## Trying to find math inside everything else

### Integral Limit Game

This year when I was in my intro to integrals unit, I tried to look back at this blog for the second integral game I know I played (besides this one), and saw I hadn’t blogged about it. I had tweeted about it, but now I’m thinking, you know, I should, uh, archive things that I only tweeted about in a more permanent place, in guess Twitter doesn’t last much longer.

Anyway, this game is based on The Product Game, with the same structure of turns – players take turns moving a token on the bottom rows, that then determine which square in the top section, where the first player to get 4 in a row is the winner. (I usually have students play in teams of 2, but I’ll keep saying “player” go forward.)

The idea here is that the bottom rows represent the limits of a definite integral. One player plays as the Upper Limit, and the other as the Lower Limit. Once both limits are placed, the player who most recently went calculates the value of the definite integral on the accompanying graph, then covers the square in the top section with the area. (Remember that if the lower limit is greater than the upper limit, the sign is switched!)

Making the function that would give a variety of answers was a fun challenge. After coming up with a graph I thought looked good, I wound up making an excel sheet to calculate all the possible definite integrals to see how balanced it was, and adjusted.

I’ll include that excel sheet as well, as it’s useful for checking answers (as a teacher), although of course each team should be checking each other. After doing a bunch of different integrals on the same function, students often realize they can use their previous work to help them find new answers, reinforcing the cumulative nature of integrals.

## Files

### BYORF

One of the other games I made this year was during our rational functions unit: BYORF, which stands for Build Your Own Rational Function. (This was originally a placeholder name, but it kinda grew on me.)

BYORF is a drafting game, a la Sushi Go or 7 Wonders. You play over 2 rounds (because that fit best in our 45 minute period – 3 rounds might be better with more time?), drafting linear factor cards to build into rational functions that match certain criteria. Here’s an example of a round between two players.

In this example, the left player used only 4 of their linear factors (as you don’t need to use all 6). Then we can compare each of the 5 goal cards, which are randomized each round. L has 0 VA left of the y-axis, while R has 2, so that is 3 points to R. L has a hole at (-2, 1/3) while R has a hole at (-1, -2), so L gets 5 pts. They both have a HA at y=-1, so both score those 4. Then we have the two sign analysis cards, which score points if you have that formation somewhere in your sign analysis. R has the first one (around x=3) and both have the second one (L around x=1 and R around x=-3). So after one round, both players are tied with 11 points.

I hope that gets the idea across. The fact that students need to check each other’s work to make sure the points are being allocated correctly builds in a lot of good practice. After we played the game, I did a follow-up assignment to ask some conceptual questions (which is where the above example comes from). I’ve also attached that here.

I hope you have some fun with BYORF!

### Letter Scramble

In our combinatorics unit in pre-Calculus, we tend to look at every problem as a letter rearrangement problem. This lets us move beyond permutations and combinations to model any problem involving duplicates. I wanted to build a game that had the students quickly calculate the number of arrangements for a given set of letters, so I came up with Letter Scramble.

The idea behind the game is that students have a set of goal cards with the number of arrangements they want to reach, and a hand of letter cards. On each player’s turn they can play a letter card to change the arrangement, and thus change the total possible number of arrangements. (They can also skip their turn to draw more goal cards, a la Ticket to Ride.)

I calculated all the possible answers you could get using 7 different letters and up to 8 slots, including if some of those slots are blank (and thus make disjoint groups), then determined which of those answers repeat at least once, and assigned them scores based on that.

One interesting thing about the gameplay is it promoted relational thinking. Instead of calculating each problem from scratch, you can based it on the previous answer you calculated. (So, for example, if the board read AABBCD, that would be 6!/(2!2!) = 180. But if you change that to AABBBD, one of the denominator’s 2! changes to 3!, which is the same as dividing by 3, so it’s equal to 60. No need to calculate 6!/(2!3!) directly.

Some examples of scored goals:

### Slopes and Lattices Game

Okay, here’s a game I came up with off the cuff today. It kinda worked, but I guess if other people tried it and gave feedback, that’d be swell.

Players: 2 (or 2 teams), each with two colors

Board: A 10×10 grid.

The game is played in two phases. In the first phase, each team takes turns placing points on the grid, until each team has placed 5 points. The origin always is claimed as a neutral point. Every point has to be on a lattice point. (In the example below, I was blue and my student was yellow.)

In the second phase, on their turn, each player may place a new lattice point and form a line with one of their original 5 points. If that line then passes through one (or more!) of the opponent’s original 5 points, those points are stricken. If one player can strike out all of the other player’s points first, they win. (If not, then whoever strikes out the most.)

There is one caveats to round 2 – when a line is drawn, determine the slope of that line and write it below. That slope can’t be used again.

