ING DURING THE MIDDLE YEARS ................................................................................20 ITEMISING ASSESSMENT IN THE SOUTH AFRICAN CURRICULUM Catherine Attard GUIDELINES: UNDERSTANDING ISSUES OF TEACHER POWER ............. 145 LEARNING BY REFLECTING – A PEDAGOGY FOR ENGAGED MATH- Willy Mwakapenda EMATICS LEARNING ..............................................................................................................29 HOORAY FOR MARTIN GARDNER! ............................................................................ 156 Berinderjeet Kaur John Gough DESIGNING MATHEMATICAL INVESTIGATIVE TASKS ....................................36 PLAYING WITH | Tin Lam TOH WOLFRAM ALPHA .....................................................................163 David Leigh-Lancaster DIFFERENTIATING MATHS TEACHING AT SECONDARY LEVEL ...............46 MATHEMATICAL GAMES: JUST TRIVIAL PURSUITS? ..................................... 170 Dr Ian Lowe Paul Swan ................................................................................................................................. GEOCACHING: A WORLDWIDE TREASURE HUNT Derek Hurrell ENHANCING THE MATHEMATICS CLASSROOM ................................................54 SOME RATIONAL NUMBER COMPUTATIONS ..................................................... 179 Leicha A. Bragg, Yianna Pullen and Megan Skinner David Leigh-Lancaster A ROAD WELL TRAVELLED ................................................................................................63 AUSTRALIAN MATHEMATICS CURRICULUM IMPLEMENTATION: Ian Thomson, Ormiston College TEACHER REFLECTIONS ON STUDENT THINKING INFORMING FU- AN EFFECTIVE NUMERACY PROGRAM FOR THE MIDDLE YEARS. .........71 TURE DIRECTIONS ............................................................................................................. 187 Yvonne Reilly, Jodie Parsons and Elizabeth Bortolot, Gaye Williams PERFORMANCE OF LOW ATTAINERS FROM SINGAPORE IN NUMERACY BUILDING RESILIENCE TO BUILD PROBLEM SOLVING CAPACITY: 79 TASKS IMPLEMENTED FOR Chang Suo Hui, Koay Phong Lee & Berinderjeet Kaur THIS PURPOSE ........................................................................................................................ 199 LESSON STUDY – HOW IT COULD WORK FOR YOU ..........................................89 Gaye Williams Susie Groves and Brian Doig TRANSFORMING SPACE ................................................................................................... 207 STUDENTS’ AND TEACHERS’ USE OF ICT Marj Horne IN PRIMARY MATHEMATICS............................................................................................99 Esther Loong, Brian Doig and Susie Groves iv v John Kermond AN INTRODUCTION TO THE ALGEBRA OF RANDOM VARIABLES John Kermond Haileybury College Senior School (Keysborough Campus) Theorems for finding the probability density function of (1) the sum, difference, product and quotient of two independent and continuous random variables, and (2) a function of a single continuous random variable are given and illustrated using simple examples. The theory might stimulate interesting ideas for the Mathematical VCE Methods Unit 4 SAC Analysis Task 2 (Probability). Introduction There are many real world problems (particularly in the field of engineering) that require application of algebraic operations to random variables (1, pp 6-9). For example, the sum of independent normal random variables arises in problems where the probability that a given number of people will exceed a specified maximum weight limit is required. While many students might naively assume that the usual rules of algebra apply in such problems, this assumption is not correct. any pattern and so are not classified, and if this occurs it is included in the feedback as well. Teachers can interview these students to establish their real level of understanding. In this example, the diagnosis is reported in 5 stages (0 to 4). These stages relate only to this topic – they are not yet linked to an external framework such as VELS levels. 1 An Introduction to the Algebra Of Random Variables John Kermond Sum of two independent and continuous random variables Theorem 1 Suppose that X and Y are two independent and continuous random variables with pdf ’s f(x) and g(y) respectively. Then the sum U = X + Y is a continuous random variable with pdf given by (1, p 47). Example 1 Let X and Y be two independent standard uniform random variables with pdf ’s given by and respectively. Then the sum Figure 1. (a) Regions of the uy-plane defined by and . (b) The pdf of . U = X + Y is a continuous random variable with pdf given by Example 2 Let X and Y be two independent standard normal random variables with pdf ’s given where by for and for respectively (2, p 146). Then the sum U = X + Y is a continuous random variable with the sample space and pdf given by: Since and it follows that the sample space of U = X + Y is . Figure 1 (a) suggests that there are three cases to consider. Case 1: . Then . Case 2: . Then . Case 3: or or . Then . Note: . Therefore Since is the pdf of a normal random variable ( and ) (2, p 145) This defines a triangular distribution and is shown in Figure 1 (b). 2 3 An Introduction to the Algebra Of Random Variables John Kermond where it follows that . Therefore Since and it follows that the sample space of U = X – Y is . There are two cases to consider. Case 1: . Then . and is recognised as a normal distribution with ( ) and ( ) (2, p 145). In general, the sum of two independent normal random variables with means and Case 2: . Then . variances and and and respectively is a normal random variable
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