Modeling the Best Location for Heli-Hydrants and Fire Stations

The context for this Pull-Out is modeling the placement of heli-hydrants and fire stations for use in fighting fires.

The Pull-Out begins with a Preliminary Reading, Using Helicopters as an Aid in Fighting Wildfires, which reports on California wildfires and the use of heli-hydrants in combating them. In the Preliminary Activity, Deciding on the Best Locations for Heli-Hydrants, students brainstorm criteria to consider in deciding where to install heli-hydrants. 

In Activity 1, Keeping Your Distance, students are introduced to two different measures of distance: helicopter distance and firetruck distance, which differ because fire trucks need to travel on streets. Students explore different criteria for determining the “best” location for installing a heli-hydrant or building a are station and and that different criteria can lead to different solutions.

In Activity 2, Developing Linear Region, students simplify the modeling problem by restricting the region to a one-dimensional number line. They then solve the modeling problem of finding the best location for a heli-hydrant under two criteria: minimizing the total distance from heli-hydrant to each fire zone center and minimizing the maximum distance from the heli-hydrant to the firre zone centers. 

In Activity 3, Returning to Two Dimensional Maps, students attempt to apply what they have learned from working with a one-dimensional region to two dimensions. Whether this is possible depends on both the criterion chosen for "best" and the geometry–firetruck distance or helicopter distance. Students are introduced to the geometric median and to the minimum enclosing circle problem (also known as the mini-max circle problem).

Some of the material for this Modeling Pull-Out was adapted from Chapter 1: Gridville, Course 2, Mathematics Modeling Our World, 2nd edition.

Note: The information below was created with the assistance of AI.

Level of Mathematics

The module is primarily designed for upper middle school to early high school mathematics, especially courses that include algebra, geometry, and mathematical modeling.

Estimated grade level:

• Grades 8–10 (most appropriate)
• Could also be used in Algebra I, Geometry, or Mathematical Modeling courses.

Why this level:

• Requires understanding of coordinate geometry.
• Uses absolute value and piecewise functions.
• Introduces optimization and modeling concepts.

Application Areas (Real-World Context)

The module applies mathematics to real-world infrastructure and emergency planning problems.

Main Application Areas

1. Emergency management / fire protection planning
• Determining optimal locations for fire stations and heli-hydrants.
2. Urban planning and public safety
• Efficient placement of emergency resources in populated regions.
3. Transportation networks
• Modeling travel distance along streets vs straight-line distance.
4. Geographic analysis
• Spatial decision-making using coordinate grids.
5. Operations research / optimization
• Minimizing distance, cost, or response time.

These contexts help students understand how mathematics informs public policy and engineering decisions.

Prerequisites

The teacher notes specify several prerequisite mathematical skills.

Conceptual Prerequisites

Students should understand:

• Distance concepts
• Distance on a number line
• Distance between two points in the plane
• Coordinate systems
• Reading and plotting points on a grid
• Absolute value
• Particularly for distance on a number line
• Median of a dataset

Recommended Technical Tools

• Excel or spreadsheets
• Graphing calculator (TI-84)
• GeoGebra for geometry exploration

Subject Matter (Mathematical Content)

The module integrates several mathematical topics across algebra, geometry, and modeling.

Algebra Topics

• Absolute value functions
• Piecewise functions
• Graphing functions
• Linear relationships
• Tables and graphs

Geometry Topics

• Distance formulas
• Perpendicular bisectors
• Midpoints
• Circles
• Coordinate geometry

Statistics Concepts

• Mean (average)
• Median

Mathematical Modeling / Optimization

• Minimizing total distance
• Minimizing maximum distance (minimax)
• Geometric median
• Minimum enclosing circle problem

Distance Metrics

Students compare two geometries:

1. Euclidean distance (helicopter distance)
   Straight-line distance.
2. Manhattan distance (firetruck distance)
   Distance along grid streets.

Major Mathematical Ideas in the Module

The lesson develops several big mathematical ideas:

1. Optimization

Students determine the best location based on different criteria.

2. Modeling Assumptions

Simplifying real-world problems (e.g., reducing to a number line).

3. Multiple Solutions

Different criteria produce different optimal locations.

4. Geometric Reasoning

Using medians, bisectors, and circles to solve spatial problems.

5. Algorithmic Thinking

Students apply Weiszfeld’s algorithm to estimate the geometric median.

Correlation to Mathematics Standards

This module aligns well with Common Core State Standards (CCSS) and modeling standards used in U.S. high school mathematics.

CCSS Mathematical Practices

The module strongly supports:

1. MP1: Make sense of problems and persevere in solving them
2. MP2: Reason abstractly and quantitatively
3. MP4: Model with mathematics
4. MP5: Use appropriate tools strategically
5. MP7: Look for structure

Relevant Common Core Content Standards

High School – Modeling

• HSM (Modeling)
  Apply mathematics to solve real-world problems.

Algebra

• A-CED: Create equations to represent relationships.
• A-REI: Solve equations and analyze relationships.

Functions

• F-IF: Interpret functions and graphs.
• F-BF: Build functions modeling relationships.

Geometry

• G-GPE: Coordinate geometry.
• G-CO: Geometric constructions.

Statistics

• S-ID: Summarizing and interpreting data (mean and median).

Interdisciplinary Connections

This module also connects mathematics to:

• Geography
• Environmental science
• Emergency response systems
• Public infrastructure planning
• Computer science (algorithms)

Instructional Goals of the Module

The lesson aims to help students:

• Understand different measures of distance
• Develop optimization strategies
• Use graphs and models to make decisions
• Apply mathematics to real-world planning problems
• Interpret mathematical results in practical contexts