- Author:
- Andre Douglas
- Subject:
- Geometry
- Material Type:
- Homework/Assignment
- Level:
- Middle School
- Tags:

- License:
- Creative Commons Attribution Non-Commercial Share Alike
- Language:
- English
- Media Formats:
- Text/HTML

# Education Standards

# Why is looking at an iguana different from looking at a parrot fish? Pythagoras has an answer....

## Overview

This middle school geometry lesson is submitted by Andre Douglas, Secondary Mathematics Teacher in the Virgin Islands Department of Education. Mr. Douglas extends this lesson beyond geometry to help students gain an appreciation of the fish and wildlife in this US territory. Teachers using this lesson plan can extend this lesson through information provided by the National Park Service website: https://www.nps.gov/viis/learn/kidsyouth/queen_parrotfish.htm

# Lesson Plan submitted by Andre Douglas, Secondary Mathematics Teacher, Virgin Islands Department of Education

**Light travels pretty fast. **In fact, in air, light travels 30 cm (or about 1 foot) in a billionth of a second. One billionth of a second is called a nanosecond. Thus we can write that the speed of light is 30 cm/nanosecond or 1 ft/nanosecond, depending on which measurement system we are using. Suppose lightning flashes 3 km away (3000m, so that’s 300,000 cm).

Remembering the formula D=RT or T = D/R, we can compute that it is only 300,000/30=10,000 nanoseconds or about one hundred-thousandths of a second before we see the flash. Sound travels a lot slower, so we wouldn’t hear the thunder until nearly 10 seconds later.

Suppose you are looking at an iguana that is 6 feet below you and 8 feet from you, as in the following figure. Use the Pythagorean Theorem to compute the distance *x *from the iguana to your eyes. The reason we see the iguana is that light travels from the iguana to our eyes. Use the formula T=D/R to compute the number of nanoseconds it takes light to travel from the iguana to your eyes.

Light travels slower in water than in air, just like us. In fact, the speed of light in water is only 22.5 cm/nanosecond or 0.75 ft/nanosecond. Suppose you are swimming and you see a Parrot fish, 6 feet below you and 8 feet away, as in the diagram below. Compute how many nanoseconds it takes light to travel from the Parrot fish to your eyes. Notice that it takes several more nanoseconds than it took you to see the iguana. Big deal. We don’t notice the difference of a few billionths of a second. Or do we?

Have you ever noticed that when you look into water, objects aren’t where they appear to be? One way to see this is to stick a pencil halfway into a glass of water. It will appear that the pencil is bent. Why is this? The answer is because light travels at a different speed in air and in water, and that light travels the path that takes the least amount of time.

Suppose you have a container of water that is 12 cm deep, and there is a mark on the bottom of the container. If you look at the mark from a point that is 50 cm away and 20 cm above the water level, there are many paths from the mark to your eyes. One such path is seen in the following figure. But light chooses the path that takes the least time. Use the Pythagorean Theorem to compute *w*, how far the light traveled through the water. Use it again to compute *a*, how far the light traveled through the air. Use these distances to compute the number of nanoseconds it took the light to travel through the water and the number of nanoseconds it took the light to travel through the air. Give each answer to 4 decimal places. How long would it take the light to travel that path from the mark to your eyes, to 4 decimal places?

Compute the length of time, to 4 decimal places, it would take light to travel the path seen in the next figure. Compare your answer to the previous answer.

Pick your own path and determine the length of time it will take light to travel that path from the mark to your eyes. Compare this answer to the other 2 answers.

Compute the time it takes light to travel the path seen in the next figure. That time will be smaller than the time it takes to travel any other possible path. Light actually travels the path that takes the least amount of time. So this is the actual path light will take.

From your perspective, the mark will seem to be further away than it actually is. In fact, it will appear to be where the fake mark is seen in the next figure.

Remark: Since light travels slower in water than in air, it tends to take a short path through water and a longer path through air. In fact, no matter where you observe the underwater mark from, the light will never take a path that exits the water 14 cm or more from above the mark. Thus, light traveling such a path will reflect back into the water. See the following figure. This means that when you are underwater looking up, you can only see outside of the water that is (nearly) directly over your head.