Activity: Constraints and Trig (Day 143)

Lesson 14-3 of the U of Chicago text is called "The Tangent Ratio." In the modern Third Edition of the text, the tangent ratio appears in Lesson 13-5. (By the way, Lesson 13-4 of the new edition is all about the Golden Ratio. This lesson doesn't appear in the old edition, but we did mention it back on January 6th, Phi Day of the Century.)

In past years, I've established the tradition of posting an activity on the day after Lesson 14-2, and I see no reason to end this tradition. And so this is exactly what I'm going to do -- even though it now means that I'm posting two activities three days apart (after I posted the Lesson 13-8 activity late).

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

Today's idea comes from Micaela, a Washington State teacher who goes by the username "Alternative Math" -- named for the alternative high school to which she is assigned.

https://alternativemath.wordpress.com/2016/02/05/geometry-constraints-and-trig/

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. 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.

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.

Returning to 2018, I admit that I gave in to the temptation of posting a second activity within a week of the first activity. Now I must give in to yet another temptation -- making a traditionalists post less than a week after the previous such post. After all, even in the part I quoted from last year, I briefly mentioned traditionalists -- and two years ago, the post corresponding to today's was a full-blown traditionalists post.

Moreover, traditionalist bloggers and commenters have been especially active this week. Barry Garelick continued to post on Sunday about the Ed Week article that I mentioned last week. This post has drawn five comments, four of which are from SteveH. Then Garelick made another post today just to highlight two of SteveH's comments.

Meanwhile, I used to link to the Joanne Jacobs website more often, back when the traditionalist Bill (Parker) was a regular commenter there. Yesterday, a post about a certain district (actually, it's the LAUSD right here in Southern California) allowing students to take Stats instead of Algebra II drew 14 comments, including three from Bill. And some of the other commenters in that thread are trying to out-Bill Bill.

Let's start with Garelick and SteveH. We begin with Garelick's Sunday post:

https://traditionalmath.wordpress.com/2018/04/08/and-just-what-are-you-after-exactly-dept/

Barry Garelick:

Just what conceptual understanding do people think is missing from middle school math?  It isn’t that students are just given problems to solve without explaining what the concepts are.  Percentages are explained, as are decimals, as are fractions, and why one uses common denominators, and even the why and how of multiplication and division.  Anyone who teachers middle schoolers or even high school students, knows that students gravitate to the “how” rather than the why.

Actually, Garelick, students also gravitate to the "when," as in "When are we ever going to use this in real life?"

There's no need for me to quote the SteveH comments, since Garelick quotes them himself in his post from today:

https://traditionalmath.wordpress.com/2018/04/12/steveh-on-problem-solving-transference-and-understanding/

These SteveH comments are long, so I'll only quote parts of them today. (Follow either Garelick link above to read the original comments in their entirety.)

SteveH:

I. Problem solving and transference

I want to say more about problem solving and transference. Polya is worthless, so what problem solving skills are transferable? You can draw pictures, label variables, and write equations to look for m=n. Everybody does that, but still, it’s difficult. You can study governing equations and their variations. This is classic homework p-set work. However, does D=RT problems and variations easily transfer to work problems? Yes and no. Both are amount = rate * time sorts of things, but there are lots of odd variations. Look at AMC test problems. You could know a lot about logarithms, but the testers are really good at finding odd problem variations. I look at some of their questions and feel really stupid. Success on the AMC is not really about general transference, but very specific and detailed preparation of problem classes for a timed test. You don’t study general problem solving skills. You study in detail every past test question you can get your hands on. Only that reduces transference distance.

II. Understanding and challenging assignments

This is from my son’s old Glencoe Algebra 1 textbook on factoring differences of squares – Exercises. (page 451)

Section 1 – Basic skill problems like this – Factor or prime?
256g^4 – 1

Section 2 – Factor and solve
9y^2 = 64

Section 3-6 various word problem applications

Section 7 – Open Ended
“Create a binomial that is the difference of two squares. Then factor your binomial”

Section 8 – Challenge
Section 9 – Find the error
Section 10 – Reasoning – find the “flaw”
a = b
a^2 = ab
a^2 – b^2 = ab – b^2
(a-b)(a+b) = b(a-b)
a+b = b
a+a = a (remember that a=b)
2a = a
2 = 1

Section 11 – Writing in Math

Section 12 – Standardized Test Practice

Section 13 – Spiral Review questions

Section 14 – Reading Math
Learn to use a two-column proof for algebraic manipulation

Actually, I recognize SteveH's Section 10 question -- recall that three years ago, I used to tutor an Algebra I student whose class used the Glencoe text, and I remember helping him on this exact "find the flaw" problem. (It was in my April Fool's Day post from 2015.)

This is what I wrote three years ago about this problem:

This problem is a classic "proof" that 1 = 2. In this proof, we are given that a = b in Step 1, and in Step 8, we divide by a - b (actually given as a^2 - ab, but factoring gives a - b). As a = b, a - b must be zero. So we actually divided by 0 -- and this is why division by zero  must be undefined, since otherwise we could prove that 1 = 2. I once tutored a geometry student (not my current student) whose teacher assigned the classic "proof" of 1 = 2 and the student was asked to find the error. It took a while, but I think the student did eventually figure out that a - b = 0, so that they were dividing by 0.

