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
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