M3a Modelling Session 3

Slide1: 

Objectives for session 3 To consider the use of models that simulate random behaviour, including those used in sample teaching unit 8.4 To examine modelling activities for the classroom that extend sample teaching units 7.4 and 8.4 To consider ways of teaching modelling, using software other than spreadsheets, such as programming and web-based simulations

Modelling with rules: 

Modelling with rules Some systems can be modelled by rules that are ‘constant’. When the inputs are the same then the outputs will be the same. Such a predictable system is known as a deterministic model. The values of the outputs are completely determined by the inputs. There is no element of unpredictability, chance or randomness.

Modelling with randomness: 

Modelling with randomness Can you think of situations in which there appears to be a strong element of chance? Do these systems show any general patterns that can be seen ‘in the long run’? Quantitative models of these systems have to build in unpredictability. These use notions of randomness and probability. We then use a random model to simulate non-deterministic behaviour of a system.

Slide4: 

Using MSW LOGO Write a procedure to print out how many rolls of a simulated die are needed before a 6 appears.

Slide5: 

To Rollto6 Make "count 0 Make "die 0 Until [:die = 6] [Make "die 1 + Random 6 Make "count :count + 1] Print :count Name of procedure Commands nested in a procedure called ‘Rollto6’ Creates a variable called count and sets it at 0. Creates a variable called die and sets it at 0 Loops until die rolls a 6 Makes a random number 0-5. Add 1 gives 1-6 Makes count go up one each time. Acts as a counter. Prints the number on the counter when a 6 is rolled. Variables Rules

Rolling a die: 

Rolling a die How many rolls of a fair six-sided die do you think will be needed, on average, before the first 6 appears? How many rolls might be needed for: a four-sided die before the first 4 appears? a twelve-sided die before the first 12 appears? How might this generalise for an n-sided die? The challenge is to write some procedures in a language such as MSW LOGO, or that of a graphic calculator, to simulate the process.

Slide7: 

To RolltoN :n To RolltoN :n Make "count 0 Make "die 0 Until [:die = :n] [Make "die 1 + Random :n Make "count :count + 1] Print :count End

Slide8: 

Additional challenge Write a procedure RollExperiment, using RolltoN. It will need the arguments: :n to specify the type of die, and :times to specify how many times to simulate the roll to :n. It should output the mean number of rolls needed.

To Rollexperiment :n :times: 

To Rollexperiment :n :times To RollExperiment :n :times Make "sum 0 Repeat :times [RolltoN :n Make "sum :sum + :count] Make "mean :sum / :times Print :mean End

Slide10: 

Weblinks For example: Modelling weather forecasts http://www.metoffice.com/weather/charts/animation.html Modelling ocean wave swell http://www.metoffice.com/research/ocean/operational/wave/swellanim.html Modelling pitstops in formula 1 http://www.formula1.com/fun/games/pitcrew-play.html (needs Shockwave) Modelling disease spread – foot and mouth http://www.hjones-sons.co.uk/fmdani.htm (Cases are yellow on the day they are discovered, red on day 2 and dark orange on day 3 onwards.) Coupon collection simulation http://www.mis.coventry.ac.uk/research/discus/prob.xls (Select coupon collection tab at bottom of screen.) Best-value comparison http://uk.kelkoo.com/