## Trying to find math inside everything else

### The Factor Draft

Last year at #TMC13, I ran a session called Making Math Games. I stared off with an overview of what makes a game a good game, while still being good math pedagogy as well. Then we spent most of the session in two groups brainstorming idea for games for topics that are somewhat of a drag to get through. The other group worked on something in Algebra 2, though I don’t recall what – I must say both groups were supposed to write up what we did and neither did. (But I do think Sean Sweeney was in the other group, so maybe he remembers.)

My group worked on a game for factoring, focusing on Algebra 1. I took the ideas from the session and made a mostly operational game. Then, about 2 months ago, Max Ray came to visit me on the day I was unveiling that game in class. He saw it and it worked out…okay, but here was definitely improvements to be made. So we talked over lunch (about many things, not just the game – he’s great to talk to!) and then tried out some changes with my lunch gang. The changes seemed to work and I went forward with the new version in my afternoon classes to great success. By the end I think I had a really wonderful game, and so I wanted to share it with you.

### The Materials

A set of Factor Draft cards includes 3 differently-colored decks. Mine, pictured here, were green, blue, and yellow. One deck (green here) is the factor cards, with things like (x + 2) and (x – 1) written on them. Another deck (blue) is the sum cards, with numbers like 10x or -4x. The last deck (yellow) is the product cards, with numbers like +36 or -15.

### The Set-up

Lay out the cards as follows: make a 3 x 6 rectangle of factor cards, a 4×3 rectangle of sum cards, and a 4×3 rectangle of product cards, all face up. Place the remaining cards in separate piles next to the playing area.

In the cards I printed, I didn’t put the Xs on the blue sum cards. Max suggested I do because it’s easy to be confused on which is which.

### The Objective

The goal of the game is to collect 4 cards that can be used to complete a true statement of the following form: (factor card)(factor card) $= x^2$ + (sum card) + (product card).

### Gameplay

Each turn, a player may select any card from the playing field and place it face-up in front of them. They then replace that card with a new card of the same color from the deck. Play passes to the left. A player may have any number of cards in front of them, and may use any four cards to build a winning hand.

The cards I collected after turn 5. There’s two possible cards I could pull to win the game – can you see which ones?

If at any point a player achieves victory, if they had more turns than the other players, they must allow the other players additional turns to attempt to tie. Upon a tie, discard the winning cards and continue play as a tie-breaker.

A winning hand.

My co-teacher, when we were testing the game, said that it felt like Connect 4, in that with each move you have to decide whether to go on the offense to try and complete your four cards, or go on the defense and block the other players’ sets. But as each player gets more and more cards in front of them, it’s hard to see all of the connections and effectively block, so the game will always eventually lead to victory.

I may need to adjust the number of cards and type of cards in the decks, but I think what I currently have works well – if you have any feedback on the card distribution, let me know. The sum cards go from -10 to +10, with the numbers closer to 0 more common. The product cards go from -60 to +60, with each product card being unique. And the factor cards go from (x-10) to (x+10), also with the ones closer to 0 being more common. (There are no (x+0) cards.)

I did a whole little analysis to determine how many of each type of card to include…but maybe that’s a post for another day.

Sum:Product Deck – The first four pages are the sum deck, the next four are the product deck, the last four are the factor deck.

Factor Draft Play Mat and Rules – Players can use these mats to place their cards and check for a win.

### Counting Calories

So my Fit Meter not only counts the steps I take, it also tells me how many calories I’ve burned based on what it measures as the intensity of those steps (whether I’m running, just walking around, going up hills/stairs, etc). This seems like it would be useful, especially because Wii Fit gives me a certain goal to meet every day. But where it gets tricky is if I want to also combine it with My Fitness Pal for tracking my eating as well. MFP makes certain assumptions about how many calories you burn in a day when they give you the goal of how much you can eat. But if I am using those assumptions, then I can’t use the calories on my Fit Meter as an accurate guide.

I’ve determined that the Fit Meter’s calculated calories only come from activity above sitting quietly, which has a Metabolic Equivalent of 1. So, using METs, I can figure out how many calories I should eat if I only sat quietly all day, or slept. (Sleeping has an MET of 0.9). If I sleep for 8 hours a day (I don’t, but better to overestimate that), and sat for the other 16, given my weight of about 100 kg I would burn 16*1*100 + 8*0.9*100 = 2320 calories per day.

So now I can accurately combine the two – if I tell Wii Fit I want to lose 1.5 pounds a week, it will tell me I need to burn 854 calories per day on my Fit Meter. So then I can finagle My Fitness Pal to let me eat 2320 calories per day, so that they align correctly – if I eat 2320 but burn an additional 854, I’m still losing. That way, if I wind up eating more, I know how much more I have to exercise to compensate.

