SBAC Practice Test Questions 17-18 (Day 169)

This is what I wrote last year about today's lesson:

Question 17 of the SBAC Practice Exam is on inscribed angles:

Use the circle below to answer the question.

The circle is centered at point C. Line segment PQ is parallel to SR. What is the measure of angle QPS?

(Other givens from the diagram: angle QRS = 68, quadrilateral PQRS is inscribed in the circle.) 

A) 68
B) 112
C) 136
D) 158

For this Geometry question, This is a job for the Inscribed Angle Theorem. (Now you can see why I said that the Inscribed Angle Theorem is the only part of Chapter 15 that actually appears in any SBAC or PARCC questions.)

Angle QRS is 68, so arc QPS is 2 * 68. Since a full circle is 360 degrees, arc QRS is 360 - 2 * 68. So by using Inscribed Angle again, angle QPS is (360 - 2 * 68)/2 = 180 - 68. This is 112 degrees, and so the correct angle is B).

By the way, I intentionally avoided arithmetic until the last step, 180 - 68. If angle QRS had been x, then QPS would be 180 - x. The opposite angles of an inscribed quadrilateral are supplementary, and this is essentially the proof. The givens that two of its sides are parallel (i.e., that the quadrilateral is a trapezoid) and that SR is a diameter are irrelevant.

Both the girl and the guy from the Pre-Calc class correctly answer B) for this question. But neither of them show their work completely. It appears that both students are simply assuming that PQRS is an isosceles trapezoid -- a correct but unproven assumption. (We're given that PQRS is a trapezoid, but not that it's isosceles.)

Question 18 of the SBAC Practice Exam is on graphing equations:

Choose the ordered pair that is a solution to the equation represented by the graph.

(The graph shows the equation y = (2/3)x + 2, but only the graph is given, not this equation.)

A) (0, -3)
B) (2, 0)
C) (2, 2)
D) (-3, 0)

Since only the graph is shown, we must look at the graph to see which of these points appears to lie on the graph. The first three choices clearly miss the graph. Point D) is difficult to tell since the step size on the graph is 2 rather than 1, but since none of the others are correct, D) is the answer.

This reminds me of the awkward method that Lesson 3-2 of Glencoe and the first lesson in Edgenuity use to solve equations. We're given an equation like (2/3)x + 2 = 0, and we're asked to solve it by graphing the function f (x) = (2/3)x + 2 and seeing where it crosses the x-axis! I'd argue that it's easier just to solve the equation algebraically than to graph it. And suppose the answer wasn't -3, but something like -2.5 or -3.5. That would be hard to discern on the graph! At least here on SBAC, the equation is already graphed, as opposed to being given a (one-variable) equation and expected to solve it by graphing in the coordinate plane.

Both the girl and the guy from the Pre-Calc class correctly answer D) for this question. But the guy appears to have crossed out A) before marking D).


Today is an activity day. I choose Exploration Questions from the two sections (Geometry Lesson 15-3 and Algebra I lesson 3-7) f the U of Chicago text listed below. The Geometry question isn't much (about camera angles), but Algebra I contains some interesting questions. One is about calendars -- which I just happened to write about earlier this week.

SBAC Practice Exam Question 17

Common Core Standard:

Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

SBAC Practice Exam Question 18

Common Core Standard:

Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

Commentary: The Inscribed Angle Theorem is in Lesson 15-3 of the U of Chicago Geometry text, and it's the only Chapter 15 Theorem that appears on the SBAC. Meanwhile, Lesson 3-7 of the U of Chicago Algebra I text is arguably related to this standard. This standard might be confusing if students are required to learn about this before solving equations algebraically, which, unfortunately, is the way Edgenuity presents this material.

For future reference, here is the following Common Core Standard, which is also relevant here:

CCSS.MATH.CONTENT.HSA.REI.D.11
Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear....