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 from her seminar 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 this year), and they develop a game of their own.

Keri Herman chose Tens Go Fish, a classic addition fluency game. As an extra feature, she demonstrates the game with Tiny Polka Dot cards. (Find them here at Math for Love.)

Keri's original game was a new idea for the seminar: she was interested in making a puzzle. I recently saw a description of a puzzle as a game for one person.  That certainly fits here. The puzzles are available on a Google doc here. What follows is Keri's story of making the game, and her ideas on why we should play games in math class.

Story of my Game

I knew when I got the opportunity to create my own game, I wanted to develop something that was related to quick multiplication facts. The reason being that my memory of learning my multiplication tables was always timed and quite stressful for me as a young student. I wanted to create a game where students could get great practice of their multiplication facts, and build in aspects of a good game; strategy, any player can win, etc. 

My first idea was a game board, moving the amount of spaces of the product. However, I was then drawn to the idea of more of a maze. I started with a small grid and filled in very small multiplication facts, students would have to find their way to the end. This turned into the development of three mazes, 5 x 6, 7 x 8, and 9 x 10, all with their own unique solution. To figure out how to design these mazes took a lot of different approaches, starting from scratch, and overall just thinking about how to make them work. I believe that the final product of these mazes will provide students with a very fun way to practice their multiplication, while being able to try to solve the maze. 

The goal was to have a large mathematical objective for the game. Students will be focused on trying to find the solution, even if they are going the wrong way, or have to start over, they are still constantly doing the math and getting practice of their multiplication facts. I think this game would be something that teachers should play with their learners because you can never have enough practice with multiplication. Especially in the 9 x 10 maze, all multiplication facts are used from 1 through 9 (not including zero). These mazes will also help students recognize patterns between multiples, factors, and products. 

These games could be used within a lesson, if students finish early, or simply just given as an opportunity for more practice, without time constraints. I also share within my video the development process of these mazes. With students who have learned multiplication facts,  I think it would be a great idea to turn this into a project or performance assessment. Students can work to develop their own maze. Not only does it take strategy, but at the same time students are able to continue working with the facts themselves and continue to recognize patterns. Overall, I am very proud of the way these mazes have turned out. I want to continue to show these to math educators and I hope that students will enjoy solving them as much as I had hoped. 

Why Play Games in the Math Classroom?

As a future math educator, incorporating games into the classroom is something that I want to use and will continue to encourage others to consider as well. It is often looked over to play games in the classroom, but the reasons as to why they are beneficial to student education should be considered. There are few specific reasons that are important to point out, including; building mathematical knowledge and skills, collaboration with peers, student engagement, critical thinking skills, and more. Each one of these reasons in its own makes games in a math classroom worthwhile. 

Building Mathematical Knowledge

Math games all are built upon their own goals and mathematical objectives. Teachers have the option to choose a game that targets the content that is being focused on. To find a game that can build mathematical knowledge, choosing a game that is relevant to your current learning goals within a classroom can help students extend their skills. There are so many aspects built within games that students can pick up on mathematically, without noticing. This can be beneficial to students because they are still learning, but without the title of class, homework, or assessments. 

Collaboration with Peers

It is important for a classroom to have communication among students that can lead to quality discussions. Discussions can uncover so many helpful aspects to student learning. In a game setting, a lot of times students will play with each other in teams, or against each other. In both cases, students are able to communicate and learn from each other. Students are able to pick up on each other’s strategies and build off of them. When playing with each other, this can help build a more positive classroom environment. This is because this type of communication is not usually seen in a regular lecture or discussion. 

Student Engagement

Oftentimes we hear negative assumptions about math and negative attitudes are common when stepping into a classroom for some students. It is important as teachers that we are able to increase student interest by engagement and participation. Incorporating math games into the classroom is a great way to develop student engagement. A lot of times, the mathematical objective of games are mixed in with aspects of interaction, surprises, and fun. A game can also change the view of many students. All students can participate and it is important to use games where any student can win. In math class, students can often point out the “smartest” students and become discouraged. When using games that are designed that anyone can win, not just based on skill, this can build a lot of confidence in students. 

Critical Thinking Skills

In many situations, students become disengaged after they reach the level of knowledge and understanding. However, it is things like analysis, critical thinking, and application that get students to really push past that level of reasoning for the content that they are learning. Math games provide a different way to push students to build upon their critical thinking skills. Having to figure out a strategy to finish or win the game is a very important tool when it comes to building these skills. With that being said, games that are chosen to play in a class should have aspects that involve strategy. 

