Session September 10, 2008
Phil presented The Locker Problem as stated in the NCTM resource: Activities for Junior High School and Middle School Mathematics, Volume 2.
A high school has 1000 students and 1000 lockers. Each year as an end-of-school celebration, the students line up in alphabetical order and perform the following ritual: The first student opens every locker. The second student goes to every second locker and closes it. The third student goes to every third locker and changes it (i.e. opens it if it was closed, or closes it if it was open). Similarly, the fourth student changes every fourth locker, and so on. After all 1000 students have passed by the lockers, which lockers are open?
The Locker Problem is a well known exploration that exposes numerous number patterns appropriate for middle school mathematics. Factors, prime numbers, least common multiples and relative primes are useful in the solution to the problem. Numerous extension questions delve deeper into numerical relationships. Students are introducted to subscript notation. Although many teachers were familiar to the basic problem, Phil's extensions and questioning skills provided a model for exquisite classroom exploration.
Dr. Grizzard also introduced us to Five-Cycle Number Patterns . Beautiful patterns can be used to practice fractions, decimals, and integers while designing interesting cycles.
Click here to access the problems used in this session.
Some of the AIMS Standards addressed include:
1.1.2 Use prime factorization to express a whole number as a product of its prime factors and determine the greatest common factor and least common multiple of two whole numbers.
1.2.2-4 Multiply and divide whole numbers, decimals, and fractions.
1.2.6 Apply the commutative, associative, distributive and identity properties to evaluate numerical expressions involving whole numbers.
1.2.7 Simplify numerical expressions using the order of operations.
5.2.1 Analyze a problem situation to determine the question to be answered.
5.2.2 Identify relevent, missing, and extraneous information related to the solution of a problem.
5.2.3 Analyze and compare mathematical strategies for efficient problem solving; select and use one or more strategies to solve a problem.
5.2.6 Communicate the answer to a question in a problem using appropriate representations, including symbols an informal and formal mathematical language.
