Math Hombre

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.

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

 I had my elementary ed class canceled for low enrollment this fall. Make of that what you will.

The replacement course is College Algebra. Ironically named, since it is mostly Algebra 2. Which is required in Michigan. Our sequence has been 097 (prealgebra) -> 110 intermediate algebra (algebra 1) -> 122 College Algebra.  It used to be + 123 (trigonometry) to go on to Calculus, but we have a nice precalc class now (124) so people needing to take calculus that don't place into it can just take 1 semester. The audience for 122 then, is now general education, and people who need courses that require it, like the basic chemistry, intro physics, and statistics. It's a 3 credit course, and my section meets twice a week.

The course has traditionally been quadratics -> polynomials -> rational functions -> exponentials -> logarithms -> light touch of statistics. So what do we want from the quadratics unit? This post is me trying to think out loud to get it straight for myself. The schedule is pretty packed, so I have 2-3 weeks per topic, 4-6 class periods.

The instructional sequence I have planned is visual patterns -> modeling (Penny Circle and Will It Hit the Hoop?) -> graphing/equation forms (Match My Parabola & Form Fix) -> solving equations (vertex form & graphing), mostly in a modeling context.

The visual patterns do a lot of work. They offer a hook, they give learners a chance to notice and wonder, they give us a chance to problem solve. They are also different from what most students have seen in algebra, sadly, so offer a way to let them know that this course might be different. I also have them read Elizabeth Statmore's post on math as a thinking class. I asked them, "What do you think the main idea is? How does this compare with your own ideas about learning math or your previous experiences?" and you can read their responses on this doc.  I think they get it. Mathematically, I think my main point is the use of variable as a relationship rather than an unknown. The transition from step number to x is very natural. Secondarily, they get to see multiple equivalent expressions. Which is one of those math ideas which many learners see as a bug, but mathematicians think is a central feature.  Part of the richness of these problems is what the old NCTM standards called the representation process standard. Tables, expressions, visual and the connections between them all move us forward. Here's a handout with four quadratic patterns. The bricks and the darts and kites are very difficult to visual make a symbolic rule for. I might have made them or might have found them at Fawn Nguyen's visualpatterns.org or it could be a mix.

Modeling is a key theme of the course, and Penny Circle and Will It the Hoop? are a good start to it. I was surprised how many learners went with an exponential form, and the reveal is the perfect way to settle it. We will be using Desmos activities a lot, and those are pretty slick introductions. The Penny Circle builds on the covariation use of variables, and the basketball leads into the graphing we'll be working on next. 

This is where we are as I write.

I'm convinced that one of the barriers for these students is understanding graphs. Thankfully, making them is easier than ever. But I don't think that many know how to think with them. Again with the representation standard, the connections between the symbolic expression and the graph is mostly taxonomical, and I want it to have meaning. Though this is a place where I could use some help. Regression supports this goal, as it brings tables into the web of connections. Activities where they vary parameteers and observe the effect on the graph help, at least in terms of taxonomy. Solving equations with graphs is an opportunity to build some of the understanding I want, as, especially for applications, the context is another piece of the representation. Writing this, I'm a little surprised by how hard it is for me to put my goal here into words. That would undoubtedly help with the teaching!

Solving equations is last for me, partly because it is so much what they perceived the focus to be in their previous math courses. I don't care especially for a lot of symbolic skill here. I don't teach solving by factoring, though the factored form in connection with graphs is something I emphasize. I do like the approach of solving from quadratic form, because it builds on a theme in math I love about doing and undoing. This leads better into exponentials and logarithms than it does polynomials and rational functions. The symbolic fluency that I want is being able to see a quadratic as series of steps. Take a number, subtract 2, square it, double it, add five is the same as  . To find what numbers make 13 from that function, we can do by undoing.  I love Graspable Math for this, as the dragging to undo seems to really help get across the idea, though it doesn't work on the balance nature of equations. Here's an example GM activity with 3 quadratics to solve.

I'm very interested in your thoughts. What are the key ideas you want in a quadratics unit? What am I leaving out that you love? What understanding do you want your learners to develop or skills do you want them to have for graphing? Why?

P.S.

Probably violating some internet rule here, but really liking the Twitter discussion about this post.

@DavidKButlerUoA: This line was very interesting: "multiple equivalent expressions... is one of those math ideas which many learners see as a bug, but mathematicians think is a central feature". I'd love to hear more about that.

@joshuazucker: My interpretation is that beginners may want there to be only one answer and experts see how useful it is to have multiple representations that make different behaviors immediately visible.

@mathcurmudgeon: When 90% of calculus (and every math course, really) is rewriting expressions in an equivalent form that we can work with more easily.

