Lesson 2.2.1: Slope (Days 31-33)

Well, it didn't take long for me to receive my first parent complaint. This morning, I meet the principal for the first time, as he inform me that one parent is upset over my use of the word "dren" yesterday (to indicate the opposite of a "nerd" -- or someone too lazy to learn basic math). Even though I was careful to say that none of the students are "drens," all it took was for one parent to overhear my explanation of the word during at-home distance learning and the complaint was made.

Because of this, I've now decided that my word "dren" has outlived its usefulness. Recall that I've posted this word a few times since the earliest days of this blog -- well before I ever used it in the classroom. I thought that the word would motivate the students to work harder at math. Some students believe that math is only for "nerds" and that it's OK not to be good at math. My goal was to turn it around, so that it's not OK to avoid learning math.

We think back to the ancient sour grapes analogy (due to Aesop) -- the fox finds that it's too difficult to reach the grapes hanging on a high vine, so he walks off muttering that the grapes are sour anyway, so that he wouldn't want any. Likewise, students who struggle at math rationalize their failure to learn it by saying that it's not useful in the real world. My use of the word "dren" was like telling the fox that the grapes are sweet (that is, that math really is useful) -- but that makes the fox feel even worse about not getting any.

Recall the famous math teacher-blogger who died earlier this year, John Berray. He was able to motivate the students to learn math without using negative words like "dren." Instead, he convinced the students that they can be successful at math after all: "You are a math person." By telling them this, Berray was telling the fox that he can get the grapes after all. I was taking the students' comments at face value -- the fox thinks the grapes of math are sour, so I must convince him that they're sweet -- when instead I should have addressed the underlying issue -- the student feels less than confident at doing math.

Therefore I will stop using the word "dren" in the classroom. I am retiring the "Dren Song," as I will no longer sing it in the classroom. Perhaps I may one day rewrite this song to use positive words instead of "dren," but that won't be until after I complete this long-term assignment. Also, "No Drens" -- a parody of TLC's "No Scrubs" -- is also defunct. There's no reasonable way to rewrite this song to be more positive.

I hope that my second meeting with the principal will be under better circumstances. This Friday, he has called an after-school meeting to address next week's hybrid reopening.

But for this last week of distance learning, recall that today only the Wednesday-Friday cohort has synchronous learning, as opposed to the Tuesday-Thursday cohort. Therefore today is basically a repeat of yesterday's lessons. I teach slope to the eighth graders and slow down the seventh grade integer lessons -- I finish addition and only introduce subtraction.

There are also issues today in periods 3 and 5 with turning on cameras vs. leaving them off in order to play around or avoid working. I'm finally added to Canvas, but my next step is to access APEX.

I was going to sing the "Dren Song" to today's cohort, but since I'm retiring that song, I replace it with my traditional second song, Square One TV's "Count on It":

Count On It

Lead vocals by Larry Cedar

Sooner or later, you’re gonna see some math
You can count on it
Sooner or later, those numbers cross your path
You can count on it

You may be hoping it will go away
But let me tell you, math is here to stay
You can count on it, hoo, yeah
You can count on it

Everywhere you look, they’re measuring the action
You can count on it
Everywhere you look, they’re even using fractions
You can count on it

They’re keeping time, and they’re keeping the score
They draw the line, and they’re running the store
You can count on it, hoo
Yeah, you can count on it

Look at the dial; look at how far
Look at how much; look where we are
Look at the gauge; look at the graph
Check out the numbers; you’ve got the last laugh

‘Cause it ain’t mystery; there’s nothing tough about it
You can count on it, that’s right
Soon you’re gonna see that you couldn’t live without it
You can count on it, hoo

Don’t take a genius or a great magician
To make a pretty good mathematician
You can count on it, hoo, yeah
Yeah, you can count on it, whoo
Oh, you can count on it, whoo

Baby, you can count on it

When I sing this song, I also mention the fifth of my resolution, "We treat people who are great at math as heroes." The use of the positive word "hero" to describe someone who's good at math is much better than the use of the negative word "dren" to describe someone who isn't.

My new blogging schedule has me posting Monday-Wednesday only. Therefore my next post will be on Monday, October 5th.

Lesson 2.2.1: Subtraction of Rational Numbers (Day 30)

Today the eighth graders begin their study of slope. This is obviously one of the more important topics of eighth grade, since it appears again in Algebra I.

Yesterday I didn't mention Sarah Carter's DIXI-ROYD mnemonic since domain/range didn't appear anywhere in the APEX lesson. But today I definitely mention her "Slope Dude" mountain analogy, including "puff-puff positive," "nice and negative," "boring zero slope," and "undefined cliff." Here's a link to the post where she introduces "Slope Dude":

https://mathequalslove.blogspot.com/2015/11/slope-dude-says.html

In the lesson, there is a brief mention of similar triangles as proof of the slope formula. This is indeed required by the Common Core Standards. But, as we know, similarity isn't taught until the G standards, while slope is considered an EE standard -- and most texts teach all EE standards before any G's. This is why, while deriving slope from similarity is a good idea, I can't see how it works in practice. Again, most students learn about slope before similarity.

Meanwhile, seventh grade is learning about subtraction of integers. I obviously don't want to rush integer arithmetic since we know how important this topic is at this level. In fact, the head of the department comes in to tell me that it's OK so slow down this topic a little. Originally, the students are to take the subtraction quiz at the end of this week, but she may delay this. She'll soon update the official pacing plan/study guide with a new date for the quiz.

I receive a message that one eighth grade girl who had previously opted out of hybrid has since changed her mind and will be opting into hybrid -- moving from Cohort WO to WF. But due to the size of the classes, she's moving from my first period to my fourth period. She becomes the tenth student in my fourth period in-person class on Wednesdays and Fridays -- without the schedule change, she would have become the fifteenth student in my first period.

Because of the schedule change, I was sent a copy of her grades -- once the change is made, her grades will be automatically erased of the system (and so the copy is for reentering the grades). But since my Canvas and other accounts still aren't working, this is the first time I've seen any of my students' grades -- and so this is an opportunity to see more about how grades will work during distance learning.

Actually, there are some non-APEX grades in the system, but still, APEX is the lion's share. Unfortunately this student has an APEX grade of D. Thanks to some non-APEX grades (mostly introductory videos and short benchmark quizzes), her current grade is C-. Still, this does show that this girl can probably use some extra help, and so I should reach out to her when I get the chance.

And I'll be seeing her tomorrow, along with the rest of the Wednesday-Friday cohort. As I mentioned earlier, this school has cohorts even during distance learning. So tomorrow I'll repeat today's lesson.

Today's song is the "Dren Song," my traditional first day of school song:

The Dren Song -- by Mr. Walker

I don't know why I take math.
I'm all caught up in its wrath.
I'd rather just be a dren.
I would be so happy then.
Tell me what would happen when,
I'm no longer just a dren.
What if I were great at math?
What would be my future path?
Customers won't think it's strange,
When I figure out their change.
Algebra and calculus,
Get me in a cool college.
Once I finish my degree,
Future employers will see,
Of my strong background in STEM.
I know that will impress them.
Reach the moon, be a hero!
I won't just be a zero!
I'll be great, or it may seem,
That this all is just a dream,
'Cause my math skills are so bad.
I can't subtract! I can't add!
I can't multiply by ten.
I will always be a dren.
Now I know why I take math.
Help me find a better path!
I would be so happy then.
But I'm just a dren.

