How many complex roots does x^22 - ix = 0 have?
This isn't a Geometry problem, and so I'm only posting it in this school-year post because it contains an error. Let's try working it out:
x^22 - ix = 0
x(x^21 - i) = 0
x = 0 or x^21 = i
So we have at least one root, 0. The other roots must be the 21st roots of i. We know that one of these 21st roots must be i itself since i^21 = i. The others all lie on the unit circle -- indeed, they are the vertices of a regular 21-gon inscribed in the unit circle with one point at i. Using Euler's notation from last week, we can write these as:
e^(i(pi/2 + 2pi n/21)) for n = 1, 2, ..., 20, 21
Thus there are these 21 roots on the unit circle, along with the root 0 at the center. Therefore the equation has a grand total of 22 complex roots -- but today's date is the 21st, not the 22nd. And so this problem contains an error somewhere.
Most likely, the error is that Rapoport is using the word "complex" to mean "nonreal." The equation indeed has one real root, 0, and 21 nonreal roots on the unit circle.
Some authors use the word "imaginary" to mean "nonreal," but others use the word "imaginary" to mean of the form ia for some nonzero real number a. Those who use "imaginary" to mean "nonreal" introduce the term "pure imaginary" to mean ia for some nonzero real number a. (By this definition, the equation above has only one pure imaginary root, i itself.)
But no author uses the word "complex" to mean "nonreal." The complex numbers C are a field, and like all fields, C contains an additive identity, 0. Here's a Metamath proof that 0 is a complex number:
(Here's how it's proved there -- i * i + 1 = 0, and the field C of complex numbers is closed under addition and multiplication. Notice that both i and 1 are complex numbers according to "axioms" -- that is, postulates. Therefore 0 is a complex number. QED)
And indeed, the equation x^22 - ix = 0 is a polynomial equation of degree 22. So by the Fundamental Theorem of Algebra, it must have 22 complex roots, unless at least one root is of multiplicity greater than one. But every root of the equation has multiplicity 1, and so there are exactly 22 complex roots.
Rapoport's error here was caused by terminology. It's correct for her to say "nonreal root" and acceptable for her to say "imaginary root," but wrong for her to say "complex root" unless she means all 22 roots in C.
This might be another situation where Anglish or "Plain English" might help us out here. (The Anglish website uses a term "dreamt" number/scoring but I don't know whether it means "imaginary," "pure imaginary," or "complex" yet. So this is something to work on and expand.)
There's some news on the coronavirus front. First of all, not only have elementary waivers been granted to Orange County schools, but there's a possibility that the county may be removed from the state watch list as soon as this weekend.
Once again, the first step is to remain off of the watch list for two full weeks. This will take us up to Labor Day weekend, allowing for the possibility of schools reopening after the holiday. But then again, this doesn't mean that they will open immediately.
Most of the county schools getting a waiver are private or charter schools. My new district is not the lone district in Orange County getting the waiver. OK -- I avoid blogging the name of any district or charter where I work, but there's no use in hiding the name of the district getting the elementary waiver (seeing as I don't work there). That district is Los Alamitos Unified.
First of all, Day 1 in Los Al Unified isn't until a week from Monday. Then there will still be a full week of distance learning, so that elementary kids won't be back on campuses until the day right after Labor Day. The elementary schools will have a hybrid plan for at least four weeks, then they might consider a full five-days-per-week reopen.
I expect there to be a similar lag for the rest of the county. Once the fourteenth day of being off the watch list is reached (over Labor Day weekend), it's likely that districts will take at least two more weeks or so to reopen. This time is to prepare cleaning supplies and discuss reopening procedures with the respective board members and unions. So we can't be quite sure when exactly my particular district in Orange County will reopen.
As slow as this might seem, San Diego County will be even slower. There are more stringent local requirements for the schools to reopen, even though the county was taken off the list earlier. San Diego Unified officials used the word "months" (not "weeks") in telling when the schools might actually reopen.
