Lesson Plan (abridged)
Title of lesson: Exploration of Geometric Relationships in a Triangle: Locating a Stadium in North Carolina's Research Triangle
Audience: 9th/10th Grade (Geometry/Pre-AP Geometry/AP Geometry)
Content Objectives: Determine the centroid, circumcenter, incenter, and orthocenter in preparation of designating a location site for a new baseball stadium out of the state of North Carolina. Using three cities (Raliegh, Durham, and Chapel Hill) that makes a research triangle, students will determine the best location based on the most cost-effective position within the triangle. Also, apply coordinate geometry and measurement tools within GeoGebra to support mathematical reasoning.
2020 Alabama Course of Study (Mathematics):
Mathematical Practices:
Establish mathematical goals to focus learning (MP1).
Use and connect mathematical representation (MP3).
Facilitate meaningful mathematical discourse (MP4).
Pose purposeful questions (MP5).
Build procedural fluency from conceptual understanding (MP6).
Support productive struggle in learning mathematics (MP7).
Geometry Standards:
G.16: Use coordinates to compute perimeters of polygons and areas of triangles and rectangles.
G.17: Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems.
G.21: Use geometric constructions and modeling to solve design problems.
Behavioral Objectives:
Students will:
Construct and identify the centroid, circumcenter, incenter, and orthocenter of a triangle using GeoGebra.
Use mathematical modeling and cost analysis to evaluate the most cost-effective stadium location among the triangle centers.
Justify their stadium site selection using geometric reasoning, proof, and real-world data (highway construction/resurfacing costs).
Prerequisites:
Familiarity with triangle types and basic constructions (midpoint, perpendicular bisector, etc.)
Basic coordinate geometry (distance formula, slope)
No prior modeling or software experience required; will be taught in-session.
Materials: GeoGebra (Classic) software; Simplified map(s) of the research triangle; Laptop; Internet access
Procedure:
Introduction:
Display the problem and research triangle.
Pose the question of “If you were asked to place a new professional baseball stadium somewhere between Raleigh, Durham, and Chapel Hill, where would you put it and why?”.
Engagement:
Define the terms (centroid, circumcenter, etc.)
Slowly work each configuration by introducing various components of GeoGebra:
Plot triangle vertices
Construct each triangle center
Label each center clearly
Lecture/Direct Instruction:
Correct their mistakes and clarify certain subjects/topics.
Introduce the mathematical modeling of cost-effectiveness via highways mileage:
New highway construction: $125,000 per mile
Highway resurfacing: $50,000 per mile
Discussion & Application:
Compute calculations for distances between each point to each city.
Classify which roads need to be built vs resurfaced
Work in pairs or groups to compare findings and justify their choices
Conclusion:
Students individually or in groups:
Recommend one stadium location
Justify with geometric reasoning and cost analysis
Present to class or write a brief report
Key Questions:
What makes each triangle center (centroid, incenter, etc.) unique in terms of its geometric properties?
Why might the centroid not be the most cost-effective location, even though it's the “balance point” of the triangle?
How does the circumcenter being equidistant from all vertices affect the location decision?
How do we determine which roads need to be built versus resurfaced from each center?
How can we use the distance formula to compare travel distances accurately?
What are some real-world consequences of choosing a location that’s cheaper but less accessible to the population?
Can this kind of modeling be used in disaster planning, emergency response, or delivery logistics?