Additional Math Pages & Resources

Monday, January 14, 2013

Rainbow Math

This double rainbow appeared outside our Excel Math office in San Diego after a brief rain. It actually seemed to be much closer and more brilliant in color than this photo shows.

A rainbow is sunlight spread out into its spectrum of colors and diverted to our eyes by raindrops. Read more from The Center for Atmospheric Research at:

Perhaps a simpler definition of a rainbow is a multicolored arc made by sunlight striking raindrops. When sun, shining from behind the viewer, strikes water droplets in front at just the right angle, a rainbow is produced.

A rainbow does not actually exist at a specific spot in the sky. Its location depends on where you're standing and on where the sun is located at that time. It's an optical illusion. The sun must always be behind the person seeing the rainbow, and the air must be both sunny and full of moisture.

You can see rainbows near other kinds of water, including mist, spray, dew, and waterfalls, as well as rain. Some scientists think rainbows may also appear on one of the moons of Saturn (called Titan).

A rainbow shows up as a spectrum of light—a band of colors that include red, orange, yellow, green, blue, indigo, and violet. Roy G. Biv is a popular name used to help students remember the order of the colors in the rainbow's spectrum.

Elementary school students sometimes use a song to help remember the order of the rainbow's colors (from top to bottom, sung to the tune of "Paw-Paw Patch" or "Ten Little Indians"). Sing the first line three times:
Red, orange, yellow, green, blue, purple;
I see a rainbow bright, bright, bright.
Red is on the outer part of the rainbow's arch, while violet is always on the inner section of the arch. Roy G. Biv includes indigo between blue and violet. Actually, the rainbow is a continuous band of colors from red to violet and even beyond the colors that the eye can see.

In a double rainbow, such as the one in our photo, water droplets reflect light twice. You can see faintly in the picture that the top rainbow has the colors of the spectrum reversed. It appears as a mirror image of the bottom colors with red on the bottom and violet on top. Read more about rainbows on

Your more advanced students may be interested in the path of one light ray incident on a water droplet:

As the light beam enters the surface of the drop at A from the direction SA, it is bent  or refracted a little and strikes the inside wall of the drop at B, where it is reflected back to C. As the light beam emerges from the drop it is refracted (bent) again into the direction CE. The angle D represents a measure of the deviation of the emergent ray from its original direction.

Descartes calculated this deviation for a ray of red light to be about 180 - 42 or 138 degrees. The ray shown here represents the ray that has the smallest angle of deviation of all the rays incident upon the raindrop. It is called the Descarte or rainbow ray. Much of the sunlight as it is refracted and reflected through the raindrop is focused along this ray. The reflected light is diffused and weaker except near the direction of this rainbow ray. Read more at

In Excel Math, students learn about rays and angles. We help them understand that the angle measure of the whole is the sum of the angle measures of the two parts. Try these exercises to teach your students how to measure angles. Click here to download Excel Math Grade 4 Lesson 70. The student page is available as part of the Individual Student Set (an entire year of lessons—155 lessons—for one student), available for purchase at

Reprinted from Excel Math Grade 4.
Click here to download Lesson 70

This slide from Projectable Lesson 70 shows how we define a "ray." Click here to download slides from Excel Math Projectable Lesson 70:
Excel Math Grade 4 Projectable Lesson 70Click here to download Grade 4 Projectable Lesson 70.

A future post will continue the rainbow math with a fun watercolor wash rainbow project.

Visit us at to learn more about Excel Math and see how these elementary math lessons can benefit your students. Still have questions? Take a tour of Excel Math or leave a comment in the box below.

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