And in fact, I'd discussed this example in more detail in an earlier post, Thanksgiving break 2014.

SteveH continues:
So what’s the problem here? Is it just that teachers only assign the problems in the first two sections?

Well, I don't recall what my student's teacher assigned him, but clearly Section 10 was included.

OK, let's move on to the Joanne Jacobs site now.

Joanne Jacobs:
Algebra 2 is the gatekeeper to college, writes Pamela Burdman, of Just Equations on EdSource Today. Many students never make it past. “A quiet revolt against the dominance of algebra” could widen the pathway to college, she writes.

In some southern California schools, students are taking a new Data Science course that builds “critical thinking skills, data awareness and positive attitudes,” UCLA researchers believe.

Now I'll post the Bill Parker comments.

Bill Parker's first comment:

The required math courses for a accredited computer science program back in 1981 was:
.
Calc 1/2
Linear Algebra
Applied stats
Diff Eqns 1/2
Abstract Algebra
Numerical Analysis

Add to that

Digital Logic 1/2
Symbolic Logic
Engineering Physics 1/2

You can’t make it through Algebra 2, you aren’t going to make in most STEM fields, period.

Bill's second comment (judging by the 1981 reference, "Bill" and "Bill Parker" are the same person):

The recommended course of study for a college-prep student when I attended high school from 1977-81 (and yes, we grouped students by ability, had tracking of students, enforced discipline, and kids could drop out at 16 and learn a trade if they wanted to, and no Plyler vs Texas to worry about) was:

3 Units of Math (Algebra I, Geometry, Algebra II/Trig)
3 Units of Science (2 of which were lab sciences)
4 Units of English (English I/II, Literature (Amer or European or World), and Composition
3 Units of History/Government (World History, US/State History, US Gov’t)
1 Unit of Foreign Language

That was to be considered ready to take college level coursework
in the first year of:

Pre-Calc or Calc
English 101/102
General Biology/Plant/Animal or General Chemistry I/II
US History/Poly Sci
Electives

What it’s like today I wouldn’t want to try and guess, given the amount of remediation needed for college freshmen entering directly from high school…

UGH

Bill's third comment:

If you have to take remedial math at the college level, you aren’t prepared to take college level coursework, period…

It’s astounding the number of students who think they can succeed in college by doing a minimum amount of work in high school (never mind the cost of college is way beyond what it should be), and when students find out the actual requirements for most STEM degrees, they’ll go for a year or so and then drop out.

It’s been proven the more remediation a student needs in college (2 or more classes), the chances of them actually completing a certificate or associate’s degree is very unlikely.

In previous posts, Bill Parker makes it clear that to him, students must master Algebra II before they can begin Statistics. Thus Stats should be considered a class beyond Algebra II, not an alternative to Algebra II for weak math students. In these comments, Parker adds that they should be successful in Algebra II in order to be prepared for college, especially STEM. Thus schools shouldn't be searching for alternatives to Algebra II at all.

(I definitely disagree with Bill's statement, "if you have to take remedial math, you aren't prepared to take college level coursework, period." After all, why would a student be unprepared to take English 101/102, US History, or even General Bio just because he or she has to take remedial math?)

As I wrote earlier in this post, some commenters in this thread out-Bill Bill. It seems that to these commenters, not only should Algebra II be an prereq for Stats, but Calculus should be a prereq too!

Casey Thompkins:

Only one problem. To truly learn statistics (as opposed to simple “cookbook” stat) one needs basic calculus to understand the proofs. Before you learn calculus you need … wait for it … trigonometry and algebra!

If they teach “cookbook” statistics (here’s the formulas, plug in the values) they won’t learn much. Do they even discuss Type I and Type II errors?

PaulT:

I wonder if the kids who lack the ability for algebra 2 will have any more success in college when not required to take it.

Statistics is a great filler class, but any further progress in statistics will require algebra 2 and calculus.

Computer Science is another field that is greatly assisted by logical thought and problem solving, the same skills taxed by algebra 2.

If Stats is actually beyond Calculus, then does this mean that AP Stats shouldn't be taken until after completing AP Calc? One commenter implies that students in East Asia do exactly this:

Malcolm Kirkpatrick:
Years ago I read somewhere that in Hong Kong and Taiwan, students take statistics in 12th grade, after 11th grade Calculus.

Here's the thing -- advocates for Statistics tell us that data and stats surround us in everything we read, so students should take Stats. But these commenters are telling us that we can't truly understand Stats without Algebra II and Calculus -- so in other words, in order to understand and analyze all the data we read about daily, we must have taken Algebra II and Calc. This makes Algebra II and Calc appear to be very useful subjects -- yet students actually sitting in those classes tend to imply the opposite when they ask questions such as "When will we use this in real life?"

SteveH is clearly a huge Calculus advocate. But he seldom, if ever, mentions Stats in his comments, so we can't be sure whether he believes that Calc should be a prereq for Stats.

But some of the posters in this thread worry in a SteveH-like manner that discouraging Algebra II and Calc in high school may hurt some students.

