Distribution Dave Pratt Institute of Education, University of London

Aims To review the previous directed work task on dice. To introduce the use of the Bowland case studies. To discuss the nature of distribution. To make sense of data by considering different shaped distributions. To make sense of data by considering parameters of distributions.

What is the most likely total for the sum of three dice? Sketch a chart showing proportion of throws of each total from 3 to 18 when the three dice are thrown many times. Find three dice (or throw one die three times) and conduct an experiment to provide evidence that your chart may be right. How would you calculate whether your chart is right? By considering the shape of the chart for two and three dice, can you see what would happen to the shape of the chart if instead you threw four dice each time? What if five dice, six dice and so on?

Bowland Series of case studies Designed to be used in secondary maths classes With a focus on making mathematics engaging for all.

Bowland: How risky is life? Stage 1: Exploring our perceptions of risk  Look at handout S1 and S2.  What things do you, your friends, your family worry about?  How do we change our lives as a result?

Bowland: How risky is life? More exploring of our perceptions.  Look at handout S3.  Put them in rank order of risk.  Estimate how many deaths per year occur in England and Wales

Further work on ‘How risky is life?’ Stage 2: Discovering the facts Stage 3: How risky is life? The big picture Stage 4: Do these estimates apply to me?

Deborah’s Dilemma Software developed to explore ideas about risk with teachers. See

SBT 4: Exploring the Bowland case studies Look at the Bowland case studies and choose one that interests you, you think would engage the students in one of your classes and fits in with the scheme of work.

Rainfall data What questions might you ask? Some methods in Tinkerplots

Rainfall data Hands-on: Try out some of these methods for yourself to try to answer some of your questions. What did you find out?

Rainfall data Trend versus histogram So the trend plots values of a variable over time. Whereas a distribution shows how often different values of a SINGLE variable occur.

Types of distribution What did the rainfall distribution look like?  Wave your arms! Worksheet from RSSCSE site:

Exploring different data sets In each case below:  What questions might you ask?  For which is distribution useful? Health  Reaction times  Child development  Body measurements Sports and entertainment  Oscars

Too heavy Body Packs (Exploring Data Starters) Average and spread Which questions would be informed by average and spread?

Data versus Probability Distribution

Summary Bowland case studies offer a rich source of unusual but engaging contexts for mathematical modelling and exploration. We must be careful to distinguish between trend and distribution. In some situations, we can:  Calculate expected frequencies and obtain a theoretical probability distribution (the total of n dice).  Compare the data distribution to those expected.

Summary Tinkerplots provides a pedagogically powerful resource for exploring data, including:  Organising data in various types of graphs.  ‘Measuring’ data as an average or spread. A new version of Tinkerplots, out fairly soon, will allow modelling distributions in addition to the above functionality.