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!

I love story problems that look like they don't have enough information but they do.

The Futility Closet, an excellent source of puzzlers, had a beauty today. I wanted to dynamicize it to make a visual, and to allow for new numbers and a mechanism to provide for guess and check strategies. Here's the applet, which is also on GeoGebraTube.



What makes this solvable?

I have older sketches of two other problems, an escalator problem, via Bowman Dickson, and a burning candle problem, originally from NRICH.

Does the visualization add to the problems for you? Or does it not make a difference? Do these kind of problems engage you? I do worry about the frustration barrier with these, and am interested in how you might scaffold them.

The GeoGebra was fun. The most complicated piece was building a piecewise function that modeled the movement of the ferries. After the numbers are randomly generated, the sketch has the times for the ferries to cross, and then it's a point-to-point parameterization modified with graph transformations. It was fun to figure out as it took a lot of wee algebra bits to fit together.

The title is a nod to this Chris de Burgh song from the heyday of music videos.

Intro
"So does this bore the heck out of you?" a student asked me.

The problem is that this was after two days of doing the Barbie Bungee Jump activity. The fabulous Barbie Bungee Jump. (Cf. Julie and Fawn) I was assisting a very nice and competent substitute teacher.

Coming on the heels of Christopher Danielson's and Chris Lusto's #globalmath session on building intellectual need, it was clear that most of these students did not have it. Nor were they looking for it.

This is a good school with good students and good teachers. What's going on, or not going on? My first brief observations:

1. Students were given all the steps to follow. Being told to do a, b and c and then doing a, b and c is not engaging.
2. There was no hook. How much of a hook depends on the lesson. This one could have used the video, a discussion about bungee jumping, etc. Going straight into 'here's what you do' gives no chance for wondering. Even if it's what the students want or are asking for.
3. There was no expectation that this was worth their time or could be interesting. There is always time to start, but this might also be about developing a culture of inquiry. Students need to learn that this is what math is, and this is what math class is like or could be like.

Recount:
Day 1: Students were given a worksheet with a table, told how to assemble the rubber bands and washers and to collect data for 1 to 6 rubber bands. Then graph all of their data and freehand a line of best fit. This is the beginning of a functions unit that will end with linear functions. Then they were asked to make a prediction for how many rubber bands they would need for the drop. We didn't have the actual heights, so they predicted for 3 m. Mostly, their prediction method was pick a number that was bigger than 6. 20 seemed nice to them, though some went with 18, since 6 rubber bands was close to a meter. Two groups found the average increase per rubber band.





We weren't clear about how to do the drop in the stairwell. We didn't have a set (or maybe even one) tape measure for long distances. Two teachers wound up determining 3 drop spots and measuring the distances, between 3 and 3.5 m.

Day 2: (After a snow day and a PD day.)
The students coming back were not much more enthralled than they were Day 1. I shared how this was the start of a functions unit, the math idea of having a rule to go from input to output. I tried to phrase the question as given the input of how many rubber bands, could they predict how far it would drop. (Not much traction, as there were already instructions on the screen.) The substitute gave each group their drop height. The two groups that had figured out the averages used this to make quite specific predictions, and one group made the complete table that this would generate. At the last second they cut 2 rubberbands off of their total, to allow for the length of the disk and acceleration. They were worried that it would be traveling faster at the bottom and that would make it stretch more. Two other groups adjusted their number a bit, but without reasoning that they could share.

We proceeded to the stairwell and groups took turns making their single drop. 2 hit the floor, including one of the more mathemaical groups, 2 got about 70 cm away, 1 was more than 1 m, and the group that had made the table got to within 10 cm. Went back to the classroom, shared the results and had the winning group describe their efforts while few listened. I talked with the mathy group that hit the floor about what went wrong. Basically they felt math failed them. Double checking their work I saw the problem was that they were computing for the wrong drop height! Their calculation would have put them quite close.

So What?
The students were pretty happy. Better than a typical math class, playing with rubber bands, leaving the classroom. The sub was okay with it, as students were mostly in control and made it through all of the steps. I felt like we missed an opportunity.

So what would you change about the lesson? What would add/create/inspire intellectual need in what is a (potentially) great activity?


Post Script:
Excellent discussion! I just want a few of the shared links to be more visible here. But many people put great thinking below so don't skimp on the comment reading.

