Math Hombre

 May I March from April?

I was supposed to post the March/April Playful Math Carnival, but it's May! May the 4th even, happy Star Wars Day to them that celebrate it.

180 is a pretty amazing math number. 18 divisors, more than any smaller number. 18 divisors also makes it refactorable, divisible by the Very abundant, as you might guess. Harshad (or Niven) also, divisible by the sum of its digits. The sum of two squares (both squares of divisors!) For Euler's totient function, 

Of course, 180 is probably most famous in math for being the sum of the angles in a triangle, or have the degrees of a full turn. How would you prove the triangle relation? (I tell my math history students that is one of the few theorems every math major should be able to prove.)

 I once again am getting to teach the math game design seminar (at some point they'll realize it's too fun to count for my workload) and I wanted to try and capture my design thinking on a promising new game.

Phil Shapiro shared on Bluesky his math pairs game. A randomized list of 1 to 100 where you find pairs that add up to 100. 

For whatever reason, that made me wonder about a game finding differences. In elementary there's often a default to subtraction=take-away (Separate Result Unknown si parlez vous CGI) (I don't speak French) So a game that focused on the difference would be a good thing. I thought, what if you roll a die and need to find a pair that is that far apart?

For kids I like a number board that has a structure, so kids can use patterns to find what they want. (Nothing against Phil's game, where the Where's Waldo feeling is a lot of the fun.) My first try was a double spiral.

It was a lovely pattern, and I liked how it put small and large numbers together. As the game play evolved, it became clear that I needed a normal grid. 

The other thing you can see here is pretty typical for me when I have a mechanic idea. Try out the mechanic and worry about the win condition as you go. The above image was my first try. I asked on Bluesky who won, and Phil responded probably yellow, since it seemed to have more territory. Biggest block of squares? Longest path? I stuck with that for a while. Eventually I realized the game is about differences, the win condition should be, too. What if the path with the biggest difference won?

Mechanically I really liked that. Then there's an advantage for the first player. And it raises questions: what's a path? I thought it should be only edge to edge, but it became too easy to cut someone off. Having squares connect corner to corner gave some of that Blokus energy. I did wonder about the sum of two paths, but that's unnecessarily complicated.

I'm still trying different play rules. Should one of your new squares have to be adjacent to one of your old squares? Currently I'm saying no, because that makes more interaction possible as well as opening up more strategy with more choice.

The board was 9x9 originally because I wanted that double spiral, so it had to be odd x odd. I can see this being on a hundred board. Great representation, and I love to have kids spend time with it. I like that +/-9 are above each other, because 10s are often comfortable already, and it feels like 9s still makes for lots of interesting patterns. It does make a game around 20 turns - which is long for 2nd & 3rd grade. Although kids play Joe Schwartz's Hundred Board Game (definitely a Best of Math Games awardee; video explaining it) is more turns, but the turns are quicker. 

Why I think this is worth developing is because as I play, I have to think! Looking for pairs, any in good strategic placement, what is possible... a lot to consider. Too much for middle elementary? I hate to underestimate the players. Towards the end of the game, there are surprisingly frequent times that you can't take the number you would first take. The number rolled makes a difference in play, as well as providing some variance that helps with surprise.

One thing that came up is what if there's a tie? Then the winner is the person with the biggest difference on their second path that doesn't overlap their first path. Maybe a second path that doesn't cross their first?

Current rules text: 

Two teams. Roll a 10 sided die (0=10) or flip a Tiny Polka Dot card. High number goes first and gets 10. Team 2 rolls and takes 90 and then 90 minus their number. 

On your turn, find two numbers whose difference is the number you rolled and color them in your color. After 10 and 90, teams can choose any pair of numbers with the difference they rolled.

Game ends when both teams have to pass because there is no pair with that difference. Both teams draw a path connecting the biggest difference that can find. Squares connect edge to edge or corner to corner. Winner is the team to have the biggest difference in their path.  For example if team 1 makes a path from 10 to 71 (71-10=61) and team 2 makes a path from 24 to 90 (90-24=66) team 2 wins! If tied, the winner is the team with the longest 2nd path that doesn't overlap their first.

