OrdOp – the math card game using Order of Operations

I played a game with my math students today. It’s called OrdOp, and we used it to practice our mental computation skills. We used standard playing cards, but the “real” version uses cards numbered 0 to 25.

If you want to try it, here are the rules (and printable cards).

OrdOp – standard playing card rules

OrdOp – custom cards and rules

OrdOp – how to play (video)

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Using video to capture quadratic motion

In my MBF3C class today we observed quadratic motion and modelled it with http://www.desmos.com, and online graphing calculator. I’ve recreated the steps here (with some fake data).

First, I went to Desmos and projected a blank Cartesian coordinate system onto the screen at the front of the classroom.

Desmos_blank
Then, I found a black rubber stopper (I teach in a science classroom) a little more than an inch across.

I asked for two volunteers who would be willing to throw things at each other. That was easy.

They practiced lobbing the stopper back and forth to each other in front of the screen, trying to get the black rubber to crest visibly near the top of the grid. Eventually they were confident they could do it.
I held my iPhone in landscape orientation and recorded a slow-motion video. After a few attempts I felt we had a successful toss, and the students returned to their seats without injury.

We scrubbed through the video slowly and recorded the x– and y-coordinates for each major tick of the x-axis.

Then we plotted the points in Desmos (using the Table feature):

Desmos_table

We then graphed a generic quadratic using the vertex form and its parameters (y=a(x-h)^2+k). Desmos provided the sliders for each parameter:

Desmos_basic

As a group the students decided to make a negative and small, to flatten the curve, then they adjusted h and k to fit:

Desmos_transformed
Some interesting stuff to note about the process:

  • Even using 120fps there were places near the edges of the curve for which it was hard to see the coordinates (blurring and gaps).
  • The vertex wasn’t on the y-axis, which was surprising to the students.
  • The glare of the projector made the grid a little hard to see.
  • We had to have the lights out to make the grid visible at all on camera, and the dim lighting made the video a bit grainy.
  • The parabola we fit to the data worked really, really well.

What’s next

I’m going to perform some more motion tasks with them to get more quadratic data, and we’re going to do some curve-fitting to model and predict things (for example, how far can you throw a ball off a 10th-story roof?).

I’d like to have a large, physical grid on the wall or something so that I can have the lights on when we record video.

I want students to record video and analyse it. Lots of them have iPhones, and I bet some of the other phones can take good, crisp video. If not, there’s some learning there too (about interpolation if nothing else).

I’ll try some other phenomena also.

A video you can use

Here’s another video we took, if you want to use it:

Physical phenomena for quadratic relations

I’m working on a quadratics unit for my MCF3M (online) and MBF3C (F2F) classes. The Ms need to be able to do a few more things, but both groups have to be able to model quadratic “stuff” using an equation.

I’ll be using desmos.com pretty heavily, and I got some great ideas from Heather Theijsmeijer (@HTheijsmeijer).

I’m trying to find some examples of physical phenomena that I can have students in either class play with to practise/demonstrate modelling. Here are my ideas so far:

Throwing or Bouncing a Ball

This is the first thing I thought of. A ball follows a nice parabolic path in the air if it’s moving horizontally.

My plan is to have students use a phone or camera to record a video or a rapid burst of images, overlay a set of axes, and fit a curve to the path. My iPhone can record at 120fps, which is great. I also found a handy post at Stack Overflow that explains how to extract images from a video, so that might be helpful too.

Pouring Water from a Hose

Set your hose at an angle, turn on the water, and snap a picture. Parabola. Beauty. Maybe put a piece of grid paper behind it, or just import it into Desmos.

Rolling a Ball Up An Incline

This one’s messy, but I think it might work.

Dip a marble in some ink or paint. Set a piece of grid chart paper on an incline (say, a piece of plywood) and roll the ball on an angle up the paper. When it crests and rolls back down, it should have left parabolic paint. On graph paper.

Other ideas?

I’m open to suggestions. I have stuff like photos of suspensions bridges, etc., but I really want something students can generate on their own.

Key Learnings from our e-Learning Collaborative Inquiry

I learned a lot over the past couple of days. My board brought in Donna Fry (@fryed) and Tim Robinson (@timrobinsonj) to guide our group’s learning, and several folks from the board’s central program team were helping out as well. Check out #elADSB for a bunch of new Twitterers, too.

Here are some prompts that we were asked to respond to, and here are my (rather brief) thoughts.

1. ​​What questions, wonderings do you have ​with respect to our problem of practice?

How do we encourage collaboration that isn’t false collaboration? How do we ensure that the collaboration is meaningful and valuable? I don’t want to force collaboration among students when it doesn’t make sense.

What does a rich task look like in a course which has a lot of technical, procedural learning? Is it enough that there is inquiry in the task, or does it need to be “authentic” and relevant? I’m concerned about fauxthenticity: forcing an unrealistic application out of a concept just so that we can say it’s “real-world”.

2. What key learnings have you had over yesterday and today?

I need to provide more structure for self-monitoring and self-reflection for my students, and then ensure that they follow through on that self-assessment. They’re still kids, and they need a firmer hand with organization and checking on their learning, or they may neglect important stuff (or even delude themselves into thinking they understand concepts that they don’t).

I need to consider including more face time in my course – maybe having “Math Chat With Mr. G” or something on certain afternoons… Even if some students can’t get me live, I shouldn’t prevent it for those who can.

3. Moving forward, as a result of your learning, how do you envision your courses/practice changing?

I need to be more careful about connecting students with each other instead of encouraging them to rely on me.

