## 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:

### The Integral Struggle

I had an extra day to fill for one of my sections of calculus, thanks to the PSAT, so I set about thinking up a game I could play. I gotta tell you, as much as there is a paucity of good content-related math games out there, it’s extra so as you move up the years in high school. I can still find a good amount of algebra and geometry games online, but Algebra 2? Precalc? Calc? Fuhgeddaboutit. Well, I’m teaching both Pre-Calc and Calc this year, so I guess I’ll just have to make them myself. (I brainstormed two more during said PSAT, so those might happen soon enough.)

We just covered function transformations and how they affected integrals, so this game hits on that topic. (Yeah, we’re doing integrals first.) Here’s how it works: there are 3 functions/graphs, each of which has a total area of 0 on the interval [–10,10]. One team is Team Positive, and the other is Team Negative. Teams take turns placing numbers from –9 to 9 that transform the function and therefore transform the area. Most importantly, they also place the numbers in the limits of integration, so they can just look at a specific part of the graph.

With three functions, it becomes a matter of best of 3 – if the final integral evaluates to a positive number, Team Positive gets a point, and vice versa. If it’s a tie at the end of the game, because one or more of the integrals evaluated to 0 or were undefined, then redo those specific integrals. (I discussed with the students whether it would be better to have it be the total value of all 3 integrals instead of best of 3, but I think that makes the vertical stretch too powerful.)

That’s it! Let me know what you think. Materials below.

### 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.

### Crossing the Transverse

Oh my god, I haven’t blogged since August! This has been a hell of a year, let me tell you. But maybe I’ll tell you in another post, because this one is about the new game I made in my Geometry class. (My first non-Algebra game!)

So the game is called Crossing the Transverse. The goal of the game (pedagogically) is to help identify the pairs of angles formed by lines cut by a transversal, even in the most complex of diagrams. The goal of the game (play-wise) is to capture your enemy’s flagship.

Here’s the gameboard:

I printed out the board in fourths, on four different pieces of card stocked, and taped them together to make a nice quad-fold board. Then I made the fleet of ships out of centimeter cubes I had, by writing in permanent marker on the pieces the letter for each ship.

Here’s the rules.

In the game, each type of ship moves a different way, which makes it feel a lot like chess – trying to lay a trap for the enemy flagship without being captured yourself.  Many of my students really enjoyed it when we played it yesterday. Today, though, to solidify, I followed up with this worksheet where they had to analyze the angles of a diagram much like on the game board. They did pretty well on it, so I’m satisfied!

## Materials

Crossing the Transverse Rules

Printable Map (Prints on 4 pages)

No Stars Printable Map (If printing the background galaxy is not for you, here’s a more barebones version.)

Zip File with Everything, including Pages, Doc, and GGB files

### Fighting for the Center

At the Math Games morning session at Twitter Math Camp 15, we’ve been created curricular games that hit on some topics that there aren’t really good games for. I came up with the idea for this one, and worked on refining it with the help of Paula Torres (@lohstorres1) and John Golden (@mathhombre).

This game is about measures of central tendency (and range for good measure). Not only do students have to determine all of those over and over as they play the game, but they can see how changing the data set changes the values, especially as the size of the data set increases or decreases. It seems really good because it drives the need to make those calculations.

All you need is two decks of cards. The game is designed as a two-player game, but it would definitely be best done as two pairs playing against each other, so they can talk to each other about their strategies and calculations. We also recommend having students keep a running tally of the values.

### How to Pack Your Boardgames

Last year, before Twitter Math Camp, I was packing and trying to figure out which games to bring with me for the game night we were having before the conference started. I basically had three attributes I was considering: how big the game was, how good it was, and how many players could play it. I wanted to minimize the first one while maximizing the latter two.

So I tried to come up with a bunch of formulas for figuring it out, but nothing was quite working out. (I used BoardGameGeek ratings for “how good it was.”) At first I tried doing ${\frac{r \cdot p}{v}}$, but it was putting some games that just weren’t very good as top choices. The problem was that the volume was having too big of an effect – games could come in thousands of cubic centimeters of volume, but max at around 8 for rating and 12 for players. (I had to use amazon.ca to look up the dimensions because I wanted to use centimeters.)

So then I tried cube rooting the volume, or doing an exponential functions like ${\frac{p \cdot e^{r}}{v}}%s=2$, or finding the geometric mean of the three numbers, but still nothing came out right.

I was basically using three games as test cases: Dominion, which is one of the best games I own but it really big; Pixel Tactics, which is one of the smallest but is only 2 players; and The Resistance, which is small-ish, really good, and can go to 10 players. I figured that any good method should tell me to leave the first two games at home, but to bring the Resistance. If they didn’t, it wasn’t right.

Eventually, after doing some research, I determined that a common technique used in psychology when comparing variables of different ranges of values is called standardizing the variables. Basically, for each attribute, I would find the mean and standard deviation. Then, for each game, I would subtract its value from the mean and divide by the standard deviation to get a standardized value. Then I just needed to add up the three standardized values and the ones with the highest score would win. And, as predicted, The Resistance came out on top.