Do you teach trig? Do you wish your students had a solid understanding of how to relate trig ratios beyond 90 degrees to the unit circle? Are you tired of kids asking which button to press?

Well … I can’t say for sure if this solves your problem, because I haven’t used it in my class yet. But someone else has! And they said it works great!

The original I was shown is simply a full-circle protractor photocopied onto a co-ordinate grid. I decided to make a fancy-looking version in Illustrator. And since I’m all proud of it, I’m sharing it with all y’all.

unit circle protractor

unit circle protractor sheet

The idea is, if you want to look up sin or cos for a given angle, just find the angle and then read the y or x co-ord of the matching point on the unit circle. Kids should be able to visually estimate something close to two decimal values worth.


Edit: You may also want to check out the matching 30-60-90 and 45-45-90 triangle cutouts that match.

Yesterday, a particularly racially-insensitive game about the slave trade started catching the attention of @mdawriter and other people who care about race and education.  The clips passed around showed that the core gameplay used Tetris mechanics, but with black bodies being loaded onto a slave ship.  There were also shots of stereotypical dialog (“I pity the fool”? seriously?).

Melinda summarized some of the Twitter discussion from yesterday by RTing the back-and-forth between a number of educators.

For those who don’t know, I briefly worked in the game industry and have followed it from a distance since.  In my time working from home on contract, I worked on a small ‘serious game’ myself (although it wasn’t *that* serious), and I’ve continued to read up on serious games and game design as a study.

The reason I sum up my own history with this is that I don’t want to come across as flippant when I say: Yesterday’s dialog looks really, really familiar.

Now, I don’t really feel like playing this game myself to find out what value it has, or whether it has redeeming factors in the gameplay.  It might!  Although I don’t think those redeeming factors excuse things like racist dialog or the choice of using a very caricature-like visual style for the people involved.  But my point here isn’t to give an in-depth critique of this particular game.

Since day one, serious games have had to face the assumption that games are primarily meant to be fun, entertaining, and, well, not serious.  Unfortunately mediocre-at-best examples like this are not going to help a lot.  So I’d rather direct people to an actually good example that made the news in game design circles about six years ago.

Brenda Romero (then Braithwaite) is a game designer with decades of experience in the video game industry. In 2009 she presented on a series of board games that she created during a time when she was oversaturated and bored with typical video games.  The bit that reminded me of this initially was the game which got the most attention at the time, Train.  However I went back and found the video of her presentation at the 2009 Game Developers Conference, and I just can’t find a way to sum this up.

So here’s the deal: if you want to at least qualify as having “read” this post, bare minimum you have to go to the video, scroll on the sidebar to ‘Train’, and watch until at least the end of the Rabbi Belzer section.  (It requires Flash, I’m sorry. I can’t fix Old Internet, all I can do is ask you to get off your iPad and find a laptop or something.)

If Black American history is specifically important to you, you should watch ‘#3: obviously planned random tangent’ first.

If you really, really can’t listen to audio right now and you want to cheat, fine, you can read the first publicity that the game received.  (If you still feel like this is horrible and how dare she have done such a thing, you haven’t qualified as having read this whole post! Sorry, doesn’t count. Go watch the video.)

Probably the entire thing is worth watching.  I’m saying ‘probably’ because I’m totally lost in this, but I’m jumping around sections a bit and already 20min late on making lunch for my family and don’t want to put off posting this until tomorrow.

Other highlights to understand this all, from ‘The not-so-obvious obvious’:

“Frankly, the topics that i was covering, it made me feel a little … afraid.”

And from “It’s just like Halo.”:

“Train says the game is over when it ends. So a completely valid playing of Train is one that a rabbi in New York City had who walked up to the game and went, ‘Oh. I know what your game is about. I don’t want to play.’ And I said, ‘You just did.’ And she said, ‘You’re right.’ And we had a great conversation after that.”

Also ‘Treatment of tokens’ is a very short, very worth hearing section.


Okay. Don’t go on with reading this until you’ve heard enough.


Okay?  Okay.


