Semester 2 Review and Next Year Preview (Day 180)

This post fulfills my posting requirement for Tina Cardone's "Day in the Life" project. It is for the special day "Last Day of School."

8:10 -- I arrive at the first of my two schools. Second period is the first of two advanced  classes.

Like yesterday, the regular teacher has set up a ten-point assignment where students must watch a video on the Food Truck competition and then answer a question. But there are two problems here -- the first, as we've seen yesterday, the sound doesn't work on the video. The other problem is that the video is 40 minutes long (a one-hour show minus commercials), but it's a minimum day -- so classes are less than 40 minutes long today.

I solve both problems at once -- I watch the video before school, then summarize it for the students so that they can answer the question. The competition takes place in St. Louis before moving to Chicago. The winning team turns out to be Pho-nomenal, the first ever all-female team champions on the show. Many students are disappointed -- especially the boys, who were hoping that the all-male Waffle Love team would win.

After yesterday's arguments in this class, today I use a motivating trick similar to the one I used during tutorial yesterday -- each time five more students submit the assignment, I sing a song. This class earns three songs, since just over 15 students complete the assignment.

As for the main song for today, I've finally updated my "No Scrubs" (from TLC) parody. Recall that my first parody of this song contained a certain disparaging word that I no longer use in class (and I don't even want to post that word on the blog any longer). Instead, the new song is called "No Zeroes" -- after all, I'm trying to get the students to submit the assignment in order to avoid getting a 0/10 score:

NO ZEROES

1. A ZERO for a guy that thinks he's fly
And is also known as a buster
Always talkin' about what he wants

And CAN'T EVEN DO BASIC MATH
So (no)

Chorus:
I don't want your number (no)
I don't want to INTERVIEW you and (no)
I don't want YOUR RESUME (no)
I don't want none of your time and (no)
I don't want no ZERO
A ZERO is a GRADE that can't get no JOB from me
Hanging out the passenger side
Of his best friend's ride
Trying to GET HIRED BY me
I don't want no ZERO
A ZERO is a GRADE that can't get no JOB from me
Hanging out the passenger side
Of his best friend's ride
Trying to GET HIRED BY me

2. But a ZERO is HIS GRADE
'CAUSE his MATH is kinda weak
And I know that he cannot approach me
'Cause I'm lookin' like class and he's lookin' like trash
WHEN HE DOESN'T KNOW BASIC MATH
So (no) (to Chorus)

Indeed, notice that the line:

A ZERO is a GRADE that can't get no JOB from me (original TLC line: "A scrub is a guy...")

In other words, I'm not calling a student a "zero" or any disparaging term -- I'm calling their grade or score a zero. This should avoid the problems I had with the previous song.

8:45 -- Second period leaves and fourth period arrives. This is the first of two regular classes.

This class didn't meet yesterday due to the seventh grade activity, and so these students are behind the second period class. Nevertheless, this class catches up due to the song incentive, and so they also get three songs today.

After the "No Scrubs" parody, I let the students choose the other two songs. For some reason, each class ends up choosing "The Big March," even though it's well past the month of March.

9:20 -- Fourth period leaves. It is time for me to depart to the second school, my long-term school. As it turns out, this school has a late start minimum day (similar to the monthly minimum days that we had from October to December on the blog). This school hasn't even started yet. Thus I have a much longer travel break than usual.

During the longer break, I decide to provide a gift for graduating eighth graders. Yesterday I sang the eighth grade graduation song "Nothing's Gonna Stop the Love of Friends" from the old TV show Ghostwriter, and so I give the eighth graders the same gift that the graduates receive on the TV show -- a new pen. In addition, I hand out the usual candy. I do this right outside of my old classroom -- in between second and third periods, since both of these are Math 8 classes.

10:55 -- Third period arrives at my long-term school. This is the second of two regular cooking classes.

At this school the sound system is working, but there's still not much time for the whole video. And so I play only the last 15 minutes of the video for them, and then I leave them time to complete the work and earn the songs.

11:30 -- Third period ends for snack break.

11:40 -- Fourth period arrives. This is the second of two advanced cooking classes.

This class is tough, because the regular teacher specifically mentioned noon as the deadline for all assignments to be submitted. Thus I play the first 15 minutes of the video, leaving them five minutes to type a couple of sentences before the deadline. I don't take attendance until the clock reaches noon.

This group is the hardest-working of all, as it's the only one to reach 20 students and four songs. Since there's only a few minutes left after taking attendance to perform these songs, I start with "No Zeroes," followed by two of my quicker songs. Then I let the students choose the last song -- and of course, this group chooses "The Big March" yet again.

12:10 -- Fourth period leaves and fifth period arrives. This is the ASB class.

There is one more end-of-the-year activity for ASB to do -- a clap-out for the eighth graders. The younger ASB students visit all eighth grade classes to lead them out to the field. I walk to the field with the ASB eighth graders, who join the rest of their agemates.

Then all of the students walk from the field to the street, with Grades 6-7 clapping along the way. I follow the graduates and sing "Nothing's Gonna Stop the Love of Friends" again.

Once the eighth graders make it to the street, they can go home. But there are still ten minutes or so left in class, I return to the ASB room with the Grades 6-7 ASB kids. I finally take their attendance, and then I round out the day with some more songs. I return to the main songs that I've been performing today, "No Zeroes" and "The Big March." For the last verse of this song, I sing the lines:

We go Big Marching in Week 38,

Congrats everyone, you survived the school year,

And we all going marching down, now it's finally,

Time to get out of school. BOOM! BOOM! BOOM!