The 10 and 90 start is trying to remove that first turn advantage. It's also is a step towards understanding strategy, which can be nice to bake into the rules.

Definitely want to try with a d20 as well. Maybe as a 4th and 5th grade variation? On a 0 with Tiny Polka Dot cards, you could be allowed to pick a single number - which definitely could be useful. Another variation for high school + players could be the sum of two paths victory rule.

Current game board. If you try, I would love to hear what you think. I'll definitely play with my games seminar, and maybe with my elementary preservice teachers &/or 2nd and 3rd graders.

Two games with the most recent rules. 10/90 start is working well. 

PS. I make some references here to the criteria I use for thinking about games. Definitely a part of my design thinking.

PPS. I also like the name, which is unusual for me, but am open to suggestions. 

Interesting time to be hosting this carnival! I feel like there's a small resurgence with blogging, and I want to be part of it. I've really missed writing informally professionally. I've been a part-time host since Math Teachers at Play 22, 14 years ago!, and a big part of the original purpose of the blog was to collect, curate and share things that delighted and supported me. If you're interested in hosting, contact Denise Gaskins, the founder of this here carnival. The January carnival will be at her blog, but I think February is open!

177 is semiprime, for which two primes? 

177 is the ninth Leyland number, of the form x^y+y^x. What are x & y for 177? They're both prime, which should be a special kind of Leyland I think.

177 is the first "non-trivial" 60-gonal number. (1 and 60 are too easy.) What is the next 60-gonal number? (Pictured) What does the sequence of the first non-trivial n-gonal numbers look like? (6, 9, ...)

177 is a Leonardo number, so named by Edsgard Dijkstra for their relation to the Fibonacci numbers. The first five are 1, 1, 3, 5, 9... can you determine the pattern?

But the coolest thing to me is that it's the magic constant of the smallest magic square of distinct primes! I'll get you started...

I'm still 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.

Ryan Brummel made a video for Math Heads, our group game as mentioned above, a game he tested extensively with his algebra students.

Ryan's original game is a super cool algebra game where students make, evaluate and solve linear equations. The rules are surprisingly simple and the game play can be pretty intense. What follows is his story of making the game, and thoughts on math games in general.

When trying to come up with a math game, I wanted something that would apply to the math I was teaching my students.I happen to be teaching linear equations to my 8th Grade Algebra class, and my 8th grade Pre Algebra classes were going to get to linear equations later in the year. I wanted some kind of game I could use in my classroom. I wanted something simple that didn’t need lots of materials or printing out so I wondered if I could make a game where you build linear equations using a deck of cards. With decks of cards having cards with numbers 1-10 using the Ace I figured I could incorporate the face cards as variables somehow.

I brought this very rough idea to my Math 496 math games class at Grand Valley. From there my professor and classmates did a great job helping me brainstorm and try to arrange my setup so that it would be as user friendly as we would get it to be. We came to the conclusion of a rough idea of a game with two teams trying to solve a linear equation and create the biggest output.

I took that idea to my Honors class and had them try it. It went over surprisingly well, The students had a blast. They found holes in the game that needed to be addressed, and they begged me to play the next week. I brought their comments back to class and we continued to playtest and mess around with the rules and setup of the game. Once I thought we had a final product I brought it back to my students and had them play it one more time. Having honed in on some of the minor issues of the game a lot better, it went very well and my students were very self-sufficient and able to play in teams of 2-3 the whole hour without my help. That is when I knew the game was pretty well set in stone.

From there the game needed a name. My students did not have any bright ideas like I thought, however my 496 class gave me the idea of “Variable Kings” as the name since the game is all about winning variables and the king cards are the ones that count as variables. From that point I did what I never thought I would really do which was create my own math game that I can effectively use in my 8th grade classroom.

Why Play Math Games?

Coming into the Grand Valley education program I was completely foreign to the idea of math games in the classroom. I have a dad who just retired as a high school math teacher and spent 30 years in the classroom. I went all throughout my 12 year educational journey from kindergarten to high school not remembering any semblance of math games in the classroom as I know of them today. However now that I have taken math education courses, taken a math games course, and have taught in my own classroom I now can see the importance of games in the classroom.

