Thursday, February 18, 2016

Activity: Constraints and Trig (Day 105)

Today I subbed in a sixth grade special education class. It was a self-contained class, so I covered all subjects, including math. Naturally, these students were well below grade level -- indeed, they were working on word problems requiring three-digit subtraction with borrowing (in other words, using the standard algorithm). We know that this is a fourth grade standard under the Common Core:

Fluently add and subtract multi-digit whole numbers using the standard algorithm.

There were also a few word problems where they had to determine whether to add or subtract, with one- and two-digit numbers. This is arguably a second grade standard:

Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

Here was one of the problems that the students had to solve:

A recycling drive lasted for 3 weeks. Troop 15 collected 796 pounds of paper, while Troop 17 collected 598 pounds of paper. How much more paper did Troop 17 collect?

The worksheet warned the students that there may be extra numbers included in the problem. As we can see, the students must subtract 796 minus 598, not 17 minus 15. Fortunately, most of the students knew which numbers they had to subtract (even if they couldn't perform the subtraction). In some ways, this isn't terrible -- I'd much rather the students know they must subtract 796 minus 598 (even if they must use a calculator) than subtract 17 minus 15 by hand and give "2 pounds" as the answer.

This takes us straight into today's traditionalists topic. Today we'll go straight back to the blog of Dr. Katharine Beals -- at which point you're wondering, what the ...? Yes, Dr. Beals made her "Math Problem of the Week" post today, but no, it has nothing to do with Geometry -- in fact, today she posted SAT problems (and we know how little Geometry appears on the new SAT). Despite this, I have two reasons for going back to the Beals website today:

-- This week's problem, while not Geometry, is definitely related to the class I actually taught today!
-- There have been more comments posted to last week's Geometry post since I first discussed it.

Let's start with this week's Beals post:

A recent article in the NY Times raises concerns about the reading demands of the new SAT questions, even in math. The article links to the following sample problems:
Extra Credit: Does anything else stand out to you about these new SAT math problems, besides the excess verbiage?

By the way, that NY Times article to which Beals links has raised much discussion. In criticizing the "excess verbiage," Beals implies that the SAT problems are too wordy. Her fear is that the new SAT will punish students with excellent math skills but weaker verbal skills. Yet some respondents to the NY Times article counter that students who have trouble reading the math problems aren't really college material anyway (since their verbal skills are weak).

Returning to the Beals post, here's a response by a "Niels Henrik Abel" -- and no, I doubt that this Abel ever proved that the quintic is unsolvable:

Niels Henrik Abel said...
Along lgm's lines: Students do need to learn to separate the wheat from the chaff. There is something to be said for extraneous numbers in a word problem, if for no other reason than to weed out the kids that are just grabbing any numbers they see and performing mathematical operations on them. Believe it or not, I have encountered such students as a tutor. These are students who have a vague idea that they're supposed to calculate *something*, so they just start punching away, without trying to sort through the information given in the word problem and see how it all fits together.

That being said, there are students who, on the other hand, completely ignore pertinent information in word problems. I tend to tell my students that generally speaking, story problems are not in the habit of giving excess information, so if there is unused data/info, they'd better look and see if they're overlooking some crucial fact. It seems like students tend to fall into the ditch on one side or the other, because they can't differentiate between pertinent and extraneous information.

Extraneous numbers in a word problem -- hey, that's exactly what the special ed students had to deal with earlier today!

And I must agree with Abel here. I remember back when I was student teaching an Algebra I class, and there would be an exponential decay question concerning Carbon-14, which is often used to date ancient artifacts. So some students kept trying to plug the number 14 into the formula -- and it was tough trying to explain to these freshmen (who haven't had Chemistry yet) what the number 14 is referring to.

Now let's get back to last week's post. An "Education Realist" made several comments in the thread about using a courtroom metaphor to explain Geometry posts:

education realist said...
Not much point to doing proofs in geometry anyway. I don't do them anymore, other than in honors classes--and even then, I'd cut it in favor of more algebra.

education realist said...
But we don't teach it. We can't. The kids are going on to Algebra 2, and most of them don't know Algebra 1. And for the most part, low ability kids don't get proofs, period. Waste of time.

education realist said...
>IOW, if everyone cannot do it, no one is allowed; individual ability and interest be dammed! [This line was originally written by another commenter, "Momof4." -- dw]

No, don't be silly. The vast majority of kids in non-honors algebra can't do it, and the ones who can can barely do three step proofs. Meanwhile, they all need tons of algebra work, much more than they would have in years past.