• Timon: http://www.101qs.com/1563-barbie-bungee Barbie Bungee video
• Ruth: http://cohort21.com/ruthmcarthur/2013/02/02/freestyle-barbie-bungee-no-steps-required/ her blogpost sharing an open-ended/low structure approach
• Pieter: http://www.researchgate.net/publication/222209242_Analysis_of_a_fatal_bungee-jumping_accident sad accident analyzed
• Julie: (blogspot ate her comment) follow up post http://ispeakmath.wordpress.com/2013/02/02/barbie-bungee-follow-up-be-careful-when-using-the-recipe/
• Robert: http://robertkaplinsky.com/lessons-learned-in-implementing-problem-based-learning/ expands on his four C's

I Love Charts, constant source of fun graphics, had a fun temperature comparison chart. temperature graph, complete with fun. I did share my graphs finally. I used GeoGebra to make them, of course, because that's the best accurate mathematical image maker. Easy enough to make all three variations. Which, of course, lead to wondering what features would you put in a dynamic sketch for it? How could you make the sketch switch between the three ranges?
But it wasn't to scale, so I started thinking about what that would look like. Just lengths? What's the norm? Then Shawn Cornally shared his oneupped

So I had to make it then. Obviously. It's on GeoGebraTube: for download or use as an applet.

Instead of checkboxes, I knew I wanted a slider to switch amongst Kelvin, Fahrenheit and Celsius as the input. I thought about adding an input box for conversion, but those still don't work on the iPad, and I'm trying to think about that more.

The idea came to make the segments for the 0 to 100 degree temperature range where the endpoints were a function of the slider. Then I just needed a particular value for three different inputs, so it would only have to be a quadratic. I started to go to Wolfram|Alpha to do the regression, when I realized that, of course, GeoGebra would do it more easily. So with the slider at values 0, 1 and 2, the range for Fahrenheit, for example, is just determined by

F0(x)=FitPoly[{(0, -459.67), (1, 0), (2, 32)}, 2]
F100(x)=FitPoly[{(0, -279.67), (1, 100), (2, 212)}, 2]
a=Segment[(3, F0(n)), (3, F100(n))]

I was struck again by how nice GeoGebra is as a source of activities for students, but also how rich the task of making things in GeoGebra is, and wonder: how do I encourage more sketch creation from students?

Having sunk the time into making the sketch, I did think about what activity could go with it, and whipped this up:


I'd be interested in feedback on the sketch. Does it have the features you'd want for your students to use? Should it have more cute doodads? What do you think about the design of such a sketch as a task? How far should the students be past the material before trying to design the sketch?

Some activities seem ageless. They work in elementary, middle and beyond. Guess my rule I have played with very young students, and this week I got to try it with my summer intermediate algebra course.

Guess My Rule
Any number of players

A rulekeeper makes up a rule that gives a number output for a number input. (It should be a function.) Players take turns giving an input, and the rulekeeper tells them the output. If a player wants to guess the rule, they tell an input and what the output would be. If they're right, they guess the rule. Then that player makes the next rule.

I don't know who originated this game; it feels ancient and right, so kudos to whomever developed it. I let rulekeepers use a calculator, and I usually start out with a few rules until the guesser feels comfortable making up a rule.

Today I started with the number times two minus three.  (Highlight to see it, or guess from the table.) Inputs came from all over. 12,  5, 10, 1... some shock at a negative answer. I encourage the students to record the results, hoping that it will lead to some ideas about what inputs to suggest. Organizing data is not a natural tendency. At one point they started asking 30, 40, 50...

inputs	12	5	10	1	30	40	50
outputs	21	7	17	-1	57	77	97

Oooh! That's a pattern. It goes up 20 when the input goes up 10. So they could predict the answer for multiples of ten. Thinking about that, they suggested a rule that worked. We looked at the rule to see from where the up by 20 pattern came.

The solver was not comfortable coming up with his own rule, so I chose another: eight minus the number. 

inputs	7	1	10	12	20	30	40
outputs	1	7	-2	-4	-12	-22	-32

\(7 \to 1 \) then \(1 \to 7\) drew an audible gasp/hmm. There was one student who loved asking for 12. The third multiple of 10 got the rule again. Subtract eight and then take the opposite of your answer. One of the reasons I love this game is that it always brings up equivalent expressions. I shared my original rule and we talked out the equivalence.


Now they were ready to suggest a few rules. A couple like my rules, then input times 5 divided by 2, which got a nice check of "oh, that is the same as the number plus half of it," and "I had the number times one and a half." Then one student came up with a stumper. When guesses were not focusing, I started taking notes on the board.










