SBAC Practice Test Questions 33-34 (Day 177)

Today I subbed in an AP Economics class. It's in my new district. Since it's a high school class that isn't math, there's no need for "A Day in the Life" today.

All three classes I see today are AP Econ -- zero, first, and third periods. Just as in most Economics classes, the students are all seniors. Notice that I, as a young high school student, took AP Econ as junior. I was in the magnet program where social science was taught in the reverse order from the standard sequence, including semester Econ/Government in freshman year. This allows students to take the AP equivalents in Grades 10-12. I took AP Econ as a junior and AP Government as a senior. (In fact, I didn't join the magnet until junior year -- and so I had to take those AP classes, otherwise I'd have had neither class required to graduate.)

In this class, the students have an interesting project -- the Greebes project. Students work in groups to create a two-minute commercial for Greebes chips, which may have special properties, such as magically building your muscles, giving you great speed or good looks, or speaking Spanish. (I'm not quite sure of the origin of the term "Greebes" -- it appears to be a "metasyntactic" product used in Econ classes, just as "foo" and "bar" are used as variables in computer coding.)

As for today's song, it reverts back to "U-N-I-T Rate!" This is because, as is often the case in high schools, seniors don't need to attend Days 178-180. Thus this is the last day for seniors -- and to celebrate, students wear college jackets for the schools they'll attend this fall. (One girl even has a Yale jacket on -- and so I congratulate her for being a future Ivy leaguer.) And so I sing my UCLA fight song parody, even though the regular teacher appears to be a former USC Trojan.

For tutorial, I sing an extra song, which happens to be Square One TV's "Ghost of a Chance." By the way, it's rather profound that, for some of these seniors (specifically the ones in third period but don't have a fifth period -- perhaps because they also have a zero period instead), I am teaching the last K-12 class that they will ever have. And for the period zero students, this is the last time that they'll ever take a class that starts that early, as most colleges don't schedule classes before 8 AM.

Today is Friday, the first day of the week on both the Eleven and Gregorian Calendars:

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

This is something that I hope these seniors will take to heart, as they move on to take their math placement tests at their new college. I hope they will pass the test and place into credit rather than remedial math.

Today I arrived at the League Finals Track meet, in time for the 1600 final. Well, I was waiting to find out what time the last CIF qualifier needed to make, and I hear it from the announcer -- it's 4:31. As it turns out, only one runner from our league hits this time -- and he wasn't from my alma mater. Two more runners, including one from my school, finish a few seconds too slow.

Meanwhile, my own times from back then are much too slow -- and even though the COVID-97 is just a What If? story, I only want to include plausible times.

And so this officially ends the Track part of my COVID-97 story. I'll probably write another post about COVID-97 and all the other COVID What If? stories at some point.

Question 33 of the SBAC Practice Exam is on writing an equation:

Mike earns $6.50 per hour plus 4% of his sales.

Enter an equation for Mike's total earnings, E, when he works x hours and has a total of y sales, in dollars.

Well, the equation almost writes itself. Just read the first sentence out loud:

Mike earns (E =) $6.50 (6.5) per hour (x) plus (+) 4% (0.04) of his sales (y).
E = 6.5x + 0.04y

The tricky part of course is writing the decimals 6.5 and 0.04 properly -- especially the conversion of 4% to a decimal.

Both the girl and the guy from the Pre-Calc class write the correct equation -- with my help, during class that day.

Question 34 of the SBAC Practice Exam is on writing and solving a system of equations:

The basketball team sold t-shirts and hats as a fund raiser. They sold a total of 23 items and made a profit of $246. They made a profit of $10 for every t-shirt they sold and $12 for every hat they sold.

Determine the number of t-shirts and the number of hats the basketball team sold.
Enter the number of t-shirts in the first response box.
Enter the number of hats in the second response box.

If we let s be the number of shirts and h be the number of hats. Then the equations are:

s + h = 23
10s + 12h = 246

Let's solve this system by substitution:

s + h = 23
s = 23 - h

10s + 12h = 246
10(23 - h) + 12h = 246
230 - 10h + 12h = 246
230 + 2h = 246
2h = 16
h = 8

s + h = 23
s + 8 = 23
s = 15

Therefore the team sold 15 t-shirts and eight hats.

The girl correctly writes and solves the system by substitution as I show above. Unfortunately, the guy leaves this problem blank.

In fact, throughout these 34 problems, it appears that the guy more easily gives up and avoids trying to work out the more difficult problems. The girl is at least more willing to attempt them.

At first, it appeared that one of the guy's strengths is factoring, as he did well on the early problems that require him to factor. But yesterday, the girl factored the quadratic function correctly while the guy mixed up two of the terms.