I introduce algebraic/geometric proofs in trigonometry.

You are apparently ignorant of geometry's purpose in high school math. It's going to be gone in a few years. No one's getting robbed. I know much, much better than you do the needs of poor bright kids.

What's this -- Geometry will be gone in a few years? I created this blog because Geometry is my favorite math class, so it's disheartening to hear that it might disappear! Yet I must admit that Ed Realist might be correct -- after all, what did I say about the lack of Geometry on the new SAT? And of course, the transition to Integrated Math often shunts Geometry to the background as well. (Even though I defend Integrated Math, one virtue of the traditional sequence is that it allows students who are weak in Algebra I to take a break from algebra by taking Geometry. The last thing they want to think about at that point is Algebra II!)

Beals saw fit to respond to this comment herself:

Katharine Beals said...
Gone for everyone? (Whose idea is that?) If true, that would be a real shame. Geometry proofs are a great introduction to proofs --for mathematically capable kids of all social economic backgrounds.

And I usually don't agree with the traditionalists over high school math, but here I admit that I agree with Beals. I don't want to see Geometry disappear any more than she does!

It's seems interesting that Ed Realist would wait until Trig to teach proofs. There aren't that many proofs in Chapter 14, the Trig chapter of the U of Chicago text. Indeed, I'd focus on proofs during the first semester of the class and move on to more activities when we reach the trig chapter -- just as I'm about to do in this post.

Before we get to the activity, I have one more thing to say about the Beals blog. In between last week's and this week's "Math Problem of the Week" posts, she wrote about her daughter's education:

Of all the even halfway decent high schools around us, it’s the closest one. But it’s far more than decent. Known for embracing quirky kids, including erstwhile homeschooled kids, its Quaker-inspired social tolerance appears to extend beyond the usual types of tolerance to... the idiosyncrasies of personality.

The most famous Quaker school in the country is, of course, Sidwell Friends -- the school that the First Daughters attend. I doubt that Beals is sending her daughter to Sidwell (as her family lives in Pennsylvania, not Washington DC), but considering how many posts I devoted to Sidwell (in order to discuss my Presidential Birthday Plan), it's interesting to see what a traditionalist like Beals has to say about her local Quaker school.

Not only that; its algebra courses use Brown & Dolciani. And, as one former homeschooler told me, it’s the only school they found that was willing, without argument, to let their child skip ahead in math.

Brown & Dolciani is, of course, a pre-Core text (which is what Beals prefers). It's a 1980's text -- I believe that I myself once took a course using Brown & Dolciani (either Algebra I or Pre-Algebra, but definitely not Algebra II).

I'm not sure what Beals means by "without argument, to let their child skip ahead in math." I mean, "without argument" almost sounds as they'll place a freshman into Calculus if the parent insists, even if the student hasn't mastered Pre-Algebra yet. Then again, we know that Sidwell Friends has three different math tracks. It appears that her daughter, an entering sophomore, will be placed into Algebra II (as Beals is so excited about the algebra texts). This would be the middle track at Sidwell.

In math, we'll continue through Weeks and Atkins Geometry, progressing from polygons to quadrilaterals. But, as soon as we heard about school’s algebra texts, we switched from my mother’s A Second Course in Algebra to Brown & Dolciani. 

In science, we’ll continue with Biology (Campbell and Reece) even though the school’s sequence is chemistry in 9th grade and biology in 10th.

Chemistry in 9th grade -- well, if my own school had had that, maybe there wouldn't have been so many problems in my class with that Carbon-14 question...

I was torn as to whether this activity should be posted today or tomorrow -- obviously, in the end I've decided to post it today. Let's describe the activity first, and then you'll see why this activity fits better today than tomorrow.

Recall that during the first week in February, several teachers posted some interesting lessons -- many of them relevant to Geometry. There were so many ideas posted that week that I could almost fill out all the activity days the rest of the year with these activities.

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.
Notice a few things mentioned in this response -- slope triangles. I've mentioned back when we were covering Chapter 11 that slope triangles are the key to understanding slope for Common Core. And we also see how this teacher uses delta-y and delta-x for the slope. I still find it interesting that those little obscure delta symbols that I once saw in a college-level chemistry text are now appearing in high school Algebra I and Geometry classes!

Indeed, the fact that we jumped directly from Chapter 11 on the coordinate plane (where I shoehorned slope into the unit) to Chapter 14 on trig is perfect for this activity. Ideally, slope should still be fresh in the students' minds, and the idea of slope will help the students understand the idea of tangent.

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.

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