Math Concepts: division as equal sharing, naming fractions, adding fractions, infinitesimals, iteration, limits Prerequisite: able to identify fractions as part of a whole
This is how I tell the story:
We have a cake to share, just the two of us. It’s not TOO big a cake, ‘cuz we don’t want to get sick. A 8 × 8 or 16 × 16 square on the graph paper should be just right. Can you cut the cake so we each get a fair share? Color in your part.
How big is your piece compared to the whole, original cake?
But you know, I’m on a diet, and I just don’t think I can eat my whole piece. Half the cake is too much for me. Is it okay if I share my piece with you? How can we divide it evenly, so we each get a fair share? How big is your piece?
How much of the whole, original cake do you have now? How can you tell?
I keep thinking of my diet, and I really don’t want all my piece of cake. It looks good, but it’s still just a bit too big for me. Will you take half of it? How big is that piece?
Now how much of the whole, original cake do you have? How could we figure it out?
[Teaching tip: Don’t make kids do the calculation on paper. In the early stages, they can visualize and count up the fourths or maybe the eighths. As the pieces get smaller, the easiest way to find the sum is what Cohen does in the video below—identify how much of the cake is left out.]
Even for being on a diet, I still don’t feel very hungry…
She stared at the diagram for a minute or two, while I bit my tongue to keep from breaking her concentration.
Then she said, “Oh, I see! It’s an infinite cake.”
It took me much longer to understand what she had seen so quickly: Imagine stacking up smaller and smaller circles in the top part of the triangle. Because all the proportions stay the same, each circle is exactly half as wide as the one below it. To find the height of the triangle, we can just add up all the diameters of the circles.
In the land of Fantasia, where people communicate by crystal ball, Wizard Mathys has been placed in charge of keeping the crystal connections clean and clear. He decides to figure out how many different ways people might talk to each other, assuming there’s no such thing as a crystal conference call.
Mathys sketches a diagram of four Fantasian friends and their crystal balls. At the top, you can see all the possible connections, but no one is talking to anyone else because it’s naptime. Fantasians take their siesta very seriously. That’s one possible state of the 4-crystal system.
On the second line of the diagram, Joe (in the middle) wakes up from siesta and calls each of his friends in turn. Then the friends take turns calling each other, bringing the total number of possible connection-states up to seven.
Finally, Wizard Mathys imagines what would happen if one friend calls Joe at the same time as the other two are talking to each other. That’s the last line of the diagram: three more possible states. Therefore, the total number of conceivable communication configurations for a 4-crystal system is 10.
For some reason Mathys can’t figure out, mathematicians call the numbers that describe the connection pattern states in his crystal ball communication system Telephone numbers.
Can you help Wizard Mathys figure out the Telephone numbers for different numbers of people?
T(0) = ?
T(1) = ?
T(2) = ?
T(3) = ?
T(4) = 10 connection patterns (as above)
T(5) = ?
T(6) = ?
and so on.
Hint: Don’t forget to count the state of the system when no one is on the phone crystal ball.
Math Concepts: sorting by attribute (card suits), counting up, counting down, standard rank of playing cards (aces low). Players: two or more, best with four to six. Equipment: one complete deck of cards (including face cards), or a double deck for more than six players. Provide a card holder for young children.
How to Play
Deal out all the cards, even if some players get more than others. The player to the dealer’s left begins by playing a seven of any suit. If that player does not have a seven, then the play passes left to the first player who does.
After that, on your turn you may lay down another seven or play on the cards that are already down. If you cannot play, say, “Pass.”
Once a seven is played in any suit, the six and the eight of that suit may be played on either side of it, forming the fan. Then the five through ace can go on the six in counting-down order, and the nine through king can go on the eight, counting up. You can arrange these cards to overlap each other so the cards below are visible, or you can square up the stacks so only the top card is seen.
Players do not need to wait for both the six and eight of a suit to be played before they begin building the fan up or down.
The first player to run out of cards wins the game.
If you want to keep score, count the cards remaining in your hand after one player goes out. After everyone has had a turn as dealer, whoever has the lowest total score is the champion.
In some traditions, play always begins with the seven of diamonds, so whoever has that card goes first.
The player to the dealer’s left may lead any card, and then all the suits must start with that number (instead of with seven) and build up and down from there.
Fan Tan Trumps
When the dealer gets to the end of the deck and there aren’t enough cards to give every player one more, the last few cards are turned face up and may be played by anyone as needed. The suit of the last card becomes the trump suit, and cards of that suit may be played on any of the fans, with the card they replace going on the trumps fan. In this case, the cards must be laid out in overlapping rows, not stacked up, so everyone can see where the trumps have gone.
For instance, if spades are trump, then a nine of spades could be played on the eight of hearts, which would leave the nine of hearts without a home—so it has to go on the spades fan.
Exceptions: The seven of the trump suit starts its own fan, like any other seven, and the last card dealt (the one that named the trump suit) must also be played to the trumps fan when its turn comes.
Deal only seven cards to each player, and set the rest of the deck out as a draw pile. The first player who cannot play must draw one, which he may play if possible. If not, and the next player also cannot play, she must draw two. If neither of those cards will play, and the next player has nothing to play, he must draw three, and so on, each player drawing one more card than the last person. When one of the players is finally able to lay down a card, this resets the draw count back to zero.
In Crazy Tan, players are allowed to lay down a run (playing several cards in a row of the same suit on a single turn). Or they may play parallel cards (cards of the same rank in different suits, all played in the same turn). Or a player may even lay down parallel runs, if the cards happen to work out that way.
Fan Tan may also be called Crazy Sevens. Like any folk game, it is played by a variety of rules around the world. If you search for it on the Internet, you may run into an unrelated Chinese gambling game called Fan Tan, which is similar to Roulette.
Do you enjoy math? I hope so! If not, browsing the articles linked in this post just may change your mind.
Welcome to the 85th edition of the Math Teachers At Play math education blog carnival—a smorgasbord of links to bloggers all around the internet who have great ideas for learning, teaching, and playing around with math from preschool to pre-college.
Math Concepts: addition to thirty-one, thinking ahead. Players: best for two. Equipment: one deck of math cards.
How to Play
Lay out the ace to six of each suit in a row, face up and not overlapping, one suit above another. You will have one column of four aces, a column of four twos, and so on—six columns in all.
The first player flips a card upside down and says its number value. Players alternate, each time turning down one card, mentally adding its value to the running total, and saying the new sum out loud. The player who exactly reaches thirty-one, or who forces the next player to go over that sum, wins the game.
For a shorter game, use only the ace to four of each suit. Play to a target sum of twenty-two.
History (and a Puzzle)
Thirty-One comes from British mathematician Henry Dudeney’s classic book, The Canterbury Puzzles. According to Dudeney, “This is a game that used to be (and may be to this day, for aught I know) a favourite means of swindling employed by card-sharpers at racecourses and in railway carriages.”
Dudeney challenges his readers to find a rule by which a player can always win: “Now, the question is, in order to win, should you turn down the first card, or courteously request your opponent to do so? And how should you conduct your play?”