Helpful way to summarize notes in MDM4U?

I made a little booklet for my Grade 12 Data Management students to help them organize and summarize their notes in this part of the course. It’s not exhaustive; it’s just to show them that it’s worthwhile to take time to review their learning and actually record it, and to give them a structure to try.

Skills Tracking – Combinatorics – booklet

So many students don’t really consolidate their new learning each day, and then the study by “reading over” their notes (which means flipping pages while Netflixing, I think).

Is this format helpful for reviewing and summarizing? Is it a good physical size (it’ll print on US Letter size paper and fold in half, yielding a 5.5″W by 8.5″H booklet)?

Semester change

<begin break from planning>

A new semester means new students! They’re great! They have a lot of names, though. I’ll do my best to be able to reliably distinguish them.

Finishing off a semester means that there are a ton of students I care a lot about that I will never get to teach again. That really struck me today, and it’s bittersweet for sure. I’ve poured a lot of myself into developing these young men and women, and it’s crazy-hard to watch them move on, even if most of them are still in the building with me. I already miss them a lot.

I’m teaching new courses! I get to try new stuff, which is pretty fun. And it’s also a ridiculous amount of work, particularly those courses I’ve never taught before.

I have my own classroom! That’s pretty sweet. I haven’t had that happen in a long time. I have to make it more inviting though. It’s a bit sterile for my taste. Nothing a Star Wars poster (or mural…hmmm…) can’t fix.

<end break from planning>

Off Topic On Purpose

This morning Gregory Taylor (@mathtans) tweeted to me:

We back-and-forthed once, which I hope displays properly here:

And he was right. So here I am, even though I’m tired, because I like to write. It might even relax me a little.

In a rut

I notice that my last four posts have all been about work. That can be all right, I suppose, but I try to keep this blog more varied than that. Unfortunately, when I decided to write a post tonight only work-related ideas popped into my head. Tragic. I’m living and breathing this job, and it’s consuming my entire brain.

I don’t want to be in this brainrut all the time; I don’t think it’s good for me, my brain, or my family.


My plan is to intentionally write some posts in the next while that are not related to my job in any way that I can perceive.

Therefore I’m signing off entirely for the night, even though I’m sure there are emails waiting for responses, because I’ve already worked 60 hours this week and it’s only Thursday. I’m going to escape into something I enjoy and turn off my teachermind until tomorrow morning.

Good night.

Solving quadratics – what’s most important?

I’m currently teaching MPM2D, Grade 10 Academic math, and I’ve been having conversations about what’s most important for students to master in the quadratics part of the course. We’ve learned about the different forms of quadratic equations (standard, factored, vertex) and how transformations apply to give us vertex form, and we’ve started factoring.

The expectations I’m looking at right now are summarized here:

  • expand and simplify second-degree polynomial expressions, using a variety of tools and strategies
  • factor polynomial expressions involving common factors, trinomials, and differences of squares using a variety of tools and strategies
  • determine,through investigation,and describe the connection between the factors of a quadratic expression and the x -intercepts (i.e.,the zeros) of the graph of the corresponding quadratic relation, expressed in the form y=a(x-r)(x-s)
  • interpret real and non-real roots of quadratic equations, through investigation using graphing technology, and relate the roots to the x -intercepts of the corresponding relations
  • express y = ax^2 + bx + c in the form y=a(x-h)^2+k  by completing the square in situations involving no fractions, using a variety of tools
  • sketch or graph a quadratic relation whose equation is given in the form y = ax^2 + bx + c , using a variety of methods
  • explore the algebraic development of the quadratic formula
  • solve quadratic equations that have real roots, using a variety of methods

Of all that stuff, what’s most important? There is time pressure in this course, and I’m trying to focus on what’s truly essential moving forward. I also want to make sure that my evaluation reflects the importance of each understanding and skill.

Have you taught this course (or the sequels, MCR3U and MCF3M)? What do you think?

Some ICS3 assignments in Java

I wrote a few practice tasks for my online ICS class for using loops and arrays, as well as a challenge task for anyone who’s interested. You’re welcome to use them in your classes if you like.




Which kind of calculator promotes good algebraic thinking?

I teach high school math. Students bring scientific calculators to class, or they sometimes have to borrow one from me. I have two types available: immediate execution calculators and formula calculators. I’ve been wondering lately whether one type of calculator is better for learning algebra than the other.

Here’s how they work (see Wikipedia for a longer explanation:

Immediate Execution

The TI-36X Solar 2004 version immediate execution calculator.

These calculators work by performing calculations along the way as you type in values and operations. For example, you can evaluate the expression

\sin(3 \times 45)

by typing 3, multiply, 45, then the sine key. As you press operations and operands the calculator will evaluate what it can according to the rules of order of operations, or BEDMAS. For binary operands (those taking two values to produce a result, like multiplication), you put the values in order. For unary operations (those taking just one value, like squaring or taking a sine), the value must be present on the calculator screen when you press the operator key. These calculators usually have a bracketing feature to allow the user to work through complex expressions without using memory storage.


