Dot Paper Generator

I’ve been using dot paper (both “square” and isometric) in my grade 9 class lately, so I put together a Java application that generates it. The image files it produces are PNG files. Feel free to use it if it’s helpful. No warranty, expressed or implied, yada-yada.

Google Drive link to download .JAR application file


Sample PNG file.

Here are a couple of PDF examples that I produced from the PNG files:



The PNG files are set to 72dpi, not the desired dpi the user chooses… I haven’t figured out a simple way to set that information in the PNG metadata. The PDF files above are both 600 dpi, if I remember correctly. 


Sloppy notation doesn’t seem to be reducing understanding of solving linear systems

A couple of weeks ago I wondered here:

Is sloppy notation for solving linear systems reducing understanding?

The TL;DR is “no, not really”. There are other problems besides notation.

Using subscripts to denote a specific point isn’t something Grade 10 students seem super-familiar with, in spite of their supposed experience with the slope formula:

m = \frac{y_2 - y_1}{x_2 - x_1}

More than three quarters of my students simply neglected to use those subscripts when solving systems. They wrote solutions without following the model I presented to them in class.

The ones who did use the notation had a stronger understanding of the concepts/strategy on average. I don’t believe the use of good notation was the cause of that improved understanding; rather, students who understood the concepts were more likely to use the [more complex] notation I presented.

There were two main barriers to understanding in this unit.

First, students do not connect the graphical and algebraic representations of lines. If presented with an equation like


most students can tell me the slope and the y-intercept. But until I ask for those parameters, or until they actually graph the line, they typically don’t visualize that line at all – it’s just a bunch of symbols.

This lack of crossover between representations means that students are not making sense of their own work and judging the reasonableness of their solutions.

Second, students are neither skilled nor fluent with solving linear equations. They do not always remember the inverse operations, and they rely on phrases and tricks to complete these processes. They have trouble because phrases like “move it to the other side and make it negative” doesn’t work well for multiplication and division, and they forget to apply an operation to each term in an equation.

It’s kind of the same problem as the first. There is a feeling of flailing about in the classroom, of trying to apply poorly understood or misunderstood rules to a fairly complex process without even being able to confidently test whether the result is correct.

So notation isn’t the issue. If you have kids in grades 8 or 9, make sure they can solve equations quickly and accurately, including those with fractions. If you have kids in grade 9 make sure they practice graphing lines and determining equations based on graphs. They’ll be in much better shape when learning the more complex techniques in Grade 10.

Stop being selfish: adult brains need helmets too

I spent part of Sunday afternoon riding bikes with my kids on the Hub Trail in Sault Ste. Marie. It was a good time, and we all enjoyed ourselves. Here we are about halfway through our 5K trip, taking a break on a bench:

We passed a few other cyclists along the way. The Hub Trail parallels Queen Street, a major road, for a portion of our route. Queen Street now has a bicycle lane, for which I’m sure many cycling enthusiasts are thankful. Unfortunately, my non-scientific survey of those pedaling people indicates they either (a) vastly overestimate their skills, or (b) vastly underestimate their mortality.

Hardly anyone was wearing a helmet.

Queen Street has a posted speed limit of 50 km/h, which means most cars are hovering around the sixty-click mark. At that pace a cyclist in the bicycle lane can’t do a whole lot to prevent injury if there is an accident, whether it is caused by a motorist or the cyclist.

Unless, of course, that prevention was enabled prior to the journey. Like, you know, putting a helmet on.

We know helmets work. We even have a law that says kids have to wear them. Why don’t adults have to wear them too? What is it about adult brains that is less valuable than kid brains? As far as I understand it, adult skulls aren’t significantly more durable when negatively accelerating due to asphalt.

Some people claim some sort of obscure “right” to not wear a helmet, in the same way that it’s an obvious right to ride a motorcycle sans cranial protection at 70 mph on an interstate. “It’s my life” and all that nonsensical garbage. Your decision to not wear a helmet is outrageously selfish. When you make a tiny mistake, or when a driver makes a tiny mistake, your decision to not wear a helmet might amplify that error from minor to catastrophic.

“I grew up riding my bike without a helmet, and I turned out fine! We weren’t so reckless because we knew we didn’t have helmets on!”

