About time arithmetic

TUESDAY, SEPTEMBER 24, 2013 

Repost January 19 2019

In the 5th and 6th grade math challenges, there are many problems about time.

This is was written on September 24, 2013 around the autumnal equinox.

The sun rose today at 7:02 AM and will set tonight at 7:06 PM..

It will rise tomorrow at 7:04 AM.

How long is the day September 24 and the night September 24-25?

Day and night should be approximately equal and approximately 12 hours each.

Answer below.

Yesterday, the sun rose at 7:02 AM and set at 7:06 PM. Today it rose at 7:04 AM. How long was the day on September 24 and how long was the night of September 24-25?

Note that the colon between hours and minutes is not a decimal point; time uses base 60. One each side of the colon, you can use decimal arithmetic. Working with time is a little tricky!

Day: use 24 hour clock, add 12 to PM hours. So, sun sets at 19:06. 19:06 - 7:02 = 12:04

Night: how much time between sunset and midnight? 12:00 - 7:06. You can't subtract :06 from :00, so "borrow" 60 minutes from the hour of midnight: 11:60 - 7:06 = 4:54. Now add the time until sunrise:

4:54 + 7:04 = 11:58.

The day is 12 hours and 4 minutes long and the night is 11 hours and 58 minutes long. These don't add up to 24 because we are looking at two different days. We are near the autumnal equinox so the day and night are approximately equal to 12 and approximately equal to each other.

As time goes on, days will get shorter and nights longer until the winter solstice, in December.

Today (this was September 24 2013) the sun rises at 7:04 AM and sets at 7:04 PM, so the day is exactly 12 hours.Tomorrow, the sun will rise at 7:05 AM, the night is longer than the day: prove this!

You will find the night is 11 hours and 61 minutes, so change 61 minutes to an hour and one minute

SOME CRYPT ARITHMETIC HINTS

SOME CRYPT ARITHMETIC HINTS

1)     READ THE PROBLEM! THEY MAY NOT BE ASKING FOR ALL THE LETTERS, OR ONLY SOME NUMBERS ARE ALLOWED.

2)     IN THIS SITUATION

 XXX

+XXX

----

NXXX N=1 (The number sticking out is 1)

3)If F + 1 generates a carry, F=9

4)EVERY LETTER STANDS FOR A DIFFERENT NUMBER, EVERY NUMBER IS A DIFFERENT LETTER. THIS CAN SOMETIMES REDUCE THE NUMBER OF GUESSES YOU NEED TO MAKE, BECAUSE SOME NUMBERS ARE ALREADY "USED."

5) SAME_NUMBER + SAME_NUMBER = 2 X SAME_NUMBER = EVEN_NUMBER. FOR EXAMPLE,9+9=18, YOU WRITE 8, AN EVEN NUMBER, AND CARRY 1.

6) JUMP TO THE LAST COLUMN, BECAUSE THERE ARE NO CARRIES TO WORRY ABOUT.

7) IF YOU GET TO A DEAD END,BACK UP!

Sunrise, Sunset for September 26-27 2013

Facts: On September 26, 2013 the Sun rose at 7:05 AM and will set at 7:02 PM. On September 27, the Sun will rise at 7:07 AM.

Question: How long is the day on September 26 and how long is the night of September 26-27?

-----------------------------
Hints: for mathematical methods for time, see the posting with title Equinox. Once you have done the calculation, the results will make sense; the day will be 11 hours and 50-some minutes, the night will be 12 hours and some minutes. Just look closely at the sunrise and sunset times.

From now until the Winter Solstice on December 21 2013, the days will be getting shorter and the nights will be getting longer.

The Daily Jumble

The Daily Jumble appears every day in the Saint Paul Pioneer Press. You can also see it at this address:
Daily Jumble for September 24 2013 (Sorry, this does not work on Apple products, only on PCs)

I have found that some students really like this kind of problem.

Hints: #1 A group of boats.
#2 A deep narrow ravine, often seen in Western movies, a gang might hang out there.
#3 One of the most popular computer styles, it is two three-letter words stuck together.
#4 What happens to a concert when the band refuses to play (verb).

You will end up with these circled letters: FEEGCHAOPACN. You are trying to make a six-letter word, a two-letter word, and a four-letter word. There are very few two-letter words, one of these is OF, so cross out O and F.  Now look at the cartoon, the words on the cartoon, and the letters that are left and guess the cartoon caption.

