Today is Nineday on the Eleven Calendar:
Resolution #9: We attend every single second of class.
What makes today interesting with regards to this resolution is that it's a minimum day. It's because today is the last day before the four-day President's Day weekend in this district. The special schedule presents its own advantages and challenges as far as enforcing classroom attendance.
8:00 -- Second period arrives. This is the first of two Chemistry of the Living Earth classes, and the only class with an aide. (Again, "first period" really means zero period.)
These students are reading an article in a Science News magazine about metals and alloys. Then they must answer questions on a worksheet about the articles they read.
Once again, even though it's not as important for me to compare high school science classes to middle school science at the old charter school, I can't help but make the comparison here. Recall that there were some math/science magazines at the old charter that I sometimes passed out to my kids, especially seventh graders during the Wednesday so-called "Advisory" period. Once again, today shows me that I could have created worksheets for them to work on, so they have something to do.
I'm not sure who produced today's worksheets. They don't look like anything a teacher hastily created in Word -- instead, they look like official worksheets from Science News itself. When I received the magazines three years ago, I didn't have access to any official worksheets. So I would have needed to create them -- but it would have been worth it, especially if they were science articles. These might have provided me the opportunity to give seventh graders some science on Wednesdays.
It happens that second period starts five minutes later on minimum days. (At another school in this district, second period starts five minutes earlier on minimum days.) Of course, I shouldn't expect the extra five minutes to slow down the tardies (especially since there are still many tardies on Monday Late Start days). And indeed, there are three late students today (out of about 15 students in each of these special ed classes). I repeat Resolution #9 to the tardy students as they arrive.
Five students complete all the assigned work. Naturally, my Valentine's Day incentive is in effect -- and this time, I do have the holiday candy in addition to the pencils. On the other hand, the aide catches three students who don't bother to work at all, so I write them on my bad list.
8:40 -- Third period arrives. This is the second of two Chem classes.
For some strange reason, this is the period with the most tardies. A whopping six students arrive after the bell in this period. While we expect first or second period to have lots of late students, sometimes there is a later period with lots of tardies without any apparent reason.
This is a great time to start my music incentive. Since today is the last day of school before V-Day in this district, of course I must sing Square One TV's "Mathematics of Love." As usual, I introduce the Roman numeral version as the opening teaser, then add an additional verse using regular numerals each time five more students complete the assignment.
I end up singing only the first verse. A few more students finish just before the end of class, but too late for me to sing the second verse.
9:25 -- Third period leaves for snack.
9:45 -- Fourth period arrives. This is the first of two Biology of the Living Earth classes. Only one guy is tardy to this class.
9:50 -- A lockdown drill begins. A siren rings during the entire drill.
10:00 -- The lockdown drill ends.
The students have another article to read in Science News, but this one's more biological -- it's all about how monsters such as Nessie, dragons, and Bigfoot are based on real animals. This time, the students have a full packet of questions to answer (which they'll finish next week). Because of this, I could have sung the "Packet Rap" (or even "Meet Me in Pomona" since the lesson is on animals), but I stick to "Mathematics of Love."
It's understandable that the long, loud siren would distract the students from working, but some still refuse to complete even the first page during the 25 minutes remaining in class. I end up writing four names on the bad list.
10:25 -- Fourth period leaves and fifth period arrives. This is the second of two Bio classes.
This class doesn't earn even one verse since only two students work hard on the assignment. And so I must spoon-feed the answers to the questions to the first page on the projector. Afterward, more students finish the second page than in the previous class, and only one girl makes the bad list.
11:10 -- Fifth period leaves and sixth period arrives. This is the only Algebra I-A class.
There are three tardies in this class -- but unlike third period, there's an explanation here. On regular days, lunch is between fifth and sixth period, and so this trio decides to go to the cafeteria even though it means arriving late to class. One guy gets only a bottle of water, while two girls -- both of whom were also in the fifth period class -- buy a full lunch.
Once again, I tell these students about the ninth resolution -- but this leads only to argument. Indeed, I think back to the old charter school, where the "special scholar" had also complained about the minimum day schedule. In both cases, the argument is that they should have a break after the same number of periods as the regular day schedule, even if those periods are shorter. Even though the minimum day ends just ten minutes later than regular lunch time starts, it just feels so much longer (as in "two whole classes" later, rather than ten minutes later).
