How to Add or Subtract Two Fractions by the Butterfly Method



The butterfly method is used in Japan to teach addition or subtraction of two fractions. Japanese schools teach this method and encourage students to "forget" the butterfly drawing as they get better using the method.

To add or subtract fractions the butterfly way,

1. Write the fractions side-by-side as usual and draw two wings along the diagonals made by the numerator of one fraction and the denominator of the other fraction and draw an antenna on each wing.

2. As suggested by the wings, that look like a multiplication sign, multiply the numbers in each wing and put the product in the antenna for the wing.

3. Think or say: “This poor butterfly needs a body.” To give it a body, connect the bottom parts of the wings with a body-like loop and multiply the two denominators it connects, putting the product inside the body.

4. Add or subtract the numbers in the antennae in keeping with what is being done to the fractions and put the result over the number in the body.

5. If necessary, reduce or simplify the result.

Just How Many Stars are There?

People have been asking this question for a long time. Back before telescopes were invented by a Dutch eyeglass maker, Hans Lippershey in 1608, the number of stars was what we could see with our naked (or unaided) eyes. This number is estimated as no more than 10,000 stars. This number varies according to how you count them, as only 1/2 the stars are visible from a single place on Earth and then only 1/2 the time. You can't see stars through the Earth and during the daytime!

After the telescope was invented, the number of stars we could see increased as we could see more of the fainter stars we couldn't see before. This number is probably about 1,000,000,000 (1 billion), but no one could actually count them all. Instead, Jérôme Lalande published the Histoire Céleste Française in 1801, which contained an extensive star catalog. The observations made were made from the Paris Observatory and so it describes mostly Northern stars. This catalog contained the positions and magnitudes of 47,390 stars, out to magnitude 9 (very, very faint), and was the most complete catalog up to that time. This would give only about 100,000 stars of magnitude 9 or better if the Southern stars could be counted.

 

What we call the Milky Way was known long before telescopes, but it was not known that is was made up of stars. The name Milky Way is from a Greek fairy tale involving a goddess and her milk. After telescopes became commonplace, following the efforts of Galileo, it was soon realized that the Milky Way is an unusually dense collection of stars, all lined up.

 

When people first started looking through a telescope, they noticed fuzzy spiral objects which were grouped with other such "nebulae" and with star clusters. These were at first considered interesting curiosities in the sky, but not to be confused with really interesting things such as comets.

 

Edwin Powell Hubble (1889-1953) demonstrated that most "nebulae" are objects outside our galaxy, using the 100-inch telescope at Mt. Wilson (in 1924). With this powerful instrument, he was able to measure the distance to the Andromeda nebula itself, observing stars within that neighboring galaxy. Thus, the enormous distance to this neighbor became clear: Hubble gave it almost a million light years. As it turned out, this was still quite a bit short of the truth. The distance is greater than 1.5 million light years. With this greater distance, the information coming in from the Andromeda could be interpreted correctly, and it turned out to be a sister galaxy, quite similar to our own.

 Andromeda Galaxy

We now know that there are about 300,000,000,000 (300 billion) stars in the Milky Way Galaxy alone. The Milky Way is larger than average, an average galaxy has about 100 billion stars.

In 2012, Hubble scientists produced the Hubble Extreme Deep Field image. The area searched was an area of clear sky that was less than 1/100th the area of the full moon. In that little rectangle, scientists were able to count over 10,000 galaxies. If this area is typical, there are 100 billion galaxies in the universe. An average galaxy has 100 billion stars, so there are 10,000,000,000,000,000,000,000 stars in the universe (or 10²² in scientific notation).



Large Numbers With Names

The following table lists names of large numbers which are found in many English dictionaries and thus have a special claim to being "real words". The "Traditional British" values shown are not used in American English and are becoming very rare in British English, but their other-language variants are dominant in many non-English-speaking areas, including continental Europe and Spanish speaking countries in Latin America.

          English also has many words, such as "zillion", used informally to mean large but unspecified amounts.

