# Daylight In the Northern Hemisphere (Modeling Module)

### Author: Marsha Davis and Floyd Vest

**Teacher notes**

The focus of this Modeling Module is using sinusoidal models to estimate the length of daylight and the sun's declination and altitude over time in various locations in the Northern Hemisphere. In Activity 1, students work with a data set that consists of day number and length of daylight for Boston, Massachusetts. Based on these data, they determine a sinusoidal model of the form

f (x)=Asin(B(x-C))+D

for estimating length of daylight given day number, x. They compare their model to one determined using a graphing calculator's regression capabilities and must grapple with an output from regression that specifies models in an alternate form, asin(bx + c) + d. In Activity 2, students determine models for estimating length of daylight at various locations using sunrise/sunset data from only two days, the winter and summer solstices. In an effort to understand Seasonal Affective Disorder (SAD), students use their models to estimate the length of daylight and instantaneous rate of change in daylight for specific dates. When graphing models in Activities 1 and 2, calculators must be in radian mode. In Activity 3, attention shifts to models that estimate solar declination (in degrees) as seen from Earth and the elevation (in degrees) of the sun above the horizon. These models involve both sine and cosine functions.

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