As I sing this song to the fourth period, a few students claim in "chat" that they were "drens." I make sure that I stay positive here -- I tell them that last week when I was hired here, I asked whether there were any "drens" at this school, and that the response was "of course not." So I reassure them that they aren't "drens" -- there are only smart and almost-smart students. I also mention the fourth resolution during tutorial: "We need to inflate the wheels of our bike," with also goes along with this idea.

Lesson 2.1.1: Functions and Relations (Day 29)

 Today is the first day of my new long-term subbing assignment. This counts as a special day, and so I'm doing "A Day in the Life" today:

8:00 -- I arrive at my new middle school. Some kindergartners and their parents arrive -- officially, the elementary school is a separate school, but they share the main office and other parts of the campus. It is a phased reopening, where one or two grades restart each day -- kindergarten today and middle school next week.

The office workers and I prepare for my first day of distance learning. We continue attempting to log in to the Canvas website, but I still can't get in. Fortunately, yesterday I spoke to the regular teacher, and he already sent me copies of the lesson via Google Drive so that I at least have enough material today.

I'm shown to my room for the new three-and-a-half months, which isn't far from the main office. I've been given a Chromebook for streaming the class, and there is also a document camera and projector.

8:45 -- It's a Late Start Monday, and so there's a teachers' meeting today. The regular teacher calls me to tell me how to log into the meeting on Google Meet. It is a math department meeting. The head of the department asks sixth, seventh, and eighth graders to discuss the major content standards that they plan on teaching during the upcoming week.

After the meeting, the regular teacher stays on to make sure that I'm connecting everything right. Our goal is to make sure that I use the document camera to display the copies of the lesson that I printed last night, and that the projected image is visible to the Chromebook camera so that students can see it.

9:45 -- First period begins. This is the first of two eighth grade classes.

This blog will focus on the eighth grade class, and so the title of my posts will be the name of the lesson that I'm teaching them. So today's lesson is an introduction to functions and relations.

As I lecture from the notes (that come from APEX, via Canvas), I don't like one of its examples, a table where some values of x are repeated, yet the relation is nevertheless a function because the corresponding values of y are also repeated. Instead, I take a dry erase packet that's already set up for a functions lesson (in reality, it was set up for a demo lesson for a different job interview) and fill in my own relations that are more typical of the examples used to distinguish functions from non-functions.

This might have been a good time to introduce Sarah Carter's DIXI-ROYD mnemonic, except that the words "domain" and "range" (the D and the R) don't appear in the lesson. Otherwise I certainly would have mentioned DIXI-ROYD.

As is common during these Google Classroom lessons, it's hard for me to tell whether the students are actually paying attention or not. Most student cameras are off, and very few of them are speaking.

10:30 -- First period ends and second period begins. This is the first of two seventh grade classes.

These students are learning how to add integers. I tell the students that comfortability with positive and negative numbers is one of the most important lessons learned in seventh grade, as it sets them up for success next year and in high school. 

This time, a few students do speak up. Some of them are having trouble finding the lesson on APEX, while others struggle to see exactly what I'm writing under the document camera. This is one of two classes with a co-teacher, and she tries to give them a link (which she posts in the Google Meet chat) to the notes, but it only seems to work for some of them.

11:10 -- Second period ends for a short break.

11:20 -- Third period begins. This is the second seventh grade class.

More students have trouble finding the lesson on APEX. Since my Google Meet chat is still open, I just post the link provided by the co-teacher again.

12:05 -- It's now time for a sort of "tutorial" period, for intervention, support, and extension.

Some students remain from the third period class. And so I continue the seventh grade lesson on integer addition, especially for the sake of those students who aren't able to see the APEX lesson.

I also use tutorial to tell the students one of my resolutions. This time I tell them the third rule:

3. We remember math like riding a bicycle.

If these students want to be successful in the future, then they should remember how to add integers just as easily as remembering how to ride a bike. One student (apparently quoting the regular teacher, since I see it written on a poster in his room) responds in the chat that the brain is like other muscles -- it must be exercised in order to stay strong. This goes along with my general idea.

12:35 -- Tutorial ends, and it's now time for lunch.

1:15 -- Fourth period begins. This is the second eighth grade class.

And unlike the other eighth grade class, but just like the seventh grade classes, there are students who can't access the lesson on APEX. This is the second of two classes with a co-teacher, and once again she provides a link that allows some of the students to see the lesson.

1:55 -- Fourth period ends and fifth period begins. This is the Math Skills class.

I tell the students that this class will be run just as the regular teacher does, where they are expected to log in to sites such as ST Math, Dream Box, or even Khan Academy. But it's impossible for me to tell whether anyone is actually using those sites or not -- and I suspect that this will be a continuing problem with this fifth period class.

2:40 -- Fifth period ends and sixth period begins. This is independent study PE for all students, and thus conference period for all teachers.

During this time, I contact the regular teacher. He tells me that he has, in fact, forgotten to activate the APEX lessons for all classes other than first period. This explains why no one in first period complains about APEX access. Of course, if my Canvas and APEX had been working properly, then I could have activated the lessons myself.

Meanwhile, the workers in the main office notice that my name is misspelled on my new email account (it appears as "Davis" instead of "David"). Originally I was going to wait a few weeks to inquire about this at the district because this is the least of my troubles -- getting Canvas working is much more important than having my name spelled correctly. But we can't rule out the possibility that the name mismatch is the cause of my problems with Canvas.

Whenever I do "A Day in the Life," it should always be followed by reflection. My biggest fear going into today was that I'd have problems with technology -- and indeed I do. But I'd much rather struggle with technology now than when hybrid starts next week and students are in the classroom. Let's hope that my Canvas will finally be working by then.

I decide to look up the stats on how many students are in each of my new classes:

• My largest class is first period, with 32 eighth graders.
• My smallest class is fifth period Math Skills, with sixteen students (with nine of them in the seventh grade). My smallest main math class is second period, with 20 seventh graders.
• In general, the classes with the co-teacher are smaller. This is probably due to their being more special ed students there, hence the need for the co-teacher.
• For hybrid next week, in general last names A-L attend school on Wednesdays/Fridays, while last names M-Z attend school on Tuesdays/Thursdays. Yet I notice a few students with surnames starting with M -- and even N -- in Cohort WF.
• Of my largest first period class of 32 students, 19 are in the Wed./Fri. group. Five of them are opting out of hybrid, leaving fourteen of them in Cohort WF proper. This represents the most students that I will have in the classroom at once during hybrid.
• The fewest number of in-person students I will have is five in Math Skills, with four of them in seventh grade. The smallest in-person cohort in a proper math class is seven, in second period.

My blog will be following the first period schedule, which turns out to be my biggest class.

Ordinarily I won't sing a song on Mondays, but today I sing a song for seventh grade only. That's because today's song fits the lesson perfectly:

Same sign, add and keep.

Different signs subtract.

Keep the sign of the bigger number.

Then you'll be exact.