Standard dictionary numbers

Name	Short scale (U.S., Canada and modern British)	Long scale (continental Europe, older British)
Million	106	106
Milliard		109
Billion	109	1012
Trillion	1012	1018
Quadrillion	1015	1024
Quintillion	1018	1030
Sextillion	1021	1036
Septillion	1024	1042
Octillion	1027	1048
Nonillion	1030	1054
Decillion	1033	1060
Undecillion	1036	1066
Duodecillion	1039	1072
Tredecillion	1042	1078
Quattuordecillion	1045	1084
Quindecillion	1048	1090
Sexdecillion (Sedecillion)	1051	1096
Septendecillion	1054	10102
Novemdecillion (Novendecillion)	1060	10114
Vigintillion	1063	10120
Centillion	10303	10600

In the list, scientific notation is used. For instance, a trillion is listed as  10^12. This is spoken as "ten to the twelth" and written as 1,000,000,000,000. Notice that when we write big numbers, we put a comma between every third digit.

The googol Family

The names googol and googolplex were introduced in Kasner and Newman's 1940 book, Mathematics and the Imagination, in the following passage:

The name "googol" was invented by a child (Dr. Kasner's nine-year-old nephew) who was asked to think up a name for a very big number, namely 1 with one hundred zeroes after it. He was very certain that this number was not infinite, and therefore equally certain that it had to have a name. At the same time that he suggested "googol" he gave a name for a still larger number: "Googolplex". A googolplex is much larger than a googol, but is still finite, as the inventor of the name was quick to point out. It was first suggested that a googolplex should be 1, followed by writing zeros until you got tired. This is a description of what would actually happen if one actually tried to write a googolplex, but different people get tired at different times and it would never do to have Carnera a better mathematician than Dr. Einstein, simply because he had more endurance. The googolplex is, then, a specific finite number, equal to 1 with a googol zeros after it.

Value	Name	Authority
10100	Googol	Kasner and Newman, dictionaries
10googol =	Googolplex	Kasner and Newman, dictionaries

If you wish to see a googolplex written out, here's a link you can click on, http://www.googolplexwrittenout.com/ . Of course it comes in multiple volumes, each containing 1 million digits (mostly zeros) of the written out number. A googolplex is so big that if you could read 1 million volumes of 1 million digits in only a 1/1000 of a second, it would still take you MUCH longer than the age of the universe to finish.

Citations:http://en.wikipedia.org/w/index.php?title=Names_of_large_numbers&oldid=665322020

Tic-Tac-Toe as a Magic Square

Here's a game that looks like a math puzzle, but is really tic-tac-toe in disguise.

In the math game, two players take turns picking a number from one to nine. The numbers can only be picked once each game. The first player who picks three numbers that add up to fifteen wins.
A game board is usually drawn up like this:


1   2   3   4   5   6   7   8   9
The players take turns circling their number picks using different colored markers. The first player that can total fifteen with three of their picks wins. A sample game could go like this:

Player 1 circles 5
Player 2 circles 7
Player 1 circles 6
Player 2 circles 4 (to block 5+6+4)
Player 1 circles 8
Player 2 circles 2 (to block 8+5+2)
Player 1 circles 1 and wins 8+6+1

This is a game that is isomorphic to tic-tac-toe, but on the surface looks completely different. Two players in turn say a number between one and nine. A particular number may not be repeated. The game is won by the player who has circled three numbers whose sum is 15.


 Plotting these numbers on a 3×3 magic square shows that the game exactly corresponds with tic-tac-toe, since three numbers will be arranged in a straight line if and only if they total 15.

You don't have to memorize every square on the magic square, although that isn't too hard to do. Remember that the diagonal is 4-5-6 and the other two corners are 2 and 8. Play your regular game of tic-tac-toe, but pick the numbers that match up with the square you would normally place your X or O.

Just for fun:
Can anyone come up with a name for this game? Put it in the comments.

How to Tell Which Fraction is Largest

The secret is to cross-multiply the two fractions. Cross-multiplication is a handy math skill to know. You can use it for a quite few different purposes. You are going to use it to find out which of two fractions is greater.

To cross-multiply two fractions:

1. Multiply the numerator of the first fraction by the denominator of the second fraction and jot down the answer.
2. Multiply the numerator of the second fraction by the denominator of the first fraction and jot down the answer.

