Intent

Students return to the sun shadow problem with a goal of identifying the key variables in the situation. The sun shadow problem will actually be solved in the activity A Bright, Sunny Day.

Mathematics

A shadow cast by the sun differs from one cast by a lamp in an important way, which students observed early in this unit: the sun shadow doesn’t change length as the object casting the shadow moves. This is because the source of the light is, in effect, infinitely far away. The length of a sun shadow is a function of just two variables: the height of the object casting the shadow and the angle of elevation of the sun.

Progression

Students discuss the sun shadow situation as a class, sharing ideas and working to identify the key variables in the model.

Approximate Time

25 minutes

Classroom Organization

Whole class

Doing the Activity

As a class, read the activity in the student book.

Discussing and Debriefing the Activity

Ask students to share their ideas on this situation. They should recognize that the height of the object casting the shadow, which has been labeled H previously, is still an important variable.

They might also mention time of day and position on the globe as variables that could affect the length of a sun shadow. Ask, Why might “time of day” or “position on the globe” affect the length of a sun shadow? Try to get students to see that these things affect the angle at which light from the sun hits an object. The goal is for them to recognize that the angle of the sun’s position is the other crucial variable.

It may help to bring out that at noon, sun shadows are at their shortest, and that toward dusk or shortly after dawn, sun shadows are at their longest.

If students have trouble picturing the desired angle, suggest that they look straight ahead and then tilt their heads as if looking up toward the sun. The amount of “tilt” is the angle in question.

Culminate this discussion by introducing a diagram similar to the one below. Introduce the term angle of elevation for the “tilt” represented by θ (the lowercase Greek letter theta) in this diagram. Students should recognize that they are looking for a way to express S as a function of H and θ. That is, they want a formula for a function g such that S = g(H, θ).

Figure 1

Key Question

Why might “time of day” or “position on the globe” affect the length of a sun shadow?