So here’s what I want to add to this:

  • Games are an expressive medium, and they can represent ideas from nearly any area of life.
  • Making a game about atrocities is possible. But making a good game about atrocities is really, really friggin’ hard.  Brenda Romero is a professional game designer with over 30 years in the industry, and she made these games as a personal exercise of love with no desire to commercialize, no external publishing or funding constraints.  She thought about every single design decision really carefully and invested herself into it.
  • These games were not made lightly, nor were they made to be entertaining.  They weren’t the result of someone saying, “Hey, let’s take these social studies standards and make them into a game! Kids like games! That’ll make it more engaging!”
  • The designer either had a personal connection with the issues she was presenting, or she was very careful to run them by someone who did.
  • Every one of these games was run by the designer herself, and run carefully and thoughtfully.  These weren’t experiences that she could simply box up and mass produce.  That does not mean that something similar can’t happen in the classroom — it can.  But it does mean that we have to see ourselves in an active role, as a co-designer of the games that we bring into our classrooms.
  • None of these were digital, but there are thoughtful digital games that do a much, much better job than the example that started this all off.  (If you really want examples, nag me.  I tried googling something I remember from a while ago but it’s failing me, and I’ll probably remember more later.)  Here’s another random non-digital example that I want to share just because it looks great.

Hopefully you can see why I have a hard time getting into this discussion within 140-char posts.

Games can express and educate us about racism.  But games that give the issues a shallow or thoughtless treatment can be just as damaging as a shallow or thoughtless book / video / whatever.

My Object Lessons essay, “How the Ballpoint Pen Killed Cursive“, is now up at The Atlantic.  (Woot!)

I’ll likely be posting some follow-up thoughts and photos that didn’t get used for the piece here later.  (I will read comments here but if you have a real rebuttal you’re waiting to pull out, might as well wait until my next blog post.)

Edit: For all of you wondering why I even thought this was worth writing about, a) you should probably read all the way to the last paragraph, and b) you should read this older stream-of-thought post where I connect my thoughts on cursive to other areas of teaching and learning.

Also, huh, my essay topped The Atlantic’s “Most Popular” list for the better part of the weekend. Thanks Internet! Apparently I’m not the only one wondering what happened to their writing style.

This post is about politics, racism, murder, and statistical numeracy.

This past week, Canada’s Minister of Aboriginal Affairs insulted the chiefs he was meeting with by pushing the blame for murdered and missing Indigenous women (MMIW) onto the Indigenous community. He used a previously unreleased statistic – because numbers can’t lie, right? *gag* – to make his point. And not long after the RCMP stepped up to confirm his statistic:

Seventy per cent of the solved murders of aboriginal women were committed by people of aboriginal descent, RCMP Commissioner Bob Paulson says in a letter to Grand Chief Bernice Martial of the Treaty Number 6 Nations.  …

Now there are all kinds of questions this should raise. Why in the world would the RCMP and Minister Valcourt be sharing this kind of data in private?  Paulson himself claims, “It is in the spirit of our bias-free policing policy that the RCMP has not thus far disclosed statistics on the ethnicity of the perpetrators of solved aboriginal female homicides.” Right. So in the spirit of being bias-free, they only discuss these statistics with politicians. Quietly. Until the politician chooses to use that statistic for political leverage.

Edit: It’s also worth noting that the 70% statistic can, necessarily, only be about those cases which are solved. It doesn’t tell us anything about the unsolved cases at all – we can’t assume they follow the same distribution. Unsolved cases could reasonably be expected to be unusual situations and so not match the patterns of solved cases at all. It could be all white guys, who knows? The data below has been corrected since not taking unsolved cases into account skewed the numbers a bit. (thanks @apihtawikosisan for the reminder!)

Seventy percent certainly sounds notable. And it is, although it isn’t really surprising given that domestic violence is the most common case for all murdered women in Canada.  But thirty percent of a Shockingly Disproportionate Number of even the solved cases is still significant. Being only 30% doesn’t let white guys* off the hook.

The RCMP already released a report on MMIW in 2014. It has some useful information, although if you need a trigger warning on racist stereotypes and victim blaming maybe don’t read it right to the end.

Using the population numbers quoted in the report, we can start filling in totals in the table below:

MMIW chart 1



Unfortunately we have some question marks! Because naturally Valcourt didn’t say anything about how many non-Indigenous murder victims were killed by fellow non-Indigenous people.  Still, that 269 total is not a small number, even without however many of the 120 unsolved cases also belong there.

Now the total number of non-Indigenous women killed is a lot higher. But that’s not surprising since Indigenous women only make up 4.3% of Canada’s total female population. This also means that comparing totals is not incredibly useful.

So on the right of the chart above, I’ve calculated how many murders there have been per 100,000 of the affected population. For example, for the 87 in the first box, I took the 628 murders where an Indigenous woman was killed by an Indigenous perpetrator, divided it by the total number of Indigenous women in Canada, and multiplied by 100,000.  (This is sort of a way of representing what the probability is that a woman in that group would be murdered**, but made easier to read and compare than writing fractions of a percent.)