Overall, math games have so many advantages when it comes to incorporating them into the classroom. Being able to play different types of games this semester has taught me so much about what a good math game should look like. Being able to develop and create our own group game, and my own game has changed my perspective on math games. Math games can help students learn in a unique, fun, and interactive way. 

 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 from her seminar 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 this year), and they develop a game of their own.

Kacy Jeffries chose Number Boxes from Jenna Laib for her first video. See Jenna's blogpost for it here. This game was really influential to the seminar this year. Corrina Campau made a high school/college focused video for it, and Jordan Burnham made a game built on that structure, Boxzee.

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

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.

Jordan Burnham selected Close to Zero, and integer addition game for her first video. Handout and original blogpost.

Jordan's original game Boxzee crosses one of my favorite classroom games, Number Boxes by Jenna Laib, with the classic Yahtzee. What follows is Jordan's explanation of the game and thoughts on why play games in math class.

Boxzee

When I was first brainstorming games, I had absolutely no idea what kind of game I wanted to make. It wasn’t until one day when I was sitting on my bedroom floor that the starting ideas of Boxzee came to me.

Originally I imagined the game to have more moving parts. I first had players each being dealt 4 cards. From there they would roll a dice twice to determine a specific operation they would be using (odds = subtract, evens = add). Then after finding out those operations you would choose 3 cards from your hand to find a largest total value for that specific round. I found that this became a little confusing and players wouldn’t necessarily be able to truly “compete” if all of their rounds operations were different than each other. If one player only rolled odd values then they would be predetermined to loose solely because the other players would have a better chance of having larger numbers if they rolled more even values. 

Moving on from here, I decided to instead come up with the number box sets. Rather than using the dice to determine operations I decided this was a more structured way that players could still affect the total value by the cards they put in without having so many moving parts. I first came up with the idea to have four different rounds. The players would both have 4 cards in their hands and needed 3 to fill into the number box sets. I also decided that they would both fill in the top box row, then move downward. After playing this a couple of times I realized it could be very common to tie. So then I chose to create a number box set that would be the final round and would use all of the cards in the players hand. I liked this much more. 

Then to incorporate more of a feel of Yahtzee, I decided that players should be able to substitute their cards into any of the top 4 number box sets of their choice in any order. This gives them more of a chance to use higher cards and lower cards when they have them for specific rows that those cards would be more valuable for each round. 

Some final touches were made after play testing with Professor Golden and my classmates. These included allowing players to chance any of the cards they have in their hand. I really enjoyed this change because it gives players more risk opportunities. The queen card was introduced as being a wild card during this time as well. I appreciated this idea because I feel like it allows players to more strategic and intentional about where they substitute certain card values into the number boxes. Finally I made a coupe of variations. I came originally came up with the addition and subtraction version of the game. I then decided to toy around with the idea of multiplication and division and made the multiplication and fractions versions.

I think that teachers should play this with their students because it makes basic operations more exciting. I think that allowing students to have so much control over placing values into expressions and solving these is something they will enjoy. I also believe that it allows students to grasp where they may rather place a larger value versus a smaller value. Since the goal is to have the largest total value for each number box set, it will look different for each set. Placing a 9 in the same value that you place a 1 or a 0 has much different affects. 

I believe that this game can be adapted and used for so many reasons. The framework of the rules and rounds is something that creates such a great skeleton to then use with multiple content areas. I have thought about creating a Binomial Boxzee and think that this would be a great next step as well.

Why Play Math Games?

Math can sometimes be a very intimidating subject area for some students. Because of this, I believe that it is important to keep the classroom environment exciting and reassuring that every student has the ability to be a mathematician no matter what level of skills they may think they have. To do this, incorporating games into the classroom can be very beneficial.

Math games are a great resource for teachers to use to introduce and practice content. When playing games in the classroom in allows students to learn content in a more relaxed environment. This allows students to feel less pressure when making mistakes. This is important because students will be more likely to try and continue trying even after making mistakes which will help them master content areas. Similarly, playing these games allows students to build their strategic and problem solving skills. They want to perform their best and win, so they are able to develop strategies that can help them succeed throughout the game.

I also believe math games are beneficial in the classroom because they can be interactive. This allows students to also help each other in teaching the math skills. By not only performing the skills needed for the game, but also using their skills to help teach their classmates they develop a deeper understanding for the content. 