I also asked for what patterns they noticed. Some good stuff.  We got to the idea of asking inputs in a pattern, and I asked them to predict the output from 2 before Edras gave us the actual. (There's some really nice research on the power of prediction in math, some by my colleague Lisa Kasmer.) We spent some time on different expressions of the rule, like ___ x ___ + \(\frac{1}{2}\) ___ x ___, (___)^2 + ( ___)^2/2 and \( 1.5 x^2\).


For a last question, I asked them to find what input would give 100, and got unexpected riches. Students really worked hard, consulting group members. Several tried to solve symbolically, but didn't know what to do with the \( x^2 \). One student had an excellent guess and check. We spent some time discussing that, including what an excellent problem solving method it is, and how when I use it, using numbers gives me a much better feel of what's going on. Maybe they've been discouraged to use it by previous teachers, but it's a great strategy. We did discuss how to make our guess more accurate and efficient. Like was 8 or 9 closer to 100 in output?

Another then shared a traditional symbolic solution. One student asked, "why did you divide by 1.5 before you took the square root?"



"Because you have to."
"It has something to do with the order of operations..."
"Doesn't PEMDAS tell you what to do?"
"But you have to reverse it maybe..."

"Let's try it." (OK that was me.) They told me the steps and I wrote it down. Amazed to find the same answer. A good moment to point out to them that are almost always multiple methods in math. Teachers may have told them that there is only one way to do things in the past, but that is bullsh*t.

I should not swear in class. (In response to my usual 'jot down one thing you want to remember' there were many repetitions of that statement. In my defense, it is a college class. And the comedic potential was ripe.)

The next activity is adapted from Pam Wells' adaptation of an activity from the excellent Mathscape curriculum.




(Here's the Word file if you want to edit - it wasn't appearing correctly in the embed.)


It dovetailed very well into the Guess My Rule, allowing us time and grist to find many connections amongst tabular, symbolic and visual representations, reiterating the equivalent expressions and multiple solutions themes, and giving us a launching point for a definition. Using the Y pattern rule, I made a table for S=1, 4, 7 and 11. The changes in output were 9, 9 and 12... what went wrong? They found that the difference in the inputs was not constant, and when we changed 10 to 11, the output pattern worked. To me, that's the moment for the generalization.

We went on to launch Linear War, which gave me a chance to share why we use the term linear and some of what this stuff has to do with lines. Next class, we play!

All in all, it was a fun class. I was impressed with their willingness to try non-traditional problems, and they're gaining rapidly in ability to work cooperatively and make conjectures. The meta-messages about mathematics seem to be gaining some traction, too.

OK. I was mildly obsessed over meeting Mr Slope Guy for the first time.

I was observing two novice teachers, and they were using Mr Slope Guy as a mnemonic for students in analyzing slope. But I think he could be fated for bigger and better adventures.  Since my son Xavier is a budding comic book artist, we put together the following (PDF of the whole comic), with some inking support from Ysabela.


The problem was whether MSG was a hero or villain, and he wound up needing to be defeated. He is a personification of algebra.

Be interested in your feedback.  Only one more thing to say: Marvel, please don't sue! Spider-man is a wholly owned trademark of Marvel, created by Stan Lee and Steve Ditko, and starring in a major motion picture this summer.

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

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



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

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


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

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

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

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

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

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

Math notes:

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

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

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

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

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

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

Transformations on a Graph
I've been looking for Geogebra applications for function transformations, and wanted to share a couple of the neat sketches I've found.

Michael Higdon, a math teacher at Kincaid, a college prep school in Texas, has a quadratic function in vertex form, y=a(x-h)^2+k, with a, h and k as sliders to study transformations.
Geogebra webpage: Transformation of Functions

Mike May, a Jesuit math teacher at St. Louis University, has a beautiful applet where you can input the function, and control vertical and horizontal shifts and scaling with sliders.
Geogebra webpage: Translation Compression

An overall great collection of interactive webpages appears at The Interactive Mathematics Classroom. It has a nice search feature and a good breakdown by area of mathematics.

My first attempt at a transformations sketch is with a cubic as the starting function. Although you can change the function.

As a webpage, and as the geogebra file.

Slope in Linear Equations
A nice collection of middle school or Algebra I activities for linear equations from mathcasts.org: Slope Explorations
Mathcasts are screencasts of writing with voice-overs. They have mathcasts for K-12, and a nice collection of interactive math activities.

Here's my first slope sketch. It tries to get at the idea of the slope being constant on a line regardless of what points are selected.

As a webpage or a geogebra file.