SBAC Practice Test Questions 15-16 (Day 168)

Today on her Mathematics Calendar 2019, Theoni Pappas writes:

Find the diameter of this circle.

[Here is the given info from the diagram: a circumscribed triangle of side lengths 24, 32, and 40.]

There are two things going on in this problem. First of all, this is a common Pappas problem where we have two tangent segments to a circle from a common point. These segments are congruent. (We had a similar problem on her calendar recently, but I never posted it because it was the weekend.) So we may call the two segments starting from one of the vertices r, two starting from the second vertex s, and two starting from the third vertex t. Each side of the triangle is the sum of two of these lengths:

r + s = 24
r + t = 32
s + t = 40

You might wonder why I chose the letters r, s, t instead of a, b, c or x, y, z. Well, here's why -- notice that 24-32-40 is a Pythagorean triple, and so the triangle is in fact a right triangle. Another theorem tells us that the radius and tangent at the same point of a circle are congruent. Thus the quadrilateral formed by the two radii and two legs has (at least) three right angles, hence a rectangle. And since two adjacent sides are congruent, it's indeed a square. Two of its sides are radii, and so all four sides are equal to the radius. That's why I chose the variable r to represent that length.

So now let's solve the system of equations. We begin by subtracting the bottom two, using the elimination method to get rid of t:

r + t = 32
s + t = 40

r - s = -8
r + s = 24

2r = 16

And we're already done, because all we need is the diameter, which is twice the radius! Therefore the desired diameter is 16 -- and of course, today's date is the sixteenth.

Notice that today's SBAC Prep is also about solving systems, albeit with just two variables. This is what I wrote last year about today's lesson:

Question 15 of the SBAC Practice Exam is on systems of equations:

A store sells new and used video games. New video games cost more than old video games. All used video games cost the same. All new video games also cost the same.

Omar spent a total of $84 on 4 used video games and 2 new video games. Sally spent a total of $78 on 6 used video games and 1 new video game. Janet has $120 to spend.

Enter the number of used video games Janet can purchase after she purchases 3 new video games.

Here is the system to solve:

4u + 2n = 84
6u + n = 78

Using the substitution method:

n = 78 - 6u
4u + 2(78 - 6u) = 84
4u + 156 - 12u = 84
-8u = -72

(By the way, if we use elimination instead, multiply the 2nd equation by 2 to obtain -12u - 2n = -156 which also leads to -8u = -72.)

u = 9
6(9) + n = 78
54 + n = 78
n = 24

So Janet purchases three video games for $72, leaving her with $48, enough for five used games. And therefore, the number students must answer with is 5.

Both the girl and the guy from the Pre-Calc class correctly answer 5 for this question. It helps that this is one of the questions I specifically show them in class.

Question 16 of the SBAC Practice Exam is on systems of inequalities:

Click on the region of the graph that contains the solution set of the system of linear inequalities.

y < (-1/2)x + 3
y > 2x - 2

The correct graph must be below the (-1/2)x + 3 (or the downward-sloping) line and above the 2x - 2 (upward-sloping) line. These intersect in the region to the left -- the correct answer to shade.

The girl from the Pre-Calc class correctly answers this question. But the guy only marks the point where the lines intersect -- which is the solution to the system of equations, not the system of inequalities as directed. Well, at least he knows that the solution to a system of graphed equations is the point where they intersect (as many students don't know this). On the actual SBAC, I'm not sure whether it's possible to plot a point, since the software might be automatically set up to shade one of the four regions instead. That's fortunate for this guy, since once he realizes that the correct answer is a region, he'll probably figure out the inequalities.

SBAC Practice Exam Question 15

Common Core Standard:

Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

SBAC Practice Exam Question 16

Common Core Standard:

Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

Commentary: In the U of Chicago Algebra I text, systems appear late -- not until Chapter 11, with systems of inequalities in Lesson 11-10 as well as systems of equations in the earlier parts of the chapter. In the Glencoe text, however, systems appear in Chapter 6. Nonetheless, systems are considered a part of second semester Algebra I. Make sure that the students know the difference between a system of equations and a system of inequalities.