And in some ways, this pandemic school year was felt like a Big March the entire year.

12:45 -- Fifth period leaves, thus ending my day of teaching,

2:00 -- The actual graduation ceremony begins. It's a drive-through ceremony -- for the next three hours, students and their families drive past the school. They take a grab bag, then proceed over to an area where the student can get out of the car to have a picture taken. They can return tomorrow to pick up these pictures.

I'm not obligated to stay for the ceremony, but I voluntarily watch the first hour for two reasons. The first is that I did teach some of these eighth graders from late September to early January, so I do have some connection to them. And the second is because I missed watching the eighth graders graduate at the old charter school four years ago. This takes us directly into the reflection part of this post.

In many ways, I think of this long-term subbing position as the completion of the year I taught at the old charter school. The length of time I spent at both schools is close to a full school year. And we can count all the minimum day posts (including today's) as part of the Tina Cardone challenge that I started that year -- the monthly posts plus the special day "Last Day of School."

I'm not quite sure what the graduation ceremony and last day of school would have looked like exactly at the old charter. I know that the final week of school was a special "Week of Service" -- which turned out to be having the students help pack away classroom materials to prepare from the move from our co-located school (sharing with a district elementary school) to our own building.

The graduation was on the Friday before that "Week of Service." I know that the eighth graders went on a field trip to Magic Mountain after the ceremony. Today's school ordinarily has a grad trip to Knott's Berry Farm, but that was cancelled due to the pandemic. It was replaced by yesterday's day of outdoor activities and games on the field.

I didn't see much of these activities, since I had to spend most of the day inside the classroom to supervise the seventh graders. It's likely that I would've had to do the same at the old charter school -- our charter was small without a lot of extra classified staff, and so someone would've had to watch Grades 6-7 while the eighth graders are enjoying their final week. In particular, I might have been cocooned in the classroom during the ceremony, and may not have been able to go to Magic Mountain.

Perhaps at the very least, the charter would have had something similar to today's clap-out, so that the Grades 6-7 kids and I could have seen some part of the celebration. And I could have seen the eighth graders at some point during the day so I could congratulate them.

All of this is, of course, describing what would have happened if I'd completed that school year. Given that in reality I didn't complete the year, I could have just attended the ceremony on my own -- then I wouldn't have had to worry about supervising Grades 6-7 since I was no longer an employee. After the ceremony, I found out that my predecessor teacher had attended the graduation -- and if he was there, then surely I could have been there.

So this is part of the reason that I wanted to watch today's ceremony -- I still feel guilty about not seeing my charter eighth graders graduate. In some ways, being there today makes up a little for my not being there four years ago.

Those eighth graders, of course, are now seniors who have their high school graduation. I have no idea where they are -- they might be at LAUSD high schools, or perhaps charter high schools (as they obviously attended a charter middle school). There's no way for me to know and find out. It would really make up for my absence four years ago if I could attend the graduations of some of those students, but I don't know where any of them are now.

Today is Sevenday on the Eleven Calendar:

Resolution #7: We sing to help us learn math.

Well, I definitely sing today -- there's no need to repeat which songs I performed.

Today on her Daily Epsilon on Math 2021, Rebecca Rapoport writes:

What is the ratio of the area of the outer hexagon to the area of the inner hexagon?

(Here are givens from the graph: The sides of the outer hexagon are all 7. The region between the two hexagons is divided into six right triangles, each with a congruent acute angle, and the side opposite the acute angles are all sides of the outer hexagon, namely 7.)

There are several ways to approach this problem. One way is to note that the six given congruent angles are all exterior angles of the inner hexagon. Since the exterior angles of a convex polygon add up to 360, each angle is 60, and so the six right triangles are all 30-60-90 triangles. This is enough to conclude that both hexagons are in fact regular hexagons.

The specific side length of 7 is irrelevant. Instead, we think in terms of 30-60-90, where those 7's are the sides opposite the 60. Each side of the inner hexagon is equal to the hypotenuse of the triangle (which is 2/sqrt(3) times the given side, which is a long leg) minus the short leg of the triangle (which is 1/sqrt(3) times the given side), so it's 1/sqrt(3) times the given side.

So each side of the outer hexagon is sqrt(3) times each side of the inner hexagon. By the Fundamental Theorem of Similarity, the area of the outer is sqrt(3)^2 times that of the inner hexagon. Therefore the desired area ratio is 3 -- and of course, today's date is the third.

I usually end my Day 180 post with my plans for next year. Well, here's one thing -- I don't plan to keep all three of my subbing districts. While subs often work in two districts, three is rare -- I only added the third district because they hired me as a long-term sub. But as of now, I haven't decided which two districts to keep.

And so my usual discussion about next year's blog calendar holidays and quirks is dropped. I don't know which districts I'm working in next year, so I can't say which calendar I'll follow on the blog.

The blog is now in summer mode. My first summer post will be next week at some point.

Cooking with the MTBoS (Day 179)

Today I subbed in a middle school cooking class. It's in my new district -- and yes, it's the same teacher whose class I covered back in my May 7th post. She teaches at two different schools -- one of which is the one where I completed my long-term assignment from September to January.