The fact the you’re alive and uninjured says nothing about all of the people who have been injured or killed in these sorts of accidents. Your random survival is a single data point. I’m happy you’re alive, but I’m so disappointed that you’re being so careless with your life and the future of that anonymous driver you might share an accident with.

While I’m talking about it, start wearing a helmet when you’re skating with your kids. I don’t care if you played hockey. Accidentally hitting your head on the ice is stupid.

(If you do a quick Google search about the efficacy of bicycle helmets to prevent injury you’ll find a lot of ideologically-driven pages saying that they don’t really work. Read a lot further than that if you want the truth. Helmets reduce injury in the event of a collision. The larger, more complex question is whether legislation is effective in preventing overall injuries.)


Life Protip: Let people like what they like


I played some music today during my math class for the first time this semester. We were completing some exam review and I thought it might be nice to have some tunes on to lighten the mood (which is sometimes a bit leaden in that class, unfortunately).

Remember that I work with teenagers, so I’m sure you won’t be surprised when I tell you that students complained about the music. I played a mixture of current “hits” of various genres. Some vocal students griped about this or that artist, claiming that “he can’t sing”, etc. I switched to music they would be less familiar with but which I knew was “classroom safe”. As I anticipated, the leaders of tomorrow didn’t like that either.

They nearly all listen to music, but they each listen to their own music, which is supreme in their eyes, and everything else is, of course, absolutely terrible.

I tried to impart words of ancient wisdom, but I don’t know if anyone really agreed with me. “Let people like what they like, and you can like your stuff,” I said. “As long as they’re not hurting anyone, it’s fine.”

This isn’t just about music, though. I have dozens of interests, and many of them are niche or “weird” for most people. That doesn’t take away from my enjoyment of them, but often I don’t have many people to share with.

It took me many years to understand that it’s great when people like something, and it’s even better when there are lots of different things that people like. Let people like what they want to. If you’re lucky, they’ll share it with you, and you can learn to appreciate it the way they do.

It’s not a snow day here

I’m fine with that. I don’t really want to lose any time in the classroom, and there’s a rescheduled NOSSA football game in the snow today.

But if it had been a snow day, what would I be doing?

I’d probably do a bunch of school work, honestly, including work for my e-Learning course. There is always more work to do than hours to do it.

But then I’d head outside with my kids, dork around in the snow, probably strap on the XC skis and have a great time.

So maybe I wish it had snowed a little bit more last night.

What would you do with a snow day?

Sprawling ramble about whether to become awesome

Audio recording here.

I have a lot of interests. Maybe a lot of people do, but I feel I might be straying to one end of the continuum. I’m not saying it’s bad, exactly, but it definitely impacts my life.

Here’s the problem: everything is interesting. Really, everything. I’m fascinated by it all.

Look, here’s my current Twitter bio: “e-Learning Contact in Northern Ontario. I also read, write, paint, play guitar, code, draw, nordic ski, run, take pictures, shoot, cook, and play with the kids.”

The first problem with that? I left a bunch of stuff out. I love Lego and video RPGs. I love editing video. I love woodworking. I even like to cut the grass. I love gardening, except for the actual creating-a-garden part.

The second problem is more obvious: that’s a very, very long list, and it’s going to grow over time. I’m 34 years old; I know I haven’t come across all of the awesome and fascinating things to learn about in life. I expect and hope to have decades of riveting years ahead of me, and I just don’t really have room.

Up until now I’ve taken the Renaissance approach: I’ll learn a little bit about everything. Mile wide, inch deep. I’ll be okay or even pretty good, but not excellent, at everything.

But I sometimes worry that this is the “wrong” approach. Perhaps it’s less fulfilling somehow than becoming awesome at something. I feel like I have it in me to be excellent at, well, anything. That’s not me boasting, that’s just how I feel we all are about most things. We can become great at most things if we commit to them. That commitment is the component that’s missing.

For myself, committing to something means the exclusion of a lot of other things. For example, if I were to decide to pursue painting in my spare time I’d probably have to give up guitar or writing or programming or blogging or… see? I don’t believe there is enough time to do it all. I already feel bad sometimes about not being a highly skilled guitarist because I don’t practice or try to learn new skills. How would I feel if I had less time for guitar because I spent my time painting? Would I feel even more guilt?

