6:55 -- Yes, this teacher has a first ("zero") period class. This is an honors class. As it turns out, this class is preparing for the SBAC next week. I wrote earlier that this week was the English SBAC, and now next week will be the Math SBAC. Students have a math packet filled with Released Test Questions -- and yes, these are the exact same problems that I covered on the blog last May and June.
Notice that only juniors need to prepare for the SBAC in California. While most Pre-Calc classes have a majority of seniors, honors Pre-Calc classes tend to have more juniors -- after all, juniors who make it to Pre-Calc tend to be honors students. These students are working at SteveH level -- many will be taking AP Calculus next year or are in the IB program. We know that both AP and IB are strongly recommended by SteveH.
There is also one sophomore in the class. She is working at Bruce William Smith level -- an advanced level one step beyond SteveH. The sophomore and all the seniors don't need to work on the SBAC Prep packet.
7:50 -- First period leaves and second period arrives. This is another honors class. Again most of the students are juniors, and there are two guys who are Bruce William Smith sophomores here.
But in addition to the sophomores and seniors, one junior refuses to do the SBAC Prep packet. She tells me that she won't be taking the SBAC. Much has been said about the opt-out movement -- parents who oppose standardized testing and thus tell their children not to take the test. I know that this movement is much stronger in other states than here in California. This girl could be opting out of the SBAC -- or she could be just opting out of doing work when the sub is here.
At the end of this class, I realize in horror that some students have written on their packets when they are supposed to use a separate sheet of paper -- oops!
8:50 -- Second period leaves and third period arrives. This is yet another honors class. This time, I tell the students to take out a separate sheet of paper before I pass out the packets. By doing so, I'm emphasizing my request that they not write on the packets.
One girl recognizes me from the vocal music class that I subbed for last week -- and of course, she's hoping that I'll sing some more math songs. Since this is actually a math class this time -- and since there is some extra time at the end of class -- I oblige her with a few songs. This time, I sing "The Dren Song," "The Big March Song" (since it's still the Big March), and the 2D version of "All About That Base and Height" (the version that appears on a YouTube video).
9:50 -- Third period leaves for snack.
10:00 -- It is now tutorial. I end up spending most of the time erasing the writing from the packets that students weren't supposed to write on.
10:30 -- Fourth period arrives. This is the only non-honors class of the day.
For some reason, these students don't have an SBAC Prep packet to complete, at least not today (though they will have one on Monday). Instead their worksheet is on the Laws of Sines and Cosines.
Indeed, I notice that the regular students are in Chapter 9 of their text, while the honors students are only in Chapter 8. Actually, these are different texts -- but even so, it's strange that the honors students would be one chapter behind the regular students even in a different text. We might even expect an honors text to have more chapters.
I don't really take time to explore the two texts, since I'm too busy erasing from the packets. I do notice that Chapter 8 of the honors text is on systems of equations. The focus is on 3 * 3 systems and matrices, but 2 * 2 systems do appear as a review. This actually fits SBAC Prep, since one of the questions requires a system in two variables..
11:30 -- Fourth period leaves -- and believe it or not, that ends my day. For some reason, this regular teacher has only four classes, Periods 1-4. (Technically, she's not a regular teacher but a long-term sub, since the regular teacher is out with a serious injury.)
The only real classroom management issue is writing on the packets. Once again, the obvious way to prevent this is for the teacher to have the students take out paper first, before they get the packets. I erase most of the packets clean, but two students have written in ink. I decide to copy two new packets for the "regular" teacher and take the two inked packets home with me. When we reach this year's SBAC Prep here on the blog, I think I'll refer to the two packets and how these two students answer each question.
I randomly choose questions to answer from the packets. I end up selecting Questions 33, 18, 15, 1, 2, 13, 12, and 21 in that order. Again, you can refer to my May and June posts to learn more about these specific questions. Notice that these are mostly Algebra I/II questions -- unfortunately I avoid all of the Geometry questions.
The questions I do from the trig worksheet, on the other hand, aren't random. Here are the four problems that I do in class:
1) To find the distance AB across a river, two points (B and C) 354 meters apart were laid off on one side of the river. It is found that B = 112.2 and C = 15.3 degrees. Find the distance across the river.
2) A rocket training facility has two radar stations A and B, placed 1.69 miles apart, which lock onto a rocket and constant transmit the angles of elevation to a computer. Find the distance to the rocket from A at the moment when the angles of elevation from the tracking stations are 29.3 (from A) and 80.2 degrees (from B), as shown in the figure.
4) To determine the distance across a lake AB, a surveyor goes to point C where he can measure the distance from C to A and from C to B as well as the angle ACB. If AC is 265 feet and BC is 188 feet and Angle ACB = 42 degrees 18 minutes, find the distance AB across the lake. (Of course, it's the 42 degrees 18 minutes that confuses the students.)
6) Circular tracts of land with diameters 900 meters, 700 meters, and 600 meters are tangent to each other externally. There are houses directly in the center of each circle. What are the angles of the triangle connecting the houses and what is the area of that triangle?
