3 times the complement of angle A is 38 degrees more than its supplement. a = ?
This is a straightforward application of Lesson 3-2 on types of angles:
3(90 - a) = 180 - a + 38
270 - 3a = 218 - a
52 = 2a
26 = a
Therefore the desired angle is 26 degrees -- and of course, today's date is the 26th. Oh, and by the way, expect a Pappas problem everyday this week. Yes, her calendar has Geometry all this week.
This is what I wrote last year about today's lesson:
Lesson 0.9 of Michael Serra's Discovering Geometry is labeled "Chapter Review." In the Second Edition, chapter reviews have their own lesson numbers, but in the modern editions (just as in the U of Chicago text), the chapter reviews are unnumbered.
At this point, we may wonder, should there be a Chapter 0 Test? If there were a Chapter 0 Test tomorrow, then this would allow us to start Lesson 1-1 of the U of Chicago text on Day 11, which would be Wednesday.
Some teachers may point out that Chapter 0 consists of just introductory activities and so it shouldn't be tested -- and besides, a Chapter 0 test would be so soon after the first day of school, when many students are still requesting schedule changes from their counselors.
On the other hand, without a Chapter 0 Test, the first test would be the Chapter 1 Test on Day 20. At some schools, grades must be submitted every quaver (i.e., twice a quarter). Day 20 would be very close to the end of the first quaver -- and at many schools, grades are due a few days before the mathematical end of the quaver. So whether or not there should be a Chapter 0 Test depends on how often a school issues progress reports, as well as whether a teacher wants to give a solid test before those first progress reports are issued.
As far as this blog is concerned, my decision is to follow what my old school did three years ago. If you recall from that year, the first test I gave my students was called a "Benchmark Test." This was, of course, a diagnostic pre-test to determine what the students already knew, and what they would need to learn in the coming year.
Therefore tomorrow I will post some Benchmark Tests for Geometry. It will preview lessons to be covered the entire year. Nonetheless, today's worksheet is based on review questions from Lesson 0.9 of Serra's text.
Here is the Blaugust prompt for today:
- Fav [math][ed] book read and take aways for this year (and beyond?)
Before I answer this question, let me announce that today is the first day of school in the district where I receive most of my subbing calls. Therefore, I expect that next week will mark my return to the classroom as a sub.
Speaking of which, I also notice that this district where I get most of my calls has already posted a calendar for next year. There's usually only one reason a district would post a calendar a year early -- big changes are coming, especially the change to the Early Start Calendar (with school starting earlier and the first semester ending before winter break). And indeed, that's exactly what's happening. Thus both of my districts will have similar calendars. They won't be exactly the same (since one district always starts on Wednesday, the other on Monday), but close enough.
In fact, it's enough for me to declare that I'm switching to the other district calendar next year. The Geometry curriculum that I post fits the Early Start Calendar better (with Chapter 3 near the PSAT and second semester starting with Chapter 8). But it's always annoying to keep posting all of those disclaimers in each post (such as, in one district it's Day 157 and the end of the seventh quaver, but in the other it's Day 150 and midway through the trimester -- or even worse, I'm subbing in a class where they're reviewing for the final, but on the blog we've already taken the final). So finally, the district where I get most of my calls will be the one whose calendar we're following.
But that's next year. This year I've already committed to following my old district. And so today is still Day 9 on the blog, even though it's Day 1 in the district where I'm usually called. (And yes, this year we still have the problem with two radically different spring breaks. So I'll post "Enjoy your week off!" and then sub four out of five days the following week without posting.)
Meanwhile, today is the 101st birthday of Katherine Johnson -- the NASA mathematician who was the main subject of the movie Hidden Figures. I can't help but think back to our field trip to the movie theater during my year at the charter middle school.
Today's Blaugust prompt partly repeats an earlier one about my favorite blogs and books, except that this one focuses only on books. Of course, in honor of Katherine Johnson's birthday, I should begin with the actual book Hidden Figures.
I've actually only glanced through the book and never read it cover-to-cover. I do know that the film really only corresponds to the last few chapters of the book. Much of the book focuses on some of the other women besides Johnson -- and some of them don't even appear in the movie. The author, Margot Lee Shetterly, also describes their work for NACA, the predecessor agency to NASA. Who knows -- perhaps one of these days, I'll finally read Shetterly's book in full.
