Friday, May 7, 2021

SBAC Practice Test Questions 3-4 (Day 162)

Today I subbed in a middle school cooking class. It's in my new district -- and indeed, I've subbed for this class before. I first mention this class in my February 24th post.

Since it's a middle school class, it's worth doing "A Day in the Life" today:

8:10 -- As I explained in my February 24th post, this school begins with second period. (Middle schools in this district have different ways to number the periods during the pandemic.) This is the first of two advanced cooking classes for eighth graders.

Unlike February, this time the regular teacher has planned it so that no cooking occurs on a sub day. So instead, the students complete three assignments on Canvas. The current unit is on pasta -- apparently, the students have already cooked pasta earlier this week. The first assignment contains a video.

Unfortunately, I do have some problems with technology this period. This teacher likes to use her Chromebook that's connected to a projector, so that she can have the video be viewed by the in-person and opted-out students at the same time. She's already logged into her Chromebook, so that everything is already set up.

But for some reason, the projector doesn't display the Chromebook screen. All I see is a dark screen with an arrow that represents the mouse position. This class has an aide, but she can't figure out why the screen won't display either.

Moreover, I couldn't show the video to the opted-out students either. Whenever I try to play the video and then share the screen, Google Meet would close within a few seconds. And so I have no choice but to ask all the students, in the classroom and at home, to play the video on their own.

As for my song, I want to sing a special Mother's Day song. A few years ago, I found myself in a music class on Mother's Day Adam. I have an old piano songbook (the same one where I found that Kelly song to perform on St. Patrick's Day this year). It has two songs that contain "Mama" in their titles, and so I played both tunes on the piano that year. And so, even though this isn't a music class, I come in hoping to sing these as Mother's Day tunes.

But unfortunately, neither song is appropriate for school. In one of the songs, the titular mother drinks lots of alcohol. And in the other -- well, let's just say that the "Mama" in that song is a "Hot Mama." I do figure this out before starting to sing in the classroom.

I frantically search the songbook and find one that mentions a "Mama." There's also a "Daddy" in this song (but school will be out before Father's Day) -- it's "Daddy Sang Bass, Mama Sang Tenor." It's arguable that this song is also slightly controversial -- it has a religious theme. But religion here is much preferable to the content of the other two "Mama" songs.

9:05 -- Second period leaves and fourth period arrives. This is the first of two regular cooking classes mostly for seventh graders.

During the passing period, suddenly the screensaver kicks in, and I can't remove the screensaver without the teacher's password. (This is weird -- apparently the Chromebook had been on overnight without a screensaver or password, and now the short passing period is enough to turn it on.)

And so I shut it down and log in myself -- and then more strange stuff occurs. The microphone stops working -- but now the projector is working, and I can share my screen on Google.

Even though this is a different level, all classes have the same assignment. Thus I'm able to play the video and show it to both in-person and opted-out students. The sound is coming from the video, so no mic is needed here. If I need to communicate with the at-home students, I type it into the chat.

I perform "Daddy Sang Bass" again, except only the in-person students can here it. Then again, I often sing songs that only the in-person students can hear (after the Zoom kids have already logged out).

Today this class is much better behaved than they were on my February visit. I remind the students to be quiet and log into Canvas when they first enter the classroom.

9:55 -- Fourth period leaves for snack. During the break, I find out that a special Teacher Appreciation Week snack is being served in the lounge, and so I grab a bite to eat.

With the transition from Level 3 to Level 4 hybrid (that is, from two to four in-person days per week), the bell schedule at this school has changed. It was decided that while more students can fit inside the classroom, lunch should still have social distancing. So now there are two lunches (with the choice of first or second lunch based on classes, not the old hybrid cohorts). Tutorial used to be right after the lone lunch, but now it's after snack.

But all of this is moot, because I'm subbing for the cooking teacher who works at two different schools, and her second school is where I completed my long-term. And so while students at both schools are attending tutorial, I'm travelling from one site to the other.

11:20 -- Third period arrives. (Recall that the rearranging of the periods in January was done just to accommodate this travelling cooking teacher -- her first school retained the old period numbers, while my long-term school renumbered the periods 1-5). This is the second of two regular cooking classes.

At this school, the technology works much better. The Chromebook is properly connected to the projector, and the microphone and sharing screen features are working. This class also has an aide (the same aide I had in two of my five long-term classes), but I don't need her help this time.

Each time I visit my long-term school, I meet students I taught during those months. And in this period, I see a girl who was online during my entire long-term assignment, but has now opted in. Again, it's strange to see the live faces of the students I've only worked with online.

I ask her about her Math 7 class, and she tells me that she has a test today. It appears to have something to do with solving equations, even though I taught an equations unit during my long-term. And so in addition to "Daddy Sang Bass," I quickly add some of the "Solve It" song for the sake of this girl (and another student who is also taking her test today).

12:15 -- Fourth period arrives. This is the second of two advanced cooking classes.

I meet another girl, an eighth grader I taught online but is now in-person. Also, this class is when I fulfill today's resolution. Today is Saturday, the second day of the week on the Eleven Calendar:

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

I also catch a girl who starts to take out her cell phone. She must make a sacrifice by putting it away in order to be successful in this class.