Finally, playing math games allow students to build a love of math. When students are engaged and having fun playing these games, this is when they will be doing the most learning. Exposing students to games that are centered around math subjects, they will be able to see that math is more than just what they may be learning to compute in class.

Now seeing some of the benefits associated with math games, it is also important to identify what makes a good game. One of the biggest things that I believe makes a good math game is having minimal time constraints. When students are practicing their math skills within a certain amount of time some may start to feel discouraged if they are not as fast as their other classmates. With this in mind, choosing games that give students the same opportunity to be successful at completing the game whether they are fast thinkers or need some extra time is very important. 

I also believe that a good math game allows for catch up. This means that even if a student is “down” in a game or is behind, there are aspects of the game that allow the players to quickly catch up and still have an opportunity to win. Since some students may not succeed right away, offering an opportunity for them to catch up and still have a chance to win this makes the game more fun for all players. This also makes students more likely to want to play and in turn allows them to practice and learn without the fear of losing. 

In conclusion, math games being incorporated into the classroom that I urge many educators to try. Not only to practice content, but also to help build up students’ love for the subject and confidence in their own skills.

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.

Leah Barber selected Greater Than for her first video, an integer multiplication game. (Handout)

Leah's original math game is a great spin on Uno called Geo. Cards & Handout. What follows is Leah's explanation of the game and thoughts on why play games in math class.

How Geo Came To Be

My idea of Geo came from Professor Golden mentioning Uno during one of our classes. I thought that Uno already included a lot of good components of a math game. This included number recognition, being able to categorize and identify different elements of a category, problem solving, catch-up factor, surprise elements,  etc. Since Uno already had strong components of a math game I decided to create a game that was based on it. At the start I was thinking about doing a game that had to do with geometry so I began thinking of ways students could categorize shapes. Initially I didn’t know if I wanted students to create their own connections between different shapes, so I considered doing a Guess Who style game. However, after trying out a draft version of it I thought Geo would not only be less complicated but it would still offer students the opportunity to practice identifying shapes based on properties and computing area. From here I decided that instead of colors and numbers, like regular Uno, the two categories would be shape and area. Then I went through and made a rough draft of the game that iterated through many revisions until I was happy with its final form. Throughout these iterations I changed things like what the special action cards would be, what shapes would be included, how many cards would be included, what the shapes looked like, and what information I would include on the individual shape cards. 

Why Teachers Should Play GEO:

There are many reasons why teachers should play Geo with their students. Geo covers different Michigan Math Standards such as: CCSM. 6G.1: Find the area of right triangles, other triangles, special quadrilaterals, and polygons and CCSM. 5G: Classify two-dimensional figures into categories based on their properties. Beyond letting students practice finding the area of different polygons and identifying shapes by their properties, Geo helps students practice integer multiplication, reason mathematically, and build problem solving skills. Due to Geo being a competitive game, students often become engaged doing math, checking the work of other students, and reasoning mathematically in order to win. This is another reason why teachers should play Geo with their students. Geo allows students to engage in math in a fun, interactive way. Many learners have anxiety around math or think that it is boring, hard, irrelevant, etc. Geo is a way to get learners engaged and have fun while doing math. 

Other Uses: 

The materials of Geo could be used outside of playing the game. Teachers could use the cards to create a memory style game where students try to match different areas or shapes. Other uses include going through the cards as examples of computing areas with students. Teachers could also play a Polygon Capture style game where students identify all the shapes they can that fit under the different command cards. Following playing Geo teachers could have a discussion with students about what they noticed or wondered when playing the game. This could start a good dialogue about different shape properties, how different shapes are related or different, definitions of shapes, etc. They could also have students discuss strategies and problem solving skills they used to try to win. 

Why Play Math Games? 

There are many reasons to play games in the math classroom. To start, math games allow students to engage in mathematics in a fun, interactive way. Students often think that math is boring, too analytical, irrelevant, etc. By playing games in the classroom students can experience math in a way that it often isn't presented to them. This can also dispel anxieties many students experience with math. Due to previous bad experiences with math, whether it be a harsh teacher, tough material, or overwhelming course load, students can develop anxiety surrounding math. This can also affect how students think of themselves. Bad experiences with math that cause students to do poorly can lead to them thinking they are dumb or not a “math person”. By involving games into lessons students can create positive experiences with math and start to dispel any anxiety or negative thoughts surrounding math.

