Introduction

In this Case, the pupils’ initial task is to construct a simple mathematical model of the situation to help them decide whether or not the umpire made the correct decision in the situation shown in the photograph.  Pupils then revisit and test some assumptions made in this initial model, make some estimates of likely uncertainty and apply their model to other situations. The mathematical skills and thinking required emerge gradually during the Case.  This Case is designed to be used to explore a range of mathematical areas and concepts and would ideally not be incorporated into a single topic. It includes ‘handling data’, ‘algebra’ and ‘measurement’ and could be used following these topics for consolidation, integration and extension as these skills are applied in the real world.

 Mathematical Content

The mathematical activities involved in this case relate to four key concepts: competence (selecting suitable mathematical tools to use during the activities); creativity (pupils using the mathematical tools they already know to create a solution to a real life problem); applications and implications of mathematics (seeing pupils engage with mathematics in an enthusiastic manner to solve a worthwhile problem); critical understanding (modeling a real life situation with graphs.) The activities engage the following key processes:  representing (pupils prepare models to make predictions); analysing (pupils define variables and make predictions based on evidence from the photo); interpreting and evaluating (pupils identify areas of error for each variable and discuss their predictions in light of these errors).  The mathematical content includes: number and algebra (accurate calculation, measurement and units); geometry and measures (use of several movement formulae, calculations of distance, error measurements); statistics (collection of primary data, interpretation of data incorporating the use of spreadsheets).  Pupils should be competent in the use of basic algebra and formulae, including substitution, before starting this Case.  A working understanding of spreadsheets would also be an advantage.

Organisation and Pedagogy

The Mathematics and thinking required in this Case becomes complex and is recommended for pupils in level 6, or pupils working at the upper end of level 5, especially those highly capable in algebra and handling data.

During each lesson class work will involve a combination of class discussion, group and individual work.  Field work during the Case may be used to assist pupils to develop a feel for the game of cricket and collect primary data.  While traditional requirements of a classroom teacher, such as developing understanding and knowledge in pupils, apply to this Case, the role of the teacher has some unique aspects in a Case such as this.  These include setting the broad context of the Case and ensuring that pupils understand the place of each activity within this context, guiding the class to generate models by posing questions and carefully running class discussions that bring out what are likely to be some new issues associated with modelling and (differently) ask pupils to justify and explain what are some counter-intuitive findings.

Homework is provided as part of each lesson plan.  Homework is mainly designed to develop necessary skills and knowledge and ask pupils to draw together and make decisions about the material covered through the lesson.  Teachers may find it useful to discuss homework tasks as a starting point for the next lesson.

Resources

Case materials are provided from a website which can be loaded onto the school intranet.  The Overview and Introduction are intended to indicate that the Case can be used with maximum flexibility. Each lesson plan contains both a summary outline with suggested timing and then a detailed lesson plan for use by teachers.  The supporting notes for each lesson provide teachers with extra advice for each activity in the lesson. The website also contains links to all materials required including photos, video clips and required cricket knowledge.  Some worksheets are provided as an optional way of having pupils record their ideas and responses as they arise during the discussions. However, teachers are strongly encouraged not to constrain the lessons by simply handing out worksheets and asking pupils to "fill them in" as doing so would be very likely to compromise the learning objectives of this Case. Each lesson contains related homework tasks. Other support for teachers includes sample calculations and outcomes for activities, and careful discussion of what can be some challenging mathematical ideas. Included is advice for extending advanced pupils as well as vignettes written by trialling teachers that raise issues of teaching and learning that arise in this Case which may be new for teachers.  ICT requirements during the Case include data projectors, Windows Media Player and spreadsheets.  Teachers may also choose to engage pupils in a game of cricket prior to or during the Case.