7:55 -- Second period (recall that in this district, "first period" = zero period) is the first of three Algebra II classes. These students are learning about quadratic inequalities.
Originally, the regular teacher is considering giving a Chromebook lesson today. But in the entire district, the internet isn't working. So she comes in before her all-day meeting to set up something else instead. After reassuring her that I am a math teacher, she tells me that I must teach the lesson -- but at least I'll have the senior TA to assist me.
As it turns out, the TA already took AP Calculus BC last year (strong Bruce William Smith level). He tells me that since there were so few Calc BC students, the class was given online. Since he's in the IB program, he didn't take the AP exam last year, and feels that taking the IB exam this year will be more compatible with AP Stats. He does tell me that he received a grade of 100% in the class, and that some of the other BC Calc juniors last year are taking Multivariable Calculus this year at the community college.
The TA and I remind the students how to graph quadratic inequalities -- if they know how to graph quadratic equations (parabolas) and linear inequalities (dotted vs. solid line, which way to shade), then they can graph quadratic inequalities.
Then the new lesson is on solving inequalities in one variable. To do so, we set one side of the inequality to zero and factor, then create a table showing the intervals where the related function is positive or negative. We don't bother showing the students how to factor -- so many Algebra I teachers factor differently, and we don't want to confuse the students with a method that their old teachers never showed them.
Some students notice that for an inequality like 2x^2 - 6 > 11x -- the roots of the related equation are x = 1/2, x = -6 -- the function always seems to go from positive on (-inf, -6) to negative on (-6, 1/2) to positive on (1/2, inf), so they can save some steps and not test every interval. The TA informs them that while this is indeed usually the case for quadratic inequalities, he's had to solve cubic and higher degree inequalities where this trick doesn't work.
8:45 -- Second period leaves and third period arrives. This is the second of three Algebra II classes.
There's a TA in this class as well, but she's nowhere as strong at math as the second period TA. And so I'm really on the spot when it comes to teaching the lesson. But I believe that I'm successful -- in fact, this is my best class of the day.
The students are really motivated to work hard in this class. And indeed, three guys even notice errors in one of the examples of graphing a system of quadratic inequalities. For one of the equations, the teacher confused y > x^2 - 4 with y > (x - 4)^2, and on the other, she calculated the vertex of the parabola as (1.5, 7) instead of (1.5, 6.25). In case you're wondering how they find the errors, they use Desmos on their phones!
Here is the grade-level distribution of this best class of the day:
Freshmen: 1 (She's at Bruce William Smith level.)
Sophomores: 21 (They're at SteveH level.)
Juniors: 7
Officially, all three Algebra II classes are honors, which explains why there are no seniors here (as seniors in Algebra II generally aren't honors students).
Some students say that I teach today's lesson much better than the regular teacher. I don't think the regular teacher is that bad, but the kids are influenced by the errors they find in the graphing problem.
9:40 -- Third period leaves and tutorial begins.
One girl, who is in the next Algebra II class, asks for help on her previous assignment. It's on word problems, and she's confused by the following problem:
A wall of uniform length surrounds a 5 ft. by 4 ft. yard. The total area of the yard plus the wall is 30 square feet. What is the width of the wall?
She doesn't understand why (2x + 5)(2x + 4) = 30 models this problem. I draw a picture and show her that if a wall of width x is added on both sides to a yard of width 4, the new width is x + 4 + x.
After I help her with the problem, I notice that she's carrying a Chemistry text. I ask her whether her Chemistry teacher told her in her earlier class that today is Mole Day -- 10/23. Her reply is no -- and after I tell her that Avogadro's number is 6.022 * 10^23, I find out that the science teacher has actually skipped the chapter on moles! (To me, that's a wasted opportunity.)
10:25 -- Tutorial ends. It is now snack break.
10:45 -- Fourth period arrives. This is the last of three Algebra II classes -- and the only one that is designated for IB students.
At this point, the internet is working again, and so students learn the material and complete the assignment on Chromebooks instead. For some reason, this class doesn't go as smoothly as third -- I suspect that some students might have sneaked on to other websites besides Google Classroom. It also seems as if there's more improper cell phone use in this class, even though phones could have been used during the internet blackout (as the Desmos trio does earlier).
11:35 -- Fourth period leaves. Fifth period is the teacher's conference period -- and this leads directly into lunch.
1:15 -- Sixth period arrives. This is the first of two Algebra I classes -- with freshmen, of course.
There is also a lesson associated with Chromebooks for this class as well. This time, the lesson is on solving absolute value equations.
Of course, this means that my song "Solving Equations" fits here. This is the second time that I needed to include verses for absolute value. After the original three verses, I add the following:
4. To drop absolute value,
Plus and minus on the right side...