After playing the first time, it became clear that much of the game came down to placing the points. If you could place one of your points so it was collinear with two of your opponents, you can strike them both with a single line. (But this only works if there is space for a 4th, alternate color point in phase 2 to form the line.) You also want to place your points defensively, with weird slopes that don’t pass through a lot of lattice points, to keep them safe. The second player definitely has an advantage when placing points, but the first player has an advantage when drawing lines, so I’m hoping those balance out.

Thoughts?

### Remote Teaching Math Games

I love playing math games with my students, but it’s been hard with remote teaching because so many games require physical objects. I’ve been able to play a bunch using a shared whiteboard, but there’s a limit to how many work in that method. I just discovered http://playingcards.io, though, which is a platform for playing card games online with anyone. They have built in games, but you can customize your own. I’ve taken several from the #MTBoS and my own blog and made them.

To use these files, first create a custom room. Then enter edit mode:

In the Room Options Menu, you can import a file. So download the file you want from here and import it there and the game is ready to go. You can then share the room code with students, and you can even make multiple rooms for different groups of students and jump between them.

For each game, click the image to go to the original blog post, and the title for the pcio file.

### Integer Deck

This last one isn’t a game so much as a resource for many other math games. It’s an integer deck, consisting of cards from -12 to 12 of each suit (and an extra 0 for each.) I colored the suits using a colorblind-friendly color palette, on top of the symbols. You can easily edit the deck (enter edit mode, then click on the deck) to remove cards from the deck or change the particular cards. It can be used for a lot of games – and helps avoid the problem of kids wondering what J, Q, and K mean. It would be a good deck to use for, say, Fighting for the Center or these Integer Games.

### Name That Solution

I was reviewing solving equations for my SAT Math class. It’s a tricky thing to do because “equations” includes linear, systems, quadratic, and exponential equations. A lot of different skills to go over in a short amount of time.

After working through the requisite problems, I wanted a little more practice, so I came up with a game that they could play, based on the Bid-a-Note sections of the old “Name That Tune” game shows. I called it Name That Solution. Gameplay goes like this:

• Start over with a simple equation, like “x = 2.”
• Each turn, a team can change the equation in one way to make it more complex. (For example, make it “x + 3 = 2” or “5x = 2”.) Only one operation and one term can be added at most per turn. The team finished by saying “I can name the solution of that equation.”
• On a team’s turn, they may challenge the other team to, in fact, actually solve it. (“Go ahead! Prove it!”) If the challenged team can, in fact, solve the equation, they earn a point. If not, the challenging team gets a point.
• First team to 5 points wins.

They played on whiteboards so they can change the equations quickly. The students quickly learned to not overextend themselves when making the equations harder, lest they find themselves challenged. So it leads to a nice exercise of constantly mentally making sure you know the steps to solve something before you take your turn, getting a lot of practice.

At the end of one of the classes, I did a big class-wide version, half the class versus the other half. But they wound up being very conservative, with neither team challenging the other and only take moves they knew they could solve. Which I guess was the point.

That final round.

I’m currently on the road back from TMC16 in Minneapolis. (Ed: See, that’s when I started this post….) This long drive back is giving us all a lot of time to process and reflect on the experience. (I guess Rachel was right about that!)

I think I approached TMC differently this year. Lots of people have spoken about the rejuvenative properties of TMC, and I think I really needed them. I mean, everyone always feels tired when the summer finally rolls around, and rest and energy makes that better, but this time I needed something more than that. And TMC provided.

It started with Descon. You can read more about that in Rachel’s post here. But when I was struggling to choose a morning session, I settled on Tessellation Nation. Both those experiences gave me a deep joy of forming questions, exploring ideas, having successes, failures, and breakthroughs. It was like doing a hard reboot on my mind.

Some things I played with in the morning session:

Here I was trying to picture creating some sort of “inversion” tile that would connect the lizards of different chirality.

With that in mind, my afternoons provided me with guidance about the upcoming school year. Really, I can sort them in what, how, and why.

What: I went to Jonathan’s session about hacking up the curriculum. His main idea was that the curriculum should not be focused around the nouns, but rather the verbs. That is, instead of having, say, a linear unit where you solve, graph, model, and then a quadratic unit where you do the same, have a solve unit where you do both types.

The approach sorta lends itself to the kind of spiraling that I was inspired by Mary Bourassa and Alex Overwijk to try, but was afraid to. So this is a step in the right direction.

How: I’ve heard about Talking Points for a while, but never had any experience with them, so I had to go to Elizabeth’s session. It was really nice to walk through the activities and see how the points can spark cognitive dissonance in their sequencing. I also enjoyed Elizabeth’s “deleted scenes” method of instructions, which reminded me of the dialogues in the Algebra Project.

