How are rainbows formed? Why do they only occur when the sun is behind the observer? If the sun is low on the horizon, at what angle in the sky should we expect to see a rainbow?
This lab helps to answer these and other questions by examining a mathematical model of light passing through a water droplet. The contents include:
- How does light travel? - Reflection - Refraction - Rainbows: Exploration - Rainbows: Analysis - Conclusion
Objectives of the lab:
- to examine the use of Fermat's Principle of least-time to derive the Law of Reflection and the Law of Refraction
- to experimentally determine the angle at which rainbows appear in the sky
- to understand geometric properties of rainbows by analyzing the passage of light through a raindrop
- to apply a knowledge of derivatives to a problem in the physical sciences
- to better understand the relation between the geometric, symbolic, and numerical representation of derivatives
This lab from the Curriculum Initiative Project at the University of Minnesota is based on a module that was developed by Steven Janke and published in "Modules in Undergraduate Mathematics and its Applications" in 1992.
On August 24th, Gordon Spence, using a program written by George Woltman, discovered what is now the largest known prime number. The prime number, 2^(2976221)-1, is one of a special class of prime numbers called Mersenne primes; it is 895,932 digits long.
An introduction to prime numbers can be found in the Dr. Math FAQ:
A project designed to challenge middle school students with non-routine problems, and to encourage them to verbalize their solutions. Responses will be read and assessed and comments will be returned; incorrect solutions will be sent back with an explanation of the error and students will be urged to try again.
The problems are intended for students in grades 6-9 (ages 11-14), but may also be appropriate for students in other grades. A variety of problem-solving techniques are encouraged, including:
- guess and check - make a list - draw a picture - make a table - act it out - logical thinking - algebraic equations
A conversation about the merits of giving grades, student motivation and interest, the need for grades for college admission, what grading does to the grader, the illusion of absolute truth, the effects of praise in the classroom, GPA, SAT and IQ tests, and the effects of testing.
Grading vs. non-grading was first mentioned in the context of grade inflation by Michael Paul Goldenberg, during a discussion of "At-risk Algebra Students":