Deirdre Mundy:

Meanwhile, many actual STEM-bound kids are already reading the admissions requirements for their college programs in 7th or 8th grade, and start planning their coursework around it. (I.e. if you take Algebra in 9th grade YOU ARE ALREADY BEHIND).

The problem for the brighter low-income kids is that they have no one in their life who can get them on the college track in Jr. High, and they get stuck in schools where so few kids can handle math that there is no way for them to GET what they need for an STEM field. 

Mundy's comments about low-income kids reminds me of SteveH, who points out that low-income kids can't afford the private tutors (from Kumon, etc.) who can prepare them for eighth-grade Algebra I and the AP Calculus track.

Another poster continues the same line of thinking:

lgm:
The idea is to make students who are capable of Calculus etc, pay for it at the University tuition rate, rather than ‘stealing’ from the K12 budget.

In other words, this hurts low-income students who can't afford the university tuition rate.

There's one more commenter to mention here, but I won't quite that commenter directly. There is someone here posting under the name "Math is like a box of...." I've seen this same poster leave comments on other articles -- both the username and the comment itself contain sarcastic remarks in every case.

Here the poster comments sarcastically about General Robert E. Lee. Considering that Bill Parker already mentioned tracking in another comment, the General Lee mention is clearly a reference to tracking and race yet again. Even though I'm protected by the "traditionalists" label, I've made too many tracking and race posts recently. So let's leave this commenter at that.

Let's conclude by returning to SteveH at the Garelick website.

SteveH:
You can get good grades in math, but still feel like you are struggling. It’s not a “zombie” issue. It’s normal. You can get poor grades and not understand, but that’s another issue. If you get good grades and really don’t understand, then probably your grades are OK, but dropping. Some honors (proper math) classes say that you can only enter if your grade is 80+ or some such thing. Gaps and weaknesses in understanding will eventually cause you to fall off the cliff. Math is tough in high school and college.

The problem is that when students don't understand, they shut down completely. Instead of struggling to get through the material, they start asking questions like "When are we ever going to use this in real life?"

And my experience tells me that this question gets asked when the students see a word that they don't understand -- words like "sine," "cosine," or "tangent." So to me, anything that gets the students to understand what sine, cosine, and tangent are -- before their mind shuts down -- will be beneficial enough to help the students get to that STEM career.

And this is exactly what Kate Nowak does with her opening assignment. The idea is to get the students to understand what exactly the trig ratios are before they see those scary words -- since from their perspective, "sine," "cosine," and "tangent" are scary words.

JK Rowling, of Harry Potter fame, said it best: "Fear of a name increases fear of the thing itself." So the idea is for the students to have a basic understanding of the concepts so that they won't fear the name -- and thus be more willing to learn the concepts themselves.

By the way, Kate Nowak, the "High Priestess" of the MTBoS and the originator of today's activity, is no longer a frequent blogger. She's working on a new text, Illustrative Mathematics, a middle school math text. But if we look at her most recent post (dated January 26th), we can easily surmise that there's nothing in the new text for traditionalists to like:

http://function-of-time.blogspot.com/2018/01/why-we-dont-cross-multiply.html

The title of Nowak's post is "Why We Don't Cross Multiply." And I can already imagine how traditionalists will respond -- Nowak's text teaches some "inefficient" method of solving proportions instead of cross multiplication. Traditionalists would rather teach only the fastest possible method, cross multiplication, so that students can be done with it quickly and "transfer" this knowledge to more complex problem (such as similarity problems in Geometry).

Actually, I don't need to imagine it -- last year, Barry Garelick quotes Ralph Raimi.

https://traditionalmath.wordpress.com/2017/02/12/ralph-raimi-on-proportionality-whatever-that-means/

Ralph Raimi:
Since learning about the current obsession with “proportional
reasoning” I have decided that the language of functions, input and output, is
the easiest way to understand such problems.  After all, Proportional is the
description of only one class of functions, and all science is the quest for
analogous relations.  What is there against the use of letters and equations
from the very beginning, when such real-life problems are first attacked? 
First learn the number system itself, then observe that in the real world there
are many relations expressed by formulas, in which an input and an output
are related by a scale factor, or rate.  Write the relation and solve. 

Well, here is Nowak's answer to Raimi's question:

Kate Nowak:
It is a jujitsu move to start with a problem that uses only whole numbers and then write a statement equating two fractions. For people who are already intimately familiar with these ideas, it is useful to represent ratios using fractions. But we are introducing this important and new concept in grade 6, here, and students have worked hard to understand in grades 3–5 that fractions are numbers (3.NF.A) and rely on that definition in their study of fractions. The standards define a ratio as a relationship between two quantities (6.RP.A.1) (and an important ratio that is equivalent to a : b is a/b : 1 (6.RP.A.2)). To suddenly assert that a ratio (2 numbers) is a fraction (1 number) runs counter to this definition of ratio and doesn’t build on the understanding of fractions from grades 3–5. To solve a problem about equivalent ratios by jumping to a statement that equates two fractions fuzzes up the definition of a ratio and the understanding of what a fraction is.

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