Math games also allow students multiple entry points to engage in math. Oftentimes this idea of not being a “math person” is due to inaccessible lessons. By including a math game in a lesson you can create many opportunities for students to participate in math. A good math game includes some aspect of luck, strategy, catch up, or surprise that allow students who are struggling to still succeed. By creating accessible activities for students they can start to think of themselves as someone who is capable of doing math. 

Getting students to reason and express themselves mathematically can be challenging. Often students don’t want to participate in discussions in math class due to a multitude of reasons. Including a math game however is a great way to get students talking about math. Due to the competitive nature of games students are more likely to reason, argue, make conjectures, and express mathematical ideas in order to win. This creates a great dialogue where students can think through material covered in class together and come to conclusions on their own. By doing this students will continue to grow their self concept as a mathematician and be able to better communicate mathematical ideas. Math games also help students build problem solving skills. A good math game has players interacting with each other and constantly trying to figure out their next move. As stated before a good math game also includes strategy. These elements allow students to build their problem solving skills as they identify what they need to do to win, how they are going to do that, executing their plan, assessing how it worked, and what they will do next time. 

Lastly, including math games in the classroom is a great idea because it is a great way to introduce, explore, or practice mathematical concepts. Teachers or parents may feel that including a game in a lesson will distract students from their learning. This however is not the case. Math games are not something that is just filler. Instead math games are great ways to introduce new concepts by allowing students to get familiar or explore with new ideas in a low stakes, fun environment. Math games can also be used to help students review a concept they already learned by applying their knowledge in a new way. 

 I'm teaching a quick 6 week Intermediate Algebra (linear/quadratic/exponential) for incoming freshman this summer. Part of my goal is to convince them that math is different than how they might have been exposed to it. On day 1, we started with Wordle. A few learners had played it before, but quickly the whole class picked up the idea, and there were several good deductions about which letters could go where. The rest of the week, we played the daily Wordle before class the rest of the week.

This week, we started with SET. A little harder to understand, but there's so much logic. The daily puzzle has up to six solutions, which seems to allow for more participation. (Kelly Spoon noted Set with Friends for online actual game play, plus variants.)

I had ideas about what I wanted to do in subsequent weeks, but I was curious what others think and asked on Twitter. BOOM, people exploded with a bevy of resources. I used to have a blog where I shared resources, where did I put that...? After Sam and Julie posted about moving to Mastodon (because of Twitter's Troubles), I tried posting there, too.

Math Online Games & Apps

• FiddleBrix suggested by Benjamin Dickman. He suggested downloading the app, then handwrite a previous puzzle. This is a super challenging puzzle, to me, but Benjamin's suggestions are gold.
• SumIt puzzle suggested by Kelly Spoon. Lots of stuff there.
• Beast Academy All Ten also via Kelly. Really great arithmetic challenge.
• Draggin Math pay app, 
• Shirley McDonald suggested a lot of great stuff: All Ten by Beast Academy (always an open tab in my browser), Number Hive (like the Product Game on a hexagon board), Skyscrapers (Latin square with clues, from a site with lots of puzzles) and Digit Party (implementation of a Ben Orlin game; also an open tab, I may have a tab problem).
• Shirley also recommended Mathigon's Puzzle of the Day. I've been playing that in an app more days than not. (I think I'm getting better?)
• Kathy Henderson suggested the NYT Connections game, which I hadn't seen yet. That is very much in the spirit of what I'm looking for!

IRL Math Games (Free and Commercial)

• David Butler has a great collection of activities, his 100 Factorial. He singled out Digit Disguises and Which Number Where
• Neal W recommended: Quixx is a great dice game and very easy to learn. My students love 20  Express. There are rules and scoresheets online.
• Tom Cutrofello suggested the excellent Turnstyle puzzle he designed for Brainwright!
• Prime Climb by Dan Finkel, suggested by Amie Albrecht. She notes, especially David Butler's human scale Prime Climb. (Which I have played and love.
• Anna Blinstein suggested Anna Weltman's Snugglenumbers, which is a great variation on a target number game.
• Pat Bellew said remember the original: Mastermind. Erick Lee has a Desmos activity implementarion of the math version, Pico, Fermi, Bagel.
• Sian Zelbo claims Jotto is better than the either Wordle or Mastermind. (Online version.)
• Becky Steele cited David Coffey for Taco Cat Goat Cheese Pizza as well as Farkle.
• Chris Conrad recommended Quarto, amazing strategy game. Amie mentioned you can play with SET cards - how amazing is that idea. Karen Campe remembered this great Aperiodical article about the game.
• Mardi Nott, Bradford Dykes and Jenna Laib vouch for Charty Party - that's a strong recommendation. Bradford also brought up this stats version of Spot It, the Graphic Continuum Match It Game.
• Ms. Morris suggested Nine Men's Morris. Interesting game idea.

