GMST 570 Modeling Change in Mathematics and Science Day One: Friday, July 12, 2002 Introduction to Modeling

The Syllabus  Who am I?  Who are you?  What is the purpose of this course?  What resources do you need to succeed?  Is this a good course for me?  What topics are we going to “cover”?  What kind of work will I need to do in this course?  How will grades be determined?  Questions?

What is a model?  Get into groups of 3-4.  Come up with five to ten examples of models.  What do these examples have in common?  What is it that makes these examples models? How would you define “model”?  What would make a model a mathematical model?

Let’s see what you know…  In your groups, you’re going to generate some models to represent a few “real world” situations.  You are free to represent your answers any way you choose to.  After you have had time, we’ll share our answers and critique the work of each group.

“Model me this, Batman!”  The height of the grass in your yard over an entire year.  Typical distribution of heights and weights in a population.  The height a ball will bounce. As you work, think about: assumptions, units/scale, and constraints.

Activity: M&Ms  A jigsaw company wants to fill your jar with M&M candies so that a photograph can be taken of the jar looking more or less full (a handful shy will not make any difference). A bag of M&M’s holds 50 candies. How many bags should they buy? You have 15 minutes to complete this project. Before you start, you may acquire any standard measuring instruments you think you will need. One constraint: do not start with more than one bag of candy (you are constrained so that you cannot fill the jar, dump it, and count the candies).

Discussion Suppose we are starting a limousine service, driving people from a particular town to the nearest airport. For the sake of illustration, suppose that this is 75 miles one way. We want to compute the cost in gasoline of these trips for budgeting purposes. Our car has an average gas mileage for this type of driving of 25 miles per gallon.

SLIME Modeling  It’s now time to get our hands dirty in groups of two.  You will need: Two transparencies Two transparencies Two pieces of paper Two pieces of paper Teaspoon measures Teaspoon measures Pencils/pens Pencils/pens  Create spheres of slime that are ¼ teaspoon, ½ tsp, 1 tsp, 2 tsp, 3 tsp, 4 tsp, 5 tsp, 6 tsp  Let the spheres “melt” on the transparencies  Record the diameter of the puddles compared to the original volume

SLIME Modeling #2  Use your data to answer the following questions: How big would the puddle be for a 10 tsp blob How big would the puddle be for a 10 tsp blob What is the analytical relationship between the initial volume and the final puddle size? What is the analytical relationship between the initial volume and the final puddle size? How much SLIME would you need to make a puddle of diameter 150 mm? How much SLIME would you need to make a puddle of diameter 150 mm? Can you create a theoretical model for the size of the blobs? Can you create a theoretical model for the size of the blobs?

Some concepts in modeling  Purpose of a model  Resolution and scale  Simplicity, KISS, and Occam’s Razor  Assumptions  Constraints  Qualitative behavior  Quantitative behavior  Ease of use  Testing and Verification  Iteration

Basic modeling process Real World Model World Model Occam’s Razor Interpreting and Testing Formulating Model World Problem Model Results Mathematical Analysis Modeling Diagram (Taken from Mooney and Swift, page 4)

The iterative nature Examine the “system” Can you formulate a model? Can you solve the model? Identify the behavior and make assumptions Validate the model Are the results precise enough? Apply results to the system Exit Make predictions and/or explanations No, simplify Yes No, refine Taken from A First Course in Mathematical Modeling By Giordano, Weir, and Fox (page 40)

Some classes of models Either this…  Time dependent  Deterministic  Empirical  Continuous  Extrapolation  Qualitative Or this…  Steady state  Stochastic  Theoretical  Discrete  Interpolation  Quantitative

More examples of models  The Solar System and Gravity From circles and four elements… From circles and four elements… To Ptolemy’s epicycles To Ptolemy’s epicycles To Kepler’s three laws To Kepler’s three laws To Newton’s calculus To Newton’s calculus To Einstein’s relativity To Einstein’s relativity  Projectile Motion  Models of the atom

Time to practice  Work on the three problems at the end of chapter zero in groups. Prepare complete answers (there is a handout for #2 to make it easier).  Use whatever tools you need that are available (including the computer!)  We will take turns presenting our work and critiquing the solutions we have produced.

Wrap up, reflection, homework  Any questions or comments?  Reflections on class – the chain reflection  Homework Journal Journal Reading Reading Two essays Two essays Compare and contrast Compare and contrast Start planning a project Start planning a project