Mathematical Modeling and Applications

          Module 1: Introduction to Mathematical Modeling
Section 1: Understanding Mathematical Models
- Lesson 1: What is a Mathematical Model?
  - Module 1, Section 1, Lesson 1: Introduction to the concept of mathematical models and their role in problem solving.
- Lesson 2: Importance of Modeling in Real Life
  - Module 1, Section 1, Lesson 2: Exploring examples of models used in everyday situations and natural phenomena.

Section 2: Identifying Variables and Parameters
- Lesson 1: Distinguishing Variables from Constants
  - Module 1, Section 2, Lesson 1: Learning definitions and examples of variables versus constants in simple models.
- Lesson 2: How to Choose Relevant Parameters
  - Module 1, Section 2, Lesson 2: Discussing how to determine critical parameters while modeling a real-world scenario.

Section 3: Translating Real-World Problems into Mathematics
- Lesson 1: Breaking Down a Problem into Mathematical Terms
  - Module 1, Section 3, Lesson 1: Step-by-step methods to understand a practical problem and represent it mathematically.
- Lesson 2: Simplifying Complex Situations
  - Module 1, Section 3, Lesson 2: Techniques for simplifying and approximating real-world scenarios into manageable models.

Section 4: Basic Graphical Representation of Models
- Lesson 1: Introduction to Graphing Data
  - Module 1, Section 4, Lesson 1: Learning basic graph types and their relevance in showing model behavior.
- Lesson 2: Plotting Simple Equations
  - Module 1, Section 4, Lesson 2: Practice with plotting linear and basic nonlinear equations relevant to initial models.

Section 5: Overview of Mathematical Tools
- Lesson 1: Essential Mathematics Tools
  - Module 1, Section 5, Lesson 1: Overview of the fundamental mathematical tools used in modeling such as algebra and geometry.
- Lesson 2: Using Technology in Modeling
  - Module 1, Section 5, Lesson 2: Introduction to basic technology tools that assist in building and testing mathematical models.

Module 2: Building Blocks of Models
Section 1: Formulating Equations from Descriptions
- Lesson 1: From Words to Equations
  - Module 2, Section 1, Lesson 1: Illustrating how everyday language can be translated into mathematical expressions.
- Lesson 2: Examples of Word Problems
  - Module 2, Section 1, Lesson 2: Working through simple word problems to practice equation formulation.

Section 2: Linear Models and Their Applications
- Lesson 1: Understanding Linear Relationships
  - Module 2, Section 2, Lesson 1: Introduction to linear equations and how they model proportional relationships.
- Lesson 2: Practical Examples of Linear Models
  - Module 2, Section 2, Lesson 2: Applications of linear models in contexts such as speed, cost, and distance problems.

Section 3: Nonlinear Models in Real Life
- Lesson 1: Recognizing Nonlinear Patterns
  - Module 2, Section 3, Lesson 1: Identifying when a relationship is nonlinear and the basics of quadratic and exponential functions.
- Lesson 2: Simple Nonlinear Modeling Examples
  - Module 2, Section 3, Lesson 2: Solving real-life problems that benefit from a nonlinear model approach.

Section 4: Role of Assumptions in Modeling
- Lesson 1: Making Reasonable Assumptions
  - Module 2, Section 4, Lesson 1: Learning why assumptions are necessary and discussing examples of common assumptions.
- Lesson 2: Evaluating the Impact of Assumptions
  - Module 2, Section 4, Lesson 2: Analyzing how different assumptions can change the outcome of a model.

Section 5: Developing a Modeling Strategy
- Lesson 1: Step-by-Step Modeling Process
  - Module 2, Section 5, Lesson 1: Introducing a structured approach to developing mathematical models.
- Lesson 2: Case Study: Simple Modeling Project
  - Module 2, Section 5, Lesson 2: Applying the strategy to a guided example project.

Module 3: Analysis and Interpretation of Models
Section 1: Testing and Validating Models
- Lesson 1: Methods of Validation
  - Module 3, Section 1, Lesson 1: Exploring how to test a model’s predictions against real data.
- Lesson 2: Identifying Errors and Limitations
  - Module 3, Section 1, Lesson 2: Understanding common sources of error in a model and strategies for improvement.

Section 2: Interpreting Graphical Data
- Lesson 1: Reading and Understanding Graphs
  - Module 3, Section 2, Lesson 1: Basic techniques for interpreting graphs that represent mathematical models.
- Lesson 2: Drawing Conclusions from Data
  - Module 3, Section 2, Lesson 2: Explaining how to derive meaningful conclusions based on graphical evidence.