Subbing as late as Day 179 on the blog calendar is rare. Two years ago, I subbed on Day 176 -- until then, I'd never subbed past Day 175 before. There are usually no subbing jobs available. But this year, I've received calls from all three districts today. The suspicion is that it's due to the pandemic and distance learning schedules, including hybrid -- in the past, teachers would gut it through the last few days of school, especially when those are final exam days (as opposed to entrusting them to a sub). This year, teachers are so tired that they feel that school needed to be over a month ago, and these days in the 170's are just dragging on. So they take days off, even if it's finals -- at least one of the calls was for a English class that presumably has a final.

Today's teacher is in fact taking two days off, Days 179-180. Tomorrow's reason is simple -- her son is graduating high school and she wants to attend. As for today, it's because she's the ASB director at my long-term school -- she wants to spend the entire day on the field with the activity rather than spend time with her cooking classes. Keep this in mind as we go through "A Day in the Life" -- I always do this for multi-day jobs like today's:

8:10 -- I arrive at the first school, which isn't the one from my long-term.

Both schools are having special activities, but since the regular teacher only does ASB at my long-term school, she's less familiar with how the activity works at the other school. Indeed, my long-term school is having activities only for eighth graders as a graduation celebration, but this first school has something special for all middle school grades. And the seventh grade activity is right away, while the eighth grade day of fun doesn't start until after I depart this school.

All activities at both schools involve carnival games out on the field, with classes rotating so that they can play all of them. My job is essentially to deal with the students in whichever grade is not having an activity at a particular time.

Second period (and I explained in my May 7th post, as well as one from an earlier visit in February, why the periods are numbered strangely) is an advanced course, so most of the students are eighth graders. The handful of seventh graders depart for their activity. To avoid passing periods (which may be confusing as some teachers have eighth grade one period and seventh the next), the eighth graders remain in Period 2 until their own activity begins (around lunchtime).

Even those these eighth graders don't have an activity, they do have a special eighth grade graduation video to watch. But I can't get the sound to work, so I summon a tech guy. He works on it, but the sound is still on the quiet side.

After this. the regular teacher has set up some videos for them to watch. The assignment is to watch a video on an old Food Network competition on food trucks (from six years ago). Today's video is the semifinal round while tomorrow's is the finals. The students comment on which food truck is their favorite and submit it to Canvas, where it automatically receives a score of 10/10.

The day is confusing for other reasons. Yearbooks are distributed today, and some students still need to turn in textbooks to the bookroom. I have them do so while the tech guy is working on the video. I tell them that they can sign yearbooks while the eighth grade video is playing, but when the food truck video begins, they must put them away.

Another problem is what students who have opted out of hybrid are doing at this time. In this second period there is only one online student, who happens to be a seventh grader (all others are absent). But I already know in advance that the regular teacher is using the same Google Meet link for online students at her second school who aren't attending that school's activity (that is, my long-term school). This school starts later than my first school, so at first the seventh grade girl is alone in the Google Meet until those from the second school arrive -- and as soon as they do, she logs out. She's clearly confused by what's going on here. An argument can be made that since she's a seventh grader, she doesn't need to stay online (since if she were in-person rather than online, she would have attended the activity).

Then another teacher from the second enters the meet -- it's the head Math 8/Geometry teacher. She starts to play some online game with the online students from her second school, and even invites those from my first school to join in. But this is right around the time that the eighth grade video ends, and so I tell her that it's time to start the food truck video.

And then one in-person girl refuses to put her yearbook away and gets into an argument. She claims that she's already finished the video, even though Canvas doesn't show 10/10 for her (oh, and she also uses the problems with the sound system as an excuse). I tell her the story of my own days as a young eighth grader -- I was working in the school library, assigned to stack textbooks returned at the end of the year. When the librarian had to take a phone call, I stacked the books nearly to the ceiling. When he returned, he grew angry and threatened to prevent me from attending the eighth grade graduation ceremony. So meekly, I unstacked the books -- and I tell this girl that just as I nearly got in trouble at the end of the year, so can she (that is, we can prevent her from being allowed at the 8th grade activity).

While all of this is going on, another teacher arrives in the room. She's there to take over to watch this second period as it's almost time for me to depart for my second school. She can already tell which girl she needs to watch out for.

By the way, the song for today is another one from Ghostwriter (not Square One TV). It comes from the series finale, when the main three characters are about to have their eighth grade graduation (since these are, after all, eighth graders). The song doesn't really have a title, but I'll name it for the line that repeats in the pre-Chorus:

NOTHING'S GONNA STOP THE LOVE OF FRIENDS

1. After so many years that we've been together,

It's hard to say goodbye, and it's hard to say whether,

We'll stick together forever, but we've got to stay strong,

'Cause time marches on and we're moving right along.

But in years to come, if we remain far apart,

I'll always carry thoughts of you deep within my heart.

I'm feelin' like this, you know is never gonna end,

'Cause you know...

Pre-Chorus: Nothing's gonna stop the love of friends,

Nothing's gonna stop, nothing's gonna stop,

Nothing's gonna stop the love of friends,

Nothing's gonna stop, nothing's gonna stop.

Refrain: It's gonna be all right, all right now,

You'll always be true.

Nothing can come between us,

I'm always there for you (repeat).

2. That's the kind of thing that lasts your whole life through,

It will stand the test of time no mater what you do.

Years will pass when we go our separate ways,

For when we meet again, it's seem just like,

Yesterday,

When all my friends seem like they're here to stay.

But now we're grown, not just kids at play,

We got to live our own life in our own way.

Pre-Chorus: Nothing's gonna stop, nothing's gonna stop,

Nothing's gonna stop the love of friends,

Nothing's gonna stop, nothing's gonna stop.