This concludes my post for today. Tomorrow we'll see whether I can get into Canvas, which will be a major factor in how well my classes go.

A Major Announcement (Day 28)

In today's post, I have a major announcement to make. I'll add the "FAQ" label to this post and then write the rest of this post in question-and-answer format, as I usually do with my annual FAQ posts:

1. What is the major announcement?

2. What is the hybrid schedule like in my new district?

3. What classes am I teaching this year?

4. What is the curriculum like?

5. Will I continue to sing songs in class?

6. Have I changed the "New Decade's Resolutions" yet?

7. How will I stay safe in the classroom during the pandemic?

8. How will I communicate with other teachers?

9. What are my plans for this blog and social media during this assignment?

10. What is my biggest worry about my new assignment?

11. Do I have any final thoughts for now?

What is the major announcement?

I am now officially a long-term substitute teacher. Starting this upcoming Monday and lasting until the first week in January, I will be covering for a middle school math teacher.

Even though this school is in Orange County, it's not in the district where I've been working as a substitute since February 2018. But the two districts do neighbor each other. Just a week ago, an advertisement for a long-term subbing position was announced -- and we see how quickly it all happened, since I'll be starting there on Monday.

As is the case throughout the county, schools in this district will be opening soon. But my new district will be starting the hybrid schedule a week later than my current district. Thus I'll be teaching a full week of distance learning only before seeing students in person during my second week.

Yesterday, I had a online meeting with the regular teacher. He told me how he has a credential that allows him to teach high school math, yet he finds himself teaching Grades 7-8 at a middle school. In some ways, I'm in the same boat -- after all, I've named this blog "Geometry, Common Core Style" in the hopes that I'd be teaching high school Geometry. Yet somehow, I too find myself teaching younger students -- first at the old charter school in 2016-17, and now here for this teaching assignment.

Then again, it's easy to see how I ended up teaching at an Orange County middle school. First of all, unlike those in LA County, Orange County schools are preparing to open in person. While this excites many parents, some teachers are worried about the coronavirus, and so some of them are choosing to take extended leaves of absence, thereby opening up long-term spots for subs like me. Indeed, the regular teacher is worried about his mother's health.

And in fact, in this new district, elementary schools will open in person next week and middle schools the following week, but high schools won't fully start the hybrid schedule until second semester. Until then, some high school students may attend campus for sports practice, clubs, or extra help (similar to Stage 2 in my old LA County district). Thus K-8 teachers are more likely to take leaves of absence now if they're worried about the virus -- which also explains why this math position became available.

It's often said that in the Chinese language, the words for "crisis" and "opportunity" are the same (and according to Homer Simpson, that word is crisitunity). And yes, the ongoing pandemic certainly qualifies as a crisis -- but it's also providing me with this opportunity. If there had been no coronavirus outbreak, then this teacher doesn't leave and the position isn't open to me. If Orange County had stayed purple and schools remained online, then this teacher doesn't leave and the position isn't open to me.

The concurrent hybrid schedule will definitely be a challenge -- after struggling to teach both math and science at the old charter school, now I'll face a new struggle to teach both online and in-person at the same time. On one hand, I wish that my next major job wouldn't have to be under this tricky concurrent hybrid plan. But on the other hand, had it not been for concurrent hybrid, I wouldn't be here -- I'd still be hoping for day-to-day assignments (which as we've seen this week, aren't as plentiful). In fact, that's why every time I mentioned hybrid, I wrote that I'm biased in favor of whatever schedule gets me jobs.

That doesn't mean it will be easy -- not by a longshot. But any job that's worth doing and pays more than minimum wage will be difficult -- otherwise anyone could do it and the job would only have to pay minimum wage. And so next week (and especially the following week when the hybrid begins), that's when the hard work will begin for me.

The coronavirus is a crisis -- and now it's time for me to make it into a crisitunity.

What is the hybrid schedule like in my new district?

There are big differences between the hybrid schedules in my two Orange Country districts. One of them is in my new district, the hybrid schedule won't simply match the August-September online schedule, as it will in my current district. This means that I'll have to learn two different schedules -- one for my first week (the last week of full distance learning) and the other for hybrid.

Let's start with the full distance schedule first. As in my current district, on Mondays all classes and all students meet for synchronous learning. But in my new district, the students have already been divided into two Cohorts -- Cohort A (Tuesday/Thursday) and Cohort B (Wednesday/Friday). Students only have synchronous learning on these specified days (in addition to Monday), and so the other two days are asynchronous. But all periods are scheduled everyday, so students attend all of their classes on their respective three days.

But then this will change for hybrid. In previous posts, I've used the phrases "Hybrid Plan #1" and "Hybrid Plan #2" to refer to two different hybrid schedules. The number in each plan refers to how many days apart the students are physically on campus. Thus under Hybrid Plan #1, one cohort attends Tuesday/Wednesday while the other attends Thursday/Friday. But under Hybrid Plan #2, one group attends Tuesday/Thursday while the other attends Wednesday/Friday.

My current Orange County district is using Hybrid Plan #1. But in my new district, the students will attend class according to Hybrid Plan #2, on the same days (Tues./Thurs. or Wed./Fri.) that they attend online during distance learning. (My old LA County district will also follow Hybrid Plan #2, as far as days students are on campus are concerned). Mondays are completely online for all hybrid plans.

But now we must consider what these hybrids look like from the teacher's point of view. Let's look at a single numbered period and determine how often this period meets under hybrid.

On Hybrid Plan #1 (current district), first period meets on Tuesday, and then again on Thursday with the in-person/online cohorts switched. So on Hybrid #1, students are on campus on consecutive days, but the days teachers cover the same period are two days apart.

But my new school is on Hybrid Plan #2, when a block schedule will begin. So that first period meets on Tuesday, and then again on Wednesday with the in-person/online cohorts switched. So on Hybrid #2, students are on campus two days apart, but the days teachers cover the same period are consecutive.

Thus the numbers in the Hybrid Plans are from the point of view of students, not the teachers. Again, #1 and #2 means that the students are on campus one or two days apart. But the days that teachers have the same periods are the opposite -- two days apart on #1 and one day apart on #2.

Since my new school is on Hybrid Plan #2, I'll see my periods on consecutive days. For clarification, let me write out the full schedule that I'll be following:

• Monday: all periods distance
• Tuesday: odd periods, Cohort A in person
• Wednesday: odd periods, Cohort B in person
• Thursday: even periods, Cohort A in person
• Friday: even periods, Cohort B in person

Oh, and there's one more thing -- "odd" and "even" periods refer only to periods 1-4. This is because Periods 5-6 are everyday in order to accommodate PE classes. In particular, sixth period will be PE for all students, taught as "independent study," so the kids just go home after fifth period everyday:

• Monday: 1-2-3-4-5 all distance
• Tuesday: 1-3-5, Cohort A in person
• Wednesday: 1-3-5, Cohort B in person
• Thursday: 2-4-5, Cohort A in person
• Friday: 2-4-5, Cohort B in person

I assume that during normal times, sixth period is not PE for everyone. (Notice that if sixth period were PE for everyone, then it would be similar to my old charter with PE at the end of the day -- but once again, I assume that this isn't the case.)