@mathhombre: it starts with fractions. All these different ways to write the same thing. One of them must be right. (Often supported by teachers insisting on one.)  But that we can transform, rewrite and tinker leads to fluency, connections, and meaning.

@mathforge: The belief that out of all the ways of writing it there must be a RIGHT way is SUCH an interesting belief. I've never thought before that people might believe this.

I suspect that this is more prevelant than we might admit. As experienced mathematicians we might chuckle at people who think that there is a "best" way to write, say, a quadratic or a fraction. But we probably fall into the same trap with ideas.

I might, to take a random example, think that there is a "right" way to think about differentiation, or Pythagoras theorem, or a topology, or the category of smooth functions. What I mean is, "this is the way I find most intuitive".

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@KarenCampe: Love the visual patterns start & modeling focus. 

When you do graphing equation forms & use match my graph/form fix you will surely cover symmetry of the graphs & how factoring gives x-ints. I like how graphing & alg manipulation of quadratics are interconnected...

Use graphing as tool to support any algebraic rearranging we might want. Look for hidden parabola that shows complex roots. Axis of symm hidden in quadratic formula.

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[In response to "Quadratics unit in a college algebra course. What goes in, what's left out? "]

@theresawills: Probably too vague, but worth saying: RICH PROBLEM SOLVING.

 

 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.

Handout: https://bit.ly/BinomialBattleship-handout 

I'm teaching Intermediate Algebra this semester, which I haven't taught in a while, and so have been rethinking the course. My two big goals are:

1. Redeem mathematics. These are students in a good university who are having to repeat content they've already seen and maybe more than once. I figure, for most of them, I can infer bad math experiences along the way.  I want them to appreciate math, to know why algebra was a big deal in the first place, and have opportunities to do math.
2. Free them from the gatekeeper. With a new disposition, I want them to understand the content at a level that will equip them for success in further math classes (which most don't take), or statistics (which 60% do take), or even reconsider majors if they had eliminated something based on math requirements.

Reality Check by Dave Whamond

So I'm going to try to think aloud about something I asked my students to do today. It's on this activity, about one of our greatest human minds (IMHO), Omar Khayyam. No reason for this guy to be less famous than Leonardo.



How would you try to solve \( x^3+x=4 \)?

So first I think about some concrete values. 0+0=0, 1+1=2, 8+2=10. OK, one solution between 1 & 2. No negative solution. Accessing calculus (which seems like cheating) derivative is \( 3x^2+1 \) so monotone increasing. I think that means this is a root with multiplicity 3, as one root, two imaginary is an 'S-curve' as my students say with the one root. (I should look into that at some point.)

Algebraically, I think about factoring as is, which seems like no help. If I'm right about \( (x-a)^3 \) for some \( a \)... that means \( x^3 - 3 a x^2+3 a^2 x-a^3=x^3+x-4 \). So is \( a=4^{1/3} \)? That doesn't work! And then there's no way for \( 3 a x^2 \) to be zero. So I take back what I said about multiplicity! (I really do have to look at that more.)

So next I would look at numerically grinding it out. Something closer to 1 than 2 and proceed from there.

To solve it with a graphing calculator - piece of cake. At least for a decimal approximation.  Here it is
on Desmos, along with Khayyam's geometric solution.

The last thing I want to do is to verify that his solution works. The question of how did he derive this is important, but I'm not going to get to it here.

So the equation of the circle is \( (x-b/2a)^2+y^2=(b/a)^2 \) and the parabola \( y=x^2/ \sqrt(a) \). So... hmm. Substituting the parabola equation into the circle gets us a quartic!

So why a circle? It gets us a right triangle, which gives us proportions.
So
\[ \frac{x}{(x^2/ \sqrt{a})} = \frac{x^2/ \sqrt {a}}{b/a - x} \\
\frac{\sqrt {a}}{x} = \frac{x^2/ \sqrt {a}}{b/a - x} \\
a(b/a-x) = x^3 \\
b - ax = x^3 \\
b = x^3 + ax \]

Sweet!

(The mathjax is displaying odd for me, \( \sqrt a \) is square root.)

I did think it was interesting that none of the students had any idea how Khayyam was drawing parabolas before graphing. Maybe we'll have to do some directrix learning.

p.s. Deborah Kent and Milan Sherman (Drake University) wrote a great extended piece on this.

Some of my favorite assessment problems from this semester in College Algebra. Each problem is labeled with the standards it mainly addresses. The main function families for our course are:

• Q - quadratics
• P - polynomials degree 3 and higher
• R - rational functions
• E - exponential
• L - logarithmic

And then there's a category of function concepts and special functions.

For each function family, there's four repeating standards:

1. Basics - vocabulary, characteristics
2. Representation - being able to move amongst table, graph and equation
3. Symbolic - traditional algebraic skills, solving and simplifying.
4. Context - solving in application. Being able to mathematize situations.