Then again, I'm not really in a hurry for the schools to reopen -- assuming, of course, that I actually pick up some online subbing jobs very soon. (Once again, we can't be sure exactly how many subbing jobs to expect during online learning.)
Oh, and there's one more thing I want to say about my new district. Recently, an elementary teacher in my district posted on Twitter that she gave her kids an assignment involving Breakout Rooms. The assignment has something to do with Pascal's Triangle -- a bit advanced for elementary school, I'd think, but it could be just about adding the numbers in the rows rather than the Binomial Theorem that is taught during Algebra II. (She doesn't mention what grade she teaches in the tweet.)
But it does show that Breakout Rooms are possible using my new district's technology, even though they weren't mentioned in the training we had last week. We'll see soon whether I get to use them on a day that I sub.
There's some news on the coronavirus front. First of all, not only have elementary waivers been granted to Orange County schools, but there's a possibility that the county may be removed from the state watch list as soon as this weekend.
Once again, the first step is to remain off of the watch list for two full weeks. This will take us up to Labor Day weekend, allowing for the possibility of schools reopening after the holiday. But then again, this doesn't mean that they will open immediately.
Most of the county schools getting a waiver are private or charter schools. My new district is not the lone district in Orange County getting the waiver. OK -- I avoid blogging the name of any district or charter where I work, but there's no use in hiding the name of the district getting the elementary waiver (seeing as I don't work there). That district is Los Alamitos Unified.
First of all, Day 1 in Los Al Unified isn't until a week from Monday. Then there will still be a full week of distance learning, so that elementary kids won't be back on campuses until the day right after Labor Day. The elementary schools will have a hybrid plan for at least four weeks, then they might consider a full five-days-per-week reopen.
I expect there to be a similar lag for the rest of the county. Once the fourteenth day of being off the watch list is reached (over Labor Day weekend), it's likely that districts will take at least two more weeks or so to reopen. This time is to prepare cleaning supplies and discuss reopening procedures with the respective board members and unions. So we can't be quite sure when exactly my particular district in Orange County will reopen.
As slow as this might seem, San Diego County will be even slower. There are more stringent local requirements for the schools to reopen, even though the county was taken off the list earlier. San Diego Unified officials used the word "months" (not "weeks") in telling when the schools might actually reopen.
Then again, I'm not really in a hurry for the schools to reopen -- assuming, of course, that I actually pick up some online subbing jobs very soon. (Once again, we can't be sure exactly how many subbing jobs to expect during online learning.)
Oh, and there's one more thing I want to say about my new district. Recently, an elementary teacher in my district posted on Twitter that she gave her kids an assignment involving Breakout Rooms. The assignment has something to do with Pascal's Triangle -- a bit advanced for elementary school, I'd think, but it could be just about adding the numbers in the rows rather than the Binomial Theorem that is taught during Algebra II. (She doesn't mention what grade she teaches in the tweet.)
But it does show that Breakout Rooms are possible using my new district's technology, even though they weren't mentioned in the training we had last week. We'll see soon whether I get to use them on a day that I sub.
Lesson 0.5 of Michael Serra's Discovering Geometry is called "Mandalas." This is the first of two sections included in the old Second Edition yet omitted from the modern editions.
But what, exactly, is a mandala? Serra explains:
"A mandala is a circular design arranged in layers radiating from the center. The word mandala comes from Hindu Sanskrit, the classical language of India, and means 'circle' or 'center.'"
As Serra points out, other cultures had mandalas, not just the Hindus. The Aztec calendar, for example, was constructed as a mandala.
Many mandalas exhibit threefold or sixfold symmetry. They are related to the regular hexagon, and so the compass and straightedge can be used to construct them. At any rate, the compass should at least be used to draw the circle that is the base of any mandala.
All the mandalas on these pages come from a Google image search. There is no project in this section, but of course "draw your own mandala" is a natural question for this section.
Here is the Blaugust prompt for today:
Tell us about a favorite activity/lesson that makes you jump for joy when you get to use it.
Well, I guess that today's mandala activity might make me jump for joy. It's an opening activity that the students can enjoy as it allows them to exhibit their creativity.
