There is something about a question not being graded that makes the students aggressive and risky. That can create the conditions for some of the best thinking. There are many days when I think grades and points and the division between problems that I will “collect and grade” versus the ones that I will not.

In the system in which I exist, sometimes bonus questions on formative assessments are the only way to really perplex a student – to push them at the risk of pushing each student beyond their current ability to reason, but still get a solid effort.

On today’s quiz, I added the following question as a bonus:

If you type “Tan 90″ into a calculator, you will get an error message. Knowing what you know about trig, discuss the possible reasons that taking the tangent of a right angle in a triangle would make your calculator show an error message.”

This isn’t something that has come up in any of our discussions. I would like to share with you some of my student’s answers.

From James: “It has an opposite which is the hypotenuse, but it has two adjacents so you wouldn’t know which one to use unless you put it in the calculator.”

From Tyler: “There is no such thing because when you plug in Cos 90 you get 0 and when you plug in Sin 90 you get 1. Maybe it is because since Tangent is TOA, it tries to add up to 90, so like opposite is 30 degrees and adjacent is 60 degrees.”

From Brianna: “Because Tan 90 would be opposite/adjacent, but the opposite side of the 90-degree angle is the hypotenuse and you can’t have the hypotenuse on top.”

From Jeremy: “It shows an error message because the right angle on a triangle doesn’t have a defined opposite or adjacent side length because the angle is touching both legs.”

From Lauren: “With tangent, you are finding opposite/adjacent. Those are the legs, and that 90-degree angle is being made by the legs.”

From Dayna: ” There could be an error because the opposite of the right angle is also the hypotenuse of the triangle.”

From Ally: It’s not clear where the negative idea comes from, but it is curious that in a Trig world of decimals and fractions, 90 in the other functions gives 1 and 0.

From Victor

Perhaps Josh’s picture says it all.

Now, the next question: If the “two-adjacent-sides-so-the-calculator-doesn’t-know-which-you-mean…” explanation wins out…

…then why don’t we get an error message for Cos 90?

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So interesting, thanks for sharing. Seems like none of the usual ways we explain these values help them sort out the situation. Wonder what they would think of the extended unit circle diagram. (Like http://mathhombre.tumblr.com/post/57703708835/visualizingmath-trigonometry-this-is-a or http://www.geogebratube.org/material/show/id/3619)

This is the better version of that trig sketch: http://www.geogebratube.org/material/show/id/98511

That one is awesome. I still don’t have a terrific understanding of how the “other 3” trig ratios (sec, csc, and cot) and the geometry interact. So, while I’m sure this is going to be a great resource for my students, I am mostly excited about it tying up a lot of the loose ends in my own mind. Thank you for sharing this!

nice!

i’m curious what would’ve happened if you’d asked them to come up with reasons for sin 90 (using their SohCahToa reasoning the 1 makes perfect sense), cos 90, and tan 90. sin and tan seem totally reasonable using their logic, but cos is the spoiler, huh?

unfortunately students have a tendency to forget to try lots of different cases when they’re looking at unfamiliar problems. when we do number types, i ask them a bunch of T/F questions and ask them to give a counter-examples if something’s false. things that routinely go wrong are statements like “the sum of two irrational numbers is always irrational.”

from their responses, i am guessing that you don’t discuss tangent as related to the angle of elevation, thereby linking it to gradient? i wonder how their responses would change if you nudged in that direction.

That’s true. In fact, when you say “linking it to gradient”, I’m not entirely clear what you mean, which means the students surely haven’t been nudged in that direction. We have only just begun discussing angle of elevation.

On an earlier formative assessment, I used the “Sin 90 = 1, why?” question that you brought up and it went much as you suggest. Hypotenuse and opposite the same, so same/same = 1. Many student made that connection pretty well.

I hope you don’t mind a rather long reply, but I wanted to explain fully what I meant by the link to gradient.

