This is my first visit to this class in just over a year. Last year, this teacher had only one period each of Algebra I and Geometry, but this year she has three Algebra I classes (with the extra two replacing a study skills class and a history co-teaching class).
Oh, and since it's special ed, let me repeat my goal from earlier this week:
2. Keep a calm voice instead of yelling at students.
7:55 -- Second period (as "first period" is zero period at this school) is the lone Geometry class.
These students have a worksheet on Lesson 2-5. I notice that Lesson 2-5 of the Glencoe text is on postulates (similar to Lesson 1-7 in the U of Chicago text). Yet the worksheet was actually on linear pair and vertical angle problems, even though angles were taught in Lesson 1-5 of Glencoe (and Lesson 3-2 of U of Chicago). Quickly reading the worksheet, it appears that the justification for angle problems on the Lesson 2-5 worksheet is that the Protractor Postulate is covered in this lesson.
(Thus Glencoe teaches, angle properties in Lesson 1-5 before they are proved in Lesson 2-5. As I wrote last week, mathematicians like David Joyce hate it when properties are taught before they can be proved. While the U of Chicago is late in that it doesn't teach angles until Chapter 3, at least our text doesn't state properties before proving them -- the Angle Measure Postulate is in Lesson 3-1, before the Linear Pair and Vertical Angle Theorems appear in the next lesson.)
At first I'm not sure whether to sing a song in this class today. Notice that this class consists of mostly juniors (since for special ed, Algebra I is a two-year course for Grades 9-10), and so the class might be mature enough to work hard on the assignment without a song incentive.
I decide that I might as well sing anyway, since I already have a song set up for angles -- Square One TV's "Angle Dance." I first posted the lyrics on the blog while at the old charter school -- on the so-called "California Snow Day" (in other words, a rainy day). As it just so happens, it's raining here today -- the first rain of the fall season.
Some of the students state that they've already completed the worksheet. I glance at the worksheet and notice that it only consists of a few notes plus three actual problems. Thus it's plausible that they have already completed everything. And so I immediately begin the "Conjectures"/"Who Am I?" game, with the questions coming from Lesson 1-5 of the Glencoe text. It works in motivating one group of guys who appeared ready to take a free period -- but it fails to motivate a group of girls.
8:45 -- Second period leaves and third period arrives. This is the first of three Algebra I classes. (All three are sophomore classes, the second year of the two-year Algebra I course.)
These students are in Lesson 7-2 of the Glencoe Algebra I text, on the laws of exponents. Their worksheet today focuses on the division law and contains 24 problems (actually a two-sided worksheet with a dozen on each side).
The song for this period is "U-N-I-T Rate," since the second verse is a special exponent verse (for my eighth graders at the old charter school). The tune is based on the UCLA fight song -- and as a loyal Bruin, I definitely wish to sing it this week, with the big rivalry game against USC this weekend.
When I subbed in this class last year, there was an exponent problem that many of the students seemed to struggle on: -2y^7 / (14y^5) -- and I mentioned it on the blog. Well, guess what -- the same exact problem appears on today's worksheet! And that's despite my teaching that lesson about a month earlier last year. (For some reason, this year's class is moving slow compared to last year's.) I use the same method that I taught last year -- keeping the "little numbers" (exponents) and "big numbers" (coefficients) straight.
This is the only problem that I do on the board. I decide that if I try to get all of these students to sit quietly while I just do problems on the board -- or try to call some students up to do them -- I'll just end up arguing or yelling at them, which is what I want to avoid.
Fortunately, there's no sign of a test coming up in the next few days. And so instead of doing lots of problems on the board, I stick to just that one. Then I go up to some students one-by-one. I see which questions they need help on then and motivate them to be successful. After each student and I do a handful of problems together, I place that student's name on the good list. (And I don't choose the students at random -- I look to see which students appear to need extra help or motivation.)
My plan works, as I'm able to avoid arguments during this best Algebra I class of the day.
9:40 -- Third period leaves and tutorial arrives.
But today isn't the typical tutorial. There is a special program for freshmen to help them fit in better at their new campus. Two girls, both juniors, lead the class. They ask the youngsters to answer a few personal questions. Then the freshmen line up alphabetically by first name. The junior leaders divide the line in half to make two lines, and then each student reveals the personal answers to the student in the line directly across from them.
