Since this is a special ed class, I wrote last week what my goal for such a class is:
2. Keep a calm voice instead of yelling at students.
8:15 -- This is the school where homeroom is not the same as first period, and all classes, including first period, rotate.
8:20 -- But as it turns out, today's rotation just happens to begin with first period anyway.
First period is one of the math classes with a co-teacher. But as so often happens, the resident teacher is out with a sub as well. (Often this happens because both teachers are at the same meeting. This meeting is on campus, and so the regular teacher does appear a few times during the day.)
These students are preparing for a test that will occur later this week. Some of the topics on this exam include order of operations, integer arithmetic, and solving one-step equations (add/subtract only).
The other sub handles the attendance, while I of course help students with the math.
9:15 -- First period ends. Second period is the regular teacher's conference period, leading to snack.
10:20 -- Third period arrives. This is the one class that isn't seventh grade math. Instead, it's an eighth grade science class -- and you know what that means. It's time once again to compare this class to the science class that I failed to teach three years ago.
This is another science class that doesn't have a printed text -- it's all online. And so the students must access the online text on Chromebooks. There is a video on the Discovery Ed website that's all about electricity and magnetism. Students must then complete a worksheet with 14 questions on the video.
Ordinarily, I would set up my good and bad lists so that students must answer all the questions to get on the good list and half (or seven) of the questions to avoid the bad list. But I take into account that this are special ed students who might not be strong at writing. And so I reduce it to ten questions for the good list and five to avoid the bad list. At these levels, four students make the good list, while one girl answers only four questions to land on my bad list.
Now let's compare this class to the situation at the old charter school. First, I doubt that the Illinois State online science text has a video component (though I can never be completely sure). Then again, if I really wanted to show my students a science video, I could have searched YouTube for a suitable video and created a video worksheet with questions for them to answer.
Today's video topic, electricity and magnetism, is an interesting one. Back when I was a young student, while eighth grade was always physical science, much more emphasis was placed on mechanics and chemistry than electricity and magnetism. In fact, it's possible that eighth graders are now required to learn E&M topics that I myself didn't learn until I was a senior, preparing for the AP Physics C exam for E&M! (Today's worksheet seemed to be just on the basics, though.)
I do suspect that E&M was included in the Illinois State physical science (online) text. Thus it's likely that, had I followed that text properly, I would have taught some E&M that year.
11:15 -- Third period leaves and fourth period arrives. This is the first math class with an aide.
These students are also studying for a test later this week. But before the test review worksheet, there is a Warm-Up on adding and subtracting decimals. The Warm-Up takes about half an hour.
This is a tricky one. Of course, I remember another rule that I wish to follow:
4. Begin the lesson quickly instead of having lengthy warm-ups.
So what gives here -- what was I thinking? Well, I return to yet another rule:
5. Engage the students in the learning process instead of lecturing excessively.
In some ways, these two rules are at odds with each other. I wrote the fifth rule above because too often at the old charter school, I simply gave the students the answers instead of letting them think about the problem and struggle for a while.
There are seven problems on today's Warm-Up (two addition, five subtraction). I could have given the students seven minutes (one for each question), then displayed the answers on the document camera when time is up. Some students might compare their answers to the displayed answers -- but others might do no work and simply copy the answers from the board. Thus this approach satisfies the fourth, but not the fifth, rule.
On the other hand, I could give the students plenty of time to work on their own. As these are special ed students, they might need more time -- so I allow 14 minutes, or two minutes per problem. And when the 14 minutes are up, I don't simply tell them the answers, but randomly call on students to tell me each step in the addition or subtraction. This way, the students are actually doing things rather than waiting for me to provide the answer. But it takes time -- I can only go as fast as the students are willing to tell me the steps. Thus this approach satisfies the fifth, but not the fourth, rule.
Which approach is better depends on the position of the class within the current unit. If the students have new material to cover that day, then it's better to make the Warm-Up quick so that there's time to teach the lesson -- otherwise the students won't learn anything during the rushed lesson (and then struggle on the next day's Warm-Up). I wrote the fourth rule above because too often at the old charter school, my Warm-Ups took time away from the introduction of new material.
On the other hand, if the main lesson is actually just review leading up to a test, then it's better to slow down and check for understanding. And since there is a test this week, I lean towards the second approach. I don't want to speed up the Warm-Up by telling the answers, which the weaker students copy -- and then those students get the decimal problems wrong on this test. Today's lesson is all about preparing for the test -- and I claim that the thirty minutes spent on the decimal problems in the Warm-Up should count as preparation for the decimal problems on the upcoming test.
But many students simply talk throughout the Warm-Up -- even if they're working, they're still so loud that it's difficult to hear the student I called on. Indeed, this to me is what makes today's lesson ineffective -- as opposed to how much time I spend on the Warm-Up and review worksheets.
And because I must repeatedly tell the students to be quiet, they're tired of hearing my voice by the time I finally get to today's song, "Solving Equations." (Since one-step equations are on the review sheet, I sing the first two verses only.) Thus these students don't like my song as much as previous classes did. The aide ultimately writes five names on our bad list.
12:05 -- Fourth period leaves for lunch.
12:50 -- Fifth period arrives. This is the second math class with an aide.
At this school, the class after lunch is silent reading. Part of this time is reserved for the students to get organized and clean out their backpacks (which occurs about once a month).
The aide is called in and out of the room during this period. But during the getting organized time, the class is so noisy that the regular teacher -- during her meeting in an interior room -- can hear and even identify the loud students.
This time, I decide to sing the song before the Warm-Up (to whet their appetites as usual). But still, it was after having to tell the students to be quiet during silent reading and getting organized, so the song situation is only marginally better than fourth period. (That's why I try to open with a song -- ideally I sing before anything negative happens in the class.)