What does this show us? Well for starters, it highlights how much more of an issue this is for Indigenous women than for the rest of Canadian women. An Indigenous woman is over four times as likely to be a murder victim than other Canadian women.

Valcourt’s dropping of the 70% bomb was intended to deflect blame. Canadian voters, I assume he assumes, aren’t interested in shouldering the nation’s responsibility for the legacy of the residential school system and other arguably-genocidal acts we’ve committed in recent history. We’d much rather just push the blame onto the Indigenous population! That’s right, it’s all your fault.

Now, there’s no avoiding that domestic violence is a real part of the equation here. This was already known and there’s some concrete data in the RCMP report. Despite the fact that a smaller ratio of the murders of Indigenous women were done by a spouse or other family vs other perpetrators than in other cases of women murdered, the risk of domestic/family violence is still the largest overall risk for Indigenous women (as with Canadian women in general).

But is the larger amount of domestic violence in Indigenous families the only issue? Is the cause that Indigenous men are more violent? If this were the case, we would expect to see significantly smaller proportions of MMIW cases being caused by people outside the family. However, the opposite is true. Numbers are higher across the board, and murders caused by someone outside the family are actually a much higher percent of the cases for Indigenous women.

Here’s a big old chart looking at data from the RCMP report (solved cases only):

MMIW chart 2

Using the per-hundred-thousand section, we can see that Indigenous women are more at-risk of being murdered by an acquaintance than a white woman is of being murdered at all. That’s already a pretty bad sign.

Now let’s look back at that first table. What can we say about the 30%, despite the blanks? Thirty-seven Indigenous women murdered per 100,000 by a non-Indigenous killer, vs. thirty-four non-Indigenous women per 100,000 murdered at all. And that’s without even including any of the unsolved cases, which I’d assume includes some (if not a lot of) non-Indigenous murderers.

Indigenous women are more at risk of being killed by a non-Indigenous person than other Canadian women are at risk of being killed by anybody at all.

Ugh. You guys, I think it’s safe to say that us colonizers are still a part of the problem.

So let’s not pretend that Valcourt’s 70% lets white-male Canada off the hook. Nor does it excuse the government from listening to the call for a large-scale inquiry into the large-scale systemic problems. That 30% still looks an awful lot like violence influenced by racism.

The wrap-up infographic:

MMIW chart 3


* Sorry, I mean, “Non-Aboriginal perpetrators”.

** I’m using total number of murders over the past 30-odd years included in the RCMP report and then using the current population totals. Potentially wonky, but unless that 4.3% has changed drastically over the last 30 years it should still be fine for making comparisons between groups.

ps. Yes, I noticed later that Paulson said that 25% are non-Indigenous, and 5% are unknown. I’m not going to bother editing for that because, meh, if it’s unknown then most likely they were non-Indigenous; remember, 4.3% of the population.

pps. Some of the “totals” listed for unsolved cases are calculated from the RCMP report’s stated percent-solved, which was a hand-wavey 90%-ish.  They’re more than accurate enough to compare situations between groups, but don’t quote them down to the last digit.

So yesterday I popped into Staples to recycle a toner cartridge and walked past this:

Staples Graphing Scientific Calculator, aka Dirt-Cheap Graphing
Staples Graphing Scientific Calculator, aka Dirt-Cheap Graphing

The sign said it was on sale for $2.97.  Next to it was a rack of TI-83’s for ten million, I mean $125.00.

One could buy an entire class set of these Staples things and still have like thirty dollars left over, instead of buying a single TI-83.

So I decided to conduct a three-dollar experiment.  Can this thing actually do everything one would need to get through the BC pre-calculus stream?

The hardware

First thing I notice when I turn it on, in the parking lot before I drive away: the screen is glitching out.

I go back in, swap it for another, okay this one seems fine.  Well, that didn’t last either, as you’ll see in the rest of the photos.

Missing a few lines.
Missing a few lines.

I have no idea if this is indicative of all of these calcs, or if this was a batch of return-for-refund (which some of them looked to be) and that’s why they were on sale, or if I just had a bad stroke of luck, or if they act weird when the batteries are low.  Speaking of which!  They use annoying large watch batteries, the ones that run out faster and cost more money to replace than this calculator did in the first place!  So points against for the cheapo build and annoying battery choice.

