SBAC Practice Test Questions 33-34 (Day 177)

Today I subbed in a high school math class. I'm not sure whether it's really worth doing "A Day in the Life," since all classes are taking a quiz today. Well, I might as well do it, since it might end up being the last math class I sub this school year.

7:05 -- The day begins with a "first period class," which really means zero period.

Most of the classes are our favorite math class -- Geometry. But today's quiz is on probability. As I've mentioned on the blog before, statistics and probability are included in the Common Core Standards, but it's up to the states to decide how to divide the standards into high school courses. Here in California, statistics is part of Algebra I and probability is part of Geometry.

Indeed, I once wondered whether I should follow the California route on the blog and end the year with probability instead of our SBAC Prep. After all, Geometry classes (including today's) typically have mostly sophomores, with only a few freshmen and (SBAC-taking) juniors. So our Geometry need more lessons on probability than on SBAC Prep.

As the test ends, I consider singing a song on probability -- Square One TV's "Ghost of a Chance" -- but I don't. (By the way, last week on the day the other district was closed, I subbed in a seventh grade class that was also studying probability, and I did sing "Ghost of a Chance" that day.)

7:55 -- First period leaves and second period arrives. These students take the quiz, but one girl doesn't have enough time to finish the quiz. So of course I can't sing any song.

8:45 -- Second period leaves and third period arrives. These students take the quiz, and this time there's enough time for me to sing "Ghost of a Chance."

But I'm hoping to sing a second song from that show -- "Combo Jombo" -- since at least one of the questions is on combinatorics. Unfortunately, I've never written down the lyrics, and Barry Carter has the lyrics to some Square One TV songs but not the combinatorics song. And so I can't sing it.

9:40 -- Third period leaves for snack.

10:00 -- Fourth period arrives. These students take the quiz, but one girl doesn't have enough time to finish the quiz. So of course I can't sing any song.

10:45 -- Fourth period leaves. It's now time for an assembly, which is all about the arrival and distribution of yearbooks. Teachers are to lead their fifth period classes to the gym for the assembly.

But fifth period happens to be my regular teacher's conference period. Teachers with fifth period conference are supposed to supervise the outside of the assembly instead, to make sure that students aren't trying to ditch school during the assembly.

11:45 -- The assembly ends and fifth period proper begins. It is now my conference period, which extends into lunch.

1:30 -- Sixth period arrives. This is the only Algebra I class of the day.

These students are also taking a test, on the Quadratic Formula. One or two students enter the room already singing Quadratic Weasel, so of course I sing the song with them before the exam as part of the test review.

2:15 -- Sixth period leaves. Teachers with first period typically don't have a seventh period -- and this includes my regular teacher. And so my day ends here.

There's not much to say on classroom management, since it's all just making sure the students are quiet -- and not using cell phones until all students have finished the quiz.

Today on her Mathematics Calendar 2019, Theoni Pappas writes:

In the figure above the side opposite the _____-degree angle is the shortest.

All that's given is an unlabeled diagram, so let me label it. There are three non-overlapping triangles, ABC, ACD, and ADE, with the following angle measures:

Triangle ABC: BAC = 80, B = 22, ACB = 78
Triangle ACD: CAD = 70, ACD = 28, ADC = 82
Triangle ADE: DAE = 79, ADE = 31, E = 70

Of course, we don't find this answer by attempting to measure anything, since the diagram is clearly not to scale. (For example, the three angles at A add up to more than 180 degrees, yet on the diagram they appear to add up to less than 180.)

Instead, we are to use the Unequal Angles Theorem of Lesson 13-7:

Unequal Angles Theorem:
If two angles of a triangle are not congruent, then the sides opposite them are not congruent, and the longer side is opposite the larger angle.

We notice that in Triangle ABC, the side opposite 22 degrees (AC) is the shorter than any other side of that triangle. But it's not the shortest side of the entire figure. That's because in Triangle ACD, the side opposite 29 degrees (AD) is shorter than any other side of that triangle, including AC. But it's still not the shortest side of the entire figure. That's because in Triangle ADE, the side opposite 31 degrees (AE) is shorter than any other side of that triangle, including AD.

Thus AE is the shortest side of all, and this is the side opposite 31. Therefore the desired angle is 31 degrees -- and of course, today's date is the 31st.

Pappas gives this sort of question on her calendar at least once per year. She will always ask the question such that an answer can always be determined -- that is, it will always involve the sides common to two triangles (AC and AD above). For example, she won't ask for the longest side of the figure because we can't find it. The longest side of Triangle ADE is DE (opposite the 79) and the longest side of Triangle ACD is AC (opposite the 82). So we know both DE > AD and AC > AD, but there's no way to compare AC and DE -- without letting one side be 1 and then setting it up with the Law of Sines. If 70 and 79 were reversed, the longest side can be found without trig.

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 -- and since there is very little Geometry in either of today's problems, let me add some Geometry to the activities.

One of the activities is based on today's Pappas problem. I decide to interchange two of the angles as I mentioned above -- now Angle E = 79 and DAE = 70. This allows both the shortest and the longest sides to be determined.

The other activity is a probability problem -- since here in California, probability must be included in the Common Core Geometry course. It's actually Question 7 of the quiz from today's subbing.

Indeed, I decide that I will also post Questions 8-12 from both of today's assessments. (Neither one is under copyright from a textbook.) The Geometry quiz starts with the aforementioned combinatorics question followed by some deck of cards probability questions. The Algebra I test includes the Quadratic Formula and completing the square.

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.