Developing an English-Arabic math glossary (MPM2D)

I’m working on this project so that ELL students will have a little bit easier time in my math classes (making the translation burden a little lighter).

I’ve just finished an Analytic Geometry unit in Grade 10 Academic Math, and I put together this glossary of terms using some online resources:

Analytic Geometry Terms – Chart

I gave the same resource to all students, not just ELL students. We filled in the chart with diagrams to help explain the terms visually.

For the next unit, though, I need some support. The online resources I have found are missing some terms that I’ll be using. I have an asterisk (*) beside the terms I’m less certain about. If you can help, please comment below with corrections and additions.

Quadratic Relations Terms


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?