The display itself is kind of a hybrid of normal scientific calculator LCD, and a full pixel-grid graphing calc display.  There’s one section on the left where graphs appear, a row along the bottom where text appears, and some other bits on the top right for random other LCD symbols or whatever.

The software

Assuming you get one of these with a working display, this is the important part.  Can it actually do everything you need it to do?

The short answer is, yes, except for one small thing, but it also does one new thing I’ve never seen in a graphing calc that’s kind of awesome.

All the stuff you’d expect from a scientific calculator is here.  Trig, inverse trig, exponents, a fraction button (that kids never seem to clue into the existence of unless you spell it out to them), logs, etc.  It seems to be able to do a couple of different kinds of storing numbers into memory, if you’re into that kind of thing.

It also has more than I expected in terms of statistical and data-analysis functionality.  You can enter data and get standard deviation, mean, sum of squares, that sort of thing.  Most surprisingly, it can do regressions!  Linear, logarithmic, exponential, power, inverse, and quadratic.  (I didn’t test these out, I’m just reading the manual here, but this is already above and beyond our province’s math curriculum anyway.  Still, neat.)

Apparently it can also do numerical integration.  So we’re entering “could complete high school calculus with this thing maybe” territory.


Now the meat of the question: does it really do graphing properly?

Pretty much, yeah.  You can’t graph as many simultaneous functions, but you can do two at a time.  The screen is a bit smaller than a TI-83’s but let’s face it, we’re comparing low-res to low-res and they both do the job about as lousy as the other.

Also sometimes it seems to forget to erase the previously graphed function.
Also sometimes it seems to forget to erase the previously graphed function.

There’s a graphical solver that is a separate button from the normal ‘Draw’ button, but which lets you enter whatever function and whatever you want to test for y being equal to.  In fact it’s sort of less annoying than the TI-83 since you don’t need to choose left and right bounds. It just finds all the solutions it can on-screen.

There’s some additional ‘sketch’ functionality that lets you do things like draw lines between two points, or more interestingly draw a tangent to a given function at a chosen point.

The only thing this is missing: you can’t find local max/min of a function such as a quadratic.  You can zoom in on the vertex and trace it manually until you’re close enough, I suppose, but it’d be a bit of a pain.

The surprise feature

The calculator has a ‘GRAPH LEARN’ button that was bewildering at first, until I read what was going on.  This thing is letting you explore function transformations visually.

Function transformations!
Function transformations!

You can choose one of a bunch of standard functions, such as x^2, square root of x, ln x, sin x, etc.  You pick whether you want to try “shift”, ie. translations, or “change”, ie. expansion / compression.  Then it graphs that function for you and you can alter the graph using the arrow keys.  As you move or change the function, it shows the updated function definition beside the graph.  So kids could practice moving a function around and seeing how it affects the equation for that function.


If I had my own class, which I don’t, and if I could get my hands on a whole set with working screens, which might be challenging, I would totally be happy with a class set of these things.  Not being able to find the vertex of a quadratic is a small disappointment, but the fact that it manages everything else for such a low price is crazy worth it.

Also keep in mind, I’m primarily thinking of BC’s PreCalc 11 and PreCalc 12 here.  We don’t have provincial exams.  If you decide to adopt this calc for your entire class, you can just assess the “with technology” version of everything that this calculator can do, and leave the “find a min/max” for problems done algebraically.  Or with Desmos on something that’s not an exam.

Now this isn’t to say the hardware issues aren’t a real concern.  If I were putting down a lot of money for calculators, I’d like to know that they’re going to last more than a year and won’t all glitch out on me.  I’d also much prefer to have something that uses AAA or AA batteries, and not this watch-battery nonsense that costs significantly more over time and is more annoying to replace.  (Although you sure can buy a lot of watch batteries for $122.)

It does make me wonder a lot harder about the other alternatives out there, though.  I’ve seen kids use Casio graphing calculators that are less than half the price of a TI.  If someone wants to send/loan me one maybe I’ll test out one of those next time.

A lot of teachers wonder why we even use TI’s when there are free apps like Desmos, Geogebra, and Wolfram Alpha out there.  I think that’s ignoring the unfortunate pressure of needing to have something that works during exams, and that it’s an extra pain to lock down iWhatevers from messaging during classroom tests.  But there’s no reason we need to keep handing TI a monopoly over the market.  This Staples cheapo is an extreme example of something low-balling and still managing to pretty much get the job done.  It might be too cheapo, but it should still serve as a wakeup call.  There’s no reason why we math teachers can’t support our students by doing the due diligence to test out and recommend alternatives to the TI regime.