Afterward, the freshmen all go to the gym for a special snack, called "Friendsgiving." This is a phenomenon that I've started to hear about in the past year or two. It's one of two new "holidays" (the other being "Galentine's Day") that's been redone so that people can have a celebration with friends rather than the traditional family Thanksgiving or romantic Valentine's Day. (For obvious reasons, I assume a school would never have a "Galentine's Day" celebration at tutorial.)
10:25 -- "Friendsgiving" ends, and it is now break time.
10:45 -- Fourth period arrives. This is the second of three Algebra I classes, and first with an aide.
There might have been one argument during this class, if it can be called an argument. (It certainly wasn't yelling.) It was about the usual -- many students asking for restroom passes after break. (There is no "emergency only" line in the teacher's lesson plan, so there's no need for extended argument.)
11:35 -- Fifth period arrives. This is the last of three Algebra I classes, also with an aide.
This is the only class where I must place a name on the bad list. It begins when a student claims that he's already finished the assignment. For some reason, even though the assignment is on a worksheet, the regular teacher still wants them to submit the assignment on Chromebooks, by taking a photo of the worksheet on a phone and then uploading the photo to Google Classroom. The student doesn't have the worksheet, so I must check the photo online. I do see that the infamous -2y^7 / (14y^5) problem is correct on his paper.
But then, the student, now having completed all his work, plays a video on the Chromebook. This wouldn't be a problem except that he plays it loud, even after the aide tells him not to. Another guy has also completed the assignment (and has a written copy of the problems for me to check) and also plays a video, but he does eventually quiet it down. But the aide must take away the first guy's Chromebook, and so I write his name on the bad list.
I wonder how I could have avoided this situation. Perhaps I should have played the "Conjectures" or "Who Am I?" game, just as I did in second period, as soon as I found out that a few guys had already completed the 24 problems. This is tricky, though, since many other students still had work to do on their worksheets, and I wouldn't want to take time away from them. I could also have given questions from the worksheet, but then the group with the two boys who finished would dominate the game.
12:30 -- Fifth period leaves for lunch, which leads into sixth period conference.
2:10 -- Seventh period arrives. This is a sport -- girls tennis. Last week, on a day I subbed for a coach, the assistant coaches watched the athletes outside -- so I was basically done at lunch.
But today, as I mentioned above, is a "California Snow Day." Due to the weather, outdoor tennis practice was cancelled, and so the assistant coach brought the students into the classroom.
With nothing much to do, I might as well sing the songs for the girls. I repeat my songs from the math classes, "Angle Dance" and "U-N-I-T Rate."
3:00 -- Seventh period ends, thus concluding my day.
There are a few more things I wish to take away from today. As I mentioned above, the reason that I already had the UCLA exponent song prepared is that exponents are part of the eighth grade course as well as high school Algebra I. Much of first semester Algebra I -- and as we see today, even a little of second semester Algebra I (exponents only) -- overlaps with Common Core Math 8. But I'm not quite sure whether my eighth graders learned much from that unit (except for square roots).
Now I'm still keeping track of whether the math classes I'm visiting use anything that resembles interactive notebooks. Not only do I see an interactive notebooks, but I also see an exponents Foldable for the notebooks that resembles something I've seen before on Pinterest.
In fact, in fifth period there is a newly enrolled student. I help him out, not by showing him today's worksheet, but by finding an extra copy of the Foldable notes -- in one day, I taught him the rules for zero exponents, negative exponents, and multiplying powers. Meanwhile, I failed to teach the eighth graders these exponent rules in a week.
Actually, I admit that the exponent unit is tailor-made for interactive notebooks. It thus makes me upset once again that I didn't use such notebooks at the old charter school.
I mentioned earlier that I like singing the UCLA fight song during Rivalry Week. I once considered singing a rivalry parody of Sam Cooke's "Wonderful World" -- I'd replace his line "But I do know that I love you" with "Yet I got into USC" (with this line following all of the subjects that Cooke doesn't know -- the intent is that USC is such a bad school that's so easy to get in even for those who don't know anything about those subjects).
But that song might be dangerous, especially in a special ed classroom. It's one thing to sing about "drens" who don't know math, but it's another to tell students misleading information about colleges just to make a joke about my rival school. If I ever write that parody, I'll never sing it in class (unless it's in a class of advanced students, with the understanding that it's all a joke about my rival).
