Problem Solving Guide

Mathematical Modeling · Axiom Academy

A comprehensive framework for tackling mathematical modeling problems systematically. Step 1: Understand the Problem Read Carefully: Identify what quantity you need to find or relationship you need to establish List Known Information: Write down all given data, constraints, and conditions Identify Variables: Determine what changes (dependent variables) and what controls the change (independent variables) Draw a Diagram: Visualize the situation with labeled sketches whenever possible State the Goal: Write a clear statement of what the model should predict or explain Simplify Complexity: Identify which factors are most important and which can be ignored Be Explicit: Write down every assumption you make, no matter how obvious Start Simple: Begin with the simplest reasonable model; you can add complexity later Check Reasonableness: Ensure assumptions do not contradict known facts or each other Document Limitations: Note when and why your assumptions might break down Choose Model Type: Select discrete (difference equations) or continuous (differential equations) based on the problem Write Equations: Express relationships mathematically using conservation laws, rate equations, or balance principles Check Dimensions: Verify all terms have consistent units using dimensional analysis Identify Parameters: List all constants and their physical meanings Set Initial/Boundary Conditions: Specify starting values or edge constraints

This is the written version of the interactive lesson above. See the full Mathematical Modeling course.