5. Isolate the bars on the left side,
Everything else on the right side...
Even though the students can see the problems on the Chromebooks, I do the same problems on the board anyway. This is not just because I want to sing the song -- it's also because with Algebra II, the students learn better in third period (when I do the examples on the board) than fourth (when I just leave it to the Chromebooks).
Here are the examples to be solved:
1a) |x| = 4
1b) |x| = -6
2a) |x| - 2 = 5
2b) |x - 2| = 5
2c) |2x - 7| - 5 = 4
2d) 23 - |3 - 4x| = 18
2e) |3x + 1| = 0
2f) -2|x + 10| = 6
As usual, different problems require different verses to be sung. Example 2b) requires the fourth verse, second verse ("Whatever you do to the left side..."), and first verse ("Just a letter on the left side..."), while 2d) requires all five verses. (Arguably the fifth verse must be sung twice, since it takes two steps to isolate the bars.) But 2f) starts with the fifth verse to isolate the bars by dividing by -2 -- at which point there's no solution -- similar to 1b). So for 2f), we can't really sing any other verses, not even the first verse, since the problem has no solution.
As we might expect of a freshman class, there are behavior problems. It takes a long time to get started -- first because fewer than half the students are sitting in their assigned seats, and then they are slow at turning in the homework packet. (Algebra II doesn't have to turn in HW until tomorrow.)
The talking continues throughout the lesson. I fear that some students will get 2a) and 2b) mixed up on the next test, and they won't understand why their answer is wrong. But of course, the talking goes on anyway.
2:05 -- Just before sixth period leaves, the regular teacher returns. I give her the name of one student who deserves to be on the bad list. It's a guy who plays with rubber bands and takes no notes because he supposedly can't log into his Chromebook.
Seventh period is the other Algebra I class. As it turns out, whatever meeting the regular teacher has today, it has an observation component. So she takes over the lesson during seventh period, while two observers sit in the back of the room.
Actually, make that three observers. I also observe the regular teacher today, and compare her seventh period to my sixth period.
For one thing, she definitely knows how to collect HW the packets quickly. Using an app called Classroom Screen, she displays, "2 minutes to staple and place packet on tower or it's late!" Of course, I don't have access to this app during sixth period, so it takes me longer to collect the HW.
Her lesson isn't on student Chromebooks. Instead, she loads up Google Classroom on her own computer and then lectures. I suspect that her observers don't want to see a lesson where all the students do is use the Chromebooks.
She also has the students show thumbs-up or thumbs-down to indicate whether they understand a lesson, or answer a yes-or-no question (such as "Are the absolute value bars isolated?").
2:45 -- The regular teacher only reaches example 2d) before the observation is complete. She allows the students to use the last few minutes to start their homework and leaves with the observers, probably to receive feedback. This leaves me to cover these last few minutes.
I don't sing during the observation, of course. But as soon as the teacher and observers leave, I seize the opportunity to sing. I still believe that this song can help students remember all the steps to solving absolute value equations.
3:00 -- Seventh period leaves, thus ending my day.
There's one thing I wish to say about this regular teacher's classroom management. She tells me that any names I leave for her will lose participation points. This reminds me of my failed participation points system at the start of my year at the old charter school. When this didn't work, I concluded that I should only have positive participation points, rather than take points for bad behavior. Yet this teacher seems to be successful with a negative participation points system.
Of course, this teacher's negative participation points lead directly to a detention. I don't know how many lost points it takes to get a detention, but it's probably not that many.
After seeing this, I now believe that it's OK to have either positive or negative participation points -- but not both. I often gave cheap points for turning in notes with a parent signature, say -- and then those extra points protect the students, so they have points to spare before getting detention. If this were my old system, the Desmos trio would have gained a point for correcting a teacher error. But then suppose those same students later on tart talking loudly or throwing objects. Once again, the students would misbehave, but can lose points without consequences.
I also wonder whether it's possible for me to improve with managing the seating chart, when the natural inclination of so many students is to switch seats without permission. Perhaps I can use the seating chart to greet the first few students as they come in -- implying that I mean business when it comes to enforcing the seating chart.
This regular teacher definitely uses interactive notebooks for the Algebra II class. The notebooks for Algebra I appear to be interactive, but the students are merely copying notes from either Chromebook or the board into their notebooks today. Still, the fact that they must copy these notes onto a specific page (17) suggests that they must glue in interactive pages at least once in a while.
I wonder what today's Algebra I classes might have looked like if sixth period had been the period of observation and seventh period were completely mine to teach. Then I would have observed how the teacher runs her class before having to manage my period.
