Mathematics Research for the Beginning Student, Volume 1: Accessible Projects for Students Before Calculus

Preface
Contents
Games on Graphs: Cop and Robber, Hungry Spiders, and Broadcast Domination
1 Introduction
2 The Game of Cops and Robbers
3 Hungry Spiders
4 (t,r) Broadcast Domination
5 Further Investigation
6 A Short Primer on Graph Theory
References
Mathematics for Sustainable Humanity: Population, Climate, Energy, Economy, Policy, and Social Justice
1 Introduction
2 Quantifying Change
2.1 Absolute and Relative Change and Rate of Change
2.2 Linear and Exponential Change
2.3 Measuring and Estimating
3 Population Growth and Ecological Footprint
4 Climate Change
5 Energy Production, Consumption, and Efficiency
6 Economic Growth (and Collapse)
7 Policy and Social Justice
References
Mosaics and Virtual Knots
1 Math and Knots
2 Gauss Codes
3 Virtual Knots
4 Mosaics
5 Virtual Mosaics
6 Further Reading
References
Graph Labelings: A Prime Area to Explore
1 Introduction
1.1 Families and Classes of Graphs
1.2 Graph Operations
1.3 Introduction to Graph Labeling
2 Coprime and Prime Labelings
2.1 Minimal Coprime Labeling
3 Consecutive Cyclic Prime Labelings
4 Neighborhood-Prime Labelings
4.1 Building on Cycles
4.2 Building on Trees
4.2.1 Further Projects on Neighborhood-Prime Labelings
5 Conclusion
References
Acrobatics in a Parametric Arena
1 Analogies to Motivate Parametric Thinking
2 Overview
3 Parametric Basics
3.1 Parametric Function 1
3.2 Parametric Function 2
3.3 Parametric Function 3
4 Function Concepts
4.1 Functions and Nonfunctions, Based on the Context
4.2 Connecting Component Functions and the Parametric Function
5 Try Some Parametric Acrobatics Yourself
5.1 General Desmos Graphing Instructions
5.2 Exercises with some Desmos Instructions
5.3 Experimentation
6 True Acrobatics: Parametric Modeling of a Flower Stick
6.1 Flower Sticks and Digitized Motion
6.2 Modeling the Left End of the Flower Stick
6.3 More on the Flower Stick
6.4 Your Work
7 Financial Acrobatics: Modeling US Wireless Subscribers
7.1 About the Data
7.2 Views of the Data
7.3 Data Projections
7.4 Linear Fits to 2004–2014 Data
7.5 Proportional Growth
8 GeoGebra to Practice 3D Parametric Equation
9 Your Project
References
Further Reading
Software
But Who Should Have Won? Simulating Outcomes of Judging Protocols and Ranking Systems
1 Introduction
2 Fundamentals of Probability
2.1 A Little Set Theory
2.2 Computing Probability
2.3 Conditional Probability
2.4 Random Variables and Probability Functions
2.4.1 Discrete Random Variables
2.4.2 Continuous Random Variables
3 Introduction to Simulation
3.1 Random Number Generation
3.1.1 Sampling from Discrete Distributions
3.1.2 Sampling from Continuous Distributions
3.2 ``If/Else'' Statements
3.3 ``While'' and ``For'' Loops
3.3.1 ``While'' Loops
3.3.2 ``For Loops''
3.4 Writing More Complex Simulation Code
4 Suggested Research Projects
4.1 Scenario 1: Objective Ranking
4.2 Scenario 2: Subjective Ranking
4.3 Scenario 3: Comparing Voting Methods
References
Modeling of Biological Systems: From Algebra to Calculus and Computer Simulations
1 Introduction
1.1 A Description of Mathematical Modeling
1.2 Building a Model with Bias
1.3 A Note About Computer-Based Simulations
1.4 Active Learning
2 Grey Squirrels in Six Fronts Park: Modeling a Changing Population
2.1 The Problem
2.1.1 Step 1: Goals, Questions, and Assumptions
2.1.2 Step 2: Build a Model
2.1.3 Step 3. Apply the Model
2.1.4 Step 4. Assess and Revise Your Model
2.2 Conclusion and Exercises
3 Non-contact Cardiovascular Measurements
3.1 Context
3.2 The Challenge
3.3 The Initial Experiment
3.4 Weigh a Bed?
3.5 Let Us Get Our Hands Dirty!
3.6 Interesting Observations
3.7 Conclusions
4 Difference Equations in Population Ecology
4.1 Introduction
4.2 Population Growth with Difference Equations
4.3 Coding Difference Equations
4.4 Difference Equations for Predator-Prey Problems
5 Modeling the Spread of Infectious Diseases with Differential Equations
5.1 Modeling the Demise of Candy
5.2 Population Growth Models in Continuous Time
5.2.1 A Basic Model of Infectious Disease Spread
5.3 Discussion
6 Conclusions
References
Population Dynamics of Infectious Diseases
1 Mathematical Models in Epidemiology
2 An Individual-Based Epidemic Model
2.1 Model Description and Physical Simulation
2.2 Computer Simulation
2.3 Section 2 Exercises
2.4 Section 2 Challenge Problem
2.5 Section 2 Projects
3 Continuous-Time Dynamical Systems
3.1 The Derivative
3.2 Dynamical Systems
3.3 Section 3 Exercises
4 Dynamical System Models
4.1 Classification of Dynamical System Models
5 Building the SEIR Epidemic Model
5.1 Quantifying the Processes
5.1.1 Transition Processes
5.1.2 The Transmission Process
5.2 The Final Model
5.3 Section 5 Exercises
6 Modeling
6.1 Identifying Parameter Values
6.2 The Basic Reproduction Number
6.3 Section 6 Exercises
6.4 Section 6 Challenge Problem
7 Model Analysis
7.1 Early Phase Exponential Growth
7.2 The End State
7.3 Section 7 Exercises
7.4 Section 7 Challenge Problems
7.5 Section 7 Project
8 Simulations
8.1 Numerical Simulation of Continuous Dynamical Systems
8.2 Implementation of Numerical Simulations
8.3 Section 8 Exercises
8.4 Section 8 Projects
Appendix: Programs
hpsr.m
HPSR_onesim.m
HPSR_avg.m
seir.m
SEIR_onesim.m
SEIR_comparison.m
SEIR_paramstudy.m
References
Playing with Knots
1 Knots: Knotted and Unknotted
1.1 What Is a Knot? (The Basics)
1.2 Knot Equivalence and Reidemeister Moves
1.3 Some Useful Knots and Links
1.4 Unknotting Operations and Numbers
1.5 Knot Invariants
2 The Knotting-Unknotting Game
3 The Region Unknotting Game
4 The Linking-Unlinking Game
5 The KnotLink Game
6 The Link Smoothing Game
7 Conclusion
References