Background: The true length of a year on Earth is 365.2422 days, or about 365.25 days. We keep our calendar in sync with the seasons by having most years 365 days long but making just under 1/4 of all years 366-day "leap" years.

Exercise: Design a reasonable calendar for an imaginary planet. Your calendar will consist of a pattern of 366-day "leap" years and 365-day regular years that approximates your planet's average number of days per year.

**Grade Level:** High School (9-12)

**Curriculum Topic Benchmarks:**
M3.4.2, S17.4.1, S17.4.2, S17.4.3

**Subject Keywords:** Leap year, Year, Calendar, Error analysis, Successive approximations, Fractions

**Author(s):** Evan M. Manning

**PUMAS ID:** 04_21_97_1

**Date Received:** 1997-04-21

**Date Revised:** 1997-11-21

**Date Accepted:** 1997-09-05

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**Comment by Theodore Schultz on January 6, 2010**

"1. In discussing corrections to the Julian calendar, I think it is more understandable if one treats each successive correction in the average year-length explicitly and in the SAME UNITS (e.g. fractions of a day per 365-day year rather than leap days per 4 years, or 100 years, or 400 years, or 4000 years). Further, if one waits until the end to add up all these corrections, then their relative magnitudes, standing side by side, are also clearer. Thus the successive corrections to the length of the average year can be most simply viewed as follows:

__ Adding a leap year every 4 years adds 1/4 to the 365-day year or increases the average year-length by 0.25 days.

__ Omitting a leap year every 100 years subtracts 1/100 = 0.01 days from the 365-day year.

__ Adding a leap year every 400 years adds 1/400 = 0.0025 days to the 365-day year.

__ Omitting a leap year every 4000 years subtracts 1/4000 = 0.000 25 days from the 365-day year.

When these are combined, the corrected average year-length is 365 +0.25 - 0.01 + 0.0025 - 0.00025 = 365.24225 days/year. This not only avoids numbers like 97 but makes the effect of each correction relative to the others quite clear.

2. As I recall, the Mayan calendar gives an average year-length of 365.2420 (although this should be verified), in which case it is more accurate than the Gregorian, despite the very primitive tools and architecture possessed by the ancient Maya. How did the Maya accomplish this, or was it incredibly good luck? Higher-grade students might learn a lot from trying to answer this question."