| Modular Arithmetic |
| 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 Modular Arithmetic, which is the arithmetic within the realm of a new type of number system. The students will learn about "mod m" and then how it creates a number system of its own. |
| Goals |
Students come to understand that there are number systems beyond the basic few and that as mathematicians they have the ability to create new systems and numeric worlds. |
| Materials
Needed |
Open Minds and Curiosity. Hand Out for students to follow during class. |
|
Procedure
|
Opener: Start talking about binary (0's and 1's). Ask that students if they know anything about it. Make sure that the discussion covers that computers "speak" in binary, computers use 1 = true and 0 = false for logic questions and that 1 + 1 = 0 (two truths make a false). Development: Elaborate by showing them the notation of (mod m) or (mod 2) in the case of binary. Explain what mod = modulo is and how it is used. Continue by moving into (mod 3), (mod 5) and (mod 7) (or other primes). By using prime numbers, addition and multiplication are preserved. If a non-prime is used, then only addition will work. Closer: Play a game with them. Have them pick a (small) prime or non-prime, call it n, and then pick two numbers between 0 and n. Then start adding them like in the Fibonacci series (add the last two in the list to determine a new number in the list). Make sure that they are using Modular Arithmetic!! How long until you come back to the original two numbers? See who can make the smallest and longest list for different numbers n. |
| Evaluation |
Class-participation |
| Extensions |
Move onto conversation about Groups and Rings and Axiomatic systems. |