Math Hombre

All this month I'll be posting games from the Fall 2023 GAMES seminar at GVSU. This senior capstone was begun by Char Beckmann. See many of the games in this YouTube playlist. Many of the games completed in my seminar are in this playlist. In the seminar, we play lots of games and math games, the future teachers make a first video to promote a class math game that already exists, we develop a group game (a monster-themed middle school Desmos escape room and Math Heads, a number mystery game), and they develop a game of their own.

This post is sharing Corrina Campau's games - she was also the lead Desmos engineer on the escape room!

Her first video was for Jenna Laib's Number Boxes. Really an all time great classroom math game, it was extra influential to this year's seminar. Like Jordan Burnham's game Boxzee.

Corrina's original game has an original deck of cards, which would have multiple uses, but is great in her Math Spoons (Cards and Rules). What follows the video is her story of making the game, and some thoughts on why to play games in math class and which games are effective.

The Story of Algebra Spoons

Whenever I take a class at GV I am always trying to see how I can use the class to become a more effective, engaging math instructor.  In thinking about what my course content entails I became enthralled with the idea of having students differentiate between different function families.  We study linear, quadratic, exponential and logarithmic functions and so this became my starting point.  I wanted a game that would allow students to think about all of the function families as a whole.  After playing some of the games in class I decided that one of the games that could work would be to design a game like SET where students must match cards based on different attributes.  I kept thinking about SET and how I felt when I played the game.  Although I like the game, I don’t always have fun playing it because I am not necessarily the fastest player when looking at 12 cards and trying to find matching ones.  John mentioned Spoons in class one day, and I thought that was a really great idea.  I have always enjoyed playing Spoons and so decided to roll with the idea.  Thus, Algebra Spoons was born.  I began to think of the number and type of cards needed.  I decided to use linear, quadratic, and exponential function families with 4 cards in a set and 4 of each function family giving me a total of 48 cards per deck.  I knew I needed to include graphs, stories, equations, and tables, but I wasn’t sure if I should choose a theme or not.  I decided to use stories that related to GV students and even chose some stories like they had modeled in class – like the equation of the water as it comes out of the drinking water fountain.  I hoped that the stories would appear somewhat familiar to them even if the story was new.  Once the stories were written then I needed to make sure that the graphs showed the important characteristics of each story so that students would be able to determine the graphs that matched the stories with relative ease.  I also examined the tables and made sure to include the portion of the table that made the most sense when trying to match the cards.  For some of the quadratic functions I used vertex form and for some I used standard form.  In retrospect, I wish I had included factored form as well.  But making these cards took a considerable amount of time and thought, and unfortunately when I thought about factored form it was too late to change.  Having finalized the front of the cards, I decided to make something on the back to make the cards more visually interesting.  Thus, the spoons motif was added.  Ten sets of cards were printed on card stock and printed out in color.  

When I played the game with two of my MTH 109 classes, I first had them sort the cards so they could become familiar with them.  After they had a chance to match all the cards, I then passed out the spoons, and they started playing the game.  The students had so much fun!  I was overjoyed to see how they embraced this game, and this was so much more fun than doing a standard final exam review.  I would encourage all teachers to play this game as it really gives students a fun, enjoyable, and deep conceptual learning of different function families.

Why Play Games in the Math Classroom and What Makes a Game Effective?

Research shows that Games Based Learning (GBL), either digital or non-digital, in education is now one of the major learning trends of the 21st century.   So, why are teachers playing more games in the classroom, and what makes a game effective as a learning tool?  

First, for a game to be effective, a game needs to meet learning targets.  Once an instructor has decided upon what the game should help students learn then a game can be found or created that allows students to meet those goals.  In thinking about LeBlanc’s Taxonomy of Game Pleasures, we can understand the eight “primary pleasures” that arise from playing games and see how these game pleasures help to make games more enjoyable and when games are more enjoyable, they are often more effective.  

A game that requires fewer materials is typically better because there is less set-up and typically less time spent learning to play the game.  Having fewer rules or simplifying the rules is also important so students are not overwhelmed before they begin playing the game.  Games where students’ interaction with other players affects their play attract different types of players and can make the game more fun to play for all players.  A game that generates different situations or has the element of surprise can be more exciting and make players want to keep playing the game, and a game where an early advantage always causes a player to win is not as fun or effective as a game that allows all players an equal chance of winning.  