Puzzles

• Kim McIntyre suggested Sarah Carter's big collection of classroom puzzles. I have learned so many puzzles from her over the years, but especially the Naoki Inaba puzzles.
• Speaking of Japanese puzzles, Gregory White suggests Shikaku.
• Benjamin Dickman and Shirley and Gayle Herrington suggested KenKen. I've used those with younger learners and college students.
• Karen Campe had several suggestions, some in this blogpost. Times UK puzzle page, StarBattle, Suko, 
• suggested Mobiles. Love those, and we do lessons based on them. Here's a challenge problem I asked them!
• Druin suggested the Puzzle Library, which I can't access for some reason. Looks like they're intentionally made for schools.
• Susan Russo linked Cryptograms, which are some cool cruptographic puzzles. I haven't tried anything like this and am curious.
• Sarcasymptote brought up Sideways Arithmetic from Wayside School, which is what I was expecting from Cryptograms, thinking it was cryptarithms. But somehow have never seen that book despite loving Wayside.
• Ms. Morris linked a Magic Square app.

Some years I'm fortunate to be able to lead a capstone seminar where future teachers research math games and develop a math game of their own.

Gretchen Zeuch developed Fraction Reaction to be a simple to learn, easy to play, fast game that works on fraction magnitude and mixed number fraction equivalence. 

She writes:

The process of making this game had many stages. The first stage was deciding what kind of materials I wanted to use in my game. I decided to use a standard deck of cards because I really wanted to make a game that was accessible to every classroom. I then had to pick the mathematical content I wanted my game to be based on. I started by just laying out all the cards in a standard deck and brainstorming different mathematical content. I finally landed on fractions because I liked the students being able to physically see it. I then decided that making the connection between improper fractions and mixed fractions would be the most helpful. I then went through a lot of trial and error by playing the game with a variety of people. This helped me decide how points would work, specialty cards, and general playing rules.

This game is great to teach in a classroom when students are learning about improper and mixed fractions. It is very easy to teach to students as well as all students will be able to play at the same time because of the accessibility of the materials. This game will help students make the connection between an improper fraction and a mixed number. They will also be able to compare the sizes of mixed numbers and improper fractions so identify which is larger and which is smaller. Overall, this game is simple to understand and helps to solidify students' understanding of improper fractions and mixed numbers.

There are a few different uses for this game in a classroom. The first use is that, while students play, you can have them record all of their improper fractions turned into mixed numbers and then have them sort them on a number line. Another use is for students to record their answers during the game and then answer some comparison questions at the end. Lastly, another in class use for this game is to have students discuss the differences between fractions and mixed numbers and how they relate to each other.

Rules - https://bit.ly/FractionReactionRules

In addition, Gretchen made a video to promote the integer game, Zero Rummy. She  writes: This is a great game to use with young children to get them working on their addition and subtraction or to help introduce the concept of negative numbers. This game should be used as a fun exercise rather than to teach a skill. The great thing about this game is that it is stimulating for children so that they are doing math without knowing they are. It is very easy to use in the classroom with minimal materials and does not take up a large chunk of time. Children really enjoy this game and it is a very easy game to play for many ages with multiple variations.

Rules: https://bit.ly/ZeroRummy-rules

Some years I'm fortunate to be able to lead a capstone seminar where future teachers research math games and develop a math game of their own.

Melanie Hanko came into seminar with a vision of making a math game inspired by Bohnanza, a collecting and trading game with a lot of strategy and a fair amount of luck. She really worked on the details for this game. Often times we focus on making games that use minimal materials, but this is much like a commercial game, with a lot of necessary components. For a teacher wanting to give it a try, I would love to see the learners get involved with making the cards. 

Melanie writes:

In the hopes of making an exceptional game, I set off looking for game structures that were simple but had a lot of potential. Then, interested in the structure of Bohnanza: The Bean Game, I started looking at mathematical content that involved some sort of sorting. Eventually, I landed with organizing shapes into hierarchies - specifically quadrilaterals. This is largely based on a 5th grade standard. polyGONE: The Shape Game is the sort of game that engages students with mathematical discourse and reasoning minus the negative attitudes about math. While players need to have a good base understanding of the hierarchy of quadrilaterals and the different types of triangles, this game will help players to create more connections between shapes and gain a broader understanding of what gives a shape its name. 