Math classes at the primary or secondary level tend to get the reputation of being very boring. As someone who was good at math, I did well in my math classes and enjoyed them but I enjoyed them more because of my classmates and friends in the class rather than the content itself and the way the classes were run. There were some teachers that had good personalities that made the classes more engaging but again, that is nothing to do with the content and most of my classmates didn’t even feel the way I did. What happens when students say class is “boring”. That means they are not engaged, and don’t have any desire to be engaged. Students who are not engaged have no chance at success. These students who tend to not be engaged, whether it be in math or any class, are the students that are the toughest to reach, but the students we have to try and reach. What I have found when using math games in my classroom is that a lot of the students that normally tune out, or misbehave, will perk up when there is a game to be played rather than the traditional notes or worksheet. I believe the reason for this is that a lot of these games that teachers use in the classroom have a very low entry point. This means that students who feel like they struggle in math or don’t want to share for fear of getting an answer wrong, are much more likely to engage in mathematical conversation during a math game. Math games invite students of all achievement levels to participate and also have fun which is something not always associated with a math class.

The engagement piece is huge when it comes to math games in the classroom. However, if I played dodgeball every day in my Algebra class I’m sure students would be engaged, but they wouldn’t be learning any math. The thing that surprised me the most about math games is that I really feel like students get more out of it. When you pick a good math game it gets students to think deeper about mathematical concepts without even realizing it. With good scaffolding and discussion facilitation students really start to notice things about math while playing games that they wouldn’t using a textbook. The more students are engaged and are invested in the activity they are doing the more they will dig deeper and get out of said activity.

Overall I think that math games are super essential to any math classroom. Not every single part of every day has to be a game, but I think that using math games in your classroom is super beneficial to the students and the teacher. With my experience, math games cause engagement and the depth of mathematical thinking to skyrocket. Both of these are things that can be lacking in traditional math classrooms. I wish my teachers and classrooms would have incorporated math games a lot more in my education experience. And I know classmates that would have benefited greatly from that!

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. 

 So what have I been up to?

The biggest project this year has been working on Teaching Like Ted Lasso. 95% or more is Dave Coffey - inspiration, planning, and production. And I get to play along! There's a YouTube channel and the home for all the audio. We get to talk to so many interesting researchers and teachers. 

One of my favorite interviews this year was Nicora Placa:

I also get to commute with my son Xavier, a first year high school art teacher, and have an occasional low-tech, barely produced podcast with him, Background Noise. It's a little art education, a little math education, and mostly just talking teaching.

Speaking of teaching, after years of a highly variable teaching load, it has settled into a couple elementary math teacher education courses, a high school math education course, a seminar where a smaller group of students develop math games, and a math history capstone. One of the elementary courses is actually in an elementary school. The future teachers get to actually teach kids every class day, feedback from great teachers, observation by me and another student (when I can get one). The framework my colleague Esther developed for this (with others) is mediated field experience. We spoke about it at a local math teacher education conference. Here's the handout. The course page for the class is bit.ly/226-W23, and the lessons I wrote for the teaching (with a wide range of influences, colleagues, articles, curricula...) are here. 

There will soon (fingers crossed) be a series of blogposts with this year's games. We actually wound up with two seminars of five each.  Good stuff from elementary to secondary. One group developed a middle school Desmos Escape Room with a spooky monster theme. As a teaser, here's the video for one of our group games, Math Heads, which has been tested with 6th grade middle school students, algebra students and college math majors. Ryan Brummel made this video. Here's the handout with the rules and a form to support players, bit.ly/MathHeads. 

I continue to futz about with math/art. 

From one, two, three, four. The Tumblr posts have variations and links to the GeoGebra for generating them.

I cartoon when I have time or am challenged like in Mathober. They're pretty geeky. Some are college + math and some are elementary.

Just a sampler... what have you been working on?

 Do you want to host the 13^2 Playful Math carnival in October? A month that had a Friday the 13th? 

Yes, please. Should have been on 10/13 instead of 10/31 but... apologies.

169 is a palindrome in two number bases less than 16. Which do you suppose?

All odd squares are centered octagonal numbers, but 169 is also a centered hexagonal. (Visualize more with Alex CHIK's GeoGebra.)

 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.