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GMST 570 Modeling Change in Mathematics and Science

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

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GMST 570 Modeling Change in Mathematics and Science

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

  2. 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? The Syllabus

  3. 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?

  4. 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.

  5. “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.

  6. 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).

  7. 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.

  8. It’s now time to get our hands dirty in groups of two. You will need: Two transparencies Two pieces of paper Teaspoon measures 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

  9. SLIME Modeling #2 • Use your data to answer the following questions: • 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? • 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?

  10. 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 Some concepts in modeling

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

  12. The iterative nature No, simplify No, simplify Can youformulatea model? Can yousolve themodel? Examine the “system” Identify thebehavior andmakeassumptions Yes Validatethemodel Taken from A First Course in Mathematical Modeling By Giordano, Weir, and Fox (page 40) Are theresults preciseenough? Apply resultsto the system Make predictionsand/orexplanations Yes Exit No, refine

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

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

  15. 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.

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

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