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.