September 20, 2009
Mathematical Games and Card Tricks presented by Dr. Nate Carlson
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| Nate introduced Criss Cross | Groups worked in pairs to solve the problem. | This is a collage of work on the problem. |
Nate Carlson is one of the original facilitators of the Tucson Teachers' Circle. He is now teaching at California Lutheran University.
He came to Tucson to present to this group. He introduced three problems.
1.
The Game of Criss-Cross:
The idea (I think) is due to Sam Vandervelde, and his writeup for the game can be found here:
http://www.mathteacherscircle.org/materials/crisscross.pdf
The idea is this: begin with a triangle formed with three dots, and place some number of other dots inside
that triangle such that no three of the dots lie on a straight line. Players take turns choosing a pair of dots
(including the dots that formed the original triangle) and connecting them with a straight line segment
such that each new segment does not cross any existing segment. The game ends when a player is unable
to make a move, and that player loses. The question is, given the initial configuration of dots inside the
triangle, which player wins?
2.
Base 3 Arithmetic:
Deal 27 cards, face up onto a table in three columns with 9 cards in each. Ask someone to choose a card but not to tell you what it is. Three times, ask in which column the card lies, and each time, collect the cards, putting the selected column between the cards in the other two columns, leaving the columns in order, and then deal the cards again by rows to make the next set of columns. After the third choice has been made and the cards re-dealt for the fourth time, the selected card will be number 14– exactly half-way through the deck.
Explain why.
3.
Magic:
Deal three piles of three cards face-down. Have someone choose a pile and turn it over to view
the bottom card on the pile. Place the selected pile on top of the other two and spell out the name of the
card as follows.
Suppose the card is the 4 of spades. Spell out the words: “FOUR” “OF” “SPADES”. With each letter,
deal a card from the top of your original pile face down onto a new pile. When the letters run out, drop
the rest of the remaining cards, in order, on top of the cards you spelled out. Do this three times, for each
of the three words in the card’s name. Finally, spell out the word “MAGIC”, where the first four cards,
corresponding to the letters “MAGI” are face down, but deal the final one, corresponding to the “C” IN
“MAGIC”, face up. That will be the selected card. Explain why.
Numerous performance and process standards were addressed through these problems.
Some of them included are:
2.1.1 Draw inferences about data sets from lists and plots
2.1.2 Organize collected data into an appropriate graphical representation.
2.1.3 Display data. Make predictions and observations about patterns or departures from patterns.
2.1.8 Design simple experiments or investigations and collect data to answer questions.
2.2.3 Use simulations to model situations involving independent and dependent events.
2.4.1 Solve network problems using graphs.
4.1.2 Visualize solids and surfaces in 3-dimensional space when given 2-dimensional representations and create 2-dimensional representations for the surfaces of 3-dimensional objects.
5.1.1 Select an algorithm that explains a particular mathematical process; determine the purpose of a simple mathematical algorithm.
5.1.2 Analyze algorithms for validity and equivalence recognizing the purpose of the algorithm.
5.2.1 Analyze a problem situation, determine the question(s) to be answered, organize given information, determine how to represent the problem, and identify implicit and explicit assumptions that have been made.
5.2.2 . Solve problems by formulating one or more strategies, applying the strategies, verifying the solution(s), and communicating the reasoning used to obtain the solution(s).
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| Nate introduced Base 3 Card Arithmetic | David Bachman-Williams and Ramarao Cheepurupalli solve the problem together. | Mike Schmidt and DeAnna McDonald worked together. |
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| Elizabeth McDonald and Robin Mosteller compare ideas. | Nate consulted with a group. | Steve Cutler and Carolyn Hollis worked on Magic. |
Photos by Ji Li.
