Multiply Who, a math game

This past semester I got to teach a senior project class. Four preservice elementary teachers working on understanding math games, game design and making their own. Sam came into class with the idea for her game. All the games were tested with kids, and went through multiple revisions and I'm really proud of their work and the games they made.

GUEST POST by Sam Bosma.

Learning to Go Forwards to Go Backwards

Many of us remember ‘learning’ multiplication in second or third grade. I remember ‘learning’ multiplication in 3rd grade through the use of flashcards. It didn’t matter what was happening behind the scenes just as long as I got the right answer/ could memorize the answer.
Too often with multiplication, we see children simply memorizing that, for example, 3 x 8 is 24 and not really understanding the ‘why’ behind how it works. The problem with this memorization is that once students move on to division, we see them getting very quickly. This makes sense since how can we expect students to go backwards (the undoing of multiplication) if they can’t even move forwards (multiplication).
In the game I have invented, Multiply Who?, my attempt is to have student thinking more deeply about multiplication instead of simply memorizing. In this game, students will have to find patterns and think about the components of each number in order to ask the most strategic questions in order to guess an opponent’s number.

If you are interested in learning more about this game, watch the video down below or click here for the instructions and rules.

John's postscript: I got to see one of the playtests of this game with a class of 5th graders and was impressed. The format of the game works really well for 2 on 2 playing. And though Sam put the sheets in sleeves, you could play directly on a printup as well. And the game is highly adaptable to level, by modifying the sheet. But the best thing to me is that this could definitely lead to students making up a sheet for playing and really getting to do the math.

Handout: gdoc, pdf

Make It, Take It

In addition to making their own game, the teachers made a video for a game of their choice that they'd like to see in a classroom. Sam chose Make It, Take It, a money game I made up a long time ago for a teacher that wanted a money game for 2nd year.

Again, highly adaptable. It's nice how just the progression of play gets players to think of new combinations.

Handout: docx, pdf

More Money

My preservice elementary teachers are preparing to do some 2nd grade tutoring, and the teacher asked for money and time.  (Really, who couldn't do with more of both?)  Those are always challenge areas, in my experience, so I thought I'd share our resources.  Here is a collection of some money activities.  It includes Change for the Better and Make It Take It which were described in an old post, which are two of my favorites.   It's been constructive to use games for practice, interesting problems to help with concept development, and explore multiple representations.  One thing that's often lacking is a visual representation to help understand the relative value of coins.

A quick easy game:  Monopoly Money Madness

Materials:  play money, 2 dice.
Math content:  addition, money recognition, unitizing (grouping into new amounts.)
Game play:  Very simple – roll two dice, and take that much money.  If you can group your money into a larger bill (for example, a five and five ones into a ten dollar bill).  First player to $100 wins.
Variation:  have players “shop” a catalog or the web for something they would like.  Play until they have enough to buy the item they would like.


Money as a context offers some nice practice for developing unitizing, the understanding that allows learners to flexibly exchange between a groups and individual members, or the ability to switch what you are considering as a unit.  (I.e. switching from dollars to quarters or cents.)  It is such a key concept for 2nd grade, as students move from single digit arithmetic to multidigit arithmetic.  Lack of understanding in this will follow them for the rest of elementary school.

From Mathematics in the City: Measuring Teacher Change in Facilitating Mathematizing, Catherine Twomey Fosnot, Maarten Dolk, et al. (link goes to a pdf of the article)

Unitizing requires that children use number to count not only objects, but also
groups—and to do them both simultaneously.  The whole is thus seen as a group of a
number of objects. The parts together become the new whole, and the parts (the objects in
the group) and the whole (the group) can be considered simultaneously. For learners,
unitizing is a shift in perspective.  Children have just learned to count ten objects, one by
one. Unitizing these ten things as one thing—one group, requires almost a negating of the
original idea of number.  It is a huge shift in thinking for children, and in fact, was a huge
shift in mathematics, taking centuries to develop. Understanding that a square in a tiled
array can represent a column and a row simultaneously also involves a construction of
part/whole relations (Battista et. al., 1998), as does the relationship between
multiplication and division. There are many more. Because “big ideas” involve
part/whole relations, they require a shift in perspective by learners.

