Comments on: Another Day In… Honors Precalculus with Trigonometry

By: TheNanoProbe

TheNanoProbe — Wed, 10 Apr 2013 18:40:52 +0000

This experience speaks to how new technology does not by default make our lives simpler. In many instances it merely shifts our point of entry, as it did with your students. Thus, the need for human creativity continues.

By: l hodge

l hodge — Mon, 08 Apr 2013 17:09:31 +0000

Makes a lot of sense. I can see how the better technology completely destroys the Foerster Lesson. Ya, two minutes to sketch is not a big deal & agree there are some advantages. I personally don’t like to copy graphs, and sometimes I let my own bias get in the way of what may be best.

Nice replication of the original lesson on Desmos. Even if everyone doesn’t have a computer that makes for a good demonstration to either preview or re-inforce the lesson. I might slide very quickly the first time, then again more slowly, & again, …

By: Michael Fenton

Michael Fenton — Mon, 08 Apr 2013 05:23:50 +0000

I love these questions. They force me to think through every part of the lesson/handout and remind me that in an ideal lesson we take nothing for granted, carefully think through our assumptions, and provide a rationale (at least internally) for each and every instructional decision.

Regarding my choice to have students sketch the graphs, rather than providing it free of charge on the handout, some quick background is relevant.

If I’m counting right, we do three things with graphs in my classroom:

(1) Plot by hand (point, point, point, etc., connect with a smooth curve)
(2) Plot on a calculator (punch buttons, then look/observe but don’t sketch)
(3) Sketch (a somewhat sloppy, yet semi-accurate depiction of the graph(s), sometimes done after plotting on a calculator)

I see all three as useful/valuable/necessary, though different contexts/goals call for different approaches. In the case of this lesson, my purpose for having students sketch the system (which took between 30 seconds and 2 minutes, depending on the person doing the sketch) was to force my students to look more closely at (i.e., make more and more detailed observations of) the graphs. If a sketch follows the plotting/observation stage, then my students tend to observe more characteristics more clearly.

Regarding graphing functions simultaneously and then pausing at the intersections, to my knowledge this is possible on the TI-84 (Foerster’s handout was designed for the TI-84), but not with the TI-Nspire (my students have TI-Nspires, not TI-84s). On the TI-Nspire, the graphs appear instantly (instead of being “drawn” the way they are on the TI-84). Does this make sense? Does it address your question? Or have I misunderstood?

Thanks again for sharing!

P.S. I love this idea: https://www.desmos.com/calculator/kslklwpqzt

P.P.S. For students with easy access to laptops (with decently large screens) this might be a nice way to illustrate the concept I’m driving at in the activity: https://www.desmos.com/calculator/o3almuwyqt

By: l hodge

l hodge — Sun, 07 Apr 2013 16:47:38 +0000

Different class sizes, abilities, and time constraints will require different approaches, so this is not directed at the task you choose for your class. I really like the short version, but don’t think I could use it with my classes either. One way to force the students hand a bit at plugging angles into the functions is to give a graph where it is not so obvious that the graphs actually meet like These Polar Graphs. How do you know if it is an intersection, or the curves are just really close together? Well you could trace and toggle back and forth between the two functions to see if the coordinates match (essentially plugging an angle into each equation). If you trace to an apparent intersection (315 deg), you get the mind boggling result that the two points corresponding to 315 deg are not even in the same quadrant. Foerster provided a graph on his handout. How much time it takes the students to sketch the graphs onto the polar graph paper, and what do they gain by doing so? How long does it take to watch as the calculator graphs, and pause at the intersections, as in the Foerster handout, and what do they gain by doing so?

By: l hodge

l hodge — Sun, 07 Apr 2013 04:48:41 +0000

I hadn’t really thought about intersections for polar equations. Interesting! I might cut down a bit on the wordiness. A bunch of possible questions come to mind, but not sure how many I could realistically get to.

For page one I would just ask them to solve the system graphically on the calculator, give rough sketch (no grid), and confirm the solution by substituting into the original equations.

