| Euclidean Algorithm |
| Subject
Area |
Mathematics |
| Age
or Grade |
High School |
| Estimated
Length |
50 minutes |
| Prerequisite
knowledge/skills |
Students should be able to perform basic operations (addition, subtraction, multiplication, division) and have an understanding of basic Algebra. Students should also understand what the basic number systems are and how they differ. |
| Description
of New Content |
Students will be introduced to the concept of a Greatest Common Divisor. This may not be new to them, but this deterministic method for finding the gcd of two numbers will be new. |
| Goals |
Students will develop their analytic skills, practice using an algorithm and enhance their understanding of how useful math can be. |
| Materials
Needed |
Open Minds. |
|
Procedure |
Basically follow the hand out yourself but make sure you explain every detail thoroughly... make some additional notes for yourself so the students will need to fill something in as they follow you through the paper. Opener: Beggin with refreshing the students minds about the above mentioned number systems. Explain how the entire field of Number Theory primarily uses the integers. Tease their minds with some examples of how something simple is harder than it appears. (Goldbach's Conjecture, Fermat's Last Theorem). Explain the Fundemental Theorem of Algebra (all integrers have a unique facatorization into powers of primes up to ordering). Make sure they know what a prime number is and that they can name the first few. Development: Take them through the explanation of what a gcd is and how it is useful to know (simplifying fractions, factorization). Have the students practice some problems with you and by themselves. Helpful for sutdetns to try some simple ones on the board, that way the seated students can help and practice the problem as well. Closer: Use a simple computer program to show the students that the Euclidean Algorithm works for very large numbers. Entice them to return the next day when you will discuss Diophantine Equations. |
| Evaluation |
Via "Dip-Sticking" throughout the lesson. Homework assignment accompanying this lesson and in-class worksheet questions. |
| Extensions |
Continue with Diophantice equations since they utilize the Euclidean Algorithm to solve. If time permits, move on to modular arithmetic and ecryption technology. |
| References | Courseowork at Boston University |