Friday, April 28, 2017

Activity: Constraints and Trig (Day 143)

This is what Theoni Pappas writes on page 118 of her Magic of Mathematics:


That's because page 117 was the last page of Chapter 4. So Chapter 5 begins on page 119, and it's called "Mathematical Magic in Nature." So in this chapter, Pappas ties math to the natural world. In fact, the picture on page 118 is of a spider web.

Taking a glance at the table of contents for this chapter, two of the sections have something to do with, um, the birds and the bees. And that will be all I'll say about this chapter until we begin it in earnest next week.

This is what I wrote last year about today's lesson:

Today's idea comes from an anonymous Washington State teacher who only goes by the username "Alternative Math" -- named for the alternative high school to which this teacher is assigned.

Here is the original post:

I have been busy planning out a unit on trigonometric ratios for my Geometry B course. I have been trying to balance the open ended exploration and project based learning that I prefer with the more typical questions that students will eventually see on state tests or future math classes.
Here is the [Common Core] standard I’m addressing with this lesson: G-MG.3 Apply geometric methods to solve design problems (with a focus on constraints).
I introduce trig with the slope ratio, proportions, and physically measuring before I ever tell them the word tangent. I’m leaning toward using [High Priestess] Kate Nowak’s Introduction to Trig and then running a few Labs where calculate heights and distances of physical things outside before offering this [worksheet].
Afterwards I might show a few ramp fails before giving them a more open ended design problem. I’m still working on the actual formatting piece, but it will be a blueprint showing a door/stoop 5 ft high, but due to size of parking lot also has a restriction on length. Students will figure out it is not possible to use one ramp in that space and will have to figure out how to use two or more ramps to fit the constraints.
Nothing too mind blowing or exciting here, but I figure it gets at what I’m hoping they understand.
Notice that this teacher attributes this activity to yet another teacher -- New Yorker Kate Nowak. Even though I myself found this activity on the Alternative Math page, in today's post I will credit Kate Nowak as the originator of the idea. This is due to the anonymity of the Alternative Math website -- it's far easier for me to write "Nowak" than "the author of Alt Math," and it's easier to write "she" (referring to Nowak) than to write "he or she" over and over. We already know who Nowak is -- I mentioned her blog that same week and explained why she's known as the "High Priestess."

This is what Nowak writes about this activity on her own website:

The children understand that sin, cos, and tan are side ratios. The children! They understand! They are not making ridiculous mistakes, and they can answer deeper understanding questions like, "Explain why sin(11) = cos(79)." I think right triangle trig is a frequent victim of the "First ya do this, then ya do this" treatment -- where kids can solve problems but have no idea what is going on. There's often not a ton of time for it, and it responds well to memorized procedures (in the short term). So, if your Day One of right triangle trig involves defining sine, cosine, and tangent, read on! I have a better way, and it doesn't take any longer.

We see how both Nowak and the author of Alt Math agree that today's activity should be given before the students learn the definitions of sine, cosine, and tangent. And so this is why I switched today's lesson with tomorrow's -- the original plan was for me to cover Lessons 14-3 and 14-4 of the U of Chicago text (where sine, cosine, and tangent are defined) before giving an activity. But I wish to honor Nowak's wishes to give this activity before defining the ratios. If I'm going to post her lesson on this blog, then I should present it the way she suggests it to be taught.

Of course, we observe that Nowak devotes a full week to this activity. She has the students work on only the opposite/adjacent ratio on the first day -- which, interestingly enough, is exactly how the U of Chicago text teaches it (in Lesson 14-3, before 14-4). Not until the fourth day does Nowak reveal the names of the three ratios.

Then again, this is one thing I don't like about the timing of the PARCC and SBAC exams. These tests are given a full month before the last day of school -- thereby forcing us to jump through the second semester material rapidly. The test on Chapter 14 must be next week in order to keep pace. If there were more time, perhaps I really could devote a full week to this activity.

I didn't communicate with Nowak herself, but I was able to speak to the author of Alt Math. Here is my comment followed by the author's response -- I cut out the part where we were commenting on a typo that has since been corrected:

This sounds like an interesting lesson. I see what you’re doing here — just telling the students, “We are going to learn trigonometry now,” leaves the students wondering why they must learn it and being resistant. This approach, on the other hand, introduces a question first, and then they discover that trig is the way to solve it.
I look forward to finding out whether this lesson was successful or not — even more so because I’m especially interested in Geometry lessons.

  1. Basically, I give them lines with an angle measured from the horizontal and have them draw different sized slope triangles. They see or remember that the slope ratio doesn’t change for that line, but any other non parallel line will have a different slope ratio. They find missing pieces given coordinate points or find delta y or a given delta x all for the same line.
    Then I have them draw slope triangles for various other combinations and have them measure the angles and do the process again. Afterwards, I introduce the word tangent and we discuss why it might have a different name, as opposed to always referring to it as slope. This leads to triangles oriented in different directions and then eventually sine and cosine when the hypotenuse is know.
But let's think about what both Nowak and the author of Alt Math are saying here. If I, as a teacher, were to go to a Geometry class and announce, "We are going to learn about sine, cosine, and tangent," imagine what the students' responses might be. We would expect questions like "Why do we have to learn this?" or "When will we ever have to use this?" to be common whenever strange sounding words like "sine," "cosine," "tangent" (or "logarithm") appear in math classes.

And now we can see how Nowak fights this. She provides an activity where students can see why these ratios are useful, and then defines the words "sine," "cosine," and "tangent." Now students are less likely to ask "When will we ever have to use this?" because they'd have already seen how the ratios are useful.

Notice that Nowak's link above itself contains another link -- this link leads to a page titled "Church of the Right Answer." This author criticizes teachers who elevate getting the right answer over understanding the process of getting the right answer or why the answer is right, by comparing them to blind adherents of a church.

I have a special name for adherents of the Church of the Right Answer -- traditionalists. And so this goes right back to the traditionalist debate. Traditionalists, like the ones I mentioned earlier in this post, oppose activities like Nowak's -- especially if they are group tasks, or any activities that span more than one day (as Nowak suggests.) They would prefer just telling the students the definition of "sine," "cosine," and "tangent," and assigning them an individual problem set whether they compute as many trig ratios as possible -- this is the best way to ensure that students get right answers when asked to solve a trig problem.

Of course Nowak is not a traditionalist -- if she were, she wouldn't have posted this activity. Most math teacher bloggers -- especially those who post activities -- are not traditionalists. I myself am sympathetic to traditionalism in the lower grades, but not the higher grades.

Here is the worksheet, which I post intact from its source at Alternative Math.


  1. I jut stumbled on your blog again while searching for some new Geometry ideas for next year, so it was a fun surprise to see this write up of an old post I hadn't seen before. Looking forward to digging deeper into some of your other reflections.

    Altmath (Also known as Micaela :) )

    1. Thanks! Obviously I thought your worksheet was an interesting way of introducing the trig functions.

      Good luck in the new school year!