I believe that the cohorts are alphabetical, but Cohort A is the end of the alphabet while Cohort B is the start of the alphabet. On the attendance rosters, the cohorts are called "WF" for Wednesday/Friday and "TT" for Tuesday/Thursday. (In fact, I should just label the cohorts on the blog as WF and TT rather than A and B, since these are their actual names.)

There are also Cohorts WO and TO for those students who are opting out of hybrid. There is one thing that confuses me a little about the hybrid here -- we know that during August/September distance learning, Cohort WF (and WO) were in fact asynchronous on Tuesdays/Thursdays -- and vice versa for Cohort TT (and TO). This means that those cohorts will still be asynchronous on those days once hybrid begins. We know that the hybrid lessons are concurrent -- the in-person lessons are streamed for the benefit of the opted-out (WO and TO) cohorts.

This makes a big difference regarding how hybrid lessons should be taught. If off-day lessons are synchronous, then Wednesday's lessons should build off of Tuesday's. But if they're asynchronous, then Wednesday's lessons should be a repeat of Tuesday's, since the Wednesday kids never see what I taught on Tuesdays. I assume this will be explained to me fully once hybrid begins.

I don't identify districts by name here on the blog. Before today's post I keep using the phrase "old district" for my LA County district and "new district" for my Orange County district, but these names are insufficient since there are now three districts to juggle, with two in Orange County. Instead, let's just call them Orange County District #1 and #2, depending on which hybrid plan is being used. My new district where I'll be subbing long-term is thus OC District #2.

By the way, my new middle school is right next door to an elementary school. This reminds me of the old charter school which was K-8 and co-located with a district elementary school. But these really are separate schools with a fence between them.

What classes am I teaching this year?

Here's a look at what my schedule looks like here in OC District #2:

1. Math 8
2. Math 7
3. Math 7
4. Math 8
5. Math Skills

OK, so I will have two sections each of seventh and eighth grade math. This district is just like my old charter school in that middle school is Grades 6-8, but I won't be teaching sixth graders. Thus this job should be a little easier in that I don't have to worry about an extra prep of Math 6 (not to mention that I won't have to teach science).

Instead, there's a class called "Math Skills" -- a second math class taking the place of the elective for students who are struggling in math. The regular teacher tells me that these students have their own online assignments to work on. The only requirement is that the students must work a hour a week -- and in fact, the only grades he gives this class is a weekly assignment out of 60 points -- one point for each minute that the student is logged in.

Second and fourth periods contain special ed students, and so these classes are co-teaching. Actually, the other teacher is the co-teacher -- I'll technically be the resident teacher.

What is the curriculum like?

Students here in OC District #2 will begin by logging into the Canvas website. Actually, OC District #1 also uses Canvas, but Google Classroom is the main platform there. In my new OC District #2, Canvas in the main platform, so I'll have to familiarize myself with this website more.

Also, Google Meets will be used for synchronous learning rather than Zoom. I did have some problems with logging into Zoom last week, so I hope Google Meets will be better. But the regular teacher tried to contact me via Google Meets earlier, and there were a few problems. Fortunately, there's a week of pure distance learning. I'd rather work out the Google Meets kinks next week than during the following week when there are students in the classroom. I don't want the kids to be goofing off while I'm struggling to make Google Meets work.

The main math curriculum will be delivered via the APEX website. Once again, I assume that before the coronavirus, there was a traditional print curriculum, but APEX is used because it is much compatible with distance learning.

APEX divides the year into ten units. Each unit is divided approximately into weeks and days, and so a lesson might be numbered as Lesson 2.3.2, for Unit 2, Week 3, Day 2.

Here is a list of the APEX units for each grade:

Grade 8:

1. The Number System (August 31st-September 25th)
2. Functions (September 28th-October 16th)
3. Linear Functions (October 19th-November 13th)
4. Linear Equations (November 16th-December 11th, January 4th-29th)
5. Semester Wrap-Up (December 14th-17th)
6. Geometry and Measurement (February 1st-26th)
7. Transformations, Congruence, and Similarity Part I (March 1st-12th)
8. Transformations, Congruence, and Similarity Part II (March 15th-April 16th)
9. Data and Statistics (April 19th-May 14th)
10. Semester Wrap-Up (May 17th-21st)

Grade 7:

1. Rate, Ratio, Proportion (August 24th-September 25th)
2. Addition and Subtraction of Rational Numbers (September 28th-October 16th)
3. Multiplication and Division of Rational Numbers (October 19th-November 13th)
4. Foundations of Algebra (November 16th-December 11th)
5. Semester Wrap-Up (December 14th-17th)
6. Geometric Figures (January 4th-22nd)
7. Geometry in Two and Three Dimensions (January 25th-February 26th)
8. Statistics and Sampling (March 1st-26th)
9. Probability (March 29th-April 30th)
10. Semester Wrap-Up (May 3rd-7th)

The regular teacher informed me that he just finished the Unit 1 Test today, and so I am to start afresh with Unit 2. And so my first lesson on Monday should be Lesson 2.1.1. Actually, the pacing plan has all of 2.1 being completed on Monday, so that Lesson 2.2.1 is on Tuesday/Wednesday.

Within each week, Lesson x.x.4 consists of some sort of activity, but the regular teacher told me that he usually just skips these. And Lesson x.x.5 is usually a weekly quiz or test.

We see that in eighth grade, the class began with NS and some of the EE standards (for exponents), which isn't far from what I taught at the old charter school. But coming up next will be F (functions, Lesson 2.1.x) as well as more EE (slope, 2.2.x). I didn't make it to these standards the year I taught at the old school.

As for seventh grade, they will start with the most important lesson of that grade -- integers (addition in Lesson 2.1.x and subtraction in Lesson 2.2.x). Once again, note that the Common Core Standards mention "rational number" operations, and we see how both integers and rational numbers are emphasized on the list above.

Notice that for the most part, seventh grade will be proceeding through the standards in the naïve order they're listed in Common Core (RP, NS, EE, G, SP). Since this is also the intended order of the standards in the Illinois State text at the old charter school, this represents the order and pacing that I should have used at the old school. On the other hand, the eighth grade standards are in a different order, most notably with some F standards appearing in the middle of EE.

Meanwhile, there is no mention of IXL, unlike that seventh grade class I subbed last week. (It goes without saying that there's no longer a reason to keep peeking at that other class -- the one whose email notifications I keep getting -- now that I have my own class to worry about.) But still, having the last two periods of the day be computer time and PE is highly reminiscent of the old charter school, where we ended each day with IXL and PE.

Will I continue to sing songs in class?

The short answer is, of course I will. I've already devoted so many posts to music in the classroom, and now it's time for me to start singing them. The plans are to sing them on block days (Tuesday-Friday), since these truly are full-length blocks of 100 minutes each. That time needs to be broken with music.

Even though there are no 100-minute blocks during the first week with distance learning, I still want to establish music break during my first week. I usually won't sing on Mondays, but I think I will sing to the seventh grade classes on Monday, because the song "Same Sign, Add and Keep" fits with the addition of integers. This song can't wait until Tuesday/Wednesday because subtraction will be taught on those days, but the song only works for addition.

My first song on Tuesday/Wednesday, to both graders, will be the traditional first day of school song that I sang at the old charter school -- "Dren Song."