Julie‘s session on giving feedback was helpful. I’ve worked on giving feedback without grades, but it can get a little overwhelming, so it was nice to get some strategies for streamlining the process. I think the most important one for me to remember as I start the year is to make space for the comments built right in to the assignment. That’ll make the whole process easier. Also, I need to remember that EVERYTHING should get a comment, not just things that are wrong. That way comments don’t become a proxy for grades.

Joe‘s session on teaching moves for implementing games was just what I needed. I can come up with some great games, but sometimes when it comes time to play them in class, it looks more like “Okay, here’s a game, go play it.” The most important one IMO was to have the students notice/wonder about the board/materials before the game is introduced. It’s a tenet of game design that a game is well-designed if players can (mostly) figure out how to play without looking at the instruction booklet. So the noticing and wondering works well with that.

Tracy’s keynote was amazing in so many ways, but she did hit on something I’ve been working on with my math coach and is now, I’m glad to see, becoming more of the thing in the MTBoS – never skip the close. Gotta work on that more.

Why: Social Justice, of course. Jose’s keynote obviously hit on those notes – as he said, students need to trust you before they can learn from you.

I went to Andrew’s session, which wound up just being a small conversation with him, me, Sadie, and Sharon. That’s where I decided my #1TMCThing – to decorate my classroom with more explicit social justice signifiers (like a rainbow flag, or a BLM poster).

Then at Annie’s session, she talked about her Mathematicians: Not Just White Dudes project where she tried to present mathematicians that identify the same as her students – even when they got super precise on her (“Is there a gay female Dominican mathematician?”) I definitely want to bring that into my class – although I would like it if, since I’ll be teaching calculus, I could get a good variety who contributed to calculus (or I guess just used it.) There as a group we also decided to start using the hashtag #sjmath (after I determined it wasn’t be used for anything else) to share social justice math resources, which Julie pulled a lot together here.

I started writing this on the ride home from TMC, but I ended it now, and I think that was actually a good thing. TMC is so early in the summer (for me) that I don’t go into vacation-mode until after. Now that I’m actually ramping up for school again, it was good to reflect and remember what I actually want to bring into my class. So my procrastination actually paid off! (For once!)

To conclude, here’s the camp song in MP3 form.

### A Boss Fight?

One of the things about arranging your grading system like a game, as well as being a math game aficionado, is that it is pretty easy to combine the two. While yes, students can take quizzes or write essays to gain levels, they can also beat me in a math game. Of course, I’m not easy to beat, so winning against me would really show some mastery. (I do, though, allow them to gang up on me when the game is more than 2 players.)

The only students that really challenge me are the ones that hang out in my room at lunch, even though I’ve offered the challenge to everyone. And it’s cute because when they do lose they get even more determined, often because they may lose by a very small margin. (This is occasionally by design.)

The only game I’ve lost so far is Blokus, where the two Kevins beat me (but my score was still above the 4th player). As a reward, I gave them a level in Visualizer, as I figured that was the most applicable skill to winning the game. Planning ahead and visualizing paths in your mind is a useful skill. That same skill is the reward if they beat me in Ricochet Robots. In that game a team of Jane and Kevin tied me, so I still gave them reward, but they didn’t win.

It’s interesting trying to match games with skills. For example, the reward for winning at 24 is Tinkerer (since you need to play with numbers and try different things to succeed). It’s easy for games I made myself: if they can win at the Factor Draft (an upcoming post, I swear), they are a master of factoring. I have considered giving some points, not quite mastery, if they win against their classmates or my co-teacher, but to be a master, you gotta beat the final boss.

I’d love to have a bigger collection of games that I can use as assessment of skills, not just algebraic skills but the Standards of Practice as well. Any suggestions?

### Intentions Change Approach (DragonBox 2 vs DragonBox 1)

So since I first had my students play DragonBox last year, We Want to Know came out with a sequel, DragonBox 2. They are now branded as 5+ and 12+, as the original DragonBox is intended to introduce the idea of algebra and solving equations to someone unfamiliar with it, while DragonBox 2 is meant to deepen the equation-solving toolbox of someone already familiar with solving equations, allowing them to deal with more complex equations.

I was trying to decide which one to use with my class this year. It seemed like DragonBox2 would be better at first glance, because I teach high schoolers: we have seen basic equations, and now we need to kick it up a notch. But I wound up going with DragonBox 1, saving the sequel for a handful of students who blazed through it and were advanced. I know I made the right choice because of situations like I tweeted about:

There were several students who could solve the first level (one of the hardest in the game), but not the second, which came later. This showed me that there was something about the structure of an equation that wasn’t getting through and that we needed to work on it.

