Friday the Thirteenth

by: Robyn Bryant / Kaycie Maddox / Kelli Nipper

Problem Solving is experiencing how mathematics is related to everyday life. According to the NCTM Standards for mathematics, students should experience a variety of mathematical ideas through problem solving. Because it fosters motivation for learning, students will become involved in the learning process. Also, learning to work non routine problems will give them confidence to solve problems in their own lives.

The development of mathematical power for all students can be a difficult tasks. The NCTM Standards suggest a couple of strategies to accomplish this. First of all, selecting interesting and challenging problems will capture their interest. Building on previous problems or knowledge will help students find connections. Employing cooperative learning experiences will enable students to discuss strategies and solutions.

An example of a problem that supports the NCTM Standards would be the "Friday the Thirteenth" problem:

Prove that there is at least 1 Friday the 13th and at most 3 Friday the 13ths in any given year.

This problem explores a non routine life situation. Working in groups, the students can discuss the problem and their solution(s). Because the problem is interesting, and easily understood, students can build their confidence.

On first reading any problem, it is important to understand the problem. Looking through recent calendars, students can check that these characteristics are true for a few specific years.

After becoming convinced that these two statements were correct for some years, the next step in the process is to plan. "If you fail to plan, then you are planning to fail!" This doesn't mean that you must stick with your original plan, but that it is important to be focused as you begin to tackle the logic involved in solving the problem.

One plan is to list the months that the thirteenth fell on a Friday for each year beginning on a different day of the week.

Non-Leap Years

 1st Day of the Year Friday the 13th in: Monday July / April Tuesday September / December Wednesday June Thursday February / March / November Friday August Saturday May Sunday January

Leap Years
 1st Day of the Year Friday the 13th in: Monday September / December Tuesday June Wednesday March / November Thursday February / August Friday May Saturday October Sunday January / April / July

From these charts, all options are covered revealing that regardless which day of the week a year starts on, or whether or not it is a leap year, every year has some combination of 1, 2, or 3 months where the 13th is a Friday. Since the least amount of times that there are Friday the 13ths in any given year is 1, and the most being 3, every year has at least 1 and at most 3 Friday the 13ths.

Students may decide to approach this from a different angle. Listing out the day of the year that the 13ths of each month fell on starting with January 1st as day 1 and December 31st as day 365 or day 366 (for leap years).

 Month Day of the Year Day of the Leap Year January 13 6 mod 7 13 6 mod 7 February 44 2 mod7 44 2 mod7 March 72 2 mod 7 73 3 mod 7 April 103 5 mod 7 104 6 mod 7 May 133 0 mod 7 134 1 mod 7 June 164 3 mod 7 165 4 mod 7 July 194 5 mod 7 195 6 mod 7 August 225 1 mod 7 226 2 mod 7 September 256 4 mod 7 257 5 mod 7 October 286 6 mod 7 257 0 mod 7 November 317 2 mod 7 318 3 mod 7 December 347 4 mod 7 348 5 mod 7

Using Modular Arithmetic, it is easy to see that all 7 possible results are found in Leap and Non-Leap Years. Also, no year contains more than 3 months in

There are many approaches and strategies to use with different problems. When working in groups, each person generally has something unique to contribute. Class discussions provide a chance for students to share and learn the diverse ways that they may have approached solving a problem. Once you build a diversity of strategies, you can solve many other problems. Class discussions help build metacognitive awareness. It not only helps students to see different perspectives, but also to understand what they did.