Speaking of which, yes, I will continue to mention the concept of "drens" in my class. The only difference is that I will try to be a little more positive. I'll emphasize that the students in my class are definitely not drens as long as they work hard in this class.

On the other hand, I don't see much opportunity to give my students Dren Quizzes this year. The regular teacher told me that assignments from APEX are automatically graded, so there's no room to have extra graded assignments like Dren Quizzes.

And that fifth period Math Skills class certainly sounds like the sort of class that could use a Dren Quiz, but the only graded assignment there is 60 minutes of online work. Moreover, the regular teacher informed me that all tests will be online, even during hybrid. Thus unless there's a way to give the Dren Quizzes online, I won't be giving them.

Wow -- I started out discussing songs and wound up with Dren Quizzes. Let's get back to songs.

I've decided that for all of October, all the songs I'll sing will be from Square One TV. Back at the old charter school, many of my students enjoyed the songs from that show. As much as I enjoy composing my own tunes (and once again, I devoted many summer posts to new scales to compose songs in), it's always much easier to perform when I already know the tune and much of the lyrics. And so this time, I want to begin my time with more of these songs.

My first Square One TV song will be my traditional second song, "Count on It." This will be followed in subsequent weeks with some other songs from that show that discuss the importance of learning math, such as "That's Math" and "Wanna Be." These songs are harder to find on YouTube, and indeed I never found them the year I worked at the old charter school, which is why I didn't sing them there.

If there is a Square One TV song that fits the curriculum, then I'll sing it. One possibility is "Less Than Zero," since the seventh graders learn about negative numbers early in the year. (Once again, the fact that I didn't sing this at the old charter school underscores how weak my instruction on this vital topic actually was.) And I'll end October with "Ghost of a Chance," my traditional Halloween song.

In November, I'll be free to sing some of my original songs -- but even then, the first block day of that month will be Election Day. Back at the old charter school, I'd had a special song planned for Election Day -- a parody of "Do-Re-Mi" that I first heard when I was a young sixth grader (which was also an election year). But then my car broke down -- I never made it to school that, and so I never sang it. So now -- four years later, in another election year -- I wish to correct that omission and sing this Election Day song.

Earlier, I wrote that if I added any tune to old lyrics that I posted over the summer without a tune, then I'd go back and edit that post to include the tune. But I can't do that because of a Blogger problem -- recently, Blogger changed its website. Whenever we compose a post, Blogger converts it to HTML so that it can display correctly on our browser. But Blogger changed the formula/program that converts our posts to HTML -- if we attempt to edit old posts, then we'll see the converted HTML format, but we can't convert it back to "compose" format so we can edit it. Only new posts that have been made since the change can be edited -- and the change was finally completed this week.

When I sang "Plug it In" in class two weeks ago, I was able to edit my July 31st post and include a tune with those lyrics, since that was with the old "legacy Blogger." (And yes, "Plug it In" is a song that I'll probably sing at my new school.) But now I can't do that any more. Thus my blog is now cluttered with posts labeled "music" that contain songs with lyrics but no tune. The best I can do is make more posts with the tunes and label them "music."

Have I changed the "New Decade's Resolutions" yet?

At the start of this calendar year, I came up with a list of "New Decade's Resolutions" -- rules for me to get the students to follow in order to be a more successful teacher. But many of these rules are already obsolete due to the coronavirus that now defines the new twenties decade.

The three rules that need replacing are:

5. "We treat the ones born in 1955 like heroes."

This rule has two problems. First, the generation born in 1955 includes individuals such as Bill Gates who developed the technology that we use everyday. But the name Bill Gates has been linked to conspiracies regarding a possible coronavirus vaccine. I'd rather avoid that controversy simply by not mentioning Bill Gates or his generation at all.

The other problem with this rule is that its intention was to get students to avoid using calculators and cell phones in class so often. But the "no cell phone" rules are difficult to enforce now. Obviously, during distance learning, students might be looking at phones the entire time and there's nothing we can do about it.

But even during hybrid, enforcing the "no cell phone" rule can become a problem. Suppose I see a student using a phone, and he refuses to put it away, so I try to confiscate it. All he has to say now is, "Stay six feet away from me!" (due to coronavirus) and suddenly I'm powerless to take the phone.

One thing I've seen as a teacher is that certain students care about certain rules only if they can use it to their advantage. That same student will probably spend his entire lunch period less than six feet away from his friends and even have his girlfriend sit in his lap. He only cares about the "six feet" rule when he can use it to keep mean teachers away from him, especially those trying to enforce unpopular rules.

Middle schools tend to be stricter with cell phones than high schools. This is because, as I've observed, younger students -- many of whom are getting their first phones and are still curious about everything that they can do with their new toys -- are more likely to be disruptive with them. High school students who break the "no phones" rule are likely to use them to listen to music or text/play games quietly. But middle school kids want to show each other pictures, use the camera to take their own pictures, play videos with loud sounds, and so on.

Therefore the best thing for me to do, as a new member of this school, is to see how more established teachers plan on enforcing phone rules during the pandemic. In any case, talking about 1955 will be counterproductive during this time.

8. "We sing to help us remember procedures."

I like singing songs to help students learn math or show them why they should learn it. At one point, I considered coming up with songs to help them remember classroom procedures. I never got around to writing such songs, and besides, some of the procedures I wanted to sing about (such as procedures for group work, turning in papers) might not make sense during the pandemic anyway.

9. "We attend every single second of class."

Attendance is awkward during the pandemic. We're still trying to come up with rules that tell us whether a student is truly present during synchronous or asynchronous learning only.

And even during hybrid, there are issues such as bathroom breaks. With 100-minute classes, it's inevitable that some students will ask to go to the restroom often. If I try to tell a student she can't go to the restroom, I might get a response such as "I need to wash my hands for 20 seconds" -- again, a rule she only cares about if she can use it to her advantage.

Again, I could come up with a restroom pass system, but passes are something that I'd have to pass out, collect, and touch. And other ideas that I've mentioned in previous posts, such as having the kids write standards (such as "I could have used the restroom, but I didn't" once for every minute of the lunch break during which they failed to go to the restroom), are also unenforceable during a pandemic.

So let's tweak those rules so that they make sense during the decade of the pandemic. Here is a new list of "New Decade's Resolutions":

1. We are good at math. We just need to improve at other things.

2. We make sacrifices in order to be successful at math.

3. We remember math like riding a bicycle.

4. We need to inflate the wheels of our bike.

5. We treat people who are great at math as heroes.

6. We ask, what would our heroes do?

7. We sing to help us learn math.

8. We follow procedures in the classroom.

9. We pay attention to math as long as possible.

10. We are not truly done until we have achieved excellence.

How will I stay safe in the classroom during the pandemic?

Today I received my annual flu shot. I never used to get flu shots until a few years ago when a former co-worker died of the flu -- he was only in his mid-forties. Also, September 25th is the earliest I've ever received a flu shot. I might have waited a little later, but I figure that I'll be busy with my new job. So I might as well get the shot today.