This class has several routines and procedures. The problem is that students often find excuses not to follow the routines on sub days, or imply that the sub is wrong to enforce such a routine. Imagine, for example, if I had told seventh period (after seeing it in sixth period in this hypothetical scenario) that they had two minutes to turn in packets or else they're late. The students would likely have made the false claim that the teacher normally gives them more than two minutes to turn them in, or tell me that I have no authority to count the work as late. Then they'll just dawdle anyway and take as long as they want to turn in the work.
After all, the actual sixth period students already break one routine at the start of class by failing to sit in the correct seats. When I tell the students to stop talking, one student falsely claims that the sound is coming from the Chromebooks, not from talking students. That leads to another problem -- I could never make my Algebra I class just like the regular teacher's because unlike her, I had to tell the students to use the Chromebooks. I didn't have access to the apps that the regular teacher used to enhance her lesson without the Chromebooks.
When a group of students in the regular teacher's period is talking, the teacher places the entire group on her participation point warning list, so they stop talking without making false excuses. (Notice that deciding what names to write is another case of "When did they cross the line?") When I try to write down the name of my one student, I find that he (or someone on his behalf) had erased it by the time the regular teacher arrives. (I end up just telling her the name orally.)
Regarding Mole Day, in past posts I wrote about how I could have celebrated it in science class at the old charter school. But because it's two weeks earlier, Metric Week might have been more important in my old class than Mole Week.
Today on her Mathematics Calendar 2019, Theoni Pappas writes:
If the sides of this triangle are integers, what is the smallest perimeter this triangle can have?
(Here is the given info from the diagram: two of the sides are 7 and 11.)
All we need for this problem are Lessons 1-9 and 8-1 of the U of Chicago text. From Lesson 1-9, the Triangle Inequality tells us that the shortest side must be greater than 11 - 7, or 4 -- since it's an integer, it must be 5. From 8-1, the perimeter is the sum of all sides -- 5 + 7 + 11 = 23. Therefore the desired minimum perimeter is 23 -- and of course, today's date is the 23rd.
Chapter 6 of Ian Stewart's The Story of Mathematics is called "Curves and Coordinates: Geometry is Algebra is Geometry." Here's how it begins:
"Although it is usual to classify mathematics into separate areas into separate areas, such as arithmetic, algebra, geometry and so on, this classification owes more to human convenience than the subject's true structure."
This chapter is all about the development of the coordinate plane -- which in some ways, can be considered both algebraic and geometric. We begin with the French mathematician Pierre de Fermat (the same guy with the famous Last Theorem):
"Fermat noticed a general principle: if the conditions imposed on the point can be expressed as a single equation involving two unknowns, the corresponding locus is a curve -- or a straight line."
One possible such curve of course is the parabola -- and today, the students in Algebra II are graphing several parabolas.
"In modern terms, Fermat introduced oblique axes in the plane (oblique meaning that they do not necessarily cross at right angles)."
Of course, the one who introduced perpendicular axes was Rene Descartes -- and his coordinate plane, now the modern standard, is named the Cartesian plane in his honor:
"For example, on a map, x might be the distance east of the origin (with negative numbers representing distances to the west), whereas y might be the distance north of the origin (with negative numbers representing distances to the south)."
And of course, it's possible to have more than two dimensions on a graph:
"We do not find the number of dimensions of a space by finding some things called dimensions and then counting them."
Instead, we merely count the number of coordinates it takes to name a point. We return to Descartes and how he used these Cartesian coordinates to draw certain curves:
"He went on to consider equations of higher degree, defining curves more complex than most of those arising in classical Greek geometry."
We now move on to a special type of curve -- functions, such as the square root function:
"This recipe requires x to be positive. Similarly the square function is defined by f (x) = x^2, and this time there is no restriction on x."
Stewart concludes this relatively short chapter by discussing the impact of coordinate geometry today as it serves to connect algebraic to geometric concepts:
"And it is those cross-connections, revealed to us over the past 4000 years, that make mathematics a single, unified subject."
The sidebars in this chapter are all about Descartes himself, his coordinates, and how to keep track of the famous Swiss family (or "mafia") of mathematicians -- the Bernoulli family.
Meanwhile, today is the second day of the Chapter 4 Review. Recall that Chapters 2 through 6 of the U of Chicago text are short, so we have an extra day to prepare for the test. I use this extra day to find resources related to the test online.
The page I found is from a Mrs. Hester, a middle school teacher. This post is dated 2013, and she hasn't posted anything new on her blog since 2015:
http://mrshester.blogspot.com/2013/09/unit-1-8th-grade-math.html
Notice that Hester begins her eighth grade year with transformations -- as we recall, Kate Nowak does the same. Again, this is due to the CCSS standard connecting transformations to slope.
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