Time and Calendars
A Lesson in the “Math + Fun!” Series
2006
May 2006
Time and Calendars
Slide 1
About This Presentation
This presentation is part of the “Math + Fun!” series devised
by Behrooz Parhami, Professor of Computer Engineering at
University of California, Santa Barbara. It was first prepared
for special lessons in mathematics at Goleta Family School
during three school years (2003-06). “Math + Fun!” material
can be used freely in teaching and other educational settings.
Unauthorized uses are strictly prohibited. © Behrooz Parhami
May 2006
Edition
Released
First
May 2006
Revised
Time and Calendars
Revised
Slide 2
A Brief History of Timekeeping
Two Grandfather Clocks
(17th and 18th Centuries)
(1st
Pocket Watch
and
Wristwatch
(19th Century)
Ancient Sundial
or 2nd Century AD)
Modern $50
“Atomic Clock”
First atomic clock,
(US National Bureau
of Standards, 1949)
May 2006
Time and Calendars
Slide 3
Activity 1: Making a Sundial Clock
1. Fill up a small bottle with sand or soil and place a straight stick in it.
Push the stick down until it is firmly planted and does not wobble.
2. Place a large piece of cardboard in a spot where the sun shines all day
(you do need a sunny day for this activity), put the bottle and stick at
the center of the cardboard, and draw a circle around the bottle’s base
(to mark its exact location, in case it moves by accident).
3. Every hour, on the hour (9:00 AM, 10:00 AM,
and so on), draw a line along the shadow cast
by the stick and mark it with the time.
By sundown, you will have a sundial clock!
North
If you want to be able to move your clock
to a different location, use a compass to
mark north on the cardboard and have
the line point to north in the new location.
May 2006
Time and Calendars
Slide 4
Activity 2: Making a Pendulum Timer
1. Take a lightweight string that is about 1 meter (40 inches) long.
Attach a heavy bob to one end and a key-ring to the other.
2. Hang the pendulum from a long nail, so that the bob is
away from the wall and close to the floor or desktop.
3. Gently swing the pendulum and time 60 of its swings from
one side to the other. The time should be about 1 minute.
If it’s a bit less than 1 min, make the string a little longer;
if it’s a bit more, shorten the string. Repeat this until
each swing takes almost exactly 1 second.
The time for one swing (in seconds) is related to the
length of the string (in meters) by the equation:
Side-to-side swing time   Length
So to make a side-to-side swing take 2 seconds,
the string length must be about 4 meters.
May 2006
Time and Calendars
Slide 5
How Mechanical Clocks Work
1. There is a lever that oscillates about an axis;
much like a pendulum, but not hanging
2. Each oscillation causes a cogged (escape)
wheel to rotate by a small amount
3. The movement of the escape wheel
is transferred to hand movements
via gears that reduce the speed
4. A spring is used to give the
oscillating lever a gentle push
so that it does not slow down
The oscillating element
can be in the shape of a
wheel (balance wheel),
but the idea is the same
May 2006
Time and Calendars
Slide 6
Converting Oscillations to Rotation
Oscillations of the blue wheel
turn the escape wheel by
pushing its teeth, as shown
in the animation to the right
Oscillations of the red wheel
cause the Y-shaped lever to
move and to push the teeth
of the escape wheel
May 2006
Time and Calendars
Slide 7
A Brief History of Calendars
The purpose of a calendar system is to keep track of time periods that
are linked to astronomical events (day = one rotation of the Earth
around its axis, month = one revolution of the Moon around the Earth,
year = one revolution of the Earth around the Sun). Week is the only
calendar time convention that is not linked to astronomical events.
1. There are 40 different calendar systems in use today.
2. Our calendar system is called the “Gregorian Calendar”; there are also
Chinese, Hebrew, Indian, Islamic, Persian, and many other calendars.
3. Ancient Egyptians were the first to use years consisting of 365 days;
early Chinese and Greek years were 354 days, as is the Jewish year;
at various times, there have been both shorter and longer years
(for example, 46 BC, nicknamed “year of confusion,” had 445 days,
because Julius Caesar added 80 days to it so that the following year
in his reformed calendar would conform to the solar year).
May 2006
Time and Calendars
Slide 8
Why Do We Need Leap Years?
1. One calendar year (Earth’s revolution around the Sun on average) is
365 days, 5 hours, 49 minutes (365.2424 Universal days).
2. After four 365-day years, we are left with 4  0.2424 = 0.9696 extra day,
or almost one day. This is why we make every fourth year a leap year.
3. But an extra day is a little bit more than 0.9696 day. The extra 0.0304
day adds up to a full day every 33 leap years, or in about 132 years.
4. It is rather inconvenient to make a correction every 132 years, so we
drop the leap year every 100 years and put it back in every 400 years.
5. When we drop the leap year every 100 years, we remove one day
instead of 0.0304  25 = 0.76 days every 25 leap years. This is ¼ day
less than what it should be. So, every four centuries, or 400 years, we
make a correction by using a leap year. This makes it just about right.