Thus if a coronavirus vaccine comes out, I'll probably take it. Of course, there are many conspiracies involving vaccines, Bill Gates, and the upcoming election. It goes without saying that earlier vaccines are unlikely to be tested properly. In past posts, I've stated that if we take the original estimate of 12-18 months (let's take the midpoint, 15 months) from the start of the pandemic (March) to the development of a vaccine, that takes us to next summer. Thus this is the earliest that a genuine vaccine will be ready, and that's just in time for the 2021-22 school year. I assume that while not everyone will be required to take the vaccine, members of some professions, including teachers, will need to do so (just as we're now required to have TB tests).

As teachers, we're required to wear masks in the classroom. And I agree with this rule, because I've personally had a problem with yelling and spitting in the classroom -- and even when I'm not yelling, I've found that a few droplets of water might spray out of my mouth anyway. I'm aware that not all teachers agree with the mask rules, but I individually need to wear one.

I've heard that in most districts, students (except perhaps the youngest students such as kindergartners) will be required to wear masks as well. And of course, it's inevitable -- someone is going to enter my classroom on the first day of hybrid and he won't be wearing a mask.

I believe that my district has a strict policy on mask-wearing -- students get a certain number of warnings about wearing a mask, after which they'll be removed from the classroom (that is, forced onto the WO or TO cohorts that are opting out of hybrid).

But for me, enforcing this rule will be tricky. I'll be a new teacher in the classroom, just starting to learn the students' names. Obviously a student who doesn't put on his mask won't just voluntarily tell me his name so I can get him in trouble. (I will have had the class for a week before hybrid, so I could theoretically learn their names from seeing them on the computer. Then again, the same students who will break the mask rules are likely to keep them cameras off during distance learning, so I can't learn their names that way either.)

Speaking of learning student names, the regular teacher has warned me that he has two sets of twins that are easily confused (along with a girl whose twin sister isn't in any of his classes). One pair of twins speaks very little English. This reminds me of one sixth grader at the old charter school, except that his language was Spanish. These girls speak mainly Chinese -- and while I know a few Spanish words, the only Chinese word I know is "crisitunity."

Other rules to enforce in the classroom involve social distancing and seating. But I can't make decisions about this until I see what my new classroom looks like.

Of course, students really need to follow the seating chart during distance learning. But once again, I anticipate that some kids will just refuse to sit in their proper seats. This is a problem -- if we try to approach them, they'll claim that we (the teachers), not they, are the ones breaking the distancing rule.

Once again, it's up to me to see how other teachers enforce the rule.

How will I communicate with other teachers?

At the old charter school, communicating with other teachers was a problem for me. It didn't help that I was at small school with no other middle school math teachers. Of course, I could speak to the other middle school teachers, but they couldn't help me with math or science problems. And sometimes I spoke with the elementary teachers, but some issues I had were unique to middle school.

As I've mentioned on the blog before, one person I failed to communicate with enough is the special support aide who helped out in my classroom. At the very least, I should have let her know all about the weekly lesson plans so that she'd know what to expect.

At my new school, my first point of contact is with the regular teacher. He said that he's willing to help me at least through the first few weeks of transition. But technically, he's on leave -- he's not required to assist me at all. I can't go complaining to him every time I have a problem.

There are several teachers I need to keep in touch with at my new school. The first and most obvious example is my co-teacher for two periods. She needs to be made aware of what we're scheduled to teach during her two classes.

The regular teacher has also told me about some of the other math teachers. This is a boon to me, considering that I come from a charter school where I was the only math teacher. He's informed me that the teacher next door to our room is the head of the math department, so she's an excellent person I can ask for help.

Once again, I can see how other teachers are enforcing tricky rules in this era, but once again, in some ways we are all first-year teachers. Not a single teacher here is experienced at enforcing a "mask" rule or a "six feet apart" rule in class.

What are my plans for this blog and social media during this assignment?

I'm returning to my old blogging schedule from the year I was in the charter school. While I post all 180 days of school as a day-to-day sub, I figure that I'll have less time as a teacher or long-term sub, and so I'll drop it to three days a week. But instead of naively blogging two days and then taking the third day off, my blogging schedule should actually fit the school's block schedule. In particular, I should pick one of my classes and then blog only on days that the chosen class meets.

Four years ago, I used the "Math 8" label and blogged mostly about my eighth grade class. I chose eighth grade because it was the closest class I had to high school Geometry, and also because there are important G (geometry-related) standards that year.

While Math 8 is again the closest to Geometry that I'm getting this year, it's unlikely that I'll even reach any G standards before the regular teacher returns (see APEX curriculum above), with the possible exception of some 7.G standards in my last week before he returns. Unfortunately, it means that on this blog with "Geometry" in its title, I'll again be posting very little Geometry.

I've decided that I will again choose Math 8 as the focus class. And as for the period whose schedule to follow on the blog, I choose first period, which meets Monday, Tuesday, Wednesday. I'd rather choose a period that meets earlier in the week so that I can reflect on what I wrote and make adjustments for the period that meets later in the week. (Yes, I know that the period meets everyday during the week of distance learning, but I want to establish the Mon.-Wed. posting pattern anyway.)

I'm also considering going back to the old monthly reflection posts (including "A Day in the Life"), as first suggested by the math teacher-blogger Tina Cardone. Four years ago, my monthly reflection day was on the eighteenth of every month. Some teacher-bloggers still do this today -- they often chose the 20th (to represent the year 2020) as their reflection day.

I was considering choosing the 28th -- this upcoming Monday. Then my first day becomes a reflection day, and then each monthly post becomes a milestone (one month completed in the classroom, two months completed, and so on). The only problem is that November 28th and December 28th are obviously during school vacations, when we're supposed to do "A Day in the Life" on each of those blogging days.

Instead, I notice on the calendar that there are minimum days throughout the year -- and there's exactly one of them each month. All of these are on Mondays. The good news is that obviously, none of these days are vacation days. The bad news is that none of these days are hybrid days, so "A Day in the Life" will describe only distance learning. Still, the minimum days are a great time to make a reflection post.

And of course, I'll follow the Tina Cardone rule of posting on "special days." The first day of hybrid -- Tuesday, October 6th -- is an obvious choice for a "special day" (even though Cardone would never dream of including it on her list of "special days"). So this blogging schedule will provide me with plenty of opportunities to reflect on my teaching.

I will now be following my new district calendar when counting days on the blog. Fortunately, my two districts have very similar, with the two calendars always within a day of each other. Today is in fact Day 28 in both districts. But (if you remember my saying this last year), my current district closes for the Jewish holiday of Yom Kippur, which is this upcoming Monday. But my new district (OC District #2, where I'll be a long-term sub) doesn't close for Yom Kippur. Thus the day count will follow my new district, and so Monday will be Day 29.

Four years ago, the math teacher-blogger Shelli suggested that I set up a Twitter account, but sadly, I didn't heed her advice. I believe that this was a huge mistake on my part -- it cost me opportunities to communicate with other math teachers at a time when I was the lone math teacher on my campus.

I'm not making that mistake again. Even though there will be other math teachers at my new school, I still want to create a Twitter account. Since I'm blogging Monday-Wednesday, my tweeting schedule will be Thursday and Friday. I'll create my Twitter account on Thursday and then link to it from this blog the following Monday.

And Twitter is a two-way street. It's not enough for me to tweet and expect others to read. I must read and respond to other tweets if I can think of an insightful response to what they've tweeted.