Section 3: Sensitivity Analysis
- Lesson 1: What is Sensitivity Analysis?
  - Module 3, Section 3, Lesson 1: Introduction to analyzing how model outcomes change with variations in input.
- Lesson 2: Practical Examples of Sensitivity Tests
  - Module 3, Section 3, Lesson 2: Simple exercises to show the effect of changing one variable at a time.

Section 4: Dimensional Reasoning in Models
- Lesson 1: Using Units to Validate Models
  - Module 3, Section 4, Lesson 1: The importance of unit consistency and dimensional analysis in modeling.
- Lesson 2: Applying Dimensional Reasoning
  - Module 3, Section 4, Lesson 2: Exercises in checking the validity of equations using dimensional analysis.

Section 5: Communicating Model Results
- Lesson 1: Presenting Mathematical Findings
  - Module 3, Section 5, Lesson 1: Techniques for effectively communicating model outcomes to various audiences.
- Lesson 2: Writing a Simple Modeling Report
  - Module 3, Section 5, Lesson 2: Guidance on writing clear and concise reports based on model analysis.

Module 4: Simulation Techniques and Computational Tools
Section 1: Introduction to Simulations
- Lesson 1: What is a Simulation?
  - Module 4, Section 1, Lesson 1: Basic concepts and purpose of using simulations in model testing and predictions.
- Lesson 2: Real-World Simulation Examples
  - Module 4, Section 1, Lesson 2: Overview of various simulation examples in science and engineering.

Section 2: Setting Up Simple Simulations
- Lesson 1: Defining Inputs for Simulations
  - Module 4, Section 2, Lesson 1: How to choose and input parameters in a simulation setting.
- Lesson 2: Running and Observing Simulations
  - Module 4, Section 2, Lesson 2: Step-by-step process to run a simulation and observe its outcomes.

Section 3: Introduction to Computational Tools
- Lesson 1: Overview of Basic Software Options
  - Module 4, Section 3, Lesson 1: Introducing simple computational tools and software that aid in modeling.
- Lesson 2: Hands-on with a Basic Simulation Tool
  - Module 4, Section 3, Lesson 2: Guided exercise in setting up and running a simulation using accessible tools.

Section 4: Analyzing Simulation Results
- Lesson 1: Interpreting Simulation Data
  - Module 4, Section 4, Lesson 1: Learning how to read and interpret the output of a simulation.
- Lesson 2: Comparing Simulation Outcomes with Theory
  - Module 4, Section 4, Lesson 2: Exercises to compare simulated results with expected theoretical outcomes.

Section 5: Improving Simulation Models
- Lesson 1: Identifying Model Weaknesses in Simulations
  - Module 4, Section 5, Lesson 1: Recognizing common pitfalls and limitations in simulation models.
- Lesson 2: Refining and Adjusting Simulations
  - Module 4, Section 5, Lesson 2: Techniques for enhancing the accuracy and reliability of simulation results.

Module 5: Applications of Mathematical Modeling
Section 1: Modeling in Engineering
- Lesson 1: Applications in Structural Engineering
  - Module 5, Section 1, Lesson 1: Exploring how mathematical models are used to design and analyze structures.
- Lesson 2: Modeling in Mechanical Systems
  - Module 5, Section 1, Lesson 2: Case studies showing the role of modeling in mechanical engineering projects.

Section 2: Environmental and Natural Systems Modeling
- Lesson 1: Modeling Population Dynamics
  - Module 5, Section 2, Lesson 1: Introduction to models used in ecology and population studies.
- Lesson 2: Simulating Ecosystem Behavior
  - Module 5, Section 2, Lesson 2: Understanding the basics of ecosystem modeling and environmental simulations.

Section 3: Economic and Financial Modeling
- Lesson 1: Basic Concepts in Economic Models
  - Module 5, Section 3, Lesson 1: Exploration of simple economic theories and their mathematical representations.
- Lesson 2: Modeling Market Trends
  - Module 5, Section 3, Lesson 2: Practical exercises in using models to predict trends and behaviors in economic systems.

Section 4: Health and Medicine Applications
- Lesson 1: Modeling Disease Spread
  - Module 5, Section 4, Lesson 1: An introduction to how mathematical models help understand epidemiological patterns.
- Lesson 2: Analyzing Treatment Outcomes
  - Module 5, Section 4, Lesson 2: Examining basic models used in medicine to simulate treatment responses and outcomes.

Section 5: Integrating Multiple Disciplines
- Lesson 1: Interdisciplinary Approaches to Modeling
  - Module 5, Section 5, Lesson 1: Discussing how models can combine insights from engineering, science, and economics.
- Lesson 2: Capstone Project Overview
  - Module 5, Section 5, Lesson 2: An introduction to a final project that integrates lessons learned from the course to solve a real-world problem.