I like having a structure, as students getting used to the Standards Based Grading is one of the big hurdles in the course. That's why all the problems are labeled with standards, too.

These problems are different than the ones I use in class or for homework, with more scaffolding, and often reminiscent of things we've tried in groups. They also choose problems on which they're ready to be assessed. They can write up longer responses at home to turn in on problems that they do not do in class.

This first one's not mine. They read a Glen Waddell post on the Exeter method, so I thought it was a nice connection to put on an assessment.  The element of the cup really made clear which students understood what the height function was saying.



We did a fun activity in class on sound frequencies that you can hear at different ages (due to hearing loss/damage) in class when we were talking about decibels. Frequency is not logarithmic, of course, but I love using a topic as arcane as logarithms to make sense of something they've heard of (get it?) as much as decibels.

One of their favorite application problems in the semester was using exponentials to model weekend movie grosses, which are roughly exponential decay. But of course the sum of geometric sequences is also a transformation of an exponential. This problem unfortunately highlighted careful reading or the lack of it, which is not super-useful, so that some people tried an A*k^x model. But it did show the people who were making sense of that not working, and the people who blindly accepted the results. Desmos reliers (as opposed to Desmos-only-if-my-calculator-can't-do-it) did better on this.

This problem was another great one for function notation. The difference between evaluation and solving was crystal clear. And it was such a nice pattern for the people who correctly interpreted the problem that there was a good payoff.

The top problem here is also not mine, of course, coming from WODB.ca. Well, originally. The task of making a function to look like it is a great representation prompt. Students who made sense of marble slides were really able to strut their stuff.
 It was the second and third problems that were really interesting to me here. They were open middle-like in the variety of different methods people used to solve them. Recursive rules, which we never used in class, really, excellent table use, some regression... such a nice mix.

I liked it enough that I wrote a follow up for the last SBAR opportunity. (AKA the final.)
One thing that we never addressed directly that we got to the last time I taught this course is that the sum of a certain degree polynomial sequence gives the next degree. Much like the differences give the next lower degree. I LOVE that structure.

For the special function types they only had to demonstrate one of the four kind of standards that they did for the main function families. Both of these offered a lot of opportunities for sense making. In particular the normal distribution question highlighted whether people understood mean and standard deviation as descriptors.



If you have feedback on writing assessment questions for SBARs, I'd love to hear it. Whether it's modifying these or a whole different direction. Here's all my assessments from the semester in a Google folder, if that could be of help or interest.

PS. Thanks Ann and #MTBoS30 - I've now blogged more in May than I did all January to April.

Whoopee means games, of course. (For the song, Ella or Ray are the best options. Though I suppose I should go with Dr. John - no relation, despite the resemblance.)

Spoiler: I got rambly here. If you're going to not read this, here's two quick takeaways: the four new games and James' start of a game/curriculum alignment.

I was delighted when James Cleveland asked if I'd do a Twitter Math Camp morning session (meaning three 2 hour sessions) on math games with him. He'd led a one hour session at TMC13, and Sebastian Speer led another at TMC14, but this would be the first one with time to really make new games.

The format seemed pretty natural and intuitive: look at some games together, introduce a few principles, and get making. Just in case, we each had some ideas for games in case people didn't have any urges, and James had a couple of neat statistics games ideas burning a hole in his pocket in particular.  I had a mechanic; my family recently discovered Sushi Go, which has a simple and elegant drafting mechanic. There was also this Tug of War that I had been discussing with Nora Oswald  based on a Daniel Solis idea.

The plan:

• Day 1 - play good games and discuss.
• Day 2 - start design.
• Day 3 - playtest.

Writes itself, eh? Materials from the first day are mostly here on the TMC wiki, and the 2nd & 3rd days in a Google doc. (Including rules for the developed games.) 

Day 1 Games
I brought Linear War  , but we didn't actually play! I considered it for day 1 because I like how the students make the cards for the game (learning part one) and then play (learning part two) working on vocabulary, concept, quick recognition and computation. We did play: (in order of complexity)

• Product Game: Illuminations, handout (original & integers), post (decimal). My nominee for best math game ever. Comes at the content from multiple ways, amazing replay value due to the deep strategy, quick to learn, structure supports students in learning the content, adaptable... The only thing missing is context, but this would distract from it as a strategy game. Teachers thought of several different uses for this immediately.
• Quod Game/Metasquares (app not currently in US app store) All you need is a grid, and the strategy is deceptively deep. Subtle approach to content, though, as there is a great mathematical structure, but it's more about noticing it than learning it.
• Factor Draft, James' great game. Interesting in that you can parallel play or interact a lot. Really requires the mathematics that it concerns. Needs its own pieces manufactured, but they can be used for multiple purposes. Great example of development in balancing the pieces for interesting play. High cognitive load game, lots of challenge.
• Domain Ranger, post 1 and post 2. Norah's serious game. It's an intense strategy game, for which you need the math ideas of ___domain and range, and the ability to compare different graphs. Participants had awesome suggestions about this. Recognizing the difficulties in learning such a complex game, they thought about doing a 1-dimensional board set up learning game. And also the great idea of doing a preset first game, Settlers of Catan style. I'll try to work one up the next chance I have to use the game.

