Thursday, 4 February 2016

Small Changes

I have taught this weird grade 12 course we have in Ontario called Calculus & Vectors forever (well, since it was introduced) so I could just walk in each day and teach based on what I did last year. There are certainly days when that is exactly what will happen, but I am trying to tweak lessons as I go through. I have a little more time to do this as I am not revamping an entire course the way I did last semester.

Day 1: I really liked the Desmos activity I created last year so I turned it into an Activity Builder activity - here is the link. I loved being able to see what equations my students were trying so that I could give meaningful feedback if they were on the wrong path. There was a lot of good mathematical talk going on as well as some struggle which some students are clearly not used to. Many expected me to just tell them the answer when they got stuck. I did not.

Day 2: School buses were cancelled due to wicked freezing rain (the roads literally had a sheet of ice on them). Our schools never close so I had to report, but as all of our students take the bus in, I had a day without students.

Day 3: Limits. We don't spend a lot of time on limits, but I think there is value in understanding what a limit is as we head toward derivatives. In the past I have done a demonstration simulating the amount of medicine in your body using a pitcher of water and some food colouring. I have also talked about Archimedes, but switched things up this year. I let my students be Archimedes (only with calculators!) and paired them up to find the area of a regular polygon. As a class they chose a radius and we talked briefly about "apothem", a word they had never come across before.

This is what we ended up with:

Those blanks are groups that just didn't get there. This wasn't as obvious as you might think as they haven't done any work with polygons since grade 9. The area for the nonagon was erased when the group saw that an area of around 110 was not following the trend. I took a little time to ask them how they had found the area and was pleased at the number of different approaches they used to find the side length of the polygon. When I asked what they noticed they said the numbers were increasing. I asked how they were increasing and then what they might be approaching. One student guessed 80 or maybe 78. Another went straight to the area of a circle. (I don't yet know who not to call upon!)

Then we talked about Archimedes before defining a limit.

At this point they had a rough idea of what a limit was and a definition, but that doesn't necessarily translate into being able to find a limit. In past years I have given them the graph below along with 12 limits for them to find. It seemed to go well, but there were always some students who clearly had not understood any of it. So I changed my approach and projected the graph for them and asked them to write the value of the limit I said aloud on their little whiteboard. Just the number. Write it down then everyone holds up their whiteboards facing me so that I can see how they are doing. It was so good! I first asked for the limit as x approached -6 from the left. The class was pretty much split between -2 and 2.5, with a few not wanting to write anything down. I didn't say anything about their answers, instead I moved on and asked them the limit as x approached -6 from the right. The class unanimously wrote 2.5. Then we talked about both of these limits before I asked for the limit as x approached -6. Hmmm. Some wrote both previous answers, some wrote only one of them, some averaged them, some said IDK. This provided the opportunity for good discussion and reflection. We moved on to looking at the limit as x approached 5. There was even a "what if we..." question that segued into looking at the limit as x approached -1. Yes!

Then I handed out the same graph with those 12 limits we have always found and they did them without hesitation. We answered questions together and they really seemed to get it.

Next, we looked at limits of various functions given their equations. Everything was tied back to the graphs. We visualized what the graph was doing as we approached whatever value. I probably asked "Is there anything weird going on?" too many times, but I think it helps them think about whether the is a discontinuity and whether that impacts the limit.

As you have gotten this far, thanks for reading my blog. I was pumped coming out of that class and wanted to blog about it. It is strange to not blog every day!


  1. Even though your activity was for area, it reminded me of this Nick Jackiw sketch on circumference

    1. That is very cool! Thanks for sharing.

    2. Hi Mary,
      Would you mind sharing the 12 limits you found? I would think they are -6, -3, -1, 2, 4, 5, positive/negative infinity but can't think of the others. Thanks.