Actually Applicable Application Problems and Brainteasers/Rain Accumulation

Overview

Students' task with these graphs is to estimate accumulation by sketching rectangles, multiplying to find each rectangle's area, and adding the rectangles' area to find the total area under the curve. (Technically this is integral calculus, though at a level to be accessible even to elementary students.)

This is an application, not just a brainteaser. The total amount of rain that has fallen or is expected to fall can be important for real world purposes like water conservation and flood safety.

General Method

1. Mark the endpoints of the interval of interest.
2. Split the interval into equal subintervals. (How many? The problem may say or you may decide.)
3. Going up from each interval, draw a rectangle whose height is given by a point on the graph. (Should you use the left endpoint, right endpoint, or midpoint or the subinterval? The problem may say or you may decide.)
   • The problem statement may use abbreviations for the number of rectangles and which point to use for the rectangle's height.
   • For example, L₆ means you should use left endpoints of six rectangles, M₃ means you should use midpoints of three rectangles, and R₈ means you should use right endpoints of eight rectangles.
4. Multiply to find each rectangle's area
   • If you know how to calculate the area of a trapezoid, you may also use a trapezoid where one side goes between the two endpoints (so is usually slanted) and the other three sides are like in the rectangles.
   • For example, T₄ would mean to find the total area of four trapezoids.
5. Add to find the total area of the rectangles. This is an estimate for the total accumulation of rain.

Problems

Situation 1

This graph is from Weather Underground predictions for a period of rain that was expected to take place in Lufkin, TX overnight from May 2-3, 2019, as accessed on the afternoon of May 2, and plotted in Google Sheets as a "smooth line chart" with the "aggregate column B" [i.e. the times] option checked.

Estimate the total precipitation predicted during this rainy period (that is, from 6 PM to 1 AM).

Make Your Own Problem

On a rainy day, use a rain gauge or straight-sided glass and a ruler to record amounts of rain at regular intervals. Then make a graph and use the same calculation strategies to find the total amount of precipitation.

You can also use weather data someone else recorded such as from Weather Underground.