Nothing's gonna stop the love of friends. (to Refrain)

10:00 -- Second period takes a snack break, and I leave the first school.

10:40 -- I arrive at the second school. where it is now third period. This is a regular cooking class with mostly seventh graders, who attend while the eighth graders have their outdoor activity.

The sound is much better here, and so it's easier for me to get these kids to work. Also, there is no yearbook distraction -- at this school, they don't get their yearbooks until after summer break begins (a problem at many schools during the pandemic).

Since these are seventh graders, I sing "Benchmark Tests" instead. Last week, the seventh graders took a district placement exam to determine whether they should take Algebra I or Math 8 next year.

11:35 -- Third period leads directly into tutorial, where these students remain. I take requests for extra songs during the tutorial, including "All About That Base and Height" (2D version) this time and "The Why of Pi" (even though Pi Day has long past).

12:10 -- The students leave for lunch.

12:50 -- Fourth period arrives. This is an eighth grade advanced cooking class -- by now, their activity is over for the day.

I go back to "Nothing's Gonna Stop the Love of Friends" for this group of graduating eighth graders.

1:40 -- Fourth period leaves. Fifth period is ASB -- which the regular teacher is running herself. The students are out on the field taking down the outdoor activity games. Thus my day ends here.

Today is Sixday on the Eleven Calendar:

Resolution #6: We ask, what would our heroes do?

In cooking class, the heroes are the ones who are running the food trucks on the video. Students write down which group is their favorite, so they are finding out what their culinary heroes are doing.

Day 179 is the post in which I typically link to other members of the MTBoS. And with my subbing today being in a cooking class, let me find sites related to both math and cooking.

John Rowe mentions a book called Classroom Chef, where teachers learn how to cook up lessons:

https://mrrowe.com/2016/07/28/exps-logs-the-mtbos-way/

And now that I'm on Twitter, I need to embrace the T part of the MTBoS, Math Twitter Blogosphere. In fact, a tweeter named Kathy H. wrote the following about two years ago:

https://twitter.com/kathyhen_/status/1155976351175725058

Ummmmmm.... we need a math cooking hashtag!! #mathandfood #MTBoSCooks #iteachmathandcook

It doesn't appear that these labels have caught on yet. Oh, and I can't write about math and cooking without mentioning Eugenia Cheng, whose first book How to Bake Pi is about math and cooking. She's not technically a member of the MTBoS, but she does have a Twitter presence:

https://twitter.com/dreugeniacheng?lang=en

SBAC Practice Test Questions 31-32 (Day 176)

Today I subbed in an eighth grade English class. It's in my first OC district. Since it's a middle school class, I will do "A Day in the Life" today.

8:30 -- First period arrives.

These students have a quiz on Latin/Greek roots. They begin with a Warm-Up game on Kahoot (45 review questions) before the actual test. The regular teacher is on campus today -- she takes today off for a grading day, which is why I'm here.

The song for today is "Ratios" -- uh, make that "Ratio Rap." I wrote in earlier posts that after I wrote that 12EDL song, I disliked it and promised that I would change it to a rap. And so I finally fulfill that promise today.

9:35 -- First period leaves and second period arrives.

This class is a bit louder than first period. Indeed, this often happens on game days, whether it's this Kahoot game or my old Conjectures/"Who Am I?" game. And indeed, some students type inappropriate nicknames for themselves, so I must ask Kahoot to select the names myself.

10:40 -- Second period leaves for break -- and this leads into third period conference.

11:55 -- As often happens in middle schools in this district, I have supervision duty when it's time for the students to go home.

12:05 -- My supervision duty is complete. Since the regular teacher is here, she takes over for the academic support, and so this concludes my teaching day.

Today is Elevenday on the Eleven Calendar. So far, I don't have many opportunities to communicate, especially not with my fellow staff members (except for the regular teacher who make sure that everything is set up for me). But that's about to change:

1:05 -- I arrive at my long-term school. After all, I still receive email messages from them -- and one of these emails revealed that there is a barbecue for teachers today. The two schools aren't that far from each other, and so once I find out that I don't need to stay for academic support, I drive directly from the school I'm covering today to my long-term school.

And that's where communication comes in on this Elevenday. While I do wave to my former students, I talk to my fellow teachers during the barbecue meal. I speak to the teacher whose class I covered in the fall as well as several others, including the Math 8/Geometry teacher who is doing all of her classes from home -- apparently she, like me, also comes to school for the barbecue only.

At my first OC district school, I learn that the eighth graders will get a drive-thru graduation ceremony, but my long-term school is planning something quite different for the kids who are getting ready to move into high school.

Question 31 of the SBAC Practice Exam is on right triangle trigonometry:

Consider this right triangle.

[The right angle is at S, ST = 21, and RT = 35.]

Determine whether each expression can be used to find the length of side RS. Select Yes or No for each expression.

                 Yes  No
35 sin(R)
21 tan(T)
35 cos(R)
21 tan(R)

Let's first look at the two involving 35 times something R. We notice that 35 is the hypotenuse and relative to angle R, the desired side RS is the adjacent side, so we need the cosine. Thus 35 cos(R) is yes, while 35 sin(R) is wrong.

Now we check out the 21 times tangent something. We see that the desired side RS is opposite angle T while 21 is adjacent to it. So the tangent of T is RS/21. Thus 21 tan(T) is yes, while 21 tan(R) is no.