I need to set up a fast, reliable, easy-for-me-to-check-in-on self-monitoring system for my students.

I will set up Adobe Connect or Google Hangout time periodically.

I’m going to work on figuring out when requiring collaboration makes sense, when students should be working independently, and when they just have a choice (most often?).

I’m going to make some plans and then ask someone else to give me suggestions about them before I implement them. :)

LaTeX Math for e-Learning in D2L

I am teaching MCF3M online this semester, so I need to be able to include math notation in my online content, quizzes, etc. I know how to write math notation using LaTeX from my days at the University of Waterloo, and I find it a lot faster than using a graphical equation editor. I’ve tried Microsoft Word’s editor, which accepts LaTeX-like input as well as graphical input, but I still find it frustrating to use.
I’m teaching in the Desire2Learn/BrightSpace learning environment, so I need to ensure my content works well in there. Last semester I taught Computer Science/Programming and used PDF files that I created in Word Online, and I considered doing the same thing again.

But D2L has an equation editor as part of its HTML editor for webpages, discussion posts, etc. Could it be all I need?

I’ve taken it for a spin before. Here’s the workflow:

Create a new page and type into the HTML editor.

latex01

Expand the toolbar so that the Equation tools are available.

latex02

Choose \∑ LaTeX equation.

latex03

Type in the LaTeX expression, using \( and \) as delimiters for inline mode (otherwise it defaults to block mode).

latex04

latex05

Looks good.

latex12

But look at the source HTML code:

latex07

Uh-oh… that’s MathML (Math Markup Language), not LaTeX. What if I want to change something in my original LaTeX?

Well, you can see at the bottom that my LaTeX code is still there, but it’s not being used. I could remove all the MathML, cut out my LaTeX, modify it, and re-insert it using the LaTeX equation editor.

Ugh.

I thought that MathJax, the rendering engine that D2L uses for math notation, could only handle MathML (since notation from both LaTeX and graphical editors are converted to MathML), but it turns out that’s not true. MathJax can do LaTeX.

So I tried putting LaTeX directly into the WYSIWYG editor:

latex10

No dice.

latex11

The trouble is that D2L has parameters on its JavaScript call to MathJax:

latex08

That config=MML_HTMLorMML bit is saying that only MathML is acceptable input (and HTML or MathML can be output).

So I added another call directly to MathJax in my own source code:

latex09

I set the parameter to be config=TeX-AMS_HTML, which will accept my LaTeX input and render in HTML/JavaScript.

latex12

Magic.

But this is kind of a pain.

I can use D2L’s editor to insert math, but I get MathML (which I find hard to edit).

I can write in LaTeX and have it be preserved, but I need to add a script call to the start of the HTML source code (a hassle, but not too serious, I suppose).

Or I can write in some other (offline) development environment, include my script call all the time, and just upload my completed HTML files to my course. This has the advantages of being independent of D2L, available without internet access, and very shareable.

So that’s what I’ve decided to do, at least for now. So I’ve learned a little CSS to make my pages less vanilla/more functional, and I’ll try to improve the look and feel as the semester progresses.

Wish me luck.

Different kinds of Thinking: Ontario Math Achievement Chart

I’m evaluating some student work today and I’m struggling with the Achievement Chart for Mathematics (see page 28). In particular, this part of the Thinking category is bothering me:

An excerpt from the math achievement chart for Ontario

Take a look at the first point in “Use of planning skills”, called “understanding the problem”, which includes “formulating and interpreting the problem” as an example of that skill.

Now look at “Use of processing skills” point “carrying out a plan”, which includes “modelling” as an example of that skill.

Are these different? In my mind (up until now, at least), “formulating and interpreting the problem” has meant representing a situation mathematically so that we can apply our other math skills to solving it. Isn’t “modelling” in the context of “carrying out the plan” sort of the same thing? Representing components of the problem mathematically? Is the difference just when it happens (i.e. formulating/interpreting is initial planning, and modelling is during the act of solving)?

I’m not trying to be pedantic here; I’m having trouble distinguishing between the different components of Thinking when I’m trying to assess and evaluate my students’ work. I could use some external thinking on this issue (and math evaluation in general, I suppose).

Please comment; I’d love to talk to you if you have ideas about this stuff.

My students like the class blog (phew!)

I reminded my MDM4U students to visit our class blog today as I had posted a video this morning with solutions to some questions from class. I was pleased to hear that a few students have been heading there regularly. I played a bit of the video on the screen so they would understand what it was like, and then I asked for input about the sorts of resources they would like me to post.

First, most students want to have the videos. They’re nothing magical; they’re just me solving some problems on paper. A couple of students said they like the idea of being able to rewind me (since that’s hard to do live, in class).

Second, a few students also want a text format of some kind (a blog post, a PDF file, or something like that). I understand that; it’s faster if you just need a short bit of explanation. It takes a lot longer to type out a solution, but I could always write one and take a photo, I suppose.

The third response I took away was that posting videos like this is unusual in their experience, and they appreciate the extra effort on my part. Several students seemed to understand how much extra work goes into creating these supplementary resources, so that was nice.

At some point I’m hoping to post more student work for them to learn from instead of creating everything myself. I think they’ll learn a lot from doing it, and then a lot from each other. Videos would be awesome…

I care a lot about their success, and I’m hoping to give my students a better-than-average chance at being great at this stuff. A lot of the responsibility lies with them, but I’ll remove as many barriers to their learning as I can. I’m glad to hear that my work on the blog is being well-received.