Part of the issue is that I don’t know why I feel guilty. To whom do I owe a guitar performance? Only to myself, I suppose. Is my own expectation for myself truly higher than what I’m willing to meet? That’s kind of a foolish way to set my personal standards.

Maybe it’s because I’ve already “bought in”. You know what I mean: I own a guitar and a terrific amplifier; I have tons of drawing paper and lovely pencils; I have dozens of tubes of acrylic paints; I have LOTS of Lego. If I were to abandon something it would be akin to admitting those purchases were wasteful. Even to myself it’s hard to admit a mistake.

I did once, though. I used to knit; my mom taught me. I still know how. I learned a lot about the craft. I listened to podcasts, watched videos, combed through blogs and pattern sites, borrowed books from the library (yes, the physical kind), and more. I bought a full set of beautiful bamboo needles, including pairs of single-points, quintets (is that right?) of double-points, and 4″ circulars. They were lovely to work with, and I enjoyed it a lot. But I found I was enjoying other pursuits more, and that knitting was an exceptionally slow craft. I could draw a stunning portrait in less time than it took to knit a pair of socks, and I had more fun at it. So I gave my implements and a truly giant stash of yarn to my mom. Don’t worry; she’s making good use of it, and she’s teaching my daughter now.

I’m digressing, but you can see the symptoms above: I want to know everything about whatever I’m learning right now. I want to know the nitty-gritty, miniscule, no-one-else-even-cares-about-them details. I probably know more about knitting than half the people in Canada and I don’t even knit anymore.

I just finished listening to “The War of Art” by Steven Pressfield on Audible. I’ll probably write a review for it once I have all my thoughts semi-organized about it. In the meantime I’ll say that the author believes in a lot of things, which kills of a good portion of his credibility with me, and he gets a lot of things right. The good part of the book is that he exhorts the author/artist/writer/entrepreneur/etc. to overcome “resistance”, the things in life that get in the way of “the work”, that thing you should be doing. Apart from his confused spirituality (angels, the Muse, and so on), I had two problems with his messages. I’ll admit, I’m possibly not the audience he was addressing, so please enjoy a grain of salt with my criticisms.

First, I got the feeling that the advice he provides is for professionals or would-be professionals. He disdains the “amateur” for not being serious enough and not loving the art or “work” enough to commit fully to it. That bothers me a lot, because he’s saying that I should pursue the one thing that is my “destiny” with single-minded zeal. This essentially precludes the possibility that there is anything short of whole-hog to consider in my pursuits.

The second and more important issue is that he contends (“assumes” is perhaps a better word) that “the work” is more important than almost anything else in life, and that the rest of life is just a distraction and an excuse to avoid doing “the work”. While my perspective would be called “resistance” in his book, I’ll share it anyway: the other parts of our lives make us richer, and the creative parts suffer without them.

So no, I don’t think I should give everything else up to be a stellar painter/programmer/guitarist/Master Builder. It’s not like I’m trying to monetize my skills; these are more “hobbies” or things I like to do for myself.

But there’s a little uncertainty there…. Why do I want to create? Many of the things I love are creative, although I don’t think of myself as an especially creative person. For whom am I creating? If no one but me was to ever view my paintings, would I still put brush to canvas? If I wrote a dozen songs that were all stunningly mediocre, would I be satisfied? What is the purpose? Is there a purpose? Does there need to be one? Do I need goals in my pursuits? Does there need to be tangible gain, or even intangible gain? Is the activity itself enough? Can I enjoy it even without a clear accomplishment?

As you can see I haven’t answered these questions, just spewed them across the end of this post. I don’t know how I should spend my time, or even whether “should” exists. I suppose I’ll just continue to obsess over this activity or that for a while until I decide what the right course of action is.

In the meantime, now that you’ve read my 1200-word navel-inspection, I want to leave you with Wil Wheaton’s words about why it’s awesome to be a nerd. This is what I identify with, and it’s a perspective that’s informing my thinking on how to spend my time. Read the entire thing, and go watch the video of the live event, but I’ll share his definition here because it so perfectly sums me up:

I think a lot of us have realized that being a nerd … it’s not about what you love. It’s about how you love it.

Thanks, and I’ll let you know if I have any epiphanies.