Both the girl and the guy from the Pre-Calc class answer the first three parts correctly. But the guy leaves the last part blank while the girl answers it correctly. Since the guy knows that 21 tan(T) is yes, it's most likely an accidental omission on his part.

Question 32 of the SBAC Practice Exam is on the graphs of quadratic functions:

Given the function
y = 3x^2 - 12x + 9,

• Place a point on the coordinate grid to show each x-intercept of the function.
• Place a point on the coordinate grid to show the minimum value of the function.

To find the x-intercepts of this parabola, let's factor the function:

y = 3x^2 - 12x + 9

y = 3(x^2 - 4x + 3)

y = 3(x - 1)(x - 3)

So the x-intercepts are at (1, 0) and (3, 0). Since the function vanishes when x is 1 or 3, it follows that the axis of symmetry, hence the vertex, is at x = 2. There are other ways to find the vertex, but it's best just to take the mean of the x-intercepts if we've already found them.

Let's now find the y-value of the vertex:

y = 3x^2 - 12x + 9

y = 3(2)^2 - 12(2) + 9

y = 12 - 24 + 9

y = -3

Therefore the vertex is at (2, -3). The SBAC only requires students to plot these three points -- once again, I'm not sure whether the full parabola can be graphed on the computer interface.

The girl from the Pre-Calc class correctly graphs these three points and tries to graph a parabola, even though her graph looks more like a V-shape. But the guy, unfortunately, makes an error in factoring:

y = 3x^2 - 12x + 9

y = 3(x^2 - 4x + 3)

y = 3(x - 4)(x + 1)

and so his x-intercepts are at (4, 0) and (-1, 0). In other words, he graphs y = 3(x^2 -3x - 4) instead of the correct graph. This counts as both a sign error as well as confusing the needed sum and product during factoring.

So what does the guy do for his vertex? For the x-value, it appears that he wants to make his parabola look symmetrical. The mean of his two x-intercepts is 1.5. But the vertex he draws ends up being closer to x = 2, which is unwittingly the correct value. He seems to choose a random value of y -- his vertex is at (2, -4), one unit below the correct vertex of (2, -3).

Ironically, the guy's graph actually looks more like a parabola than the girl's graph. But the girl is the one who correctly find the intercepts and minimum.

SBAC Practice Exam Question 31

Common Core Standard:

CCSS.MATH.CONTENT.HSG.SRT.C.8
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

SBAC Practice Exam Question 32

Common Core Standard:

CCSS.MATH.CONTENT.HSF.IF.C.7.A
Graph linear and quadratic functions and show intercepts, maxima, and minima.

Commentary: The trig problem should be straightforward provided that the students know the definitions of the trig ratios. The parabola graphing will be tricky for the students who have trouble factoring the quadratic function.

SBAC Practice Test Questions 29-30 (Day 175)

Today I subbed in a high school math class. It's in my LA County district. Since it's a math class, I'll definitely do "A Day in the Life" today.

8:30 -- Second period arrives. This is the first of two Pre-Calc classes.

These students are learning about polar coordinates and graphs. I've mentioned this on the blog -- there seems to be more emphasis on polar graphs now than when I was a young Pre-Calc student. I see some old colorful graphs on the walls -- these were created by students from the pre-pandemic year. It appears that while this might be interesting, this isn't a pandemic-friendly activity.

Choosing a song for this class is tricky -- I don't have very many songs for higher math (that is, Algebra II and above). I end up choosing "U-N-I-T Rate!" with the justification that some of my students are seniors who are a few months away from college hence the UCLA fight song parody. Then again, it's technically an honors Pre-Calc class, so there are many more juniors than seniors in this class.

9:40 -- Second period leaves for snack break.

9:55 -- Fourth period arrives. This is an Algebra I class.

Like many Algebra I kids, these students are currently solving quadratics. And I've also mentioned on the blog that as a young Algebra I student, the Quadratic Formula was the last lesson of the year. These students have a Chapter 9 Test coming up (before, presumably, finals week). Since the test is soon, I call on the in-person students to solve one equation from the practice test on the board.

This time, there's an obvious song for me to perform -- "Quadratic Weasel."

11:05 -- Fourth period leaves for second snack.

11:20 -- Sixth period arrives. This is the second of two Pre-Calc classes.

Both Pre-Calc classes have a large number of absent students. I wonder whether there's an AP exam going on -- obviously it wouldn't be the AP Calculus exam for these Pre-Calc kids, but it could be US History or another exam that juniors are more likely to take.

Because of the absences, there's only one in-person student (and a handful of opted-out kids). I tell her that she just happens to have the same first name as the lone in-person girl in one of my long-term subbing classes (the Math Skills class during my final week there).

12:30 -- Sixth period leaves. As it turns out, the regular teacher hasn't set up the Zoom for academic support, which means that I don't need to stay for it.

Today is Tenday on the Eleven Calendar:

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

In the Pre-Calc classes, I remind the students that they must finish taking notes from the lesson before they can log off after their half-hour of Zoom.

Today on her Daily Epsilon on Math 2021, Rebecca Rapoport writes:

Find x.

(This is another problem where all the givens are in an unlabeled diagram -- and it's a transversal problem, so it's especially tough to describe and visualize. One of these days, I need to come up with a better way to describe such problems. All that matters is that two lines are cut by a transversal, and one of the obtuse angles formed is 154 degrees and one of the acute angles formed is x. They aren't exactly same-side interior angles as one of them is exterior.)

