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Now You See It, Now You Don't

Module by: Interactive Mathematics Program

Intent

Students will use the principle that the angle of approach equals the angle of departure to analyze the line of sight to an object through a mirror.

Mathematics

When viewing the reflection of an object in a mirror, the line of sight follows the principle that the angle of approach is equal to the angle of departure. In this activity, students will locate these lines of sight and note that similar triangles are created.

Progression

Students investigate the two situations in this activity individually and share their results in class.

Approximate Time

15 minutes for activity (at home or in class)

15 minutes for discussion

Classroom Organization

Individuals, followed by whole-class discussion

Materials

Protractors

Rulers

Doing the Activity

Have students read the first question, and ask what they initially think might be the answer. Many will assume that only one letter can be seen.

Discussing and Debriefing the Activity

Discuss students’ findings. Question 1 should be fairly straightforward, with students connecting point A to each end of the mirror and drawing the reflection lines.

For Question 2, ask, How did you get your point in Question 2? Watch for students who assume that the halfway point must be the point of reflection.

Which similar triangles are involved? How do you know the triangles are similar?

Students should be able to identify the similar triangles RST and VUT in a diagram like the one below (with spiders at points R and V) and prove their similarity by virtue of two equal angles: the pair of right angles as well as angles RTS and VTU, which are equal by the principle of light reflection.

Figure 1
Figure 1 (graphics1.jpg)

What fraction of the way from point S to point U is point T? Try to elicit the explanation that, because of the similar triangles, the lengths ST and TU must be in the same ratio as RS and VU.

Key Questions

How did you get your point in Question 2?

Which similar triangles are involved? How do you know the triangles are similar?

What fraction of the way from point S to point U is point T ?

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