Standing and Talking: a first attempt

Note: Google Camp 5.0 will be Saturday, Nov. 4. Registration is open for TDSB teachers on K2L. This event always sells out, so register soon! If you're interested in presenting, submissions are open until Monday, Oct. 2. The TDSB's Renewing Math Summit will be Friday, Dec. 1; you can still submit a proposal until Sept. 30. Yes, that's today. Hurry!

When I did my physics honours specialist with John Caranci way back when, he told us that one of the easiest ways to become a great teacher is to try or adopt one new technique per month. Well, I'm still working on that (I probably average 3-4 a year), but this year I'm going to really make the effort to try them several times per month.

I've already made the first change by getting the students used to grouped tables -- a bit challenging in one of my classrooms which has fixed benches, but I'm trying to make it work -- and I started my October technique a bit early because I couldn't wait.

I was inspired by this blog post by Sara Van Der Werf to try a Stand and Talk with my grade 11 mixed math students last week. To summarize, the old-style "share with your neighbour/elbow-partner/TPS" doesn't really work most of the time. Sara has found that getting the students to stand up and walk across the room to talk to another person and giving each pair a paper with something to look at with the instruction "notice 10/20/50 things about this" really increases student engagement. Her post is excellent, with specific instructions on how to make it successful and a lot of math examples to use.

(By the way, the link to the "rumors" group learning routine at the end of Sara's blog post would be great for the prediction part of POE or for review.)

I thought mapping diagrams would be a good place to try this. We'd looked at domain and range and function/NAF. I prepared this picture for them to look at and notice at least 10 things (yes, it's supposed to be a big number).

This is my revised version

Did it work? Mostly. I wound up grabbing the wrong folder and left the students' copies of the diagrams in my office, but I did put them up on the screen. Not ideal, because on my original version the arrow heads were not as obvious and I used too small a font for the sets of points and the labels, so they were a bit hard to read from the back of the room. There was a bit of "I don't know what she wants, do you know what she wants?" at the beginning, but after I encouraged them to go for the obvious first and used Sara's prompts ("I should see you pointing," "What do you wonder?" "Everything on the screen is there for a reason. What else to you notice?"),  I heard some good discussions. And once we were talking as a class, I had volunteered suggestions right away instead of the usual silence.

Some of the suggestions:

• there are circles on the page
• there are numbers in the circles
• the numbers go from negative to positive in both of the left circles
• there are no negative numbers in the right circles
• the numbers go in order
• there are 4 numbers in one left circle and 3 in the other
• both right circles have 3 numbers

I was a bit surprised that nobody mentioned the arrows, but that could be because the arrow heads wee small and didn't really register, but when I pointed out that there were arrows, more suggestions came in thick and fast:

• an arrow goes from the -3 to the 3
• another arrow goes from the -2 to the 1 (etc)
• two arrows go to the 3 in both right circles
• there are two arrows going from the -1 in one circle, but all the rest have only one arrow

Nobody noticed the connection to the coordinate pairs above the diagrams, but I think that is because the font was too small and they didn't really notice it. Once I asked "do you see a -3 anywhere else on the page?" the penny dropped.

• Oh! The arrow goes from -3 to 3, and there's a -3 and 3 together above. 
• Same with the -2 and 1.
• That first circle is all the first numbers and the second is all the second numbers

At this point I switched to Socratic questioning, and we established that the left circles were the x's, or domain, and the right circles were the y's, or range; none of the numbers were repeated and were in order from most negative to most positive; that one was a function and the other wasn't; and that you could tell whether it was a function or not by the number of arrows coming from each of the points in the domain. I then told them these were called mapping diagrams and had them create some from sets of points.

We stood the whole time we did this, and nobody complained. This was very surprising to me because there are a few students in that class who complain as a matter of principle, but who were actually mostly engaged in the activity and even offered a suggestion or two.

So will I be using stand and talks again? You bet. I'm already scheming my next picture. I love the way I could work concept attainment^* into the notice and wonder. I need to make I also focus on the "what do you wonder" questions. The diagrams do require a bit of thought first, so I'm aiming to do two per month in my math classes to begin with and work up to once a week in all classes. I'm already planning on trying this as a way to introduce B-R diagrams, chemical formulas, and circuit diagrams later on in grade 9 science; and more immediately, rational vs irrational numbers, polynomials,  like vs unlike terms in grade 9 math; standing waves in physics; and different forms of the quadratic function in the mixed math. That will do to start with, I think!

^*I did my math honours specialist final project on concept attainment, and I keep meaning to work it into lessons whenever I can. Perhaps I'll do a blog post about it so I will remember to use it.

Updated: Fidget spinner math

Update: I've added a link to the data in Desmos and TI lists below.

Yes, I jumped on board the "if you can't beat 'em, join 'em" boat.

Inspired mostly by Harry O'Malley's site, I brought a fidget spinner^* and my phone to grade 11U math class one day and the students modelled the spin. The above graph is the one of my student's results. It's a bit wonky because in the two days since I had bought it, one of the end caps had fallen off, which made the central bearing ring shift off centre. This made it slow down a lot; the above graph shows a few tries at spinning. The student at the top of the post took an average of the cycles while other students just looked at the first.

To make a good video, mark one of the edges of the spinner so you have a reference point to track. The app I used is VidAnalysis Free for Android (for Apple fans, Vernier has an app for LoggerPro that lets you do the same stuff, if not more). Don't spin it too quickly unless you have much better equipment than I do -- I tried to analyse that lovely first video and got goobledeegook because it's spinning too fast for the video to capture properly.