### Aligning Vocabulary and Notation

At my math department meeting yesterday I wanted us, as a department, to get some alignment on the various ways we talk about certain topics. I often get students in the complain “But that’s not what my middle school teacher said!” And so I don’t want them to arrive in Geometry and shout “But that’s not what Mr. Cleveland said!” So I want to tighten things up vertically.

But as we sat there, it was hard to think of specific examples of things we can change. One was the “slope formula,” which could be $\frac{y_2 - y_1}{x_2 - x_1}$ or $\frac{y_1 - y_0}{x_1 - x_0}$ or just $\frac{\Delta y}{\Delta x}$. We can to the conclusion that it should be the third across the board, but that’s maybe the only agreement we made. (Glenn’s vertex form idea was also brought up, which I convinced 2 of the 3 other teachers about.)

I did bring up Nix the Tricks, and I think I convinced my principal to buy us all copies, so that we can get some uniformity that way. But I need some feedback before the next meeting – what are some things (not necessarily tricks) that we could be more consistent about? As Glenn said in that post, if we’re going to be arbitrary [about notation and wording], at least let us be arbitrary consistently.

### The Evil Queen Steals Hearts

Part 3 of my Disney Analysis.

Despite my love for Disney films, I acknowledge that not all of them are, well, good. But after working on the last two posts, I wondered what effect those characteristics have on how well a movie is rated. Do critics share Disney’s sense of justice and just want to see those villains die? Do audiences actually prefer movies with male protagonists, as much of Hollywood seems to believe? To find out, I collected the tomatometer scores of all of the movies from my list on RottenTomatoes.com – both the critic score and the audience score as, though there is a correlation between the two, it’s a moderately weak one.

I mostly included this because I wanted a graph that wasn’t a box plot.

So first, let’s look at the fates of the villains vs how they scored. I create two plots: one for the critical scores and one for the audience scores.

From these I can conclude…that there’s not much connection between the fate of the villain and how audiences react. We can say that there is a slight audience preference for movies that actually have a concrete antagonist, and we may also be able to say that critics and villains have a slight preference for movies where the antagonist is merely thwarted, but it’s not a strong connection.

What about gender, though? How much does that have an effect? Let’s look at villain gender first.

The conclusions I can make? People love those lady villains! Despite the fact that 70% of villains are male (or maybe because of that fact), audiences and critics agree that the movies with female villains are better movies across the board. (The audience preference for female villains is not as strong as the critical one, but it’s still there.)

And as for the protagonists?

Conclusions: There’s a clear critical preference for movies that have male AND female protagonists, to give access points to all viewers, whereas audience members merely as less likely to think badly of those. There’s a slight preference for female protagonists among both audience members and critics as well, though it’s very slight.

So what could Disney learn from all this? Well, that clearly we want to see a movie with a pair of heroes, male & female, that face off against a female villain and defeat her without killing her. So make that happen, Disney.

(Oh, what? The next movie has a male protagonist and a male villain? Go figure.)

###### The Data Set

Disney Data

Part 2 of my Disney Analysis.

While doing my research yesterday, I had this exchange:

So I thought about what effect gender might have on things. Let’s take a look.

First, let’s look at the gender breakdown of the movies in general, both of the antagonists and the protagonists. (Some movies don’t have only one protagonist, so some movies are labeled as having both male and female protagonists.)

As we can see, despite Disney’s princess movies, Disney animated films are overwhelmingly male (much like most of Hollywood). Interestingly, the rates of movies with male villains and male protagonists are the same, about 70%.

I also looked at how the genders match up – do female villains only face off against female protagonists, for example?

Here we see that there’s no strong associations with male villains – either gender of protagonist can face a male villain. However, there is a strong dissociation of male protagonists to female villains – in fact, there’s only two movies that have male protagonists and a female villain – The Emperor’s New Groove and Meet the Robinsons, both in the current century. (And the “female” in Meet the Robinsons is a robot.) Female villains will also go up against an ensemble of protagonist that includes males, but in general they must go against a female main character. We also see that those man vs self and man vs society movies are literally “man” – only one female protagonist out of the 8.

Now, what about my idea that gender affects their fate? First, let’s check the gender of the villain.

Turns out I was wrong – there’s no association between gender and death (or banishment). 57% of all villains die, while 56% of male villains die and 60% of female villains do. There’s a slight association with male villains being imprisoned while female villains are merely thwarted, but the sample size for those is much smaller. But while the distribution of genders for the villains is lopsided, how they treat those villains is pretty equitable.

What about the heroes? Do the male heroes cause all the death?

No, not really. The death stats are pretty close to the overall stats. I do see an association with male protagonists banishing their foes while female ones imprison them, though.