A lot of the pieces of the game are designed with specific purposes, either to clear up misunderstandings or confusion in early versions or to clear out some of the underlying confusion. The part of the game with the most meaningful design, is the deck of cards. These cards are created to broaden player’s understanding of shapes. Included in the cards are traditional and non-traditional shapes. Different cards show different attributes of a family, like parallel lines, congruent lines or angles, and even lines of symmetry. Different cards show different looking shapes - for example both a concave and a convex kite. This differentiation within the cards, will broaden player’s understanding of shapes and relationships between shape families.

Another purpose of the design of the cards is to increase their usefulness. With all of the cutting and printing, the cards better be usable for multiple occasions. Since there is so much differentiation between the cards, you can easily use them in a sorting or a matching activity. Even before playing the polyGONE, you could match cards based on if they have certain attributes. For example, matching cards that have two pairs of congruent sides. The cards can be used in explorations of the “rules” for each shape family. For example, deciding if a right angle is necessary for a trapezoid, or if it is something that only occurs in some trapezoids. These and other activities can be easily supported with these cards and will help to broaden students’ understanding of shapes and the shape hierarchy. 

The teachers also make a video for a good math game which they would like to promote. Melanie found one of Kent Haines' games that is a very good Nim variant. She writes: 

The 100 Game is a part of the math game genre of nim, which are mathematical strategy games in which players take turns removing objects from distinct piles or groups. Not only does the 100 Game require almost no materials and setup, but it is a fun game full of mathematical reasoning. In the forefront, the game makes practice subtracting within 100 enjoyable. Behind this practice, players strategize how to not be the last person to take away from the total. This requires deductive reasoning, an important mathematical skill. Besides the math, this game is quick to learn and engages players quickly - even unwilling players. 

Mathematical Applications: practice subtraction, strategy and deductive reasoning

Materials: paper and pencil, two players

Object of the Game: Players start at 100 and subtract any number 1-10 from the total. The goal is to NOT be the last person to subtract a number. So you want to subtract the second to last number from the total.

How to Play:

• Player one will start the game by saying “100 minus [blank] equals [insert new total]. You can only subtract numbers from 1-10.
• Then both players will write out the subtraction sentence player one just said out loud.
• Now, it’s player two’s turn. This player will pick a new number to subtract, say the subtraction sentence, and both players will write down the sentence.

Example Play: Here is an example of what each player would say for a few turns. Remember that BOTH players are writing down the subtraction sentences as well.

• Player One (P1): “100 minus 5 equals 95”
• Player Two (P2): “95 minus 10 equals 85”
• P1: “85 minus 7 equals 78”
• P2: :78 minus 9 equals 69”
• ...
• P2: “23 minus 9 equals 14”
• P1: “14 minus 3 equals 11”
• P2: “11 minus 10 equals 1”
• P1: “1 minus 1 equals 0”

In this game, player one lost because they were the last person to subtract a number from 100.

Notes: After you play this game a few times, you might start to develop a sure strategy. In fact there is something special about the number 12. Finding this strategy is what engages players in deductive reasoning. Some questions you might want to ask yourself or your students/children include the following:

• What should your strategy be?
• How can you ensure that you will win?
• At what point in the game do you need to start using your strategy?
• Does it matter who goes first?

Be sure to check Kent's blogpost for more ideas.

 Some years I'm fortunate to be able to lead a capstone seminar where future teachers research math games and develop a math game of their own.

This is Alaina Murphy's game, Sorry, It's Fractions. She was really persistent in the playtesting for this game, and did a lot of work to make it fun while keeping the math content front and center in a natural way.

She writes:

When coming up with this game, I knew I wanted to make a game that dealt with some aspect of fractions. In my opinion, fractions are one of the first aspects of math that students begin to lose interest, lack understanding, and start to hate this subject. So fractions it was. Next, I wanted the game to peak their interest, while having some mechanics that they might be familiar with. Thus, I chose to utilize a board game that many kids have played at some point in their life - SORRY. This would allow kids to focus more on learning the math of this game in comparison to first trying to figure out how to play the game. So, I had the content area and the mechanics. The next step was deciding how this was going to work. I wanted to make sure that thirds and fifths were included in this game because I believe these are the scary fractions to students. I find that students have an easier time with even numbers, but give them an odd denominator and they are out. The best denominator for including halves, thirds, fourths, and fifths was 60. So what better way to help students understand the numerical value of fractions and become more comfortable with them than using a clocklike numberline! 