When I play games in my classroom, I look for games that yield the best results in the least amount of time.  I ask myself – what game can I play that allows students to understand, apply, analyze, evaluate, and create?  Games always make learning fun and interactive, so when I tell students we are going to play a game there is always some excitement in the atmosphere.  Games, if set up correctly, can provide low risk competition and meet learning targets in a manner that is more motivating for students.  The structure of the game allows students to engage in problem solving in a way which is typically more enjoyable and more effective.  Games create a more engaging learning environment and cause more students to pay attention to the teacher’s lessons, and they help students understand the concepts and retain the material better.  Games are also able to reach students of all levels and function as confidence builders.  In addition, game play encourages and deepens strategic mathematical thinking.  Playing games in the classroom also allows educators to easily include active learning in the classroom.  

Spending time creating games or selecting games that are already made is time well spent and worthwhile for students and a very effective way of presenting concepts, creating deep thinking, and motivating and encouraging students, and GBL should be included in every classroom.

Reference

Hui HB, Mahmud MS. Influence of game-based learning in mathematics education on the students' cognitive and affective ___domain: A systematic review. Front Psychol. 2023;14:1105806. Published 2023 Mar 28. doi:10.3389/fpsyg.2023.11058

We had a really interesting week in my number theory class. We are really a seminar, seven teachers and I investigating elementary number theory together.  I hope they're learning half as much as I am.

This week we were exploring primes and modular arithmetic. The first day we were thinking about the \(4x \pm 1\) and \(6x \pm 1\) structures, and the results that there are an infinite number of each type of prime.

To gain modular arithmetic practice, we played Modular Skirmish. (Cf. this post on Gauss.)

Then we started looking at this GeoGebra sketch:

The numbers increase from the bottom left corner up the column. My first attempt was a growing square, but that let you see asymptotic distribution of primes more than the modular structure.

We put this sketch up on the front screen, and advanced n. Teachers noticed the empty top rows (multiples of the modulus) and how some values separated the primes into rows: 2, 4, 6, 8, 10, 12... while others seemed to form diagonals: 3, 5, 7, 11, 13... We wondered about which were the most consecutive primes or gaps in a row, and whether that would change as m increased. (Personally I got wondering about where are the largest square gaps.) Teachers connected many of the patterns to the rows in 6. For example, in 7:
The diagonal really means the next prime is +7 -1 or 6 apart.So it goes back to that 6 structure.

Modulo 10 is really just looking at the last digits. We noticed that no digit seemed more or less common out of the four possible. Also, no consecutive dots more than 2. Is that always true?

The two coolest structure theorems are with respect to four and six. I think these helped in understanding why primes are of the form \(4x \pm 1\) or \(6x \pm 1\). Which may have also helped with the proof that there are an infinite number of primes of the form \(4x - 1\) (or \(6x - 1\) ).


We did find a modulus where there was a row of 8 consecutive primes, but I can't rediscover it!

Understanding the six structure also helped us understand a diagram that we were looking at the previous week, from a designer who was really impressed with a 12 structure. (Source in reddit/r/mathpics. The picture isn't super precise, but did offer a lot of making sense opportunities. And colorful!)

Rather than make the course a tour through the great theorems of number theory, my hope is that it can be an opportunity to do math ourselves. So instead of necessarily illustrating a theorem, I'd rather find a way to notice things that might lead to the theorem. Since we're interested in K-12 applications, divisibility tests and primality tests are of interest; that means exploring the ideas in Fermat's Little Theorem.

So the idea came - given the success of the modality/primes visualization - to visualize exponential patterns in the modular context. This sketch is what I came up with.
Oh! The patterns they found!





Definitely a lot of things that I had not noticed. Not, interestingly, Fermat's Little Theorem, but there were many observations that will lead there.

A lot of our discussion was about pairs of cycles. The visualization made it clear when two different bases created the same path, up to direction. Eg. \( 2^m \mod 5\) and \( 3^m \mod 5\).





















Furthermore, they noticed this awesome pairing within the cycles. Here's the nicest mod 13 pair.

Look back at the other data... there's a lot to notice. And it definitely has me wondering. (Copyright, trademark and kudos to Max and Annie from the Math Forum.)

