Read more...
Saturday 23 January 2010
Tuesday 9 June 2009
magic2
Below the second challenge, our *FIRST* "for real" post. E-Mail
coordinators, please make sure your teams have NTP's (Net Team
Partners) and exchange information and disscuss *HOW TO* solve
the challenges.
Good Luck
MrH
********************************
Level K-3
********************************
Pole Climbing and the Coordinate Plane...
There is a very important idea called coordinate geometry. It is
a way to find places and mark them for others to see and find.
It is used on maps and diagrams and to show pictures of
formulas. Here is how it works:
Draw a pole going up from the level ground. Starting at the
ground level, every time you go up 1 unit, there is a step to
rest on. Lets call this starting pole, Pole 0. Now, 1 unit to
the right is another pole, Pole 1. It looks just like pole 0.
And, 1 unit to the right of this pole is another identical pole,
Pole 2, and so on. It looks like this:
| | | | | |
4+ + + + + +
| | | | | |
3+ + + + + +
| | | | | |
2+ + + + + +
| | | | | |
1+ + + + + +
| | | | | |
--------------------------
0 1 2 3 4 5
Tina is an expert pole climber. She uses a special notation to
show which pole she will climb and how far she will go up that
pole.
For example, If she wants to go to pole 2 and climb up 3 steps,
she writes (2,3) and it looks like this, where "@" marks the
spot!
| | | | | |
4+ + + + + +
| | | | | |
3+ + @ + + +
| | | | | |
2+ + + + + +
| | | | | |
1+ + + + + +
| | | | | |
-------------------------
0 1 2 3 4 5
See if you can figure out these problems--Draw your own poles on
the chalkboard or bulletin board, or make a poster if you want
to.
1) How would you write these locations using Tina's notation:
A) go to pole 4 and climb 2 steps
B) go to pole 2 and climb 8 steps
C) go to pole 0 and climb 3 steps
D) go to pole 3 and climb 0 steps
2) Draw poles like Tina's and draw the locations indicated below:
A) pole 3, step 1
B) pole 1, step 5
C) (4,2)
D) (3,2)
E) (2,6)
3) John goes to (1,3) and Tina goes to (5,3). How far apart are
they?
4) Next, Bill goes to (2,1) and Tina goes to (2,6). How far
apart are they?
5) Finally, Bill goes to (5,2) and John goes to (1,7). Tina is
at (5,7). Who is closer to Tina--is it Bill or John? Explain
your answer.
**************************************
NOTE: Please print this file using a monospaced font such as Courier
Read more...
magic
**********************************
MathMagic Cycle 94-1 Level K-3 R
**********************************
LET ME COUNT THE WAYS-
1) Tina has 3 shirts-red, blue, and yellow. She also has 2 skirts-
black and white. Each day, she wears a different shirt with a
different skirt.
A) For how many days can she do this before she must repeat and
Draw pictures of Tina and color if you wish.
There is also a chart called a TREE that can help Tina to know
which outfit is next to wear. It looks like this:
|-----Black
|---Red-------|
| |-----White
|
|
| |-----Black
|--Blue-------|
| |-----White
|
|
| |-----Black
|--Yellow-----|
|-----White
So, you can see all the possibilities and even count them.
C) Uh oh, Tina accidentally ripped her yellow shirt and cannot wear
it anymore. How many outfits does she have now? Draw a tree.
2) When Billy comes home from school each day, he gets a snack
and a drink. His Mom lets him choose either a piece of fruit, a
cookie, or a peanut butter sandwich. He can drink juice,
milk, or a soda.
A) Can you make a TREE that explains all the possibilities of
snacks and sodas? How many snack and drink combinations are there?
B) How many combinations are there if the drink is MILK?
**********************************
MathMagic Cycle 94-1 Level K-3 H
**********************************
BALANCING ACT
A scale can be used to measure the weight of various items, by
comparing the item to a known weight. This cycle's problem will
require you to do some empirical work (empirical is a scienttific
term that means you will experiment and observe and then
report what you see). First you will need to make a scale using a
ruler (the kind with 3 holes), 3 pieces of string(all the same
length) and 2 baskets (like for strawberries, milk , etc that are
the same size). Tie the 3 pieces of string to the ruler through the
3 holes. Attach the two end pieces to the baskets and tie the middle
string to a support. Hopefully, the scale will be balanced--if
not, attach weights (paper clips, clay) to one side or the other
until the scale is balanced. Before you begin, explain the
differences (if any) between weight and mass.
Experiment 1) Place 4 or 5 common items that are identical (pencils,
erasers, etc) in basket #1. Decide on another, different item
(marbles are good or paper clips)that can be used to balance
them in basket #2. Then try to balance basket #1 with basket #2.
Did it work right off--If not, add more to basket #1. Explain your
results. Use your results to estimate the weight of 1 basket #1
item in terms of the basket #2 item. Check your results.
Experiment 2) How much does one piece of paper weigh, compared to
a paper clip or a pencil? Explain how you did this experiment? Which
weighs more--500 sheets of paper or 500 paper clips?
Experiment 3) You don't necesarilly need your scale for this. Suppose
6 oranges balance with 4 apples. And suppose 4 apples balance with
8 bananas. How many bananas will be needed to balance 9 oranges?
Experiment 4) Suppose now that 12 pencils balance with 4 erasers,
and that 6 erasers balance with 12 sharpeners. How many pencils would
be needed to balance 6 sharpeners?
Read more...
game theory
Symmetric and asymmetric
| E | F | |
| E | 1, 2 | 0, 0 |
| F | 0, 0 | 1, 2 |
| An asymmetric game | ||
A symmetric game is a game where the payoffs for playing a particular strategy depend only on the other strategies employed, not on who is playing them. If the identities of the players can be changed without changing the payoff to the strategies, then a game is symmetric. Many of the commonly studied 2×2 games are symmetric. The standard representations of chicken, the prisoner's dilemma, and the stag hunt are all symmetric games. Some scholars would consider certain asymmetric games as examples of these games as well. However, the most common payoffs for each of these games are symmetric.
Most commonly studied asymmetric games are games where there are not identical strategy sets for both players. For instance, the ultimatum game and similarly the dictator game have different strategies for each player. It is possible, however, for a game to have identical strategies for both players, yet be asymmetric. For example, the game pictured to the right is asymmetric despite having identical strategy sets for both players.
[edit] Zero-sum and non-zero-sum
| A | B | |
| A | –1, 1 | 3, –3 |
| B | 0, 0 | –2, 2 |
| A zero-sum game | ||
Zero-sum games are a special case of constant-sum games, in which choices by players can neither increase nor decrease the available resources. In zero-sum games the total benefit to all players in the game, for every combination of strategies, always adds to zero (more informally, a player benefits only at the equal expense of others). Poker exemplifies a zero-sum game (ignoring the possibility of the house's cut), because one wins exactly the amount one's opponents lose. Other zero-sum games include matching pennies and most classical board games including Go and chess.
Many games studied by game theorists (including the famous prisoner's dilemma) are non-zero-sum games, because some outcomes have net results greater or less than zero. Informally, in non-zero-sum games, a gain by one player does not necessarily correspond with a loss by another.
Constant-sum games correspond to activities like theft and gambling, but not to the fundamental economic situation in which there are potential gains from trade. It is possible to transform any game into a (possibly asymmetric) zero-sum game by adding an additional dummy player (often called "the board"), whose losses compensate the players' net winnings.
http://en.wikipedia.org/wiki/Game_theory
Read more...
Sunday 17 May 2009
Subscribe to:
Posts (Atom)