1:05 -- Fourth period leaves for lunch. During the break, I find out that a special Teacher Appreciation Week lunch is being served in the lounge, and so I grab a bite to eat. That's right -- at the start of the week, I was wondering whether there would be special meals for staff this week. For most of this week I didn't see any such food, but then I make up for it in spades today.

1:45 -- Fifth period arrives. This is an ASB class.

As usual, the students have their own projects to work on, including posters. Indeed, this week there are Teacher Appreciation Week posters on the doors of all classrooms, celebrating the teachers as stars.

For this class, I like to sing an extra song in addition to "Daddy Sang Bass." This time I choose "Burger Pattern" -- this is the first time I perform any Square One TV Fat Boys raps at this school.

2:35 -- Fifth period ends, thus completing my day.

Question 3 of the SBAC Practice Exam is on solution sets:

Click the table to indicate whether each equation has no real solution, one real solution, or infinitely many real solutions.

  • 5/(20x) = 1/(4x)
  • 3x = 4 + 5x
  • sqrt(2x + 3) + 6 = 0
Of these three equations, only one is undoubtedly a first semester Algebra I problem. It's easy to solve this linear equation:

3x = 4 + 5x
-2x = 4
x = -2 (one real solution)

The first equation could be taught in first semester Algebra I. Even though it's not linear, it's easily transformable into a linear equation using cross multiplying, which may be studied in first semester:

5/(20x) = 1/(4x)
20x = 20x (infinitely many solutions)

Notice that the solution isn't "all real numbers" -- x = 0 can't be a solution, since x appears in the denominators in the original equation. But the SBAC question indicates "infinitely many solutions," which is not the same as saying that all real numbers are solutions.

The last equation is definitely not a first semester Algebra I problem. Notice that just last week, I subbed in an eighth grade Algebra I class that was studying radical equations -- and last week was clearly in the second semester:

sqrt(2x + 3) + 6 = 0
sqrt(2x + 3) = -6
2x + 3 = 36
2x = 33
x = 16.5

But let's check this answer:

sqrt(2 * 16.5 + 3) + 6 = 0
sqrt(36) + 6 = 0
6 + 6 = 0
12 = 0 (no real solution)

This is an extraneous root. In fact, it should have been obvious at the step sqrt(2x + 3) - 6 that the principal square root of (2x + 3) can never be -6.

Let me write out how exactly I'd teach solving the second equation to my students:

3x = 4 + 5x

Let's get all the x's on the left side and all the numbers on the right side. We want to get rid of the 5x on the right side, so instead of adding 5x, let's subtract 5x:

3x - 5x = 4 + 5x - 5x
-2x = 4

So this is -2 times x. But I don't want -2 times x -- I want just x. So instead of multiplying by -2, let's divide by -2:

-2x/-2 = 4/-2
x = -2

Both the girl and the guy from the Pre-Calc class correctly answer "infinitely many solutions" and "one real solution" for the first two parts. But both answer "one real solution" for the third, which as we see is incorrect. The girl shows her work -- she clearly falls for the extraneous root trick. Thus our students need more work checking for extraneous roots.

Question 4 of the SBAC Practice Exam is on adding and subtracting polynomials:

Enter an expression equivalent to (3x^2 + 2y^2 - 3x) + (2x^2 + y^2 - 2x) - (x^2 + 3y^2 + x) using the fewest number of possible terms.

This is definitely a second semester Algebra I problem. Let's take care of the subtraction first:

(3x^2 + 2y^2 - 3x) + (2x^2 + y^2 - 2x) - (x^2 + 3y^2 + x)
= 3x^2 + 2y^2 - 3x + 2x^2 + y^2 - 2x - x^2 - 3y^2 - x
= 3x^2 + 2x^2 - x^2 + 2y^2 + y^2 - 3y^2 - 3x - 2x - x
= 4x^2 - 6x

This is a straightforward combining like terms problem, except that all y^2 terms are eliminated.

Both the girl and the guy from the Pre-Calc class correctly answer 4x^2 - 6x for this question, although the boy also factors it to x(4x - 6).

Meanwhile, this year I'm supposed to be posting Desmos activities -- and that doesn't stop just because we're now reviewing for the SBAC. In the following Desmos, Julia Anker uses graphs to explain why some radical equations have extraneous solutions:


This activity can be given in addition to the two worksheets on today's SBAC problems. (I provide a Desmos for Question 3, not 4, since most students won't struggle with the latter.)

SBAC Practice Exam Question 3
Common Core Standard:
Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

SBAC Practice Exam Question 4
Common Core Standard:
Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

Commentary: Equations with variables on both sides appear in Lesson 6-6 of the U of Chicago Algebra I text, while proportions appear in Lesson 5-7. For the square root equation, my Pre-Calc students last week pointed out that -6 is also a square root of 36, yet they fail to notice x = 16.5 isn't a solution because the sqrt() symbols indicate the principal square root only. It's also possible for two wrongs to make a right here -- some students claim that 2x = 33 has no solution because it's impossible to divide an odd number by 2!


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