I decided to revisit old blog posts to find books that I've written about. My problem is that most of the books I read come from libraries -- including the books that I purchase from their book sales. Many of the math books that I described and blogged about are old textbooks -- and I doubt those are what Shelli had in mind when she came up with this Blaugust prompt.
During the first two years of this blog, I wrote about some books that I'd acquired not from my local library, but from my high school library back when I was still a young student attending there. One of these books was a textbook on coding in Pascal, which I no longer own. The other I still have to this day -- I'm glancing through it as I type. It's Introduction to the Theory of Sets by Joseph Brewer. This is technically a textbook (from 1958, during the "golden age" of textbooks when they were much smaller than they are today). I still remember being fascinated by Georg Cantor's theory that there are different types of infinite sets, "denumerable" and "non-denumerable."
(I mentioned these books together because there's a connection between them. In the coding text, the author was explaining algorithms. If someone is thinking of an integer, an algorithm to find the number is to guess 0, 1, -1, 2, -2, 3, -3, and so on -- eventually we'll guess the right integer. Then in Brewer's book, this exact same sequence is used to prove that the set of integers is denumerable.)
Of course, I can't post about books and not mention Glen Van Brummelen's Heavenly Mathematics. I did spend two summers (2017 and 2018) discussing this book on spherical trigonometry. But even that book is, at its heart, just another textbook.
So you might wonder, do I read any books that aren't textbooks? Once again, I do have to mention Ottaviani's series of graphic novels on famous scientists. And I remind you that I do wish to obtain and blog about his moon landing book before this semi-centennial year is over.
Once again, we notice that Shelli's prompt mentions "favorite [math][ed] book." Most of the books read by Blaugust participants are probably "ed" books. This includes Geoff Krall's Necessary Conditions (foreword by Fawn Nguyen). It's the type of book that I probably need to read more often if I wish to improve my teaching skills. I know -- by now you're likely tired of reading how I wish to read this book someday. My problem is that I don't often spend full price on books -- instead, I like to get my books cheaply at library book sales. It'll probably be years, if ever, before I ever find Krall's book at the local library.
Have I ever read any ed books? In the past, I did allude to Harry Wong's The First Days of School, along with another ed book by Fred Jones. Indeed, I blogged a little about Wong's book during the summer before I taught at the old charter school.
Did reading Wong's book help me at all during the year I taught there? Perhaps it helped a little -- but I might have focused more on what he wrote if I had, say, let his book be the side-along reading book that summer. Instead, I wrote more about spherical geometry (not from Van Brummelen, but from Legendre's old writings I found online). Of course, spherical geometry was completely irrelevant to this class I was about to teach.
Of course, even if I had read and blogged more about Wong's book, there's a difference between reading about Wong's principles and following them when I'm standing in front of students. For example, today I reread old posts and noticed what I wrote about my Warm-Ups. I wanted to make sure that I strictly enforced the rule that students must show their work on Warm-Ups, especially Pappas-style problems where the answer was the date (which should have been Exit Passes). But that failed miserably -- one day early in the year, most students wrote only the answer, and I didn't want to give lots of zeros to more than half of the class. And the rest is misery -- the "you must show work" rule was soon neutered.
Wong stresses the importance of enforcing the rules during "the first days of school." This time of year I even call the "Wong unit" (or "Willis unit" after Paul Willis, another author). But once again, it's easy to forget everything I read when I'm actually standing in front of students. Since I never made Wong's principles (enforcing rules and procedures early in the year) into habits, I promptly failed in the classroom. It's not enough to read or even blog about Wong -- I needed to figure out how to convert his principles into action.
Today we return to Blaugust participant Cheryl Leung -- the sixth grade teacher who wants her students to "be kind" and "be brave" in her class:
https://matheasyaspi.wordpress.com/
https://matheasyaspi.wordpress.com/2019/08/26/what-mathematicians-do-2/
In this post, Leung describes a quick Exit Pass that she assigns her students today:
During the last five minutes of class today, I asked my students to write. I asked them to write about what mathematicians do. Here are a few of the things that I saw when I was peeking over shoulders.
Leung lists nine student responses. I'll quote three of them here:
Explain their thinking.
Find creative ways to solve a problem.
Persevere.
And it's easy to figure out what this is leading to -- Leung wants her sixth graders to explain their thinking, find creative ways to solve problems, and persevere in her math class.
