I choose to do "A Day in the Life" today rather than Tuesday because I didn't want to keep discussing the English Performance Task. Actually, a few students in each class are still working on these, though most have finished. Recall that the teachers are out today in order to grade those tasks, so I hope they submitted them to Google Classroom in time for the grading.
8:15 -- The day at this school begins with homeroom and announcements. Here HR is the same as third period, not first period as some other middle schools. And all the periods rotate, so HR/third period doesn't always start the day. Indeed, today starts with sixth period.
There is an eighth grade ASB rep in the class. She keeps track of everyone participating in this week's Spirit Week dress-up. For Valentine's Day, the students are to wear red today. She actually includes yours truly in the count, since I'm wearing red too.
8:25 -- Sixth period arrives. This is the first eighth grade class of the day. It was the best eighth grade class of the day on Tuesday, and so I give these students a reward of V-Day pencils and candy.
The main assignment for all classes is an article on cell phones in the classroom. The article contains both the point (phones can enhance learning) and counterpoint (they are distracting). The students must answer nine questions about the article and then do a current event on the same article.
9:20 -- Sixth period leaves and first period arrives. This is the second eighth grade class of the day.
10:15 -- First period leaves for snack.
10:25 -- Second period arrives. This is the first seventh grade class of the day. It was the best seventh grade class of the day on Tuesday, and so I give these students a reward of V-Day pencils and candy.
I'd purchased a bag of 25 candies (Kit-Kats, Hershey's, etc.) for the reward along with an equal number of pencils. There are eleven 8th graders and only seven 7th graders present today in the two classes that had earned the rewards. And so there's just enough for me to give the seventh graders a second chocolate and second pencil.
11:20 -- Second period leaves. But as it turns out, it's raining today. I've explained what rainy days entail here in California -- rainy-day schedules. Instead of one lunch for all, students attend one of two possible lunches. My class is assigned the first lunch. The rule is that all students go to third period for attendance and to drop off their belongings, and then two minutes after the tardy bell is when the dismissal bell rings for lunch.
12:05 -- Third period arrives after lunch. This is the second seventh grade class of the day.
1:00 -- Third period leaves. It is now conference period.
2:00 -- Like many special ed teachers, today's regular teacher has four classes of her own and then co-teaches one class, which in her case is fifth period. This is a seventh grade English class. Like all English teachers, this resident teacher is out grading the Performance Tasks, and so the class ends up with two subs.
Normally, at this school, there is always silent reading built in to the schedule after lunch. On rainy days this is dropped (since "after lunch" is confusing when there are two lunches). But the resident teacher decides to specify silent reading for all periods in her lesson plans.
After silent reading, the students complete a crossword puzzle on academic vocabulary.
2:55 -- Fifth period leaves. But once again, the teacher I'm subbing for has been assigned after school duty this week. Thus I must watch the area between two buildings -- in the rain.
3:05 -- My duty is over, and so the four-day President's Day weekend in this district begins.
For multi-day subbing assignments, I automatically default to the first focus resolution:
1. Implement a classroom management system based on how students actually think.
An aide takes care of most of the management today. The same two students as on Tuesday are the most disruptive. Unlike Tuesday, today the seventh grade boy was slightly more talkative than the eighth grade boy.
My main management issue occurs during a certain two-minute stretch -- you guessed it! It's the two minutes when third period must return before going to the rainy lunch. The aide doesn't stay for these two minutes -- she just leaves right after second period.
The seventh graders try to go directly to lunch instead of stay for the two minutes. Actually, all of them have the same history class next door, and so they arrive at my room early. So they must wait out all of the four-minute passing period and then the two minutes begin.
This leads to arguments, of course. With the aide out (and for some reason, the history teacher must not have told them about the two minutes), I am the one who must inform them of the schedule. So they blame me -- they see me as the mean teacher who's stopping them from going to lunch even though I'm not the one who wrote the rainy day schedule.
(This isn't the first time when I was blamed for a schedule I didn't write. Two years ago at the old charter school, students complained when I didn't let them go to nutrition on a minimum day. The schedule specified that students go to break after the third, not second, class on minimum days.)
Meanwhile, my left literally falls apart, likely because of all the rain. (The aide is the wise one, as she's wearing boots.) Since the students are already upset with me because of the two minutes, they start making fun of me. One guy claims that I'm "breaking the dress code" by having only one shoe, while a girl starts hoping that my other shoe would fall apart too! (I end up purchasing another pair of shoes during the lunch break.)
When we return after lunch, the same guy starts complaining about the assignment -- that's there's not enough time to do it because "there's two lunches" and he's "not smart." The time claim is false -- since there's no silent reading time, this period is actually one minute longer than on a dry day! And besides, many students in previous classes had enough time to finish.
But I know by now that arguing this point doesn't work. Instead, I reassure him that yes, I believe that he really can finish the assignment. With the aide's help, he's the first in the class to finish. Afterward I tell him that I knew all along that he could do it -- after all, he's the only eighth grader in the class, so he should be able to show the seventh graders how it's done! (Presumably he's in this period only because it somehow fits better in his schedule.) He replies, "I'm smart!" That's more like it!
One thing that's difficult for me as a teacher is to avoid taking what the students say personally. The students get upset with me not because they hate me, but because they think that they should be allowed to go directly to lunch -- and I am the one preventing them. They make fun of my shoes falling apart because of this. But when they return after lunch, they stop making fun of me -- and start making fun of each other.
Two boys start complaining that their legs are hurting. It's likely that the injuries are caused by slipping in the rain during lunch. The other students respond by laughing at them.
I believe that laughing at the boys is worse than laughing at me. After all, I'm able to buy new shoes, but the guys can't buy replacement legs that don't hurt. Indeed, I'd prefer that the students make fun of me instead of them.
Once I learn how not to take student criticism personally, I can take it to my advantage. When kids start laughing at each other, I can encourage them to make fun of me instead. I remember one day at the old charter school when the special scholar was struggling with her Dren Quiz. Some other students wanted to make fun of her. So I got them to laugh at me instead. Back then, there was some phrase that I incessantly repeated. (I think it was, "Now here's the problem!") And so I started saying that phrase intentionally so that they'd make fun of me and not the special scholar.
This is the sort of thing that I can do more often. Some students naturally enjoy laughter, and unfortunately they often direct that laughter at an injured or struggling classmate. I should then do something silly so that they'll laugh at me instead. This is all part of the first resolution -- managing a classroom based on how students actually think.
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 last year's eighth grade class. 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.
As usual, since today is Thursday, I must create a new worksheet. I decided to include Question #9, because students are asked to prove that EFGH is a rectangle -- and rectangles, unlike parallelograms or right triangles, are explicitly 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).
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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