Of course, that's probably not what Shelli meant when she posed this question. She obviously wants us to mention an activity or lesson that we actually taught in the classroom. This requires me to think back three years to my charter middle school classroom.
One day in November that year, I wrote about Benchmark Testing Week -- a very stress-filled week, since there were both print and online Benchmarks to take in both math and English, and they took hours to complete. But I was able to complete some activities during that week. I suppose that these activities made the students and me jump for joy -- if only because anything that wasn't a Benchmark Test made us jump for joy that week. (This week at my old school is probably the early-year Benchmarks, but here I'm referring to the end-of-first-trimester Benchmarks in November.)
Of the sixth grade class, here's what I wrote about a lesson I gave that week:
Yesterday I had a Bruin Corps member present during the sixth grade block. So here's what I did -- I began the class with a Warm-Up division problem from Illinois State -- 2400 / 51. Then I divided the class into two groups -- those who got the right answer and those who didn't. I gave the higher group to my Bruin Corps member, so he could help them move on, just as he and his fellow Bruin have helped with the higher grades this week.
But this led to problems. First of all, some students decided not to answer the Warm-Up. I remember one boy who chose not to answer, so I seated him with the students who didn't know division. Then when I assigned a division question, he answered it quickly -- meaning that he already knew all the steps and was just too lazy to do the Warm-Up. He complained that he had to sit with the students who didn't know division when he already knew it.
There were probably also some students who cheated and started copying the answers when they saw who was getting them right. I reckon that some teachers get around this by simply handing out colored cards rather than telling them where to sit. Then the students can't tell as easily who's getting the right answer, making it harder to cheat. Many of these problems persisted into the homework, where some students either skipped completely or wrote in nonsensical answers, such as "55 divided 8 is 55 remainder 3." And remember -- this is all despite the history teacher giving them time to do the math assignment in his class!
I tried to give the inspiration example I've mentioned earlier -- the Cubs. The North Siders have failed to win the World Series for 108 years before they won it all this week. They didn't let their past failure hinder their present success. Yet the students who can't divide aren't thinking like Cubs -- that is, like champions. They fail to divide, and so they keep coming up ways to avoid division rather than think, I don't know to divide now, but if I work hard (like the Cubs), I will.
Today during the IXL time, the struggling sixth graders continued to struggle. Indeed, I can't say that I've taught a single student to divide -- the stronger students already knowing division and the weaker students coming up with excuses. It doesn't help that I'm trying to crack down on students who don't remember their IXL passwords by telling them they can't use the computers. Weaker students who wish to avoid division just claim that they don't know their passwords!
As I reflect on this day four years later, this was probably one of the few times that I was able to implement Learning Centers that year. For about half the class, this was a good day, but for others, it didn't work. There were too many opportunities for students to avoid the work -- from the one boy who knew how to divide but didn't attempt the Warm-Up to the kids who tried to cheat during the Warm-Up to the ones who claim their computers didn't work. The whole idea of Learning Centers would have worked much better if I'd had tighter classroom management to make sure that students didn't attempt those tricks.
The seventh graders probably has the most enjoyable lesson of the week:
In fact, I realize that the seventh grade lesson is so easy that today, I actually go back and have the students work on the Orienteering STEM project from the Illinois State text. This decision is easy to make, since my support staff member and Bruin Corps member are both present during the seventh grade math block. So I take one group outside, give them compasses, and have them create a map to hide the "treasure" (the textbook), and then another group takes the compasses and map and uses them to find the treasure.
While all of this is going on, my Bruin Corps member watches the rest of the class. She sees that they've already mastered the concept of opposites, so she has them do some general addition of integers that are not opposites. She has them play a game for points similar to the "Who Am I?" games that I played as a sub last year, and buys pizza for the winning group. The winners turn out to be the same group that hides the first treasure -- that is, they are outside for part of the game, yet they still come back to win. And this is more amazing because the smartest student in that group is in fact absent today, so it's not as if he's doing all the work.
