Teaching an Introductory Course in Mathematical Modeling Using Technology - PowerPoint PPT Presentation

Teaching an introductory course in mathematical modeling using technology
1 / 51

  • Uploaded on
  • Presentation posted in: General

Teaching an Introductory Course in Mathematical Modeling Using Technology. Dr.William P. Fox. Francis Marion University Florence, SC 29501 wfox@fmarion.edu. Agenda. Mathematical Modeling Process Modeling Course and book Modeling Toolbox Explicative Models Model Fitting

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

Download Presentationdownload

Teaching an Introductory Course in Mathematical Modeling Using Technology

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

Teaching an introductory course in mathematical modeling using technology l.jpg

Teaching an Introductory Course in Mathematical Modeling Using Technology

Dr william p fox l.jpg

Dr.William P. Fox

Francis Marion University

Florence, SC 29501


Agenda l.jpg


  • Mathematical Modeling Process

  • Modeling Course and book

  • Modeling Toolbox

  • Explicative Models

  • Model Fitting

  • Empirical Models

  • Simulation Models

  • Projects and Labs

Modeling real world systems l.jpg

Modeling Real World Systems

The modeling loop l.jpg

The Modeling Loop

Modeling process l.jpg

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 l.jpg

Purpose of Mathematical Modeling

  • Explain behavior

  • Predict Future

  • Interpolate Information

Modeling textbook l.jpg

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 l.jpg

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 l.jpg

Tool #1 Explicative Modeling

  • Proportionality and Geometric Similarity Arguments

Proportionality l.jpg


Proportionality definition l.jpg


Proportionality graphical l.jpg


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

Geometric similarity l.jpg

Geometric Similarity

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

Scale models l.jpg

Scale Models

Formula s l.jpg


  • Area  characteristic dimension 2

  • Volume  characteristic dimension 3

Archimedes l.jpg


  • A object submerged displaces an equal volume to its weight.

  • So, under this assumption

  • W Volume

Terror bird l.jpg

Terror Bird

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

Bird data l.jpg

Bird Data

Dinosaur data l.jpg

Dinosaur Data

Assumptions l.jpg


  • 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.

Build the model and compare results l.jpg

Build the model and compare results

Model fitting l.jpg

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 l.jpg

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 l.jpg


  • 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 l.jpg

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 l.jpg

Empirical Modeling (The data speaks)

  • Simple One Term Models

  • LN-LN transformations

  • High Order Polynomials

  • Low Order Smoothing

  • Cubic Splines

Simple 1 term models l.jpg

Simple 1 Term Models

1 term models l.jpg

1 Term Models

Ln ln transformation l.jpg

LN-LN Transformation

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

High order polynomials l.jpg

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 l.jpg

Smoothing with Low Order

  • Use Divided Difference Table for qualitative assessment.

Slide33 l.jpg


  • 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 l.jpg

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 splines35 l.jpg

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 l.jpg

Projects and Problem Sets

  • More sophisticated problems and projects with real world orientation.

Simulation modeling l.jpg

Simulation Modeling

Simulation modeling a method of last resort l.jpg

Simulation Modeling-A method of last resort!

Monte carlo methods l.jpg

Monte Carlo Methods

Deterministic methods area under non negative curve l.jpg

Deterministic Methods-Area under non-negative curve

Y x 3 0 x 2 l.jpg

Y=x^3, 0<x<2

Using simulation approximate area is 4 106 l.jpg

Using Simulation: approximate area is 4.106

Simulation comparisons l.jpg

Simulation Comparisons

More complicated problem l.jpg

More Complicated Problem:

Area via simulation l.jpg

Area via Simulation

Volumes via simulation l.jpg

Volumes via simulation

Example volume of surface in 1 st octant l.jpg

Example:Volume of Surface in 1st Octant

Stochastic simulation labs l.jpg

Stochastic Simulation Labs

Projects l.jpg


  • 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 l.jpg

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.

Philosophy l.jpg

Share our vision

Recruit students

Develop teachers

Empower faculty



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.”


  • Login