The Daily Jumble is a good exercise in problem solving. I have worked out a bunch of tricks for solving them. The tricks involve permutations and logic, both math topics.

If anybody tries this, send me an email at margolis.stephen@yahoo.com and I will send you a bunch of tricks for solving this kind of problem. If you have an Apple computer and can't open the link, tell me your room and teacher and I will give you a paper copy.

EQUINOX PROBLEM SOLUTION

Yesterday, the sun rose at 7:02 AM and set at 7:06 PM. Today it rose at 7:04 AM. How long was the day on September 24 and how long was the night of September 24-25?

Note that the colon between hours and minutes is not a decimal point; time uses base 60. One each side of the colon, you can use decimal arithmetic. Working with time is a little tricky!

Day: use 24 hour clock, add 12 to PM hours. So, sun sets at 19:06. 19:06 - 7:02 = 12:04
Night: how much time between sunset and midnight? 12:00 - 7:06. You can't subtract :06 from :00, so "borrow" 60 minutes from the hour of midnight: 11:60 - 7:06 = 4:54. Now add the time until sunrise:
4:54 + 7:04 = 11:58.

The day is 12 hours and 4 minutes long and the night is 11 hours and 58 minutes long. These don't add up to 24 because we are looking at two different days. We are near the autumnal equinox so the day and night are approximately equal to 12 and approximately equal to each other.

As time goes on, days will get shorter and nights longer until the winter solstice, in December.

Today, the sun rises at 7:04 AM and sets at 7:04 PM, so the day is exactly 12 hours.Tomorrow, the sun will rise at 7:05 AM, the night is longer than the day: prove this!

You will find the night is 11 hours and 61 minutes, so change 61 minutes to an hour and one minute.

In the 5th grade math challenge, there are many problems about time.

This is written on September 24, 2013 around the autumnal equinox.

The sun rose today at 7:02 AM and will set tonight at 7:06 PM..

It will rise tomorrow and 7:04 AM.

How long is the day September 24 and the night September 24-25?

Day and night should be approximately equal and approximately 12 hours each.

Answer tomorrow.

KEPLER'S LAWS

Kepler's laws are very important

Here is a link to a 5-minute movie about them:LINK TO KEPLER MOVIE

Here is a written summary:

German astronomer Johannes Kepler was born in December 1571, and throughout his 59 years of life, he contributed immensely to science. He is most well-known, however, for his three laws of planetary motion. This work stemmed from a collaboration with Danish astronomer Tycho Brahe. Kepler mathematically analyzed some 20 years of precise planetary observations that Brahe collected. He determined how the solar system’s planets move the way they do and laid the foundation for Isaac Newton’s theory of gravity. What we now call his three laws are theories that are universal, verifiable, and precise — and they don’t just govern the motion of the planets, but also comets, asteroids, and other minor bodies orbiting the Sun.

Kepler’s first law says that the orbit of every planet is an ellipse, with the Sun at one of two focus points. A circle is a special type of ellipse that has just one focus, which is located at its center. All other ellipses look like flattened circles, and have two focus points. If you take any point on the ellipse, the sum of the distances to the focus points is constant. 

Before Kepler’s work, astronomers tried to describe the motion of the planets via interconnected circles, but they struggled to match observations. They also could not predict where planets would appear in the sky. Kepler’s theory changed that and showed how elegantly the planets moved.

Kepler’s second law says that a line joining a planet and the Sun sweeps out equal areas during equal intervals of time. Thus, a planet moves fastest when it’s closest to the Sun (at a point called perihelion) and slowest at its farthest point from the Sun (known as aphelion). You can see this property best with objects that have longer elliptical orbits, like comets. As they near the Sun, they travel much faster than when they are more distant. 

Nine years after Kepler published his first two laws, he determined the last one. Kepler’s third law says that the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit. The semi-major axis is just a term for half of the longest length of the ellipse. If you measure the period in Earth years and the orbit’s semi-major axis in astronomical units, the equation simplifies to period squared equals semi-major axis cubed. Astronomers use this relation to figure out the orbit of planets around other stars. They directly detect the world’s orbit period and can then figure out how far from the star the planet orbits.