The Sharp EL-510R formula calculator.

These calculators work by waiting until the user has typed in a complete expression to evaluate, then evaluating the entire expression. The order of button-pushing is pretty much as the symbols are written in the expression, making them easier to use for a lot of folks. Once a value is calculated, it’s stored in an “answer” variable in case it’s needed for the next evaluation.

Algebraic Expressions and BEDMAS

When we write out algebraic expressions, we have a number of conventions to follow. The most important convention is order of operations, which people usually learn to remember with the mnemonic BEDMAS or PEDMAS:

  • Brackets (Parentheses)
  • Exponents
  • Division and Multiplication
  • Addition and Subtraction

When evaluating (simplifying) an expression, you first simplify the smaller expressions inside brackets. Then you evaluate exponents, then division and multiplication in the order they appear, and finally addition and subtraction in the order they appear. It’s useful to think of brackets as isolating sub-expressions, which then follow the same rules. It’s also useful to think of this order as the “strength” of the operation: multiplication is a stronger operation than addition, so it holds its operands more tightly together, and it gets evaluated first.

When a student is learning order of operations, it often feels like a set of arcane rules. There is no reason, from the student perspective, that it has to be this way. In fact, it didn’t really need to be this way, but the convention was established and now it’s important to abide by it (if you want to be understood, that is).

How a calculator helps (and hinders) learning arithmetic

People often lament that today’s youth can’t perform basic arithmetic in their head. It’s unfortunately true; I often see students reach for their calculator to evaluate 35 \div 5 or even 4 \times 6 . These are facts which prior generations had drilled relentlessly and now have available as “instant” knowledge. Younger people typically haven’t spent enough time practising these computations to develop facility with them. This is partly because the calculator is so readily available.

(Aside for parents: If you have kids, please do make them practise their age-appropriate facts. It’ll help them in the same way practising reading makes things easier)

This will draw a lot of heat, I’m sure, but I think calculators do have a strong place in even K-6 learning. They let students explore quickly without the burden of computation getting in the way of non-computational learning. It’s the same effect that web-based, dynamic geometry software can have on learning relationships between figures, lines, etc. (if you’re looking for awesome dynamic geometry software, try GeoGebra – free and wonderful).

But calculators are a hindrance when students are learning to compute fluently. They allow a student to bypass some of the thinking part of the exercise. Don’t let students (or your kids) use a calculator when they don’t have to. Only use them when students need the speed for the task they’re completing.

How a calculator helps (and hinders) learning algebra (?)

Here’s the part I don’t know about, but I’m speculating about.

I think immediate execution calculators require students to understand the algebraic expressions we write, where formula calculators bypass the thinking part of evaluating expressions.

As with arithmetic, if practising evaluating expressions is not part of the learning, and might be getting in the way of the goals for learning, then either type of calculator is fine.

But as students are developing their understanding of algebra and the order of operations, the immediate execution calculator displays the results of operations as they are evaluated, while the formula calculator obscures the evaluations in favour of a single result.

When a student types 5, add, 6, square, equals into an immediate execution calculator, they see the value 36 as soon as they press the square button. There is a reminder that the square operation is immediate. Similarly when a student wants to evaluate \sin(30+45) they must type 30, add, 45, equals*, then sine, emphasizing that the bracketed portion has to be evaluated first (i.e. before the sine function is applied).

*A student can use brackets, which is equivalent to pressing equals before sine. Also, I hope anyone using the sine function knows that 30+45 is 75 and doesn’t need a calculator’s help for the addition.

Is there research?

I perused the InterTubes to find research into this question, but either it’s not out there or I’m not skilled enough to find it.

I want to know whether one calculator is better than the other for a student who is learning to evaluate expressions.

Has no one looked into this? Help?

I feel guilty writing this blog post

I spent about 13 hours working on school stuff today. I planned lessons, collaborated with colleagues, taught math concepts, worked one-on-one with students, developed practice materials, communicated with online students, searched out supplementary resources, marked tests,….

It’s 12:28am. I finished the school work that my brain can safely manage a little while ago, and I’m getting ready for bed. I’ll be up again at 6 o’clock to restart the cycle, and I’ll still be tired. 

I like my job, I love my students, and I desperately want them to succeed. But I’m also exhausted already and feeling guilty for taking ten minutes to tap this into my phone. The work is so important, but letting it take precedence over everything else in my life is unhealthy and is unfair to everyone, including me. It always takes up more than its fair share of my brain’s CPU cycles, and it’s taking up too much of my personal time as well. 

If anything eludes me this year it’s balance. I need some real down time, real soon.