At the old charter school, the reason for the schedule difference is easy -- it was a K-8 school, and we'd have to move the elementary lunches up earlier as well to give middle school an earlier lunch, which would be awkward.
As far as high school is concerned, this is tricky. It's possible to add a 40-minute lunch to the minimum day so that school is out at 1:20 instead of 12:40. Some students might prefer this -- they get to eat, and 1:20 is still a nice time to get out compared to 3:00 on regular days.
On the other hand, some students would prefer to have no lunch at school and get out earlier. Indeed, many seniors who only need five classes to graduate schedule themselves for periods 1-5, so that they can leave at lunch. (This is also why the cafeteria still serves food today after fifth period -- it's for the benefit of low-income/free-lunch seniors who leave after fifth.)
The idea of more breaks vs. getting out earlier appear when I constructed the Eleven Calendar (except I was considering days in the week as opposed to hours in the day). The basic eleven-day week consists of three days on, one day off, three days on, four-day weekend. We could have eliminated the midweek off day and have a five-day weekend, but I assume that we wouldn't want to work six straight days without a break. Also, with so many days off per week, we must eliminate the long summer break, replacing it with three vacation weeks, one each in winter, spring, and summer.
Now imagine the perspective of a native Eleven Calendar user trying to convert to our Gregorian Calendar and standard school weeks. One such calendar user, accustomed to working no more than three days in a row, might balk at our five-day weeks without a midweek break and prefer to take Wednesday or Thursday off, even if it means working on Saturday. Another Eleven Calendar user might prefer to take longer weekends even it means losing our long summer break. (I suspect the ultimate compromise would be four days on, three days off, leaving seven total weeks of vacation.)
That's enough about my calendar -- let's get back to the schedule argument. As usual, the only way for me to end arguments is to sing, and that's exactly what I do next.
This time, it's not "Mathematics of Love," but the Elle King parody "x's and y's," since the lesson in this Algebra I-A class (a special ed class that goes slower than Algebra I) is on graphing linear equations in standard form.
Since part of the assignment is on Chromebooks, the music incentive ends up being for putting them away and charging them. When all the Chromebooks are put away, I finish "Mathematics of Love" for the sake of the students who have me both periods 5-6 and missed the song after fifth.
11:55 -- Sixth period leaves. Seventh period is the teacher's conference period, and so this essentially starts my four-day President's Day weekend.
After the students leave, I notice one object remaining on the desk -- a water bottle, almost full. That's right -- it's the guy who's tardy because he goes to the cafeteria to buy the water bottle, then ends up drinking hardly any of the water. He could have simply gone to the drinking fountain and had an equal amount of water during passing period -- not only would he have avoided the tardy, but he would have saved money since the cafeteria now charges (starting this school year) for bottled water!
By the way, I'm also monitoring restroom visits on ninth resolution days. Fortunately, no more than two students leave during any of the periods, with the possible exception of third period. It's possible that three students go during that period -- two of them right when the period starts. But only one of them tells me her name (so I know not to mark her late) -- the second leaves without permission. As the students arrive, the second student arrives with a large group of late students, and none of them tell me that they were in the restroom, so I mark them all tardy. The (possible) third student asks for a pass during the period.
In fourth period, one student simply arrives late (but before the lockdown) and claims that he was in the restroom. Since he never asks for permission, I mark him late (and indeed, he could have gone during snack without missing any class time). The lockdown itself serves to reduce the number of passes right after snack -- only one more guy asks for water, and so I make him wait until after the lockdown is over. Thus only two total students go to the RR/water during fourth period.
Lesson 11-1 of the U of Chicago text is called "Proofs with Coordinates." In the modern Third Edition of the text, proofs with coordinates appear in Lesson 11-4.
Coordinate proofs are mentioned in the Common Core Standards:
CCSS.MATH.CONTENT.HSG.GPE.B.4
Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).
In Lesson 11-1, we are given the coordinates of the vertices of a polygon, and we are asked to prove that the polygon is a parallelogram, right triangle, or rectangle. The key to these coordinate proofs is to find and compare the slopes of the sides.