The rest of designing this game involved play testing to decide how exactly I would apply the mechanics and actually designing the game board. The best way to get students to want to do the math and find the most reduced fraction was to make the fractions they landed on special, rather than the cards. I wanted to ensure that the materials of this board game would be resources a teacher could acquire. So, the board can be printed or they can have students make their own, place markers can be anything - sticky notes, erasers, beads, paper clips it doesn’t really matter - and I either wanted to use dice or playing cards to move around the board. By using a deck of playing cards, students would be able to draw larger numbers and make it further around the board to larger fractions, because the probability of getting a card with a higher value is higher than if they were to roll dice. Plus, the probability of getting any value is equivalent between cards where it is not when rolling dice. In order to make the game faster for classroom use, I incorporated four entrances to home that all players can enter and reduced the number of place markers to two, requiring only two pawns to make it home for the game to end. I incorporated a lot of DRAW AGAIN fractions as a way to make it further around the board and as a catch-up mechanic. Bumping, swapping and sorry’s are also catch-up mechanics and they make the game more competitive, creating more interaction and discussion. Lastly, I wanted to use the colors of SORRY, but I also wanted to create a board similar to Prime Climb where the colors have meaning. So based on the factors of 60 I wanted to color coordinate the prime denominators.

• ½ is blue which is a primary color because 2 is a prime number.
• Thirds are red which is a primary color because 3 is a prime number.
• Fourths are a dark blue because 4 = 2 x 2 so it is the combination of two blues, producing a darker shade.
• Fifths are yellow which is a primary color because 5 is a prime number.
• Sixths are purple because 6= 2 x 3 so it is the combination of blue and red, producing purple.
• Tenths are green because 10 = 2 x 5 so it is the combination of blue and yellow, producing green.
• Twelfths are a dark purple because 12 = 6 x 2 = 3 x 2 x 2 so it is the combination of red and two blues or red and a dark blue, producing a dark purple.
• Fifthteenths are orange because 15 = 3 x 5 so it is the combination of red and yellow, producing orange.
• Twentieths are teal because 20 = 10 x 2 = 5 x 2 x 2 so it is the combination of yellow and two blues or green and blue, producing teal.
• Thirtieths are gray because 30 has many factors so it is a combination of many colors but one less than 60 making it gray.
• Sixtieths are black because 60 also has many factors so it is a combination of many colors and they are irreducible so I wanted it to be the same color as the outline. 

This is a great game for all types of learners to become more comfortable with fractions. Visual learners will be able to utilize the clock model and color scheme, hands on learners will be able to use the structure and game aspect, auditory learners will be able to use the discussions and verbal addition and reducing, and if teachers had students make their own boards it would be useful for those who learn from writing. 

This game is a great way to get students excited about adding and reducing fractions while becoming more familiar with factors of 60, exploring prime numbers, and ultimately improving their understanding of fractions. Other applications of this game would be to refine subtracting fractions skills by playing the game counter clockwise and subtracting the value of the drawn card, rather than adding. In order to incorporate more unlike denominators, the game board could be labeled in the most reduced form (i.e. rather than 30/60, label it as ½) and the students would add the cards in the same way. This board could be used at a younger age range to better understand adding or subtracting and number sense by labeling the board with whole numbers and playing in a similar way - this variation could be useful for learning to read a clock as well! Lastly, this game could be modified to the unit circle with pi/12 radians or 15 degrees and played with dice - here it would be beneficial for students to create their board as they go using trig to come up with the value of each position. 

Some problems that apply to this context:

• Reduce 24/60
• Reduce 13/60
• Which fraction is closer to one, ⅔ or ⅗? 
• If there are 60 people at a party and 12 are vegetarian and 4 have a nut allergy what fraction of people at the party have a dietary restriction.
• If it takes me ⅚ of an hour to get ready for school and the bus leaves in 48 minutes, do I have time to make it to the bus if it takes me 1/15 of an hour to walk to the bus stop? If not, how much time do I have to get ready?
• If I am 3 minutes away from the bus stop and it takes the bus 1/10 of an hour to get to my stop, and my sister walks 11 minutes home from school. Who will get home first? What fraction of an hour will it take each of us to get home?