When I start my trig unit, I ask the students to make sure to have a protractor and a calculator for the first day. I start off by asking them to draw a right triangle so that the hypotenuse has a steep gradient of their choice and measure the angle of elevation, then draw one with an “unsteep” gradient and measure the angle of elevation. I have them compare with their neighbors and come up with a hypothesis. I then also ask them to draw a right triangle with an angle of elevation of 45 degrees and one with an angle of 60 degrees and find the gradient of the hypotenuse.

In our Pythagoras unit earlier in the year, they did an investigation to develop rules for 45-45-90 triangles and 30-60-90 triangles, which I gently remind them. Still it’s a real challenge to get the right gradient for the angle of elevation for 60 degrees.

We go over a few of their selected triangles; they give me their chosen gradient and what they measured as the angle of elevation. I tell them how accurate they are using my magic calculator (which is also a nice time to talk about margin of error and why you can’t be exact when measuring an angle with a protractor). Then I ask for their hypotheses. They told me that there is a link between the angle of elevation and the gradient and that a gradient of less than 1 gives an angle of less than 45 and a gradient of more than 45 gives an angle of greater than 45. The best thing was that during the first part there was a discussion between two students about what would happen if you had an angle of elevation of 90 or more than 90, but unfortunately I didn’t make it into a whole class discussion (aaaah, bad teacher!) later and they decided on their own that it wouldn’t be possible because then you couldn’t have a right triangle anymore. What a lost opportunity.

Then I introduce the concept of tangent as the special relationship that they’ve already defined. so we start out by saying tan theta = gradient = rise over run. They each confirm their original gradients with the angles they picked (they can see that they’re a little off), which then of course leads them to ask how they can go backwards from the gradient to the angle, so we start off with inverse tan right away.

We then do a bunch of problems with angle of elevation and depression, both finding sides and finding the angle. and then I ask them what if they needed to find the other angle. they first suggest that i find the angle of elevation and then subtract it from 90 (ok, that’s a lie, literally no student says that. they say “add it to 90 and subtract the sum from 180,” which is so inefficient it makes me grr), but i ask if there’s not an easier way. someone always suggests rotating the triangle so that that angle is the angle of elevation.

and that’s when i introduce the idea of opposite over adjacent. then sine and cosine. so i come into trig through gradient and try to make a really strong link there. i also have a project connecting trigonometry with gradients on ski slopes, with trains, etc.

i wondered if students had a strong gradient connection would they give the reason that tan 90 is undefined because at 90 degrees you have a vertical line, so rise over run gives you something over 0, which is undefined? (maybe not — that’s a pretty big intuitive leap, but it gives another insight into and also reinforces the idea that vertical lines have an undefined slope AND that you can’t divide by zero, things I find I can’t go over often enough).

sorry for the extreme comment spam. hope this was interesting to you.

I never considered the idea of connecting slope to tangent. That is awesome. Do you have any handouts that you use? Or specific activities or sites that you use to start the conversation? Or do you just lead the conversation with a series of questions?

the latter. but probably any book you have has angle of elevation problems, so it’s just flipping the script to start there. I do have some nice (read: wicked) problems involving angle of elevation.

i’m sending you my investigation on special right triangles via email in case you were interested in that. 🙂

Thank you for sharing this … what a great opportunity for thinking (and later, discussion).

I like it. I’d always had trouble grading “why do you think” questions as assignments, but allowing them to be done for ungraded curiosity’s sake might be more useful.

The downside of an ungraded bonus question, is that a good chunk of students will excuse themselves from it, which doesn’t rule out any subsequent discussion, but is something I have to keep in mind. It can be used as a means for differentiating instruction as well. And students who produce quality answers do get some points for that, but because they are seen by the students as extra credit, it plays like an ungraded question. (and because they are seen as extra credit, like you said, the trouble with grading them is pretty well eliminated. Students don’t squabble EC points).