Since Leung's post is so short, let's return to Benjamin Leis who also posted today:
http://mymathclub.blogspot.com/
http://mymathclub.blogspot.com/2019/08/cardanos-method.html
Leis discusses a sixteenth century mathematician who knew how to explain his thinking, persevere, and find creative ways to solve cubic equations: Girolamo Cardano:
This all starts with a fantastic new video from Mathologer: 500 years of NOT teaching the cubic formula.
(I might as well include the video that Leis links to!) I've mentioned the Cubic Formula in previous posts -- Mario Livio's book The Equation That Couldn't Be Solved is all about the nonexistence of a quintic formula. (Oh, and we can add Livio's book to my original Blaugust list regarding math books that I've read.)
But the Cubic Formula definitely exists. And so this entire post asks an interesting question -- why do we teach the Quadratic Formula in high school, but not the Cubic Formula? Leis writes:
Several memories flashed in my brain after my first read: the kids breaking into the quadratic formula song during one Math Club Session where we needed to derive the golden ratio and another time when one of them put part of the cubic formula on a poster we were making to bring to MathCounts.
(By the way, the golden ratio Phi is the topic of another Livio book that I've read and enjoyed.)
But Leis doesn't necessarily like the Quadratic Formula song (by which I assume he means the one whose tune is "Pop Goes the Weasel") either:
The quadratic formula song troubles me too because I worry it hides a lack of conceptual understanding. I have other memories of dragging kids through a problem that required completing the square where it was clear their mastery was incomplete. And yet I don't look at its existence as either an admission of defeat by teachers or a lack of motivation behind it. Instead my intuition is that it and many of its often reviled mnemonic kin like keep/flip/change are the symptoms of a problem not the cause of it: Mathematics is hard. Its hard both to teach it well and its hard to learn. Often a jingle like the song remains after the lesson that was crafted to carefully explore and motivate the formula has faded away.
Notice that any argument in favor of teaching the Quadratic Formula can also be used to justify teaching the Cubic Formula. For example, if we argue that the Quadratic Formula can be used to solve interesting area problems, then similarly the Cubic Formula can be used to solve interesting volume problems.
And going the other way, any argument used against teaching the Cubic Formula can be used to justify not teaching the Quadratic Formula either! Leis writes:
This also brought to mind another online exchange to the effect that we spend too much attention on quadratic and polynomial functions in general and that they don't occur naturally much beyond the physics of trajectories. And that line of thinking is my worry about the natural result of tilting at the first windmill too long.
And he goes on to mention technology -- if we shouldn't teach the Cubic Formula because we can just use technology to solve cubic equations, then likewise we shouldn't teach the Quadratic Formula because we can just use technology to solve quadratic equations.
Therefore Leis doesn't reach a definitive answer as to why the line between "essential for high school" and "inappropriate for high school" has been drawn at the quadratic-cubic boundary.
I suspect that part of the answer involves the perception that it's reasonable for a student around the age of Algebra I/II to know and understand how to solve a general quadratic, but it's not reasonable for such a student to know and understand how to solve a general cubic. In other words, what we decide to teach at each level -- especially at the secondary level -- is what we expect students to know and understand, irrespective of how useful such knowledge or understanding is in the real world.
I agree with what Leis says about songs such as Quadratic Weasel -- it remains in the mind after the main lesson has faded away. If it's time to review for the final and the students claim that they've never been taught how to solve quadratics, the few seconds it takes to sing the song immediately dispels that notion. This is what I aimed to accomplish with the daily music break at the old charter school -- the ability to use songs to job students' memories.
But Leis implies that songs such as Quadratic Weasel decrease conceptual understanding in a way that completing the square (as well as factoring, I presume) doesn't. This is debatable -- and I'll save this for another post.
Leis writes:
Lets go back to the original question of "Why don't we teach the cubic as well?". Interestingly enough last spring I listened to a podcast with Sam Vandervelde: podcast link where he talked about doing exactly that with kids. It sounded fascinating at the time, and I remember playing around on a plane to decide how I might do it. The main problems for me were, I would need all the kids to be past Algebra 2 and it probably would work best taught over several adjacent days not once a week. I could definitely see the formula being used as a capstone project for a motivated student(s) and perhaps I should followup and see if I can get a copy of the curriculum Sam used to brainstorm some more about this. In theory, though I could actually show the video above to the kids and conceptually touch on how things work without really delving deeply. The animated algebra Mathologer uses, works really well that way.
In this way, teaching the cubic formula at the end of Algebra II is akin to teaching spherical geometry at the end of high school Geometry -- a project to which I've devoted several posts over the years (especially after reading Legendre and Van Brummelen).
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