The freshman tutorial class I see today is also interesting. Sometimes, I wonder whether this is what my so-called "advisory" class at the old charter school should have looked like. That year, "advisory" was on Wednesdays, between seventh grade music and lunch. Thus the seventh graders were the ones in my classroom during advisory. But the administrators never told us what exactly the teachers were supposed to do during advisory (except "give them extra math practice" or something like that).
During those last thirty minutes before lunch, I could have had activities similar to the ones that I'm seeing today. Perhaps I could have even had a "Friendsgiving" snack just before Thanksgiving. (I did check to see that Friendsgiving has been celebrated for the entire decade, so it's conceivable that I could have done it in my 2016 classroom.) The one problem is that we had Parent Conferences Week just before Thanksgiving, and so there was no "advisory" on the schedule.
Other schools, of course, have legitimate advisory classes. Today's high school isn't one of them, but this special freshman tutorial serves the same role as an advisory. The goal is to teach those soft skills, and convince the students to get along with each other and make friends.
Even though the seventh graders weren't new to our campus (and neither were the sixth graders, since it was a K-8 charter), I've mentioned before why the seventh grade was personally important. A genuine advisory might have generated better behavior in that class -- and maybe I might not have needed to leave that school.
This is what I wrote last year about today's lesson:
Normally, I'd be posting today's worksheet, but today is just Day 2 of yesterday's activity. I feel guilty for making a school-year post without a worksheet. But then again, I felt even guiltier for never posting multi-day activities and thereby never giving students an opportunity to continue a worksheet before posting the next one.
So there's no worksheet for me to post today. The students should continue working on yesterday's "Corresponding Parts in Congruent Figures" assignment.
And now I feel even guiltier because I mentioned very little Geometry at all in today's post -- most of the math I wrote about today is Algebra I. Let me make up for it by at least linking to some Geometry. Here is a blog post I found by retired teacher Henri Picciotto on this week's Geometry topics -- glide reflections and congruence:
2019 update: Even the Google cache for this website is no longer available. But the blog I link to here is still active, and there are some interesting posts there, including this one that's all about Geometry class in general:
https://blog.mathed.page/2019/02/20/in-defense-of-geometry-part-i/
But returning to the post I originally wanted to refer to:
In my previous post, I introduced glide reflections, and explained their importance from the point of view of congruent figures: in the most general cases, given two congruent figures in the plane, one is the image of the other in a rotation or a glide reflection. (In some special cases, one is the image of the other in a translation or a reflection.) Another way to state the same thing, as commenter Paul Hartzer pointed out, is that if the composition of two of the well-known rigid transformations (rotation, translation, reflection) is not one of those three, then it is a glide reflection.
In this post, I will give an additional argument in defense of the glide reflection: its importance in analyzing symmetric figures. The Common Core does not have much to say about symmetry (see my analysis.) This is unfortunate, because symmetry provides us with connections to art and design, as well as to abstract algebra, and is very interesting to students.
Symmetry is deeply connected to rigid transformations, and can be defined in terms of those: a figure is symmetric if it is invariant under an isometry. (In other words, if it is its own image in an isometry.) In the most familiar example, bilateral symmetry, the isometry in question is a line reflection. Another well-known symmetry is rotational symmetry. In these examples (from my Geometry Labs), the stick figure is its own reflection in the red line, and the recycling symbol is its own image in a 120° rotation around its center:
But what if we have a figure that is its own image under a translation? That is the case for this infinite row (or frieze) of evenly spaced L's. It is its own image under a translation to the left or to the right by a whole number of spaces:
...L L L L L L L L L L L L...
A frieze can be thought of as an infinitely wide rectangle, with a repeating pattern. A symmetry group is the set of isometries that keep a figure invariant. As it turns out, there are only seven possible frieze symmetry groups. In the example above, translation is the only isometry that keeps the group unchanged. But look at this one:
It is invariant under the composition of a horizontal translation and a reflection in a horizontal mirror. In other words, a glide reflection. If you want to analyze frieze symmetry, the glide reflection is absolutely necessary.
Likewise, if you want to analyze wallpaper symmetry. In this Escher design, for example, the light-colored birds are images of the dark ones in a glide reflection (the reflection lines and translation vectors are vertical.)
Do all students need to know this? Probably not. But to some of us, this is a lot more interesting than many of the "real world" applications of math I have the opportunity to present.
--Henri
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