# Proximity: Modeling Rainfall

### Author: Marsha Davis

The context for this Modeling Pull‑Out is estimating the amount of rainfall in a certain region based on rain gauges spread across the region.

This Pull‑Out begins with a Preliminary Reading, *Colorado Needs Rain Until It Gets Too Much*, which introduces excerpts from news articles citing the amounts of rainfall in different locations in Colorado. This reading proposes a modeling question of how to measure rainfall amounts over an entire region as opposed to a single city. The Preliminary Activity, *Measuring Rainfall*, asks students where to place eight rain gauges in Colorado to estimate the amount of rainfall for the entire state. When the locations of the rain gauges are changed, students consider how this change might affect their model for estimating rainfall amounts.

Activity 1, *Rainfall Averages*, begins with converting units of measurement for depth of rainfall, areas of regions, and volume of water. Given data from rain gauges that represent different‑sized subregions, students discover that a weighted average may be a better model for estimating the depth of rainfall than a simple arithmetic average.

In Activity 2, *Dividing into Subregions*, students decide how to divide regions into subregions so that all points within each subregion are closer to the rain gauge within that subregion than to rain gauges outside that subregion. Once they have created the subdivisions, they use weighted averages to estimate rainfall in the entire region given rain gauge readings from each subdivision.

Activity 3, *Voronoi Diagrams*, introduces basic Voronoi diagrams terminology. In addition, students develop of method for accurately constructing Voronoi boundaries and discover some basic properties of the Voronoi boundaries.

In Activity 4, *Searching for a Model*, students develop an algorithm for drawing Voronoi diagrams where there are three centers of influence. They expand their procedure to four centers of influence. The activity ends with a final project in which students draw a Voronoi diagram for eight rain gauges in Colorado. Then they find weights for their model and apply it to estimate rainfall depth based on the rain gauge data from the eight rain gauges.

**Prerequisites **

Students should understand the concepts of depth, area, and volume. They should be familiar with finding the average (or mean) of a group of numbers. They should have experience finding angles made by intersecting lines and know what it means for two lines to be perpendicular.

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