I also add another song -- "Same Sign, Add and Keep" ("Row, Row, Row Your Boat") -- since integer operations appear on the review worksheet. Since this class is slightly smoother than fourth, I am able to reach some of these integer operation questions.
2:00 -- Fifth period leaves. Sixth period is the second class with a co-teacher.
Once again, I help the students with their review worksheet. That's one difference between this and the aided classes -- there is no Warm-Up, so we go directly into the review worksheet.
This time, after doing so in fifth period, I also sing "Same Sign, Add and Keep" for the students I'm helping when they reach an integer addition problem.
2:55 -- Sixth period leaves, thus ending my day.
It's hard to say whether I truly fulfilled my goal of following the second rule. I had to raise my voice a few times during periods 4-5 to get those students quiet. But since I lack a true teacher tone, raising my voice is tantamount to yelling, which I must avoid.
I think back to last week's history class -- I didn't need to raise my voice at all. All I had to do was add an extra video note (six notes instead of five) and the guy's phone disappeared. But the way today was set up, it's difficult to come up with such an immediate consequence that would get the students to behave without needing to raise my voice.
I wonder whether I could have set up the Conjectures/"Who Am I?" game here. The last time I played this game was a failure -- but that was in an English class, on a writing assignment for which my game wasn't designed. In general, my game works well for test review (although I originally designed it for preview, not review).
The tricky part would have been incorporating the Warm-Up into the game. Suppose I start today by dividing the class into groups of four and asking my usual age and weight questions. Then I might move on to the seven Warm-Up questions. If I ask questions in a race format, then some students who lose the race might not even bother to fill in the answers. And for the questions that I expect everyone to answer, it make take a long time for students to answer, and so might give up -- and this is for a Warm-Up which the students must turn it. The students might get confused why I am not following the standard Warm-Up procedure by making it into a game.
I also want to compare the math I observed today to what I taught three years ago. Notice that today (approximately one trimester into the school year), the seventh graders appeared to have studied the NS and EE Common Core Standards (perhaps up to the middle of EE 4a, on solving equations, since two-step equations also appear in EE 4a).
The text this school uses is probably the Big Ideas text for seventh grade. Neither the Illinois State traditional text nor the STEM text start with NS and EE at the top. The traditional text places RP before NS/EE (since RP naively appears first in the Common Core Standards), while the STEM text follows the standards in some haphazard order.
Since the NS and EE standards are so critical to future study in math (Algebra I and beyond), I wish that the Illinois State "traditional text" actually resembled a traditional text -- with approximately 12 chapters, each covering a variety of standards. The STEM text should correspond directly to the chapters of the traditional text, with approximately two STEM projects per chapter.
I wonder whether there's anything that could have encouraged me, in 2016, to emphasize NS and EE during the first trimester of seventh grade. If I skipped RP for sixth and seventh grade, then I could have started with NS in all three grades (as eighth grade has no RP strand) -- a not insignificant consideration when my biggest fear was juggling six preps, math and science. But then again, I just as easily could have started with RP for Grades 6-7 and F (Functions) for eighth grade, then be ready to teach NS to all three grades after the RP/F strands. (In some ways, RP makes sense after EE anyway, since now the students might be ready to solve proportion equations instead of rely on line/tape diagrams to solve proportions.)
Today on her Mathematics Calendar 2019, Theoni Pappas writes:
If this is a convex regular hexagon and AB = 4 fourthroot(3), what is the hexagon's area?
(Here is the given info from the diagram: A and B are opposite vertices of the hexagon.)
This is where we must think of the hexagon as being inscribed in a circle. Then AB is actually the diameter of this circle. The radius is half of this, or 2sqrt(sqrt(3)). This is also the side length of each of the six equilateral triangles into which the hexagon can be divided, and so the perimeter of that hexagon is six times this, or 12sqrt(sqrt(3)).
Also notice that half of the side-length, sqrt(sqrt(3)), is also the base of a 30-60-90 triangle (half of the equilateral triangle) whose long leg, sqrt(3)sqrt(sqrt(3)) = 3^(3/4) is the apothem of the hexagon.
So we use:
A = (1/2)ap
A = (1/2)3^(3/4)(12)3^(1/4)
= (1/2)(12)3
= (6)3
= 18
Therefore the desired hexagon area is 18 square units -- and of course, today's date is the eighteenth.
Notice that the awkward fourth root of three is needed to make the area a whole number. If the diameter were a whole number, the area would contain a factor of sqrt(3). And if the diameter were a whole number times sqrt(3), the area would still contain a factor of sqrt(3). Only including a factor of the fourth root of 3 makes the area a whole number.
Lesson 6-6 of the U of Chicago text is called "Isometries." In the modern Third Edition, we must backtrack to Lesson 4-7 to learn about isometries.
This is what I wrote last year about today's lesson:
What, exactly, is a glide reflection? Well, here's how the U of Chicago defines it:
Let r be the reflection in line m and T be any translation with nonzero magnitude and direction parallel to m. Then G, the composite of T and r, is a glide reflection.
Just as reflections, rotations, and translations have nicknames -- "flips," "slides," and "turns," respectively -- glide reflections have the nickname "walks." The U of Chicago gives the example of the isometry mapping the right footprint to the left footprint while walking as a glide reflection. Another name for glide reflection is "transflection," since it is the composite of a reflection and a translation.
I once tutored a geometry student who had a worksheet on glide reflections. The student had to use a coordinate plane to perform the glide reflections, which were given as the composite of a reflection and a translation. But the problem was that on the worksheet, the direction of the translation wasn't always parallel to the reflecting line! In fact, in one of the problems the translation was perpendicular to the reflecting line. That would mean that the resulting composite wasn't truly a glide reflection at all, but just a mere reflection!
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