As we look at state content expectations, it's pretty clear they are focusing on skills.  Fosnot and Dolk (in Math in the City, their books on Young Mathematicians at Work, and their curriculum Contexts for Learning) have a nice way of organizing content into skills, ideas, and models, and then representing them on a landscape.  The preservice teachers took their information and tried to do the same for money.  Here's what they got: (as a pdf)

Money Problems

Excellent variation on the "how many ways to make change problem" at the New York Times today. The Freakonomics column is reporting the work of Patrick DeJarnette. I can see giving this problem from 4th grade to linear algebra!

Click on the money tag to see my previous money games. Click on the cartoon to go to the excellent Non Sequitur website.

Money Games

Here are my two favorite money games. Change for the Better is based on a James Ernest design. He's the genius behind Cheapass Games (don't be put off by the name), and the game this is based on, Fight, he used to have on his business card. It really has some non-trivial strategy and thinking to it. Later I made the connection - or one of my preservice teachers did - with Smart, the excellent poem by Shel Silverstein. The other game I think I invented, Make It Take It. The idea was from a group of teachers who wanted students to be forced to find non-standard combinations of coins, instead of always taking 27 pennies, for example. That suggested a dwindling resource game to me. It's poissible to combine both with some visual representations of money, which is a nice support for struggling students. The handout is here, if you like worksheets or some of the representation support.

Smart
by Shel Silverstein

My dad gave me one dollar bill
'Cause I'm his smartest son,
And I swapped it for two shiny quarters
'Cause two is more than one!

And then I took the quarters
And traded them to Lou
For three dimes -- I guess he don't know
That three is more than two!

Just then, along came old blind Bates
And just 'cause he can't see
He gave me four nickels for my three dimes,
And four is more than three!

And I took the nickels to Hiram Coombs
Down at the seed-feed store,
And the fool gave me five pennies for them,
And five is more than four!

And then I went and showed my dad,
And he got red in the cheeks
And closed his eyes and shook his head--
Too proud of me to speak!

Change for the Better

Materials: Each player needs 1 quarter, 2 dimes, 3 nickels, and 4 pennies.

Rules: Play in groups of 2 to 6. Each player takes a turn. On their turn they put in one coin. They can take out a combination of coins that is less than the value of what they put in. For example, if you put in a dime (10¢) you can take back up to 9¢ – if it is there. Play continues until only one person has money left.

Instruction: Beginning players should just concentrate on the moves of the game. After students have gained some experience with the game, they can try recording their games to translate to symbolic representation. The data collected can then be examined for patterns.

Make It, Take It
a money game for 2 players or teams

Materials: Play coins or coin pictures or cards, amount cards. Record sheet if desired.

Play: Put the coins in the center. Shuffle the amount cards and make a stack. Players each turn over an amount card, and the player with the smaller amount goes first. On subsequent turns, players turn over an amount card, and see if they can make that amount with the coins. If they can, they take the coins. If they can not, it’s the other player’s turn. Play until all coins are gone, or both players in a row can’t make their amounts. The winner is the player with the biggest total value of coins they collected.

Variations:
Recommended starting amounts – 4 quarters, 6 dimes, 8 nickels, 10 pennies. Other amounts can be used. Teachers can add amount cards for more complicated amounts.
Players can roll two dice to determine the amount. (Note the dice variation requires more pennies.) Advanced play allows people to make change with the coins they’ve collected. For example, trading a dime from the center with two nickels they have taken before.
Players can use dollar value charts to keep a running total.

Example:
Bill and Keenya have been playing for a few turns.
Bill turns over 12 cents and takes two nickels and two pennies.
Keenya turns over 25 cents, but there are no quarters left. She takes five nickels.
Bill turns over 50 cents and can not make it.
Keenya turns over 6 cents and takes a nickel and a penny.
Bill turns over …

Instruction:
As with most games, it is recommended to play a game with teacher vs. the whole class to launch the game. Emphasize the variation in ways to make an amount by soliciting other possibilities from the students. Ask questions like “what card would be good to turn over next?” or “what card would leave me with no possibilities?” If someone is stuck, encourage good sportsmanship in helping them figure out a way to make the total. If that doesn’t seem to be working, or you are worried about their ability to make the amounts, students can play in a team of two vs. another team of two.

Many students will try a place value approach first, taking dimes and pennies. This will rapidly run them out of one or the other, forcing them to find other amounts. The amount cards concentrate on values that can be made with one, two or three coins, though several can be made with many more coins.

In summary, the teacher may wish to have students share their strategy for figuring their total at the end of the game. It is important to summarize by having students describe how they knew if they could make an amount or not. Another interesting discussion to start is if there is a strategy for better ways to play the game – is there an advantage to using fewer or more coins to make your moves?