It became clear pretty quickly that this was going to be a good couple of days. We picked these games because they all are content focused. I do not have any problem with review games that fit any content. (In fact, here's my list.) Instead of that, we were looking to design games for specific content. Where the game play was the learning activity. Day 1 was promising because the group as a whole was really able to focus on what aspect of the content the games addressed, and where in the lesson/unit on that  content it would be appropriate.

Harvey Mudd had CHALK boards. Deja view.

Day 2 Design
We started off this day with a look at Decimal Pickle, maybe my best game, with a focus on desing thinking, the mathematical goal, and how the mechanic works in the game. One of the most interesting parts about preparing with James was thinking about classic game mechanics are use of them in math games.

 
There's a lot of room for addition in there. I'd also like to hear your thoughts about what's missing. Even just writing this I got thinking of Farkle and Yahtzee. (King of Tokyo is an example of a tabletop game that uses that great Yahtzee mechanic. I have an upper el math game that's a direct rip off adaptation of Yahtzee, too.)

For design principles, I have this goofy list of 9 I use as a framework (adapted from Mark Rosewater of Magic fame). We emphasized just a few-
 
What's really promising from the prep for this day, though, is James' start of a spreadsheet for curriculum aligned games. Here's  a Google spreadsheet version - open for editing. If you know of things to fill in, PLEASE DO. If you have a hole you especially want addressed, let us know on Twitter.

People got designing pretty quickly. We divided into 4 groups, working on statistics (James was in this group), Fraction operations, Arithmetic Sequences, and Unit Conversion. I floated amongst the groups. This was a bit of a breakthrough for me. I design mostly in isolation. But (like for most things) collaboration was energizing, powerful and fast. Between this part of day 2 and some wrap up on day 3, we finished four good games. 2-3 hours of work. My contributions floating were questions, connections with other game experiences, and the occasional idea.

Day 3 Playing
Also today, we took time to do some rule writing:

1. Rules. For me this comes late; kind of a synthesis step as you think about how to communicate the game. It will often result in design revision, though.

 James knew of a good blank template for rules writing.

People needed a little time to finish. We had a good Skype chat with Nora, who shared her experiences playtesting, took people's questions, then discussed some of their interesting feedback on Domain Ranger. Dave Chamberlain (participant) shared his published game of Team Up! which is a 4-12 common core review board game, and some of his process. Also what it took to get it in commercial finished form. 

James gives a good write up of the statistics game, Fighting for the Center. Use playing cards, players build a data set that meets some goals (measures of central tendency) hidden from the other players. It's great at making players think about how changing a data point affects those measures. Lots of interaction, since you both are playing on the same data set, catch up is not an issue, and students will find more means and medians than they ever would in a homework set.

The fraction game is about addition and subtraction, modeled a bit on the Connect 4/Product Game framework. The board is really interesting, by asking for ranges, which really leads students to using representation (on the fraction cards we had). The teacher may want a way to get students to add precisely. I think there's some more playtesting to do here, too, as the placement of the various squares was more about coming up with them. 








Honeycomb Madness, unit conversion game. This is a classic board game, and the closest to being a general review game. You start in the outer ring, and are trying to get to the inner ring by answering questions. The ring level serves as a kind of rubric, though, and might support some kind of awareness in students as to different levels of understanding of the material. There's a nice bit of randomness that's reminiscent of Trivial Pursuit. I liked that it is not the kind of game I might design; I think it might be quite popular with students, too.



Arithmetic Sequence Game. This one is right up my alley, though. Deck of playing cards. Deal three: starting value, common difference, step number. That determines a target. Each player is dealt 5 cards, and tries to get as close as possible. Then the idea that complete changes it: you bet on your play, 1-6 points. Closest gets the points everyone bet. 2nd closest gets their bet back.  Wow. Plays great. I'll be trying this out, next algebra class for sure. I made a GeoGebra sketch to help with the calculation and to practice. 

Thanks to James and all the participants. I feel like I learned a lot about collaborating in game design, and broadened my tastes a bit, too. This more than ever makes me want to get students designing games, so if you're in the area and wouldn't mind a mathematician in the room...