As we approach finals week, all of the recent Rapoport problems plus our SBAC questions can make up our own finals review.

The important thing here is that x + 154 = 180, so x = 26. Therefore the desired angle is 26 degrees -- and of course, today's date is the 26th. (The diagram also contains an irrelevant 87-degree angle.)

Actually, this week is already finals week -- League Finals week, that is, for Track. The last day that leagues can hold finals is this Friday, but my alma mater's league has always held Prelims a few days earlier, on a Wednesday.

I attend my League Prelims today -- well, since this is still technically "A Day in the Life":

4:00 -- The 800's at League Prelims begin.

I try to identify certain runners, but it's tough because the announcer gives confusing information concerning how many heats there are in each race. In particular, he announces that there are three heats for the boys Frosh Soph 800, but the team member whose season we've been following on the blog (only because his times are similar to my PR's) appears to be in the third race (and he's a junior, so he can't be in the Frosh Soph race). And indeed, his time appears to be around our usual 2:20-something.

The winning time in the first few 800's (the Frosh Soph races) is around 2:12, with top times in the other races approaching 2:00. Once again, it's difficult for me to tell how many runners in each race advance to the League Finals on Friday due to the announcer's confusing info.

Returning to my COVID-97 What If? story, notice that I did say I'd focus on the 1600, not 800. I don't see any 1600 race today -- and if I recall from my actual season, the 1600 and 3200 are run only as finals, not prelims, and so the longest distance races won't be contested until Friday.

We'll continue discussing my COVID-97 Track season in Friday's post, since that's the day of the race.

Let's get back to SBAC Prep/Final Exam Prep.

Question 29 of the SBAC Practice Exam is on angles of elevation:

Emma is standing 10 feet away from the base of a tree and tries to measure the angle of elevation to the top. She is unable to get an accurate measurement, but determines that the angle of elevation is between 55 and 75 degrees.

Decide whether each value given in the table is a reasonable estimate for the tree height. Select Reasonable or Not Reasonable for each height.

                Reasonable  Not Reasonable
4.2 feet
14.7 feet
24.4 feet
33.9 feet
39.1 feet
58.7 feet

Once again, let's use h for the height again. Angle of elevation problems usually depend on the tangent ratio, where h is the height and 10 is the distance to the tree:

tan theta = h/10

But we don't know what the angle of elevation theta is, except that it's between 55 and 75. So let's try solving the problem for both of the extreme values:

tan 55 = h/10
h = 10 tan 55
h = 14.3 feet

tan 75 = h/10
h = 10 tan 75
h = 37.3 feet

And the true height of the tree can be anywhere in between. We thus choose Reasonable for Emma's three heights in this range -- 14.7, 24.4, and 33.9 feet -- and Not Reasonable for her other three -- 4.2, 39.1, and 58.7 feet.

Both the girl and the guy from the Pre-Calc class correctly answer this question. Both of them draw right triangles to help them. The girl writes her work for 55 degrees, but not 75 degrees, while the guy starts to use sine, then corrects himself to tangent. Most likely, both of them enter 10 tan 55 on their calculators, so it was obvious that they needed to enter 10 tan 75 without writing out the work.

Question 30 of the SBAC Practice Exam is on modeling with linear equations:

Emily has a gift certificate for $10 to use at an online store. She can purchase songs for $1 each or episodes of TV shows for $3 each. She wants to spend exactly $10.

Part A
Create an equation to show the relationship between the number of songs, x, Emily can purchase and the number of episodes of TV shows, y, she can purchase.

Part B
Use the Add Point tool to plot all possible combinations of songs and TV shows Emily can purchase.

Since each song is $1 and each episode is $3, it's clear that the equation is 1x + 3y = 10. Notice that I include the coefficient for x even though it is 1, because the SBAC interface for Part A requires a coefficient for both variables.

For Part B, there are four possible solutions -- (1, 3), (4, 2), (7, 1), and (10, 0). These solutions are discrete, but I suspect that the SBAC interface automatically connects the points to form a line -- the graph of the linear equation 1x + 3y = 10.

Both the girl and the guy from the Pre-Calc class correctly answer Part A. But the guy's graph isn't linear, because he miscounts and graphs (6, 1) instead of (7, 1). The girl's graph is linear. But both of them miss the solution (10, 0) -- which is valid, as Emily could have bought 10 songs and no shows.

SBAC Practice Exam Question 29

Common Core Standard:

CCSS.MATH.CONTENT.HSG.SRT.C.8
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

SBAC Practice Exam Question 30

Common Core Standard:

CCSS.MATH.CONTENT.HSA.CED.A.2
Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

Commentary: Both the tangent ratio and angles of elevation appear in Lesson 14-3 of the U of Chicago Geometry text. Meanwhile, Lesson 8-8 of the U of Chicago Algebra I text is called "Equations for All Lines." In that lesson, linear equations in standard form Ax + By = C are given, and it's stated that lines in standard form often arise naturally from real situations.

SBAC Practice Test Questions 23-24 (Day 172)

Today I subbed in a high school math class. It's in my LA County district. This is definitely the day to do "A Day in the Life":

8:30 -- Second period arrives. This is the first of two Algebra I classes -- except that they're classes for sophomores (or older students) who have previously failed Algebra I as freshmen.

The class is currently solving quadratic equations, including the Quadratic Formula. They have an assignment to complete on Quizizz (a familiar website to us) and will have a weekly quiz on Monday. The class has two or three TA's -- older kids who serve as mentors to the Algebra I students. I divide the class into breakout rooms on Zoom, so that the mentors can help their assigned students.

