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Teaching an Introductory Course in Mathematical Modeling Using Technology

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Teaching an Introductory Course in Mathematical Modeling Using Technology

Agenda

- Mathematical Modeling Process
- Modeling Course and book
- Modeling Toolbox
- Explicative Models
- Model Fitting
- Empirical Models
- Simulation Models
- Projects and Labs

Modeling Process

- Identify the Problem
- Assumptions/Justifications
- Model Design/Solution
- Verify the model
- Strengths and Weaknesses of the model
- Implement and Maintain Model

Purpose of Mathematical Modeling

- Explain behavior
- Predict Future
- Interpolate Information

Modeling textbook

- “A First Course in Mathematical Modeling”, Giordano, Weir, and Fox, Brooks-Cole Publisher, 3rd Edition, 2003.
- Labs available on the course/book web site using
- MAPLE, EXCEL, TI-83 Plus calculator

My Process

- Show mathematical modeling tool
- Provide development, theory, and analysis of the tool
- Provide appropriate technology via labs
- Maple
- Excel
- Graphing Calculator
- Project to tie together the process

Tool #1 Explicative Modeling

- Proportionality and Geometric Similarity Arguments

Proportionality-Graphical

- We want to estimate the plot of the transformed data as a line through the origin. W versus x^3.

Geometric Similarity

- 1 to 1 relationship between corresponding points such that the ratio between corresponding points for all such points is constant

Formula’s

- Area characteristic dimension 2
- Volume characteristic dimension 3

Archimedes

- A object submerged displaces an equal volume to its weight.
- So, under this assumption
- W Volume

Terror Bird

- Problem Identification: Predict the size (weight) of the terror bird as a function of its fossilized femur bone found by scientists.

Assumptions

- Geometrically similar objects (terror bird is a scale model of something)
- Assume volume is proportional to weight under Archimedes Principal.
- Characteristics dimension is the femur bone.

Model Fitting

- * Linear Regression (least squares)
- Minimize the largest absolute deviation, Chebyshev’s Criterion
- Minimize the sum of the absolute errors

Solution Methods for each

- Least squares—calculus or technology
- Chebyshev’s—Linear Programming
- Minimize Sum of absolute errors—nonlinear search method like Golden Section Search.

Errors

- Least Squares: smallest sum of squared error
- Chebyshev’s: minimizes the largest error
- Minimizes the sum of the absolute errors

Least Squares and Residuals (errors)

- Concern is model adequacy.
- Plot residuals versus model (or independent variable)—check for randomness. If there is a pattern then the model is not adequate.

Empirical Modeling (The data speaks)

- Simple One Term Models
- LN-LN transformations
- High Order Polynomials
- Low Order Smoothing
- Cubic Splines

LN-LN Transformation

- Try to linearize the data in a plot. This plot does not have to pass through the origin.

High Order Polynomials

- N data points create a (N-1)st order polynomial.
- Problems exist with high order polynomials that provide possible strong disadvantages.
- Advantage: passes perfectly through every pair of data points.
- Disadvantages: oscillations, snaking, wild behavior between data pairs.

Smoothing with Low Order

- Use Divided Difference Table for qualitative assessment.

Goal

- To find columns that qualitatively reveal one of the following:
- f(x) linear, 1DD is constant, 2 DD is zero.
- f(x) quadratic, 2 DD is constant, 3rd DD is zero.
- f(x) cubic, 3rd DD is constant, 4th DD is zero
- f(x) quartic, 4th DD is constant, 5th DD is zero
- Then fit with least squares and examine residuals.

Cubic Splines

- Put data for x in increasing order and carry along the y coordinate (x,y). Fit a cubic polynomial between successive pairs of data pairs. For example given three pairs of points (x1,y1),(x2,y2), and (x3,y3):
- S1(x)=a1+b1x+c1x2+d1x3 for x [x1,x2]
- S2(x)=a2+b2x+c2x2+d2x3 for x [x2,x3]

Cubic Splines

- Natural: end points have constant slope but we do not know the slope.
- Clamped: end points have a known constant slope.

Projects and Problem Sets

- More sophisticated problems and projects with real world orientation.

Projects

- Sporting Event: Baseball game, horse race (based on odds), consecutive free throws contest, racketball games
- Gambling Events: Craps, Blackjack
- Real World: Queuing problems (emission inspections, state car inspections, traffic flow, evacuating a city for a disaster, fastpass at Disneyland,…

GOALS of a Modeling Course

- Dynamic, connected modeling curriculum responsive to rapidly changing world.
- Blend of excellence: Mentoring relationship between faculty and students and between college and school faculty.
- Enthusiastic teachers empowered to motivate, challenge, and be involved.
- Effective teaching tools and supportive educational environment.
- Competent, confident problem solvers for the 21st Century.
- Continued development and student/teacher growth.

Share our vision

Recruit students

Develop teachers

Empower faculty

Listen/learn/lead

Mentor/counsel/care

Selfless service

Blend of Excellence

Positive first derivative

We are successful since “we have a challenging, important mission, enough resources to do the job, and good people to work with.”

Philosophy
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