Mathematical Modeling &Integrated Problem Solving

Module 1: The Language Revolution: From Arithmetic to Algebra

• Core Theme: Arithmetic Thinking vs. Algebraic Thinking

• Key Content:

  1. Language Translation Training: Precisely converting word problems into algebraic language (equations, inequalities).

  2. Unified Model Building: Establishing general algebraic frameworks for classic problems: “Motion (Meeting/Chasing)”, “Work (Cooperation/Rate)”, “Profit & Loss (Allocation/Comparison)”.

  3. Project Practice: “Plan a Class Trip Budget & Itinerary” – using systems of equations to solve for total costs (transport, tickets, food) and per-person shares.

Module 2: Change & Relationships: Entering the World of Functions

• Core Theme: Identifying and describing patterns of change.

• Key Content:

  1. From Table to Relationship: Analyzing data to identify dependencies, distinguishing functions from non-functions.

  2. Linear Function Modeling: Investigating tiered pricing problems like “Mobile Phone Plans (Monthly Fee + Data)” and “Taxi Fares”.

  3. Project Practice: “Analyze Household Monthly Utility Data” – creating line graphs and discussing trends and possible causes.

Module 3: The Art of Optimization: Quadratic Function Modeling

• Core Theme: Finding maxima/minima to solve optimization problems.

• Key Content:

  1. Optimization in Parabolas: Understanding the real-world meaning of vertex coordinates (max profit, min cost, max range).

  2. Real-World Modeling: “Maximizing Rectangular Area with Fixed Fencing”, “Product Pricing vs. Sales Volume (Bulk Discount Model)”.

  3. Project Practice: “Design and Simulate the Optimal Launch Angle for a Parabolic Track & Field Event (e.g., Javelin)” using GeoGebra for dynamic demonstration.

Module 4: Geometry as a Measurement Tool: Congruence & Pythagoras

• Core Theme: From Proof to Application.

• Key Content:

  1. Principles of Indirect Measurement: Using congruent triangles to replicate inaccessible distances or angles.

  2. Field Applications of the Pythagorean Theorem: Measuring building heights, diagonal lengths of a field.

  3. Project Practice: “Create a Precise Scale Drawing of an Irregular School Area (e.g., Garden Bed)” – integrating measurement and scale.

Module 5: A World of Proportion & Scale: Similarity Models

• Core Theme: Solving “Inaccessible Measurement” Problems.

• Key Content:

  1. Similar Triangle Modeling: Using shadows/mirrors to measure height, using poles to measure river width.

  2. Maps & Scale: Calculating real-world areas and distances, understanding the area ratio in scaling.

  3. Project Practice: “Build a 3D Scale Model of Our School Campus” – determining the scale and calculating material needs.

Module 6: Circles & Loci of Motion

• Core Theme: Circular Paths and Positioning in Reality.

• Key Content:

  1. Applications of Circle Properties: Wheel motion, maximum storage in circular silos, satellite signal coverage.

  2. Circles in Coordinates: Locating points that satisfy distance conditions (equidistant from a fixed point).

  3. Project Practice: “Design a Lighting Plan for a Circular School Plaza” to ensure full coverage.

Module 7: The Power of Data: Statistics & Investigation

• Core Theme: Evidence-Based Decision Making.

• Key Content:

  1. The Complete Statistical Process: Questionnaire design, sampling, data organization (using spreadsheets), choosing and creating charts.

  2. Data Analysis: Calculating and interpreting the contextual meaning of mean, median, mode, and variance.

  3. Project Practice: “Conduct a Survey on ‘Students’ After-School Mobile Phone Usage Habits'” and write a brief report with actionable recommendations.

Module 8: Probability & Risk Assessment

• Core Theme: Prediction and Decision Making.

• Key Content:

  1. Probability Calculation: Analyzing the fairness of simple games (raffles, dice rolls).

  2. Simulation Experiments: Verifying theoretical probability through large numbers of random trials (using programming for simulation).

  3. Project Practice: “Design a Fair Lottery or Tournament Bracket for a Class Activity”.

Module 9: Interdisciplinary Modeling Challenge

• Core Theme: Mathematics as a Universal Tool.

• Key Content:

  1. Math in Physics: Uniform motion (s=vt), density & mass, relationships in simple circuits.

  2. Math in Economics: Simple vs. compound interest basics, discount & profit margin calculations, cost-benefit analysis.

  3. Math in Geography: Latitude & Longitude (coordinates), time zone calculation, scale and contour map interpretation.

  4. Integrated Project: “Plan a ‘One-Day City Tour'” – optimizing budget, route (coordinates & distance), and schedule.

Module 10: Capstone Project: From Problem to Solution

• Core Theme: End-to-End Integrated Practice.

• Key Content:

  • Capstone Project Launch: e.g., “Optimize the Layout and Operation of School Waste Sorting & Recycling Stations”.

  • Task Breakdown: Student groups must: define the problem and collect data (statistics), conduct spatial measurement and planning (geometry), perform cost-benefit analysis (algebra & functions), visualize the solution (charts/graphs), and present a final report.

  • Final Presentation & Defense.