Since the lesson is on the Quadratic Formula, the obvious song to sing is "Quadratic Weasel." But the breakout rooms trip me up -- after attendance, I send the students directly to the breakout rooms, and then the mentors dismiss them directly from those rooms after the requisite half hour. Thus I end up singing "Quadratic Weasel" only to the in-person students.

9:40 -- Second period ends for snack break.

9:55 -- Fourth period arrives. This is the second of two remedial Algebra I classes.

After failing to sing to the Zoom students in second period, I plan on performing to this class right after attendance, before sending them to breakout rooms. But the plan is foiled when the microphone suddenly stops working again. I restart the computer, hoping that the mic would start up, but to no avail.

So then I just open up the breakout rooms -- hopefully, the mentors would have working mics, and the students can hear them. But then there's another problem -- one of the mentors is absent, and so students are sent to a breakout room without a leader (since this teacher already preset who's in which room).

Once I realize this, I frantically start assigning these students to a room with a mentor. But while I'm moving some of the kids, one girl just leaves Zoom completely. Imagine what this class looks like from her perspective -- first I'm restarting the computer and logging in and out of Zoom, and then I'm sending her to a breakout room with no leader. And all along she can't hear me because of my mic. (I believe there is a way for me to use the chat feature with students in breakout rooms, but that would have taken time for me to figure out -- and time was precisely what I didn't have.) So I can see why she'd simply give up, believe that there's no class, and log off of Zoom.

I mark three students absent -- the mentor, the girl who logs off early, and another guy after another mentor informs me that he logged off early without doing any work. I leave a note for the regular teacher to explain what's going on with these particular students.

There's one more problem with the faulty mic and breakout rooms in this period. About a half hour into the class, a dean comes in wanting to speak to one of the students, so I send him to the breakout room to which this student has arrived. He sees the student he wants there, and so he returns and tells me to place the two of them in a breakout room. I realize that I can just place them in the breakout room of the absent mentor since no one's there now. But unfortunately, the few seconds it takes for me to figure this out is just enough time for the mentor of that room (the one the desired student is already in) to dismiss the students, "Have a nice weekend!" as they just now complete their half hour Zoom requirement.

Once again, typing into the chat would have taken too long anyway. Instead, I go up to one of the mentors who is in-person, and use her Chromebook to explain to the dean what is going on.

During all of this, the other in-person mentor finally figures out how to fix my mic -- but by now it's too late, since most students are getting ready to log out of Zoom. I thank him -- at least now I know how to make sure my mic is working in sixth period. (I wonder whether his mic fix would have helped me for fifth period AP Physics back on Wednesday.)

So once again, only the in-person students get "Quadratic Weasel."

11:05 -- Fourth period leaves for second snack.

11:20 -- Sixth period arrives. This is a Geometry class.

Yes, it's our favorite class on this blog. These students also have a Quizizz to work on, as well as an Edpuzzle to complete over the weekend to prepare for their Monday quiz.

I double-check my mic, and there are no mentors in this class, hence no breakout rooms. Thus I'm able to sing to all the students, in-person and online. The Geometry lesson today is on volume, and so you know what that means -- "All About That Base and Height."

12:30 -- Sixth period leaves for lunch, followed by academic support (which isn't worth describing here on the blog).

Today is Fiveday on the Eleven Calendar:

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

The Algebra I repeaters definitely have negative feelings toward math. I'm hoping that they can look up to the mentors as heroes -- these older students are great at math, surely well enough to help them. I can see that they have a strong grasp of the Quadratic Formula. The presence of the mentors means that I don't teach much math today, except to sing the "Quadratic Weasel" song.

This is what I wrote two years ago about today's lesson:

Question 23 of the SBAC Practice Exam is on comparing rates:

Nina has some money saved for a vacation she has planned.

• The vacation will cost a total of $1600.
• She will put $150 every week into her account to help pay for the vacation.
• She will have enough money for the vacation in 8 weeks.

If Nina was able to save $200 a week instead of $150 a week, how many fewer weeks would it take her to save enough money for the vacation? Enter the result in the response box.

First of all, let's ignore the given value $1600 completely, since it has nothing to do with the solution of this problem. What we want to know is, how long will it take Nina to save the same amount of money at $200/wk. as she does at $150/wk. for eight weeks? Thus this is the equation:

200w = 150(8)

200w = 1200

w = 6

So it takes her six weeks to save the money as opposed to eight weeks. Therefore it will take her two fewer weeks (that is, 8 - 6) to save the money -- students should enter the number 2.

Sometimes students might forget that even though w = 6, they must enter the number 2, since the question is not how many weeks will she save, but how many fewer weeks. Sometimes I avoid this problem by letting x = 8 - w, so that as soon as I find x, I have the number I need to enter:

200(8 - x) = 150(8)

1600 - 200x = 1200

-200x = -400

x = 2

I often solve Pappas problems this way -- using the variable x only for the final value that I need to find rather than any intermediate values. Also, if there's a system of equation and Pappas asks us to solve for x, I might eliminate y even if x is easier to eliminate, so that I find the actual asked-for value more directly.