Yes, this project from two years ago came from the Illinois State text. But since only some of the students are working on the project, it ended up turning into Learning Centers again.
As for the eighth graders, I finally attempted to teach them some science that week:
Well, I tried to start online Benchmarks yesterday during the math intervention time usually devoted to the other online program IXL. But the problem was that there was a power outage! I'd already charged the laptops before the blackout, but it was impossible to access WiFi during the outage, so the students couldn't access the online Benchmarks. The blackout began at the start of lunch and ended right at -- you guessed it -- 2:25 (that is, P.E. time).
The English teacher suggested that I have the eighth graders finish the written Benchmarks for her own class, since the students couldn't finish them during English class. Well, the students refused to work on them, and when I told them that they were supposed to be working on the essay, one girl called out, "Well, you're supposed to teach us science and you're not doing that!"
"Okay, then," I replied, "let's start the science assignment now."
It's a good thing that I purchased that Common Core Science book last month, since yesterday was my first opportunity to use it -- after all, science was the last thing on my mind with all of this Benchmark stuff. (Notice that there is no science Benchmark, even though eighth graders are supposed to take the NGSS science test.) So I just jumped into Chapter 7 of that book, which as I wrote in my October 10th post, is on Matter and Its Interactions. I just had the students start writing the definitions of some vocabulary from that chapter, but they only got through the first four terms after all the arguing about the essay and writing. Still, at least I got some science in at a time when I thought I'd have very little time for science.
Yes, I admit three years later that this science lesson didn't work as well as I wanted. But then again, this was a spur-of-the-moment science lesson caused when a power outage prevented me from giving the online Benchmarks. The class didn't have a science text because copies of the Illinois State science texts were still a few weeks away from arriving. In the meantime, we had to use the online science text -- which we couldn't because of the blackout.
I could have made that lesson more enjoyable by changing it into a game, similar to the impromptu game played by the seventh graders the following day. But as I wrote earlier, even writing words and definitions was enjoyable compared to taking the Benchmarks.
If you prefer that I answer Shelli's prompt with a Geometry lesson based on the U of Chicago text (outside of my one year of teaching), then one of my favorites is on the area of a circle, which I first posted in 2018:
If G can be divided into A unit squares, then G' can be divided into A squares each of length k. And the area of a square of length k is clearly k^2, so the area of G' must be Ak^2. For the circle problem k is the radius r, and A is the area of the unit circle or pi, so the area of a circle is pi * r^2. We can do this right on the same worksheet -- there's already a circle drawn of radius 10 times the length of a square, so instead of the length of each square being 1/10, let it be 1 instead. Then the area of the circle of radius 10 is equal to the number of shaded boxes, or 314, since the old unit square has been divided into 100 unit squares.
I wrote about several activities related to the circle area in that post -- which includes activities created by other teachers. Unfortunately, I've never actually taught this lesson in the classroom -- I was planning on giving my seventh graders a pi lesson for Pi Day, but I didn't quite make it to March 14th that year.
Only one Blaugust participant posted today -- Denise Gaskins:
https://denisegaskins.com/2020/08/21/how-mathematics-works/
In this post, Gaskins explains her ideas on how math works:
Make a conjecture. A conjecture is a statement that you think might be true.
For example, you might make a conjecture that “All odd numbers are…” How would you finish that sentence?
Make another conjecture. And another. Does thinking about your conjectures make you wonder about math?
Can you think of any way to test your conjectures, to discover if they will always be true?
This is how mathematics works. Mathematicians notice something interesting about certain numbers, shapes, or ideas. They play around and explore how those relate to other ideas. After collecting a set of interesting things, they think about ways to organize them. They wonder about patterns and connections. They make conjectures and try to imagine ways to test them.
And this idea fits with the Serra Geometry text. His text is all about making conjectures and testing them to see whether they are true.And mathematicians talk with one another and compare their ideas. In real life, math is a very social game.—Denise Gaskins
"All odd numbers are...," well, the only word that someone might try to place there is "prime" -- but unfortunately, "all odd numbers are prime" is a false conjecture.
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