SBAC Practice Test Questions 1-2 (Day 164)

Today I subbed in a Physics class. There are three sections of Honors Physics and two sections of regular Physics. Both classes are learning about electricity and magnetism -- honors is studying resistors in series and parallel circuits, while regular is learning about capacitance. Since this isn't a math class, there is no "Day in the Life" (even though math is needed to determine some answers, as is typical for Physics).

Today on her Mathematics Calendar 2018, Theoni Pappas writes:

P is the circle's center.

And not only are all the other givens diagrammatic, but also no point is named other than P. So let me name four other points on the circle -- A, C, B, D (in clockwise order). Then here are the rest of the givens, plus the goal, converted to symbolic form:

P is between A and B, angle PCA = 76, arc BD = 30, angle ABC = x. Find x.

(By the way, I chose the clockwise order A, C, B, D rather than A, B, C, D so that the angle to find in the goal has its vertices in order, ABC.)

It's clear that the Inscribed Angle Theorem of Lesson 15-3 (the chapter we just concluded) must be involved somehow. There are several ways to use the theorem. We notice that Triangle APC is isosceles, since two of its sides, PA and PC, are radii. So by the Isosceles Triangle Theorem, we find that angle PAC = angle PCA = 76.

Since A-P-B, angle PAC is the same as angle BAC = 76. Also, since A-P-B, AB is a diameter, and so arc ACB is a semicircle. Thus angle ACB is a right angle, so triangle ABC is a right triangle. One of the acute angles of this triangle is angle BAC = 76, so the other acute angle is ABC = 90 - 76 = 14. So therefore x = 14 -- and of course, today's date is the fourteenth.

Notice that one of the givens -- arc BD = 30 -- has nothing to do with the solution. It's very tempting to assume that arc AC = arc BD = 30, and then use the Inscribed Angle Theorem to conclude that angle ABC is 15 degrees. But the assumption that arcs AC and BD are congruent is unwarranted.

Today is the first day of my annual review for Common Core state testing. This year, I've decided to post the SBAC released test questions for high school. Notice that SBAC, unlike PARCC, only tests once in high school, and so the test doesn't distinguish between Algebra I (or II) and Geometry. But fortunately, this is a blessing in disguise -- the Algebra I questions on the SBAC might be able to prepare me, as a teacher, for my upcoming summer Algebra I class. (But don't worry -- Pappas has several Geometry questions this week, so there will still be Geometry in these posts.)

By the way, you might wonder how beginning my test review today actually fits the district whose calendar the blog is following. Well, in all high schools, this is the second week of AP testing, and the tendency in most schools is to avoid giving the SBAC during AP time. Thus, unfortunately, most high schools have already given the SBAC. I would have needed to given the last tested lesson (Lesson 15-3, used for today's Pappas question) by Day 139 in order to beat the SBAC. (The English test is given even earlier -- it was still the third quarter when it was given.) Reaching Lesson 15-3 by Day 139 is a bit tricky -- but of course I would have tried much harder to reach it if I were teaching a class with many juniors (the tested grade).

Also, notice that following the digit pattern with the new Third Edition of the text might prepare students for SBAC better than my old Second Edition. There are only 14 chapters, and the Inscribed Angle Theorem appears much earlier in the text (Lesson 6-3.) And most of what remains in Chapter 14 on circles doesn't appear on the SBAC, so it can be saved for after state testing. Anyway, let's begin our SBAC review.

Question 1 of the SBAC Practice Exam is on factoring:

1. Select the equation that is equivalent to (m^2 - 25).
A) (m^2 - 10m + 25)
B) (m^2 + 10m + 25)
C) (m - 5)(m + 5)
D) (m - 5)^2

Yes, this is an Algebra I question -- but it's not a first semester Algebra I question. Recall that the summer Algebra I class that I'm teaching is the first semester only. As a rule of thumb, the first semester of Algebra I is linear and the second semester is nonlinear. Thus as soon as we see an exponent, we know that it's from the second semester of Algebra I (if not Algebra II).

Notice that two of the answers, A) and B), aren't even logical. All we did there is add an extra term, either -10m or +10m, for no apparent reason. So I hope most students will choose either C) or D).

And of course, the key is factoring the difference of squares. Choice D) isn't a correct factorization of the difference of squares -- in fact, choice D) is the factorization of choice A). And so the correct answer is C).

2. Select an expression that is equivalent to sqrt(3^8).
A) 3^(1/4)
B) 3^3
C) 3^4
D) 3^6

Again we have an exponent, so this isn't a first semester Algebra I problem. Indeed, I suspect that rational exponents -- the idea that the nth root of x^m is x^(m/n) -- doesn't appear until Algebra II.

Once we define rational exponents, the question is easy -- sqrt(3^8) = 3^(8/2) = 3^4. Therefore the correct answer is C).

We know that the SBAC tests up to Algebra II. The idea is that most freshmen start with Algebra I, which gets them to Algebra II by the year of the test. Of course, traditionalists are upset that there's no Calculus on the test (but even if Common Core encouraged eighth grade Algebra I, that's only Precalculus by the time of the 11th grade test). Some traditionalists take it a step further and don't even accept the level of Algebra II on the SBAC, calling it "pseudo-Algebra II."

Anyway, our first two questions are beyond first semester Algebra I. Oh well -- let's hope that first semester Algebra I appears in tomorrow's questions.

SBAC Practice Exam Question 1
Common Core Standard:
Use the structure of an expression to identify ways to rewrite it. For example, see x4 - y4 as (x2)2 - (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 - y2)(x2 + y2).

SBAC Practice Exam Question 2
Common Core Standard:
Rewrite expressions involving radicals and rational exponents using the properties of exponents.

Commentary: In Question 1, students are likely to confuse the difference of squares with a perfect square trinomial. These special factoring methods are studied in Lessons 10-7 and 10-9 of the U of Chicago Algebra I text. In Question 2, students are likely to forget the definition of rational exponents and how it can be used to solve the problem. Also, they might forget that a root without an index is a square root, with index 2.

By the way, let me mention one more closely related Common Core Standard:

CCSS.MATH.CONTENT.HSA.CED.A.4
Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm's law V = IR to highlight resistance R.

I mention this one only because I subbed in a Physics class today, and many of the Honors Physics students actually use the rearranged formula to calculate resistances today.