It's hard to imagine that introducing a theorem and sharing a proof would have resulted in building any more understanding, and there's no way it would have led to doing any more math. And this will make the theorem so much more meaningful when we get there. If we do, with such a fine boatload of conjectures to explore.

So many things to write about to catch up... but it's been a while since I posted a game, so with an impending Math Teachers at Play at Math Mama Writes. The submission form seems to be wonky, so submit directly to Sue. Plus this kind of fits with U.S. Independece Day, as we have been known to fight a war or two.

This game is good already, but could be great. So if you have feedback, let me know, please! As is, it's probably best used as a review game, but I'll comment afterward about how it could be used as a framework for a unit.



Set up: Make your own deck: 11 lines. Each line should be drawn so that it passes through at least two points with integer coordinates, such as (-2,4) or (5,5).

Claim your deck! Mark each line card on the graph side with your insignia. Initials, emoticon, math symbol, etc. – your choice. Tip: make your cards NICE and personalized. Decorations and alterations that do not obscure the line or the math are not only permitted but encouraged.


War: 2-4 players. Each player needs a deck of 11 face down cards, shuffled or not – it’s up to you. Set aside any extras, make one more if you need it.

Players roll the die for the combat. (2nd roll and beyond, the winner of last battle rolls.) Flip over the top card of your deck and follow the combat rule. On a tie, flip over one more card to determine the winner of the battle. If more than two are playing, this is only on ties for best and only the people who are tied.

Play through the deck once. The winner is the player at the end with the most cards. Give cards back to the owner. Except for the Spoils of War.

Spoils of War: Out of the cards the winner captured, they take one card from the opponent’s deck to keep. Add your mark and cross out theirs. This may mean the loser needs to make a new card for their deck for the next game.

Example: The first roll is a 2. Least slope. -2 < 1

(The X and lemniscate are the players' personal marks.)

Math notes:

• Use the cards for sorting activities before playing.
• Have players keep track of hard to determine battles.
• Discuss card design strategies.
• What about undefined and zero slope lines?
• What other combat rules could you have?

Handouts: as a Scribd file, and the graphs template. At the right is an image (larger when you click on it) that you could also print for the graphs.

Discussion: Ted, one of the excellent summer grad student/teachers, tried this cold as an end of year activity with a small group, and they struggled with it.  He felt like it had a lot of promise, but that the math requirements kept students from the game since they were rusty with it.

Trying this with teachers convinced me of it's potential, as it even uncovered math for them to discuss, and generated situations they had to think about. 

I could see this game being at the end of a linear unit, where students have been generating graphs as examples as they go through the topics, using them for activities like finding slope, sorting from least to greatest x-intercept, y-intercept and slope. Use them to construct tables or find equations.  Non-contextual, but strong on representation.  What do you think?

I'm trying hard not to use too many unlicensed images but this is too perfect. God bless you Bill Waterson, wherever you are.

Yeah! I have rarely been so excited to get an email.  "We just pulled your Graphing Story out of the oven!" The only downside is that my name is prominent, when I really just incited other people to make them.  But I'm still geeked. Huge props to Dan Meyer and BuzzMath for doing this.

First, the Balloon.  Filmmaker - Anna Minnebo, Balloonist - Gregg Minnebo.






Second, The Towers of Hanoi. Stacker/Graphist - Eric Thuemmel; Camera - Monica Leneway.  In the original he just got the fourth stack complete on the 15th second, but it just barely gets cut off here. He was quick.





Still to come: four simultaneous tower builders... can understand why the graph for that is a special problem.

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)

Amazing flash animation where you choose the scale with a slider and it zooms from quantum foam to the entire universe. Spectacular.


Reminds me of the classic logscale comics at XKCD. Making their own scale picture is an excellent assignment.

Working with the 4th graders last week, the objective was just to develop multiplication facts, as they are struggling with the multi-digit multiplication.

I'm a big believer in automacity vs strict memorization, as I believe it leads to fluency and solid pre-algebraic thinking, as well as deepening operation understanding. The lesson was pretty simple, but a good place to start.

Objective: TLW see connections between adjacent multiplication facts, and use those connections to help computation.

Materials: unifix cubes, graph paper with a 5x5 structure (link goes to a 2 page pdf graph paper, so it can be printed both sides easily), blank multiplication chart.