Twitter is, of course, the "T" in MTBoS (Math Twitter Blogosphere). Once again, I'll be looking out for members of MTBoS who are middle school math teachers -- and this is tricky, since as we've seen, the majority of MTBoS members teach high school. Fawn Nguyen is the most famous middle school member of the MTBoS, but she hasn't posted since May -- and she's now a coach anyway, not a teacher.

Here are some other middle school math teachers who have blogged more recently than Fawn Nguyen:

Algebrainiac

Ilana Cyna

Anna Blinstein

Kim Charlton

Anne Mayre

Amy Zimmer

Note that Ben Leis works at a middle school, but as a math club advisor rather than a teacher -- and what he teaches his students is well beyond what I'm learning. And of course Barry Garelick, the traditionalist, also teaches middle school, but his blog is more about traditionalism than what he teaches in his class.

What is my biggest worry about my new assignment?

I worry that I won't have access to all the websites I need in time for the first class on Monday.

The problem is that setting up access to all the sites I need is a multi-step process. No step can be completed before the previous step:

1. I turn in the paperwork to the district.
2. The district sends me a link granting access to the sub caller system.
3. I create an account for the sub caller system.
4. The district informs me that my name is in the system, so I can create an email account.
5. I create an email account and request access to Canvas.
6. The district grants me access to Canvas.
7. The regular teacher adds me as an extra teacher (co-teacher) of his classes.
8. I open up Canvas, find the Google Meet link, and start teaching the class.

Step 4 occurred at the end of the school day today -- I was informed that I could now create my district email account. So in the late afternoon/early evening I did, only to find out that I still needed to request access to Canvas.

And I assume that no one is there to give me access to Canvas over the weekend, so the best possible scenario is that someone is there early Monday morning to give me access. The worst possible scenario is that access is granted not by the district, but by Canvas itself (that is, the district must contact the Canvas company and the company gives me access). It might take several days for a large company to give me the access -- meaning that I won't have the access I need to teach on Monday.

All of these problems are occurring because of the unique nature of this subbing assignment. The opening wasn't even posted until a week ago today, and I applied to the job that night. So all actors are working quickly to get me filled in this position.

Also, in an ordinary situation, if a new sub is hired, it might take a day or two to get an email account for that sub -- but the sub wouldn't necessarily need an email account the first or second day. The sub shows up at class and passes out the pencil-and-paper assignment to the students in the classroom. It's only because everyone is online due to the pandemic that I need access to email and other accounts such as Canvas from my very first day.

And the fact that this job was filled on such short notice doesn't help. For example, a new teacher might be hired over the summer, and so there's much more breathing room. The fact that each step in the process takes a day or two doesn't matter since there's more time -- the teacher will have access to all the accounts she needs by the first day. It's only because the school year has already started and the regular teacher requested indefinite leave on such short notice that I'm having problem accessing all the websites I need.

Technically, at this school Monday is considered to be a Late Start day -- I don't know whether this is normal or only during the pandemic, but note that this is similar to some high schools in my current district (OC District #1). Therefore, Monday will start with a teacher PD meeting (which will be held completely online). This will buy me some time for the district to grant me Canvas access.

Otherwise my first period will be chaos, as the students will have no idea why they won't be able to log in to class. Second period might be slightly better, as it has a co-teacher who presumably already has access to Canvas (and hence Google Meet).

Do I have any final thoughts for now?

Oh, and before you ask, no, my upcoming posts will not consist of endlessly comparing my new classes to those at the old charter school from four years ago. As a day-to-day sub, this comparison was helpful, but now my focus needs to be only on my new school, not my old school. Only on monthly reflection days will I even mention the old charter school. (Also, no discussion of traditionalists will occur during this assignment.)

Back when I was a young second grader, I dreamed of becoming a teacher. I was influenced by two of my favorite teachers -- my current second grade teacher, and the middle school math teacher who began sending me Pre-Algebra work for me to study independently.

But the following year, I no longer wanted to become a teacher. That's because my teacher, who was out a lot that year, had plenty of subs, and I didn't like how my fellow third graders seemed to talk a lot and disrespect the subs who were there. I knew that I'd have to become a sub first before becoming a regular teacher, and I didn't think I could handle all those troublemakers. That teacher was out a lot due to doctor appointments during her pregnancy -- at the end of the first semester, she would ultimately go on maternity leave. This was the first time that I'd ever seen a long-term substitute teacher.

I made a mistake, easily forgiven as I was just a young third grader. What I didn't realize was that any job that was worth doing, and pays more than minimum wage, was difficult -- otherwise, anybody could do it and the job would be worth only minimum wage. If I no longer wanted to be a teacher when I grew up, anything career that I'd have instead would also be difficult. The only way to make a living is to do things that others find too difficult to do themselves.

My third grade self would be dismayed to find out that not only am I a sub, but I've been a sub much longer than I've been a regular teacher. Now I'm about to become a long-term sub, just like the sub I had the second semester that year. And now I'm facing challenges that none of us would have imagined even thirty weeks ago, much less thirty years ago.

As for that long-term sub, she stuck around -- by the time I entered the fourth grade, she'd been hired as a regular third-grade teacher. I must remember her as I proceed during my own long-term assignment.

Lesson 2-7: Terms Associated with Polygons (Day 27)

Before we begin today's lesson, let's continue to take a peek at that seventh grade class I subbed for last week -- the one whose Google Classroom notifications I'm still getting in my email.

Well, today another IXL lesson was assigned. This time, it's due tomorrow. So we now know that this teacher usually assigns two IXL lessons per week. Recall that back at the old charter school, IXL time was likewise scheduled twice per week.

Lesson 2-7 of the U of Chicago text is called "Terms Associated with Polygons." (It appears as Lesson 2-6 in the modern edition of the text.)

This is what I wrote last year about today's lesson:

Lesson 2-7 of the U of Chicago text deals with polygons. Notice that this lesson consists almost entirely of definitions and examples. But this chapter was setting up for this lesson, since a polygon is defined (Lesson 2-5, Definitions) in terms of unions (Lesson 2-6, Unions and Intersections) of segments:

A polygon is the union of three or more segments in the same plane such that each segment intersects exactly two others, one at each of its endpoints.

It follows that this section will be very tough on -- but very important for -- English learners. I made sure that there is plenty of room for the students to include both examples and non-examples of polygons. The names of n-gons for various values of n -- given as a list in the text -- will be given in a chart on my worksheet.

The text moves on to define a polygonal region. Many people -- students and teachers alike -- often abuse the term polygon by using it to refer to both the polygon and the polygonal region (which contains both the polygon and its interior). Indeed, even this book does it -- when we reach the chapter on area. Technically, triangles don't have areas -- triangular regions have areas -- but nearly every textbook refers to the "area of a triangle," not the "area of a triangular region." Our text mentions polygonal regions to define the convexity of a polygon -- in particular, if the polygonal region is convex (that is, if any segment whose endpoints lie in the region lies entirely in the region), then the polygon itself is convex.

The text then proceeds to define equilateral, isosceles, and scalene triangles. A triangle hierarchy is shown -- probably to prepare students for the more complicated quadrilateral hierarchy in a later chapter.