Both the girl and the guy from the Pre-Calc class have trouble with this question. In fact, the guy has only written:

150x = 1600

on his paper, with no final answer. The girl, on the other hand, has written a little more:

1600 = 150w

150(8) = 1600

200

and then she gives 6 weeks as her final answer. Yes, it does take Nina six weeks to save the money -- but that answer isn't justified by anything on the girl's paper. Actually, I'm starting to wonder whether I've attempted to help her during class, perhaps by telling either her only or the entire class that it takes eight weeks to save the money at the slower rate or six weeks at the faster rate. So she tries to include the numbers 8 and 6 somewhere on her paper.

Once again, I wonder whether there's anything to do to make this question easier for the students to understand and answer correctly. Earlier, I wrote that 1600 has nothing to do with the solution -- so let's restate the problem with the 1600 line left out:

Nina has some money saved for a vacation she has planned.

• She will put $150 every week into her account to help pay for the vacation.
• She will have enough money for the vacation in 8 weeks.

If Nina was able to save $200 a week instead of $150 a week, how many fewer weeks would it take her to save enough money for the vacation? Enter the result in the response box.

Now look at those first two lines again. The lady will save $150 per week for eight weeks -- that is, she will save a total of $1200. The problem is less confusing because we now know that the target amount is $1200, not $1600. No student would write an equation with 1600 at all, and so more students are likely to get the right answer.

But what is a teacher to do, since we can't make that 1600 line disappear from the computer we're using to administer the SBAC? Well, we can ask the students to look for extraneous information -- for example, we read the following to them out loud:

Nina has some money saved for a vacation she has planned.

• The vacation will cost a total of $1600.
• She will put $150 every week into her account to help pay for the vacation.
• She will have enough money for the vacation in 8 weeks.

Question for the class: how much money will she have saved in eight weeks?

The hope is that that point, the students see that 150 * 8 = 1200, so it's $1200. At this point, the teacher tells the students that so far, 1600 has nothing to do with the problem. The phrase "enough money for the vacation" is another way of saying $1200. This is not the same as the total cost of the vacation, $1600, since presumably Nina already has $400 in the bank before saving begins.

If Nina was able to save $200 a week instead of $150 a week, how many fewer weeks would it take her to save enough money for the vacation?

And since the phrase "enough money for the vacation" is another way of saying $1200, we can rewrite the question as:

If Nina was able to save $200 a week instead of $150 a week, how many fewer weeks would it take her to save $1200?

And now it becomes obvious that at $200 a week, it takes six weeks to save $1200. It now remains for the teacher only to emphasize fewer -- it's not how many weeks, but how many fewer weeks (than eight) it takes for her to earn the money. Therefore we must subtract 8 - 6 = 2.

Some people criticize the Common Core over wording in questions such as these. Indeed, yesterday and today we have two consecutive Questions 22-23 with the phrases "average rate of change" and "enough money for the vacation" -- and both are likely to trip students up.

But I'm not sure how this question can be improved. We might really be in a situation where we want to save money for a trip, but already have money in the account. Thus we must realize to consider that we already have some money, and so it's the remainder that we need to save.

Question 24 of the SBAC Practice Exam is on quadrilaterals:

Consider parallelogram ABCD with point X at the intersection of diagonal segments AC and BD.

Evelyn claims that ABCD is a square. Select all the statements that must be true for Evelyn's claim to be true.

• AB = BD
• AD = AB
• AC = BX
• Angle ABC isn't 90
• Angle AXB = 90

Here's some Geometry. I'll look at the three length equations first. AD = AB is obviously true, since the sides of a square must indeed be congruent. On the other hand, AB = BD is definitely false -- the diagonal of a square is sqrt(2) times the length of a side, not the same length as a side. And AC = BX is false as well -- the diagonals of a square (or any rectangle) are congruent, and so AC can't be half of BD (which is what BX is, as the diagonals of a parallelogram bisect each other).

Now let's look at the angle statements. The statement that ABC isn't 90 is obviously false -- the angles of a square must be exactly 90. The other statement that AXB = 90 is true, since the diagonals of a square are also perpendicular. So students must select two correct answers, AB = AD and AXB = 90.

Notice that if all Evelyn knew about her parallelogram is AB = AD and AXB = 90, that would not be sufficient for her to conclude that it's a square. A rhombus, after all, also has congruent sides and perpendicular diagonals. (In fact, given that ABCD is a parallelogram, just one of these two is sufficient for her to conclude that ABCD is a rhombus.) On the other hand, adding the falsity of the claim that ABC isn't 90 (that is, the truth that ABC = 90) to either of the two selected statements is enough for Evelyn to conclude that ABCD is a square.

Both the girl and the guy from the Pre-Calc class correctly answer this question.

Today is a Desmos activity day. Perhaps I ought to find a Desmos that would help students with Question 23 -- find one that has a word problem with "how many fewer" in it. But of course I can't find any such activity.

So instead, I search for a Geometry Desmos on parallelograms. I found the following activity, created by Mark Kreie last year:

https://teacher.desmos.com/activitybuilder/custom/5e37be8bc801c66b02607322

This activity is in addition to the two posted worksheets below. I link to this Geometry activity since this is, after all, a Geometry blog, even though I spend most of today's post discussing Algebra I (first the Algebra I classes from today, and then the algebraic Question 23).

SBAC Practice Exam Question 23

Common Core Standard:

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

SBAC Practice Exam Question 24

Common Core Standard:

Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

Commentary: The first equation I wrote for w can be solved as early as Lesson 4-4 of the U of Chicago Algebra I text, although the distributive property in the x equation must wait for Lesson 6-8 a little later on. Meanwhile, the properties of rectangles, rhombuses, and squares are covered in Lesson 5-4 of the Geometry text.