In DragonBox 1, you only really have four abilities: you can combine inverses into 0, you can divide a card by itself to get 1, you can add a card from the deck to the game (one on each side), and you can attach a card from the deck to another (multiplication/division), as long as you do it to every card in the level. In DragonBox 2, you can do new things like flip a card from one side to the other, divide a night version by a day version (leaving negative 1), combine like terms, factor out common terms, and treat complex expressions as single units to multiply/divide by.

Those are all good things to do, and someone proficient in algebra should be able to do those things. But I backed away from using it in class because it lacked the why. At the end of the first DragonBox lesson, I compile the notes students took while playing to make a comprehensive list of rules and abilities you have in the game. The one student who played DragonBox2 insisted that, in the game, you can slide a card from one side to the other. No matter how much I pressed him, he didn’t see that the card wasn’t sliding over, it was flipping/inverting.

And that’s what I was afraid of by using DragonBox2. These tools are important, but they have to be earned by understanding them. DragonBox2 gives them to you by completing previous levels, not necessarily by understanding how. At the least, in DragonBox 1, because you are stuck with the basics, you have to grapple with where the solutions come from. They can’t magically appear.

So while DragonBox2 is rated as 12+, I wouldn’t give it to any student who didn’t already have a firm grasp on the concept of equality. Maybe post-Algebra 1. Or at least not until much later in the year.

### Math Games

Back in January I participated in a panel on Math Games over at the Global Math. I meant to write this follow-up post shortly after, but January was a hell of a month for me and it slipped to the wayside. See my talk here, at the 2:55 mark.

I sorta hit the same point over and over, using six different games as examples, but that’s because I truly believe it is the most important point in both designing math games as well as choosing which games to use in your classroom. If the math action required is separate from the game action performed, then it will seem forced and lead students to believe that math is useless.

This can be fine if you want. Maybe you want to play a trivia game, where the knowledge action is separate from the game action. But if you pretend that they are the same, then you have problems.

This is the same essential argument as the one against psuedocontext. It may seem like you could say “It’s just a game,” but students see it as a shallow way to spice something up that can’t stand on its own. (I’m not saying review games and trivia games don’t have their place, but they can’t expand beyond their place.)

Below are the six examples I gave, with the breakdown of their game action and math action. I hope to use what I learned in this process to have us make a new, better math game in the summer, during Twitter Math Camp.

### Example 1 – Math Man

A Pac-Man game where you can only eat a certain ghost, depending on the solution to an equation.

If we apply the metric above and think about what is the math action and what is the game action? Here, the math actions are simplifying expressions and adding/subtracting, but the game actions are navigating the maze and avoiding ghosts. If I’m a student playing this game, I want to play Pac-Man. The math here is preventing me from playing the game, not aiding me, which makes me resentful towards that math.

### Example 2: Ice Ice Maybe

In this game, you help penguins cross a shark filled expanse by placing a platform for them to bounce over. Because of a time limit, you can’t calculate precisely where the platform needs to go, so you need to estimate. That skill is both the math action and the game action, so that alignment means that this game accomplishes its goal.

Verdict: Good

### Example 3: Penguin Jump

Here you pick a penguin, color them, and then race other people online jumping from iceberg to iceberg. The problem is that the math action is multiplying, which is not at all the same. The game gets worse, though, because AS the multiplying is preventing you from getting to the next iceberg, because maybe you are not good at it yet, you visibly see the other players pulling ahead, solidifying in your mind that you are bad at math, at exactly the point when you need the most support. A good math game should be easing you into the learning, not penalizing you when you are at your most vulnerable point, the beginning of your learning.

Verdict: Terrible

### Example 4: Factortris

This is a game that seems like it has potential: given a number, factor that number into a rectangle (shout-out to Fawn Nguyen here in my talk), then drop the block you created by factoring to play Tetris.

Again, the math action is factoring whole numbers and creating visual representations, which are good actions. But the game action is dropping blocks into a space to fill up lines. As Megan called it, though, we have a carrot and stick layout here, and often in many games. Do the math, and you get to play a game afterwards. (Also, the Tetris part doesn’t really pan out, because all the blocks are rectangles, which is the most boring game of Tetris ever.)

### Example 5: Dragonbox

I’ve written about Dragonbox before, so I won’t write about it too much here. The goal of Dragonbox is to isolate the Dragon Box by removing extraneous monsters and cards. The math actions include combining inverses to zero-out or one-out, or to isolate variables. The game action is to combine day/night cards to swirl them out, or isolate the dragon box. The game action is in perfect alignment with the math action, which makes the game very engaging and very instructive.

Verdict: Good