Both students, of course, have already long completed the SBAC. I wonder how well these two students fared on the state test, but I most likely will never know.

Today is our last activity day. In the past I forced probability into the activity, since probability is part of the California Geometry curriculum. Here's an intro to probability by Lisa Emerson:

https://teacher.desmos.com/activitybuilder/custom/5ffd8fb43bb8230d4c9db01b

By the way, if you remember that Geometry class that I covered for a week in January (and continue to receive emails for the rest of this year), Chapter 12 of that class is probability. Indeed, the teacher gave her Chapter 12 Test last week. Once again, many students lost points from their computer score for failing to show work, but she also added some points, when a student entered a decimal where the computer expected a percent, or vice versa. (See -- I was able to tie today's Question 33 to today's Desmos after all.)

SBAC Practice Exam Question 33

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.

SBAC Practice Exam Question 32

Common Core Standard:

CCSS.MATH.CONTENT.HSA.CED.A.3
Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

Commentary: Both of these word problems should be straightforward, but the first is tricky for students who don't remember percents. Students are likely to give up on the conversion step and never reach solving the system for the last problem.

Monday is Memorial Day, and so my next post will be on Tuesday.

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 25-26 (Day 173)

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

In triangle ABC, AB = 6 and BC = 8. What is the maximum value of the area?

This is yet another one of those problems that straddle the line between Geometry and trig. If we use trig and teach the formula A = (1/2)xy sin(theta), then we know that the maximum value of sin is 1. Thus the max value of the area is:

A = (1/2)xy = (1/2)(6)(8) = (3)(8) = 24

So the desired maximum area is 24 -- and of course, today's date is the 24th.

Now we ask, is there a purely geometric way to solve this problem? Consider the following -- we begin with segment AB of length 6. Now draw a line parallel to line AB that is exactly 8 units away. The locus of all points 8 units from B (that is, the location of all possible points C) is a circle of radius eight. It is tangent to the new line. The maximum area thus appears at the point of tangency -- otherwise C lies in between the two lines and so the height (and thus the area, half the base times the height) is less than that said maximum.

This explanation is a bit off -- it could probably be made more rigorous. The Radius-Tangent Theorem -- that the radius to a point of tangency is perpendicular to the tangent line -- likely comes into play.

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

Question 25 of the SBAC Practice Exam is on inequalities:

A student earns $7.50 an hour at her part-time job. She wants to earn at least $200.

Enter an inequality that represents all of the hours (h) the student could work to meet her goal. Enter your response in the first response box.

Enter the least whole number of hours the student needs to work to earn at least $200. Enter your response in the second response box.

This is a first-semester inequality problem.

7.50h > 200

h > 26 2/3

So the inequality is 7.50h > 200 ("at least" = "greater than or equal to") and the smallest whole number value that satisfies it is 27 hours.

Both the girl and the guy from the Pre-Calc class fail to write the correct inequality. Both of them are confused by "at least" and write "7.50h < 200." Actually, the guy writes 200 > 7.50x, which is equivalent to the girl's inequality. The guy doesn't bother solving his inequality. The girl solves it by writing "at least 26.67" hours -- she missed the words whole number. Notice that no inequality needs to be written in the box -- all that's necessary is the single whole number 27.

Question 26 of the SBAC Practice Exam is on comparing statistics:

Michael took 12 tests in his math class. His lowest test score was 78. His highest test score was 98. On the 13th test, he earned a 64. Select whether the value of each statistic for test scores increased, decreased, or could not be determined when the last test score was added.

(The possible stats are standard deviation, median, and mean.)

Stats appears in the last chapter of Glencoe Algebra I, and so this is a second-semester question.

Anyway, the mean must decrease because Michael's last test score is lower than that of any previous test that he has taken. The standard deviation must increase when that low value is added. But we can't be sure about the median. The median of 12 values is the mean of the 6th and 7th value, but the median of 13 values is the 7th value (the old 6th value before the 64 happened). So median could decrease if the original 6th test is less than the 7th -- but the median could stay the same if the original 6th and 7th scores were equal. (It's impossible for the median to increase here!)

The girl from the Pre-Calc class gives the correct responses for the mean and median, but she incorrectly writes that the standard deviation decreased. Then again, most students are less familiar with standard deviation than with the measures of central tendency. But at least she tries -- the guy just leaves it blank.

This lesson was at one point given as an activity. Of course, our activity day this year will be later on.

SBAC Practice Exam Question 25

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 26

Common Core Standard:

Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.

Commentary: The inequality I wrote for h can be solved as early as Lesson 4-6 of the U of Chicago Algebra I text. Meanwhile, stats isn't covered in the text at all, except for Lesson 1-2 where mean and median appear, but not standard deviation.