I might provide a copy of the graph of the polar equations on the handout and ask them to use it to estimate r & theta for the intersections. This is a pretty elaborate graph to copy onto paper. Also, I don’t want to tell them to graph it on the calculator, but not use trace.

Then, I would ask them to graph on the calculator, trace the first function to the intersections, and record theta & r for each intersection. Then do the same for the second function. Now just ask them to compare their table of estimates with the two tables of calculator values and describe or explain any differences. Maybe ask them to label points where there is a discrepancy with the different polar coordinates that were produced and if they notice anything. Maybe ask whether two people might disagree on the number of points of intersection for the graphs. Maybe ask whether two people might disagree on the number of solutions to the system. Maybe ask if changing the domain would change the number of intersections, the number of solutions, both, or neither. Maybe ask if it is possible that a polar system has no solutions but an intersection; a solution but no intersections; both; or neither. I really would rather not directly ask them to plug the points into the equations.

I think the curiosity is that polar coordinates can be used to describe point in more than one way. I think they have plenty to chew on at this point, and I am not sure how much understanding I would get with the auxiliary graphs, so I would omit it.

By: Michael Fenton is a boss. | Productive Struggle

Michael Fenton is a boss. | Productive Struggle — Sun, 07 Apr 2013 02:03:16 +0000

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By: Michael Fenton

Michael Fenton — Sat, 06 Apr 2013 06:16:40 +0000

Thanks for the feedback. Great stuff. I’ll update the handout with some of your suggestions and share a link when ready. The reason I used a rectangular grid on page two… Never thought to do otherwise (until now!).

Thanks again!

By: Michael Fenton

Michael Fenton — Sat, 06 Apr 2013 06:12:54 +0000

Awesome! Thanks for the amazingly speedy double comment. I appreciate your comment on (2) in particular.

By: Joshua Zucker

Joshua Zucker — Sat, 06 Apr 2013 06:06:27 +0000

Oh yeah, and I should be sure to answer your actual questions, too.
1) There’s some handholding and some room for discovery, so thinking about some of the schools I’ve taught at (among the top in the country, where my honors precalc class is not unlikely to have some 10th grader who is taking the USAMO or something) it would be a bit too handholdy, but overall it seems reasonably balanced. You know your students better than I do.

2) The specific goals are not all that exciting, but it’s always good to do some things that show the power of big, general strategies. So what I’d emphasize with my students here would be that we’re seeing yet another example of the power of having multiple representations for things, in this case the polar graph of r as a function of theta vs the rectangular graph of the same thing.

3) It was certainly clear enough for me!

4) I thought the layout into three pages made things hang together very well, so that you could see the differences between each page very clearly. I wonder if there’s a way to help them see the similarities between page 2 and 3 more sharply, to make sure they really understand that there’s a deeper sense in which they are “the same”.

By: Joshua Zucker

Joshua Zucker — Sat, 06 Apr 2013 06:01:32 +0000

Love the handout!

I’m surprised that you give them rectangular graph paper to draw their polar graphs on, though. Is there a reason for that? I think it might help make the distinction between the three pages sharper to have a polar grid laid out on page 2 in contrast to the rectangular grids on the other pages.

The Big Idea here is definitely that polar coordinates take the theta coordinate and wrap it around, so you can only “see” values of theta modulo 2pi, and even then there’s another difficulty because of the possibility of negative values of r. The next page “unwraps” all that so you can see the whole range of theta.

I wonder if it’s best to call the variables on page 3 (x,y) or if it’s better to call them (theta, r) and label the axes clearly. Of course with the calculator technology they’re going to have to call things (x,y) there, but still, maybe it’s better for them to see what the variables “really” stand for before munging them into a form that the calculator will graph appropriately for them. The way I see it, on the first page you’re graphing (x,y) in a rectangular grid, on the second page you’re graphing (r,theta) in a polar grid, and on the third page you’re graphing (theta, r) on a rectangular grid.

And this is the first time in my life that I’ve ever wondered why, if theta is the independent variable and r is customarily a function of theta, we write them in (r,theta) order instead of matching the (independent, dependent) order that we usually use. Weird. Why?