But here's another Common Core Standard:
CCSS.MATH.CONTENT.8.EE.B.6
Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
So students are supposed to use similarity to prove the properties of slope. David Joyce, whom we mentioned throughout Chapter 9, also endorses the use of similarity to prove slope -- and indeed, he has harsh words to say about the treatment of coordinate geometry in most Geometry texts:
In a return to coordinate geometry it is implicitly assumed that a linear equation is the equation of a straight line. A proof would depend on the theory of similar triangles in chapter 10. At this point it is suggested that one can conclude that parallel lines have equal slope, and that the product the slopes of perpendicular lines is -1. The only justification given is by experiment. (A proof would require the theory of parallels.)
And in our text, similar triangles don't appear until Chapter 12. Thus to follow the Common Core and David Joyce, we should wait to teach Chapter 11 until after Chapter 12. Students need to have mastered similar triangles before they can begin learning about slope.
This has been the source of many headaches in my blog posts over the years. First of all, I'd start with Chapter 12, then go back to teach slope and some other Chapter 11 topics -- but I'd never actually reach Lesson 11-1.
The problem, of course, is that slope is an Algebra I topic. High school students are thus going to see slope well before they ever see similarity, because they take Algebra I before Geometry.
I also wrote extensively about Integrated Math courses. But even Integrated Math usually covers slope before similarity -- indeed, slope is a Math I topic, while similarity is a Math II topic. I once tried to devise my own Integrated Math courses that teach similarity before slope, but I failed. It's difficult to justify teaching similarity (from the second half of Geometry) before slope (which is from the first half of Algebra I).
In fact, we notice that the Common Core Standard requiring students to use similarity to prove the slope properties is an eighth grade standard, not a high school Geometry standard. This now makes sense -- students are introduced to slope in eighth grade in order to prepare to study it in more detail in Algebra I.
I think back to the eighth grade class from three years ago. Of course, student behavior and classroom management were issues. But another problem was that I began teaching translations, reflections, and rotations -- and rotations, understandably, confused some students. The extra time spent on isometries meant less time on dilations -- and dilations are the bridge to similarity and slope.
I now sometimes wonder whether it's better to teach only one of the isometries -- perhaps reflections, since they generate all isometries (i.e., all isometries are the composite of one or more reflections) -- and then skip directly into dilations. But this contradicts the Common Core Standards that explicitly mention translations and rotations -- and these might appear in PARCC or SBAC questions.
At any rate, if the connection between similarity and slope is covered in eighth grade, then it doesn't need to be introduced in high school Geometry. And so we can write about slope in Chapter 3 without having to prove anything about similarity first. As I mentioned before, Chapter 3 is a great time to teach slope, since it's a review topic from Algebra I, and Chapter 3 is often taught right around the time of the PSAT (where slope questions will appear).
When David Joyce wrote about slope and similarity, he forgot that there's a class called "Algebra I" where students learn many things about slope and coordinates without proving everything. In the end, I did say that this year I'd adhere to, not Joyce's suggestions, but the order of the U of Chicago text.
And by the order of the text, I mean the order of the old Second Edition. Earlier, I wrote that Lesson 11-1 appears as Lesson 11-4 of the new Third Edition. So what exactly appears in the first three lessons of the modern version?
Well, Lessons 11-1 through 11-3 of the new text correspond to Lessons 13-1 through 13-4 of the old version of the text. Indeed, the new Chapter 11 is called "Indirect and Coordinate Proofs." You might recall that Chapter 13 of the old text has been destroyed, and its lessons are now included as parts of different chapters. And so the first half of the old Chapter 13 now forms the first part of 11. (There are now only three lessons instead of four because the old Lesson 13-2, "Negations," has now been incorporated into the other three lessons.)
Otherwise Chapter 11 remains intact in moving from Second to Third Edition. Chapter 11 of the old edition has six lessons, and these correspond roughly to Lessons 11-4 through 11-9 of the new text.
Let's finally take a look at the new Lesson 11-1 worksheet. We begin with the two examples from the text -- the first problem lists four ordered pairs and asks us to prove that they are the vertices of a parallelogram, while the second lists three pairs that may be the coordinates of a right triangle. In each case, students are to calculate the slopes of the sides formed by adjoining vertices, and show that these slopes are either equal or opposite reciprocals.
CCSS.MATH.CONTENT.HSG.GPE.B.4
Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).
The second part of that standard, on circles, will have to wait until later in the chapter. To complete the rectangle question, students must calculate the four slopes, and show that slopes of opposite sides are equal, while slopes of adjacent sides are opposite reciprocals.
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