For example, suppose you have these two fractions:

 3              4

---  and    ---

 7              9

When you cross-multiply, you get these two numbers: 3 X 9 = 27 and 4 X 7  = 28

Make sure you start with the numerator of the first fraction. To find out which of two fractions is larger, place the two numbers you get, in order, under the two fractions. The larger number is always under the larger fraction.

 3              4

---            ---

 7              9

27            28

so 4/9 is larger than 3/7 (or 4/9 > 3/7).

What do you do if you have more than 2 fractions?

Answer: You take them two at a time.

For example, we have the fractions 5/13, 7/17, and 8/19. Which is the largest?

Take the first two and cross multiply:

 5                                7

---             and           ---

13                              17

5 X 17 = 85               7 X 13 = 91

so 7/17 is larger than 5/13.

Now take the larger of these two, 7/17 and compare it with the third, 8/19.  

 7                                8

---             and           ---

17                              19

7 X 19=133              8 X 17=136

so 8/19 is larger than 7/17 is larger than 5/13.



How to Never Lose a Game of Tic-Tac-Toe

There are many places on the internet that show you how to never lose a game of Tic-Tac-Toe. One of my favorites is Quora. You can click on it for FULL directions, including pictures. I'll go over some highlights.

IF YOU GO FIRST...Avoid placing your first piece on an edge square, and keep it on the center or a corner square. Placing it on an edge square will give your opponent the advantage.

IF YOUR OPPONENT GOES FIRST...Unfortunately, if your opponent goes first and uses all the above techniques, there's no way that you can win. In fact, the only way you can win is if his/her first move is an edge piece.

After the first moves by both players, you can follow the list below. Do the first thing on the list that you are able to do.

Win: If you have two in a row, you can place a third to get three in a row.

Block: If your opponent has two in a row, you must play the third to block your opponent.

Fork: Create an opportunity where you have two threats to win (two non-blocked lines of 2).

Blocking an opponent's fork:
        Option 1: You should create two in a row to force your opponent into defending, as long as it doesn't result in them creating a fork. For example, if "X" has a corner, "O" has the center, and "X" has the opposite corner as well, "O" must not play a corner in order to win. (Playing a corner in this scenario creates a fork for "X" to win.)
        Option 2: If there is a configuration where the opponent can fork, you should block that fork.
Center:  Mark the center. (If it is the first move of the game, playing on a corner gives "O" more opportunities to make a mistake and may therefore be the better choice; however, it makes no difference between perfect players.)
Opposite corner: If the opponent is in the corner, the player plays the opposite corner.
Empty corner: The player plays in a corner square.
Empty side: The player plays in a middle square on any of the 4 sides.

Notes:
The above list is the programming for the first computer tic-tac-toe game. In 1952, OXO (or Noughts and Crosses, the English name for tic-tac-toe) for the EDSAC computer became one of the first known video games. The computer player could play perfect games of tic-tac-toe against a human opponent.

Bonus:
Can anyone comment on what a nought is?

Chomp

This game is called Chomp. It is normally played with just a table of squares on a piece of paper between two players, but I find it easier to understand by thinking of a chocolate bar like this:


POISON>>

             My favorite Chomp playing board

To play Chomp the first player chooses a square on the board, and then takes away everything above and to the right of it (you are taking a bite out of the top right corner of the chocolate bar). The second player then does the same thing with another remaining square. This process keeps continuing until all that remains is the bottom left square. The bottom left square is poisoned and the player who has to take it loses.

To understand how to play the game, click here to go to UCLA and play against their computer. It can be proven mathematically that the first player can always win, but the proof doesn't show how to do it. See if you can win at UCLA and come back and tell us how you did it.

Warning: This game is not really a game, but rather a unsolved proof that mathematicians have been trying to solve for quite a while. It is HARD. Do not spend too much time on it.

Credits: The game was originally stated by Fred Schuh in 1952 as his "game of divisors". David Gale reinvented this game. His version used an m-by-n (any size rectangular) chocolate bar. The name Chomp was invented by Martin Gardner in Scientific American.