Rules: https://bit.ly/SorryItsFractions-rules

Board: https://bit.ly/SorryItsFractions-board

The teachers also made a video for a math game they wished to promote. While there are other videos for games called Guess My Rule, Alaina wanted to share her own take. I heartily endorse this, and have used it myself from 2nd grade to university. She writes:

There are various reasons why Guess My Rule should be used in your classroom. First of all, this game requires little to no materials - no printing, cutting, or random pieces needed. As long as students have a way to record numbers they will be set. Games, such as this one, will get students thinking about math in a fun, hands-on way that encourages collaboration and critical thinking. With this version of the game, students are encouraged to explore functions and identify patterns that will allow them to predict outputs and eventually deduce a rule. This game will give students an opportunity to experiment with expressions, practice solving equations, and familiarize themselves with symbolic representations. 

If you are not convinced yet, there are so many ways that we can apply the framework of this game to learn and practice math!  If you plan to use this game in an algebra class you will not be wasting your time, because it can be applied to any algebraic function and even graphs. In geometry this game could be used for guessing what axiom a figure or statement applies to or for learning terminology by grouping correct shapes. It can also be used with younger kids to learn simpler arithmetic. Lastly, we can extend this problem to higher level learners and explore various rules at the same time, not limiting the rule keepers to linear functions but allowing them to pick from any range of functions. So why not use this game?

Standards: 

• CCSS.MATH.CONTENT.8.F.B.5 Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear).
• CCSS.MATH.CONTENT.8.F.B.4 Construct a function to model a linear relationship between two quantities. Determine the rate of change  and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
• CCSS.MATH.CONTENT.8.F.A.2 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). 
• CCSS.MATH.CONTENT.8.F.A.1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.1

Rules:

• Rule Keeper makes a rule
• Rule Guessers take turns giving an input
• Rule Keeper records input, calculates output (secretly), and records the output
• Rule Guessers continue to one by one give inputs until they feel they have found the rule
• ON THEIR TURN, Rule Guessers must say I would like to guess, then they must give an input AND predict the output of their given input.
• Rule Keeper informs the guesser if the output is correct
• If the output is CORRECT, the Rule Guesser guesses the rule
• If the output is INCORRECT, the next Rule Guesser continues giving an input or they can choose to guess.
• If the Rule Guesser successfully guesses the rule, they will become the next Rule Keeper and the current Rule Keeper becomes a Rule Guesser

Link to John's version of the game.

Some years I'm fortunate to be able to lead a capstone seminar where future teachers research math games and develop a math game of their own.

One such is this high school algebra game from Lucas Pohl. He writes about this in what follows.

When thinking about creating a math game myself, I knew I had a couple goals in mind. We had done multiple readings about what makes a good classroom game, and obviously I wanted to fulfill those criteria. Things such as being engaging, strategic, and grounded in coursework were very important to me. I had two initial thoughts: at first, I wanted to do a game that is based on statistics. Statistics is one of my favorite areas of math, and I think that it could lead to a great board game. However, I ended up going to my second thought, which was an adaptation of Battleship.

The initial idea was that the coordinate system used in battleships reminded me of different methods I had seen to multiply polynomials together. In school I remember myself and classmates having trouble multiplying polynomials together, so I thought that would be a good context of the game. Luckily, making an adaptation of a game checks some game design criteria for you. Because of this, I felt like I could focus on the subject area of the game. After trial and error, I had figured out the best setup for the game. Each team gets two grids, an attack and defense grid. The attack grid had the binomials on the sides, and the attackers would have to calculate the trinomials to attack, however, the defense grid was completely filled out. The sequence and fluidity of gameplay was then discovered through playtester feedback.

I think that teachers should want their learners to play this game because it is very effective at its job. Even creating the game, I became much more efficient multiplying binomials together. There is very little to suggest that playing this game is off topic, or unuseful. The game essentially is essentially getting students to do homework level repetitions, but in a context that makes it more competitive and fun. Another reason for teachers to implement this game is the opportunity for variations, and classroom connections. I feel this game has great flexibility and potential to be implemented in not only a lesson plan, but even lecture, or assessment questions. For example, teachers could use this game to get into conversations about common factors, and factoring trinomials. The game could become more engaging by letting students choose their own binomials for the grid.

These are just a few examples of the advantages of implementing Binomial  Battleship into the classroom. The truth is, this game is very young, but the potential it has to advance student learning is very high.