But let's get back to the SBAC Prep packets. In fact, there's one thing that confuses me about them -- their very existence. You see, two years ago at the old charter school, I tried to print out some questions from the SBAC Prep website, but I was unable to. There was no print button on the screen, and pressing Control-P (or Command-P on Macs) did nothing. I assume that the print command is automatically disabled by the SBAC site itself, out of fear that students or teachers might try to print questions from the real test (as opposed to the practice test). Yet somehow, today I see a packet of SBAC released test questions.
Two years ago, I wrote that my experience at the old charter seemed to go downhill the moment that SBAC Prep time was implemented. Sixth and seventh graders regularly talked throughout my discussions of the SBAC questions, while eighth graders got into arguments with me about anything to do with SBAC Prep -- from not being able to see the questions on the board to forgetting how to answer them to wishing that we didn't have to do it at all.
I suspect that if I could have had printed worksheets, everything would have gone smoothly. Some students could try the questions independently while I helped others. In the eighth grade class, my support aide was there -- and she could have helped the students answer questions from the worksheet rather than wait for me to write them on the board. And of course, the complaint about not being able to see the questions on the board instantly disappears if there's a worksheet. Instead, the lack of worksheets brought only frustration on my part as well as the students' part -- and that frustration eventually led to my leaving the school.
In short, if only I could have printed the SBAC Prep worksheets, I might still be working there at the old charter school today! Yet I was unable to do such a simple task, since pressing Control-P on my computer didn't work and thus I assumed that printing them was impossible. The existence of today's SBAC Prep worksheet finally proves my assumption wrong two years later.
What should I have done to get my SBAC Prep worksheets? I could have tried asking other teachers whether it was possible to print from the SBAC website. I should start with the English teacher, followed by the elementary teachers (in the testing grades -- third and above). If I saw her during the Common Planning meetings, I could have asked my counterpart at the sister charter. Finally, I could have asked math teachers around the country on the Twitter account that I should have created.
But instead, I never printed the SBAC Prep worksheets, and I eventually left the school.
Lesson 14-1 of the U of Chicago text is called "Special Right Triangles." In the modern Third Edition of the text, special right triangles appear in Lesson 8-7 (as I explained in yesterday's post).
This is what I wrote last year about today's lesson:
Chapter 14 of the U of Chicago text is on Trigonometry and Vectors. Here's the plan:
Today, April 5th -- Activity (includes Lesson 14-1: Special Right Triangles)
Monday, April 8th -- Lesson 14-2: Lengths in Right Triangles
Tuesday, April 9th -- Lesson 14-3: The Tangent Ratio
Wednesday, April 10th -- Lesson 14-4: The Sine and Cosine Ratios
Thursday, April 11th -- Lesson 14-5: Vectors
Friday, April 12th -- Activity (includes Lesson 14-6: Properties of Vectors)
Monday, April 15th -- Lesson 14-7: Adding Vectors Using Trigonometry
Tuesday, April 16th -- Activity (as explained in last year's posts)
Wednesday, April 17th -- Review for Chapter 14 Test
Thursday, April 18th -- Chapter 14 Test
So the plan for this chapter is straightforward. I've noticed how many texts, including the U of Chicago, discuss the tangent ratio in a separate lesson from sine and cosine. I suppose that in many ways, sine and cosine are alike in a way that tangent isn't. The sine or cosine of any real number is between -1 and 1, while the tangent can be any real number. Therefore the graphs of sine and cosine resemble each other. The tangent ratio involves two legs, while the sine and cosine ratios involve one leg and the hypotenuse. Even the name "cosine" includes the word "sine," while the name "tangent" doesn't include "sine."
But that's for next week -- how about today's lesson? Lesson 14-1 of the U of Chicago text is on Special Right Triangles -- that is, the 45-45-90 and 30-60-90 triangles. The text emphasizes how these triangles are related to the regular polygons. In particular, the 45-45-90 and 30-60-90 triangles are half of the square and the equilateral triangle, respectively. We can obtain these regular polygons, in true Common Core fashion, by reflecting each right triangle over one of its legs. The regular hexagon is also closely related to the 30-60-90 triangle.
The questions that I selected from the text refers to these regular polygons and using the triangles to measure lengths related to the regular polygons. I mentioned today how I like to watch baseball over summer break -- well, a baseball "diamond" (really a square) appears on the worksheet. Also, a honeycomb, with its hexagonal bee cells, also appears.
The review questions that I selected are also preview questions. Two of the questions involve similar right triangles in preparation for geometric means in Lesson 14-2, and the other one is about how to simplify radicals, so we can explain in Lesson 14-4 why the sine and cosine of 45 degrees are usually written as sqrt(2)/2.
Today is an activity day. In past years, I wrote that the teacher who created this activity stated that it should be presented before sine, cosine, and tangent are taught. But scheduling this activity was awkward since there was no Friday between the start of Chapter 14 and the tangent lesson.
But this year, Chapter 14 begins on a Friday. This means that I can give the activity right now -- before the trig ratios, just as its creators intended!
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. 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 soon in order to keep pace. If there were more time, perhaps I really could devote a full week to this activity. [2019 update: And as we find out this week, schools are giving the SBAC math test next week!]
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.
As we begin Chapter 14, notice that the Laws of Sines and Cosines might appear in the Pre-Calc classes that I taught today, but not most Geometry texts.
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