You then pick reference points (known length and origin of coordinate system) and track your mark. You can skip forward and backward in time to get to the section you want to analyse. I goofed because I forgot that I had made a spin without my finger in the way; by the time I remembered, I had already invested too much time getting the other data. Hence the starts and stops.

I didn't want to take up class time getting the analysis ready, so we discussed what equation we were likely to see, and then worked on other problems. Before the next class, I made the analysis and turned the data into graphs in Google Sheets. The next class, I put the x-distance graph up on the screen and got the students to figure out the model. When they had an equation, I graphed it against the data (if you do this, remember that spreadsheets do trig with radians, not degrees).

It was a really good exercise, considering it's the first time I've officially used the VidAnalysis in class. We had some great discussions about the vertical translation (did I deliberately make the coordinate system off-centre? no, but I will next time because that led to interesting math); how to deal with the increasing period; how the amplitude of our function compared with the actual measured distance.

I've since shifted the centre bearings back and made another, better video analysis. The screen shots are below:

		What I like about this, mathematically, is a) how it shows the spinner slowing down; b) how it shows that I didn't hold the camera completely still -- notice that the "zero line" of the equation shifts up (nice for composition of functions!); and c) the x- and y-distances are essentially translations of each other (sin vs cos). I could have really expanded on this activity and got them to break the functions into different domains.
		The velocities show the same math effects as the distances; this could be used to show that the derivative of sinusoidal functions are still sinusoidal (and how). If only Google Sheets would get their act together and let us connect points in scatter plots.
		More screenshots. What I really like is that you can upload the data as a csv file to Drive. Copy-paste makes it simple to create a spreadsheet. Note that the Free part of the app means ads. I was still giving it a trial run, but I think I will upgrade to the premium version because it's a great little app.

I started the trig functions section by creating a periodic wave using a salt-shaker pendulum (an idea I cribbed from someone on Twitter -- I can't remember who it was, but I'd love to give her the credit). I now wish we had filmed the pendulum at the same time so we could compare our rough model to the data. Future ideas!

I have a graphing calculator assignment that grabs tuning fork data from a microphone; I get the students to model the function and work backwards to determine the frequency of the fork. We didn't get to it this year, but it would go well with this exercise.

If you'd like to use my data, feel free to make a copy of fidget spinner 2, or download the csv file. I'm going to import this into a Desmos activity and graphing calculator lists at some point; when I do, I'll update this post with links. Update:Here are links to the data in Desmos and as TI lists.

^*I meant to use one of the students' spinners in my MCF3M class, but ran into difficulties because that student wasn't in class the day we were supposed to do the model. Since I wanted a permanent mark on the spinner, I decided to not buy trouble and get my own. Plus, they're fun.

I did this exercise with the 11M students as well, but we wound up doing it as a class instead of individually.

Happy accidents

I've been teaching for a good while now, but I'm happy to know that there are still things I can learn, because it keeps me sharp. Also? Happy accidents become teachable moments and an exercise for one class turns into several exercises for three different classes.

I'm always on the lookout for "real-world" examples of math and physics that aren't the usual boring cell phone/well bore/cannon ball stuff. I came across this video of the water fountain at Detroit International Airport and was struck by one image that was filled with different parabolas, thanks to the initial velocity of the water and the perspective of the shot. I turned it into an exercise and assignment for my MCR3U where they had to find the equations of two parabolas, and then the equation of three lines, one which was a secant to one parabola, one which was a tangent to the other, and the third was a secant to one and a tangent to the other.

Naturally, I wanted to modify it for use with my MCF3M class. I came up with an exercise where they find the equations of two parabolas: one in root form and one in vertex form. For their assignment, they'll have to turn each equation into the other form algebraically (plus standard form for good measure).

I tried it out three days ago. To help them prep, I got them to pick the axes and a parabola to look at (noting that each water jet is actually two parabolas). We measured the roots and used the y-intercept to find the a value. Pretty straightforward, and I thought determining the vertex form would be a snap.

Except that when we calculated a, we got a completely different number. Not "we're off by a few decimal places" different, but -0.19 vs -1.1 different. These students are still struggling a bit with vertical stretches and compressions, so a discrepancy like that is not on. 

I asked a colleague to verify my calculations, and he figured out that my calculations were fine. The problem was probably that for the parabola the class chose, the y-intercept and vertex were so close together that a=-1.1 was within the accepted error. I used another point far from the vertex and got a=-0.18. Much better.

[By the way, we I did also make some heinous measurement and calculation mistakes, but since all mistakes I make are intentional (ahem), this just gives me an opportunity to talk about making sure our values make sense. More happy accidents.]

Unfortunately, I had the DLL PD today, so I wrote this all down on the board and hope they got it during today's class. I'll review on Friday when I next see them. They need to have the equations (and domains and ranges) ready for next Thursday's assignment.

That discrepancy is really interesting. I will have to modify this worksheet to tell the students to make sure their points are not too close together. Plus, I may have accidentally stumbled on a realistic way to teach uncertainty in my grade 11 physics class. A happy accident indeed. I'll keep you posted.

Once I get my act together, I'll create a page where I will share my various worksheets and handouts. For now, check out my course webpages (link up top) under "Handouts and Assignments". My class notes for both (all three?) lessons will be posted under the "Notes" section at the end of the month.