Lesson:
Start out with a small set of cubes, such as two stacks of three cubes. What multiplication problem is this? (You might choose if you're going to make an issue of order or not. To me, this is 2 of 3, making it 2x3.) This is 2x3 and 2x3 is 6. We're going to pass the cubes around our group.

When you have the cubes, each person can either add a cube to each stack, or add a stack of the same height. Then you say the new multiplication and what the answer is. I add a cube to each stack and say it is 2x4, which is 8.

As the stacks went around, I saw students slowly gaining an idea of figuring out the next problem by adding on to what they knew before. It took a little bit to get the idea of what multiplication computation it was, but they got the idea of what moves were allowable immediately. Soon, several of the students were adding to get the next fact.

We restarted with 3 stacks of 1. The students were much more fluid. There was a bit of an issue with the cubes being distracting with them. If I had thought about working with students who hadn't used the cubes much, I would have given them time to play first, setting up multiplication problems of their choice.

The next phase of the lesson was to move to graph paper. We drew a 2x3 rectangle, and the students were comfortable with thinking about that as 2x3. We did one together, where the group decided which side to add squares to. 2x4, 2x5, 3x5, ...

Then each student got their own graph paper and started building a chain of rectangles, with the new dimensions filled in and the result. I saw several students using the adding strategy. One student didn't get the idea of what the connection was, and just drew rectangles and filled in the area. But maybe that was what she needed to attend to.

As it was time for students to go, we summarized by looking at a multiplication chart. Filled in a fact they agreed on, 6x5. I led them through how to use that to go on, by adding to get to 6x6 or 7x5. They went back to class with their own chart and a page of graph paper. As I saw them working on the charts in their free time, some were using patterns as they had seen them before, some were using them for the first time, and one student asked for how that worked. A couple of examples got her started.

The next week: The week after this I tried to get the students to help me develop a game. A couple of them found it less than engaging, but Mrs. B mentioned that all the students were antsy. Day before a long weekend? Cabin fever? The game is designed to become obsolete, but I don't think that's the issue. I picked it because I've wanted to work this out, and the other thing they've been working on in class is area and perimeter of compund rectangular shapes.

Break Up (In development)

Two players or teams.
5-structure graph paper, pen, optional dice.

Game play: Determine the size of a starting rectangle. This can be done through choice, each team choosing a side length, or rolling three dice, or rolling four dice, or rolling 2 dice plus 10. If dice rolling, each team should roll one side.

On your team's turn, you either divide a rectangle, or calculate an area, or do both. Your team gets a point whenever an area is filled in. After all the areas are filled in, the team's whose turn it is next gets to try to find the total area. To emphasize using known facts, you can only fill in a rectangle if you know the area as a fact.

Examples: you determine a 12x15 rectangle. The first time divides the 12 into 10 and 2, and fills in 10x15=150. The second team divides 15 into 10 and 5, and fills in 2x10=20. The first team fills in 2x5=10. The second team finds the total, 150+20+10 and gets 210. So 12x15=180.


You determine a 9x12 rectangle. The first team sections off a 6x9, and doesn't know that as a fact. (There was one student who loved dividing in half.) The second team split off 5x9 and filled in 45. The first team filled in 1x9. The second team filled in 6x6 as 36. The first team (fudging a bit) figured 3x6 with 12+6. The second team mis-added 45+9+36+18 (hard sum!) and the other team got 108.

Notes: I thought the game would be better as a cooperative game, but the kids wanted to try it with points. They thought about scoring the area (as I have) but that gives the first team too big an advantage. They looked forward to scoring points, but didn't seem to care much about winning. They thought maybe you should keep track of points across multiple games. It did strongly encourage mental computation.
There's not much strategy to this game. It's about tic-tac-toe level that way. I could see this turning into kids designing their own board of compounded rectangles, that might be interesting. But it definitely encourages relational thinking for multiplication facts, which is worthwhile. If anyone has ideas for improving the gameplay, I'd love to hear them.

EDIT:
Sue VanHattum, from Math Mama Writes, was reminded of a game called Raging Rectangles from a North Carolina instructional resource packet. See the comments for details.

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?

Excellent graphical portrayals of percentage at this site:
The World of 100.

I'm imagining using them both to have students think about comparisons with their own community, and looking at the designs as mathematical representations to see if they represent or how they represent the numerical information.