However, as Elena said above, most Disney deaths are not directly caused by the heroes – they are often accidental or caused by the villain themselves. By my reckoning, there are only 5 villains that are directly killed – Maleficent, Ursula, Scar, Shan Yu, and Captain Rourke.

Now, you may think, “Well, James, 3 of those movies have female protagonists and 2 male, so there’s no association, right?” Well, yes…but then, think about who actually deals the killing blow: Prince Philip kills Maleficent, Prince Eric kills Ursula, the Hyenas kill Scar, Mushu kills Shan Yu, and Milo kills Rourke. Yes, even Mulan does not actually land the final killing blow, though she arranges all the circumstances of that death and should be credited with it.

That’s right, these two are literally lady killers.

### Disney Justice

Part 1 in a 3-part series of Disney analysis.

I was out on Rob’s balcony this morning and my stream of consciousness was something like this: I shouldn’t stand so close to the edge, I might fall, no I won’t, that’s silly, where did this irrational fear of falling come from, maybe it’s all those Disney villains I grew up on, all the villains always fall to their doom, hmm, you know, in Frozen the villain doesn’t die at the end, that seems pretty unusual to me, but is it?

So I decided to do some research and determine just how often Disney villains die. Below are my results, and some other conclusions.

(Notes about the data set: this only includes the animated features created by Walt Disney Animation Studios, not a subsidiary. This list also does not include any film that is not one continuous story – this leaves us with 43 films total. I also has to make some decisions between focusing on villains and antagonists. Some characters are villainous, like Mad Madam Mim from The Sword in the Stone, but she’s hardly a major antagonist in the film. Other characters, like Aunt Sarah in The Lady and the Tramp, are antagonistic but hardly evil. I’ve decided to focus just on antagonists.)

I categorized the fates in four ways – death, imprisonment (not always in an actual prison), banishment (or being driven off in some way), and thwarted (where the hero wins but nothing really bad happens to the villain, such as in Cinderella).

8 of the films have no real antagonist (as opposed to man vs man, their conflict would be classified as man vs society or man vs self. [And man vs nature in the case of Bambi.]) But in a majority of the remaining films that do have antagonists, the antagonist dies by the end. (Often by falling – 7 villains fall to their deaths.)

Another question then arose – has Disney always been this swift with the death penalty, or has that changed over time? So I made some box plots of the years for the different fates.

Though the very first Disney movie, Snow White, has the villain die, it’s a clear outlier – as is Sleeping Beauty. Most of the villain death occur during what is known as the Disney Renaissance, aka my childhood. The time of villains being defeated but without really changing their status quo harks back to an earlier time, whereas banishment and imprisonment as more universal. Interestingly, the films without villains all come from either the 40s or the 2000s. Neither is thought of as a big time for Disney films.

Below is my data set (spoilers). Perhaps more analysis will come in the future.

### Renaissance Man

My coworker, the history teacher, is teaching the Renaissance right now, and she used me as an example of a Renaissance Man. That day, in every class, they came in, “Mr. Cleveland! Ms. Bradford said you are a Renaissance man! Mr. Cleveland! You’re a humanist!”

In one of the classes, one of the students told another they were drinking toilet water. I told them that all water is toilet water, since toilet water is just tap water. When they asked why they can’t drink it, I said that it’s only because of touching the bowl, but that if they wanted, they could drink the water in the tank and, in fact, if there was some catastrophe where they needed drinking water and couldn’t get it, that would be an excellent place to get some. “Wow, you really are a Renaissance man! Do you know…how long the school is? In inches?” Sure kid, 3408 inches. (Nice how such a specific number makes it sound really accurate.)

One of my sections, though, did not believe it, because I had previously told that that all teachers lie. (That came after several “My middle school teacher said you can’t do that!” “Well, actually….”)

### Classroom Research

Back in grad school, instead of one big thesis we had to do two research projects – one math research and one classroom research. I don’t think my math research is particularly noteworthy (it was about automating the instrumentation of a harmony given a melody, in the style of John Williams), but I did like my classroom research. Towards the end of the program I was talking with my of my classmates and we said how we both enjoyed it and could see going back into a program to do research in the future. I don’t know if that’s still true for me, but the future holds many possibilities.

The focus question for my research was, “Given the three standard methods of solving systems of equations, which methods do students prefer and why?” I had some ideas going into it that held true, some that were thrown out, and some interesting other ones. For example, some of the students preferred a certain method just because it was the one they learned first, even if they knew it wasn’t the best one. Visually-inclined students did not prefer graphing, as I expected, but rather elimination, because of the way the numbers lined up. Some students changed which method they used in order to avoid something – one student had trouble solving equations with x on both sides, so the method they used was the one that didn’t lead to that scenario.

You can see more of the findings, and the whole paper, below if you are interested. And yes, if you read it, you’ll notice that the pseudonyms I chose for the students were all based on Doctor Who companions.