## Problem Set

# Leap Day Math

### Overview

Every four years, during a leap year, the imperfect match between the length of a calendar year and Earth's orbit result in an interesting calendar adjustment known as a leap day. In this problem set, students calculate the difference between the calendar year and Earth's orbital period, determine how much extra time gets added to our calendar and identify which years omit leap years.

### Materials

### Management

- This problem set is ideal for before, during or after leap day, but it can be used at any time of any year.
- Students can complete the problem set independently, in pairs or small groups, or together as a class, depending on student abilities.

### Background

The length of a year is based on how long it takes a planet to revolve around the Sun. Earth takes about 365.2422 days to make one revolution around the Sun. That's about six hours longer than the 365 days that we typically include in a calendar year. As a result, every four years we have about 24 extra hours that we add to the calendar at the end of February in the form of leap day. Without leap day, the dates of annual events, such as equinoxes and solstices, would slowly shift to later in the year, changing the dates of each season. After only a century without leap day, summer wouldn’t start until mid-July!But the peculiar adjustments don't end there. If Earth revolved around the Sun in exactly 365 days and six hours, this system of adding a leap day every four years would need no exceptions. However, Earth takes a little less time than that to orbit the Sun. Rounding up and inserting a 24-hour leap day every four years adds about 45 extra minutes to every four-year leap cycle. That adds up to about three days every 400 years. To correct for that, years that are divisible by 100 don't have leap days unless they’re also divisible by 400. If you do the math, you'll see that the year 2000 was a leap year, but 2100, 2200 and 2300 will not be.

### Procedures

- Discuss the approaching leap day, ask students if they know why or how often we have leap days/years. Optionally, have students read What is a Leap Year?
- Explain the reasoning behind the addition of a leap day every four years.
- Have students complete problems from the student worksheet in a manner appropriate for their abilities. (See Management.)
- Earth revolves around the Sun in approximately 365.2422 days. How long is that in days, hours, minutes and seconds?
- After the Earth orbits the Sun four times, how many extra hours, minutes and seconds need to be balanced out by a leap year?
- What is the difference in hours, minutes and seconds between what is added to the calendar every four years with leap day and what is actually needed?
- With the extra, unneeded time that has been added to the calendar every four-year leap cycle, how much extra time would be added to the calendar every 400 years?
- Because of the extra time that gets added to the calendar over a 400-year span, years that are divisible by 100 omit leap day unless they are also divisible by 400. Which years in the past 500 years and the next 500 years have omitted leap day?

### Discussion

*What would happen if Leap Days weren't inserted into the calendar every four years?**What would happen if Leap Days weren't omitted from the calendar every 100 years?*

### Assessment

- Earth revolves around the Sun in approximately 365.2422 days. How long is that in days, hours, minutes and seconds?
- 0.242 * 24 hours = 5.8128 hours
- 0.8128 * 60 minutes = 48.768 minutes
- 0.768 * 60 seconds = 46.08 seconds
**Answer: 365 days, 5 hours, 48 minutes, 46 seconds**

- After the Earth orbits the Sun four times, how many extra hours, minutes and seconds have built up that need to be balanced out by a leap year?
- 4 * 0.2422 days = 0.9688 days
- 0.9688 * 24 hours = 23.2512 hours
- 0.2512 * 60 minutes = 15.072 minutes
- 0.072 * 60 seconds = 4.32 seconds
**Answer: 23 hours, 15 minutes, 4 seconds**

- What is the difference in hours, minutes and seconds between what is added to the calendar every four years with leap day and what is actually needed?
- 1 day minus 23 hours, 15 minutes, 4 seconds
- Count on from 23 hours, 15 minutes, 4 seconds to get to 24 hours.
**Answer: 44 minutes, 56 seconds**

- With the extra, unneeded time that has been added to the calendar every four-year leap cycle, how much extra time would be added to the calendar every 400 years?
- 44 minutes, 56 seconds * 100 = 4400 minutes, 5600 seconds
- 5600 seconds / 60 = 93.33 minutes
- 4400 minutes + 93.33 minutes = 4493.33 minutes
- 4493.33 minutes / 60 / 24 = 3.12 days
**Answer: 3.12 days**

- Because of the extra time that gets added to the calendar over a 400-year span, years that are divisible by 100 omit leap day unless they are also divisible by 400. Which years in the past 500 years and the next 500 years have omitted leap day?
**Answer: 1700, 1800, 1900, 2100, 2200, 2300, 2500**

### Extensions

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