Thursday, September 3, 2015

math games class catch up

It has been a while since I posted a summary of our math games, so this is just a quick catch up to summarize what we've been doing:

Grade 1

Addition war
2 players
pack of playing cards (A to 10)

Deal out all cards to both players and keep the cards face down in a stack. Each round, both players turn over the top two cards and add their values. The player with the higher sum wins and collects the cards in their points pile.

If there is a tie, those 4 cards are kept to the side as a bonus for the winner of the next battle. Repeat this with ties until there is a winner for one round.

After playing through the original stack, look to see who has collected more cards in their points pile. That person is the winner.

For a more challenging version, use face cards and assign values J = 11, Q = 12, K = 15.

Solo addition bridge
2 - 4 players (we played with 3)
pack of playing cards (A to 10 for beginner game, add face cards for extra challenge)

Deal out 5 cards to all players. They pick up these cards to form their hand. Proceeding clockwise, each player lays down one card from their hand, going twice around the group. Each player adds together the value of the two cards they played. The highest sum wins and collects all the cards played. That is one "trick."

After each trick, deal out 2 cards to each player to refill the hands to 5 cards.

The player who won the last trick is the first to play a card on the new round.

When there aren't sufficient cards to deal equally to all players, deal the hands equally (all players start each round with the same number of cards) and keep the remaining cards as a bonus for the player who takes the last trick.

We played with 3 players and, for the advanced game added 2 jokers to make a deck of 54 cards. Based on popular consensus, the jokers were assigned a value of 1,000,000. Interestingly, the extremely large value meant that the players were reluctant to play their jokers and, twice, both players kept them to the final trick so that the winner was actually decided by the higher of the second card!

Group addition bridge
4 players working as pairs (partners) with the partners sitting opposite each other
pack of playing cards (A to 10 for beginner game, add face cards for extra challenge)

Generically, play is the same as solo addition bridge, but each round the partners each play one card and the team that has the higher sum wins the trick. While there are still reserve cards in the deck, hands get refreshed up to 5 by dealing a single card to each player. The leader for each trick rotates clockwise so that everyone gets a chance to be first (second, third, and last) to play.

Second and Third Grade

Multiplication Blind Man's Bluff
3 players
pack of playing cards using A to 10 (A counts as 11)

One player deals a single card to each of the other players. They hold that card up to their forehead. The dealer announces the product of the two cards. Then, the two players try to figure out the value of the card on their own forehead.

Role of the dealer rotates after each round.

We played this as a cooperative exercise. To make it competitive, you can award points to the first player to get their card value.

There are two ways to make this more difficult. Adding face cards with made up values is one way. Instead, we had the dealer give one player two cards, add those, then multiply that sum by the value of the other card. 

Yet another step is to deal each player two cards, then multiply the two sums.
When playing this version at home, J1 came up with the idea of giving clues to figure out the value of the two individual cards. This was a really interesting activity because it got him to think about what characteristics help specify the two cards and which clues actually don't provide new information.
For example, if I know the sum of my two cards is 11, does it help me to know that I have one odd and one even number?

To make a standardized version, the second round of clues is to tell each player the product of their values. 

Largest Difference/Smallest difference
many players (at most 9 per deck of cards, fewer with advanced versions)
pack of playing cards A to 9

Deal out 4 cards to each player. They then form two 2-digit numbers and subtract the smaller from the larger. The player with the greatest difference wins that round.

For slightly greater challenge, deal out 6 cards (for two 3-digit numbers) or more (forming 4 or 5 digit numbers). Again, the aim is to form two numbers with the same number of digits that have the greatest difference.

For a much more interesting game, we shift the goal: now, we try to find two numbers with the smallest difference (larger minus smaller). After playing a bit, we had some good conversations about what the students noticed, what strategies they used, and whether there was always a unique answer.

Multiplication Pig (variation of addition Pig)
2 dice (we used 2d6)
2-3 players (or more, grouped into teams)

Players start with 200 points and try to work to 0 (or below).

Each turn, the player rolls both dice. If neither is a 1, they multiply the two values and add this to their score for the round. They can either choose to roll again or take their score for the round and subtract that from their cumulative score.

If two 1's are rolled, then  their overall score is set back to 200. If one 1 is rolled, then their score for that round goes to 0 and they lose their turn.

Variations come from varying to characteristics of the game:
- Start with 0 overall points and, each round, add the points for your round to try to break a target (practices addition instead of subtraction in forming the overall target)
- Add the two dice instead of multiplying (shifts the practice to addition instead of multiplication)
- Use dice other than 2d6, possibly more dice or differently shaped dice (note: the overall target and/or penalty conditions might require some adjustment)

Some PIG observations

Dice games are loud games, compared with card games. I think this is because the value of the dice is revealed to everyone at the same time.

Based on expected values, the optimal decision whether to keep rolling to bank the points for that round depends on how many cumulative points you have and your score for that round. However, we observed that the students chose to bank their points very early, relative to an expected value maximizing strategy. I think this is because their experience with the game causes them to over estimate the likelihood of rolling a 1 (or two 1's) and/or to underestimate how many points they can earn on a single roll because of the multiplication.