Many math teachers who write blogs say that they sometimes show YouTube videos in class. Here is one that gives a song about the three types of triangle. It comes from a TV show from my youth -- a PBS show called "Square One TV." This show contains several songs that may be appropriate for various levels of math, but I don't believe that I've ever seen any teacher recommend them for the classroom. I suspect it's because a teacher has to be exactly the correct age to have been in the target demographic when the show first aired and therefore have fond memories of the show. So let me be the first to recommend this link:

Another song from Square One TV that's relevant to this lesson is "Shape Up." Notice that many geometric figures appear on the singer's head -- though not every shape appearing on her head is a polygon:

Last year right after Lesson 2-7, I posted activities for the Daffynition Game and Jeopardy, and so I repeat those activities today. This is what I wrote last year about these activities:

And now I present my worksheet for the Daffynition Game. Remember that only one of these worksheets need to be given to each group -- in particular, to the scorekeeper in each group. The students write their guesses for Rounds 1-4 (or 5) on their own separate sheet of paper. I recommend that it be torn into strips so that they are harder to recognize. And the teacher provides the index cards, one for each student. Make sure that the students give back the index cards so you can reuse them for the next period. The students may keep their "guess cards," so there should be one for every student in every period.

The second page is for the Jeopardy game -- just as with the Daffynition game, there should be index cards, with the number of points on one side and the question (um, the answer, since the response is the question) on the other. In my class the questions were taped to the front board. In this version of the game, the categories correspond to the four lessons covered earlier this week. Of course, some lessons, such as Lessons 3-1 and 3-2 on angles, are tailor-made for Jeopardy, but unfortunately we haven't quite covered the lesson. Of course, we'll get there next week. I didn't include a Final Jeopardy Question, but here's a tricky one:

Final Jeopardy Category: Types of Polygons

If two points lie in the interior of this type of polygon, then the segment joining them lies in the interior.

Lesson 2-6: Unions and Intersections of Figures (Day 26)

Before we begin today's lesson, let's continue to take a peek at that seventh grade class I subbed for last week -- the one whose Google Classroom notifications I'm still getting in my email.

Well, today another IXL lesson was assigned. The due date isn't until next week, so it appears that this teacher is already looking ahead to next week. I notice that while one class only has to reach a maximum score of 80, the other class has to reach a score of 90 -- the honors class. It takes a minimum of nineteen questions to reach a score of 90 if all of them are correct. As we already know about IXL scores, getting from 80 to 90 is harder than going from 70 to 80, yet easier than getting from 90 to 100.

A new lesson for next week has also been posted -- one-step equations, multiplication and division. As we may recall, in the year at the old charter school, I didn't reach one-step equations until February. I sang the song "Solving Equations," and the important verse for one-step equations is, "Whatever we do to the left side, the same done to the right side. A letter alone on the left side, a number alone on the right side. That's all you have to do to solve it!"

Meanwhile, let's finally get to that first episode of Neil DeGrasse's science show -- the one I missed six months ago on National Geographic but didn't watch until last night on FOX.

Episode 1 of Cosmos: Possible Worlds is called "Ladder to the Stars." Here is a summary:

• Humans were originally hunters and gatherers, until we became voyagers to new worlds.
• Our "ship of the imagination" can take us anywhere through space and time.
• NASA's Voyager 1, the most distant object in space, is currently traveling in interstellar space.
• Our first destination is a pair of black holes that will cause a huge collision that will slow time.
• We can watch Cosmos through our devices because we can manipulate electromagnetic waves.
• If we could manipulate gravitational waves, we'd understand the start and future of the universe.
• On the Cosmic Calendar, life began (on September 15th) four billion years ago.
• DNA -- the ladder of the double-helix -- evolved from materials formed in the middle of stars.
• On the Cosmic Calendar, mammals (on December 26th) developed a neonatal cortex in the brain.
• About 9,000 years (20 seconds) ago, the first cities were built, yet society was still egalitarian.
• The first maps were devised, and humans were able to locate "home" on their maps.
• In the 17th century, Enlightenment philosophers began to study the world around them.
• Dutch-Jewish philosopher Baruch Spinosa wrote The Book of Nature -- to him, God is nature.
• There are two kingdoms in the world -- those who build the world and those who destroy it.
• Plants and insects developed a symbiotic relationship that lasts for millions of years.
• Every third bite of food we eat ultimately goes back to insect-pollinated plants.
• It's up to us to prevent the sixth great extinction event, the Anthropocene.
• A few decades from now, a fleet of starships will depart from the driest Atacama desert.
• Their destination is Alpha Centauri -- our first exploration of a world outside our solar system.

As the first episode of the season, this is in some ways a summary of the season to come. We've already watched episodes about the Cosmic Calendar, and the Anthropocene extinction event last spring, and traveling to Alpha Centauri is the topic of the second episode -- which, unfortunately, was not the second episode that aired on FOX last night.

Instead, for some reason FOX aired Episode 11 last night, "Shadows of Forgotten Ancestors." I already described this episode on the night it aired on National Geographic, in my April 13th post.

This is one reason why I like to watch episodes when they first air, rather than be on the mercy of networks who don't rerun their shows properly. I didn't watch the first two episodes in March because the schools were still open that week (and Cosmos was something I decided to watch during the lonely weeks of the coronavirus shutdown), and now there's still an elusive Episode 2 that I haven't seen.

And who knows when Episode 2 will finally air on FOX? It certainly won't be this upcoming Tuesday, because that's when the first of the Presidential Debates will air (and there was a commercial for those debates during Episode 1) -- oh well!

Lesson 2-6 of the U of Chicago text is called "Unions and Intersections of Figures." (It appears as Lesson 2-5 in the modern edition of the text.)

This is what I wrote last year about today's lesson:

Lesson 2-6 of the U of Chicago text focuses on unions and intersections. This is, of course, the domain of set theory.

In many ways, set theory is the basis of modern mathematics, and so many textbooks -- including higher math such as Precalculus and beyond -- mention set theory early on. Of course, the focus in this text is on unions and intersections of geometric figures. In particular, unions are used to define both polygon and angle, while intersections are used to define parallel lines.

The first three examples in the text, where the underlying sets contain natural numbers, real numbers, and points -- are OK. But I didn't like the fourth example, on airlines. I've decided to throw this one out -- if we want a non-mathematical example, why not just use sets of letters, such as {a, e, i, o, u}, the set of vowels?

One of the most important sets in mathematics is the null set, or empty set. According to the text, this set can be written as either { } or an O with a line through it (often called O-slash by students). Once again, since I can't represent that symbol on Blogger, let's use the strikethrough instead: O.

Now the text mentions that the intersection of two sets might be the empty set. But it doesn't mention what happens when one finds the union, or intersection, of the empty set and another set. As it turns out, the union of the empty set and another set is that other set -- so the empty set acts as the identity element for union, just as 0 is the identity for addition and 1 is the multiplicative identity. But the intersection of the empty set and another set is the empty set -- so the empty set acts as the absorbing element for intersection, just as 0 is the absorbing element for multiplication.

One question students often ask is, if { } is the empty set and O is the empty set, what's {O}? When I was young, I once heard a teacher point out that this is not the empty set because it's no longer empty -- it contains an element.