Very infrequently (every 1000s of years), we need more corrections.
Interesting fact: Some years have a leap second added to them to
correct for variations in timekeeping by atomic clocks around the world.
May 2006
Time and Calendars
Slide 9
Finding the Day of the Week for a Date
Between 1901 and 2099
Example: On what day is July 4, 2006?
Month
Divide year number by 4,
ignore the remainder
2006 / 4 = 501,
with remainder 2
January, Leap Year
–3
January, Nonleap
–2
Add the year
501 + 2006 = 2507
February, Leap Year
0
February, Nonleap
1
Add the month code from
table to the right
2507 + (–3) = 2504
March
1
April
–3
Add the day of the month
2504 + 4 = 2508
May
–1
Divide by 7, keep only the
remainder
2508 / 7 = 358
with remainder 2
June
2
July
–3
The remainder gives you
the day in the table below
Tuesday!
August
0
September
3
October
–2
Code
0
1
2
3
4
5
6
November
1
Sun
Mon
Tue
Wed
Thu
Fri
Sat
December
3
May 2006
Time and Calendars
Slide 10
Activity 3: Your Birthday in Different Years
Month of birthday: _____
Day of birthday: _____
Year of your 18th
Birthday: _____
Your birthday in
the year 2050
Divide the year number by
4, ignore the remainder
_____
/ 4 = _____
with remainder ___
_____
Add the year
_____ + _____ = _____
_____ + _____ = _____
Add the month code from
table in the previous slide
_____ + _____ = _____
_____ + _____ = _____
Add the day of the month
_____ + _____ = _____
_____ + _____ = _____
Divide by 7, keep only the
remainder
_____
/ 7 = _____
with remainder
_____
/ 4 = _____
with remainder ___
/ 7 = _____
with remainder
__
__
The remainder gives you
the day in the table below
May 2006
0
1
2
3
4
5
6
Sun
Mon
Tue
Wed
Thu
Fri
Sat
Time and Calendars
Slide 11
Activity 4: Making a Calendar for the Year 2020
1. Figure out whether 2020 is a leap year.
Remember that a year is a leap year if it is divisible by 4, but not if it is
also divisible by 100, except if it is also divisible by 400. For example,
the year 1900 was not a leap year (divisible by 100 but not by 400)
but the year 2000 was a leap year (divisible by 400)
2. Find the day of week for January 1, 2020; then complete the calendar.
January
February
March
April
May
June
S M T W T F S
S M T W T F S
S M T W T F S
S M T W T F S
S M T W T F S
S M T W T F S
July
August
September
October
November
December
S M T W T F S
S M T W T F S
S M T W T F S
S M T W T F S
S M T W T F S
S M T W T F S
May 2006
Time and Calendars
Slide 12
January
February
March
April
May
June
S M T W T F S
S M T W T F S
S M T W T F S
S M T W T F S
S M T W T F S
S M T W T F S
July
August
September
October
November
December
S M T W T F S
S M T W T F S
S M T W T F S
S M T W T F S
S M T W T F S
S M T W T F S
January
February
March
April
May
June
S M T W T F S
S M T W T F S
S M T W T F S
S M T W T F S
S M T W T F S
S M T W T F S
July
August
September
October
November
December
S M T W T F S
S M T W T F S
S M T W T F S
S M T W T F S
S M T W T F S
S M T W T F S
May 2006
Time and Calendars
Slide 13
Perpetual Calendars
There are only 14 different
calendar types: seven types
depending on the day of the
week for January 1, and two
variations for leap and
nonleap years. If you print
these 14 calendars and call
them 0, 0*, 1, 1*, . . . , 6, 6*
(where the number is the
day of week for January 1
and * indicates a leap year),
the table to the right shows
you which calendar to use
for any given year.
Year
Type
Year
Type
1947, 1975, 2003
3
1961, 1989, 2017
0
1948, 1976, 2004
4*
1962, 1990, 2018
1
1949, 1977, 2005
6
1963, 1991, 2019
2
1950, 1978, 2006
0
1964, 1992, 2020
3*
1951, 1979, 2007
1
1965, 1993, 2021
5
1952, 1980, 2008
2*
1966, 1994, 2022
6
1953, 1981, 2009
4
1967, 1995, 2023
0
1954, 1982, 2010
5
1968, 1996, 2024
1*
1955, 1983, 2011
6
1969, 1997, 2025
3
1956, 1984, 2012
0*
1970, 1998, 2026
4
1957, 1985, 2013
2
1971, 1999, 2027
5
1958, 1986, 2014
3
1972, 2000, 2028
6*
0
1
2
3
4
5
6
1959, 1987, 2015
4
1973, 2001, 2029
1
Sun
Mon
Tue
Wed
Thu
Fri
Sat
1960, 1988, 2016
5*
1974, 2002, 2030
2
May 2006
Time and Calendars
Slide 14
Next Lesson
Date Unknown
May 2006
Time and Calendars
Slide 15