SBAC Practice Test Questions 13-14 (Day 167)

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

Question 13 of the SBAC Practice Exam is on solving equations:

Emily is solving the equation 2(x + 9) = 4(x + 7) + 2. Her steps are shown.

Part A
Click on the first step in which Emily made an error.

Step 1: 2(x + 9) = 4(x + 7) + 2
Step 2: 2x + 18 = 4x + 28 + 2
Step 3: 2x + 18 = 4x + 26
Step 4: -8 = 2x
Step 5: -4 = x

Part B
Click on the solution to Emily's original equation.

-10.5  -6  -2  0  2  4.5  8

Ah, so this is definitely a strong first-semester Algebra I problem. We must check each step carefully to find an error. For Part A, we see that in Step 2, Emily must combine the like terms 28 and 2, but in Step 3, she writes the sum as 26. (Clearly, she was thinking of 28 - 2 instead of 28 + 2.) Therefore the first step that contains an error is Step 3.

For Part B, let's start from the last correct step (namely 2) and proceed with the sum 28 + 2 = 30:

Step 2: 2x + 18 = 4x + 28 + 2
Step 3': 2x + 18 = 4x + 30
Step 4': -12 = 2x
Step 5': -6 = x

So students must click on the value -6.

My students are likely to make mistakes like Emily's, and so I must be able to correct them. If Emily were a girl in my class, I could help her by praising her last correct step (namely 2) then and ask her, "What's 28 + 2?" Hopefully, she'll realize that it's 30, not 26, and then she can finish solving the equation.

Both the girl and the guy from the Pre-Calc class correctly answer Step 3 and -6 for this question.

Question 14 of the SBAC Practice Exam is on explicit and recursive functions:

Match each recursive function with the equivalent explicit function.

Recursive functions:
f (1) = 18; f (n) = f (n - 1) + 6; n > 2
f (1) = 18; f (n) = f (n - 1) + 12; n > 2
f (1) = 1; f (n) = 6f (n - 1); n > 2
f (1) = 1; f (n) = 12f (n - 1); n > 2

Possible explicit functions:
f (n) = 6^(n - 1); n > 1
f (n) = 12 + 6n; n > 1
f (n) = 12^(n - 1); n > 1
f (n) = 6 + 12n; n > 1

The first two recursive functions are arithmetic sequences, so they might conceivably appear in the first semester of an Algebra I class. The last two recursive functions are geometric sequences. These used to appear in Algebra II, but since advent of the Common Core, I've seen geometric sequences and exponential functions pushed down into the second semester of Algebra I.

Let's look at the two arithmetic sequences first. The first term of each sequence is 18, but in the first sequence the common difference is 6, while in the second sequence it is 12. The coefficient of n in the explicit function is the common difference, so we already know that the first sequence must match to the "+6n" function while the second matches to the "+12n" function.

Even though we don't need it, it's helpful to know how to determine the constant term in each of the explicit functions. Some texts tell the students that the explicit function is f (1) + (n - 1)d (where d is the common difference), and so they must expand (n - 1)d. The U of Chicago Algebra I text has the students work backwards to find a "zeroth term," and this is the constant. So in the first sequence, the zeroth term is 18 - 6 = 12, and in the second, it's 18 - 12 = 6. Therefore the first explicit function must be f (n) = 12 + 6n while the second is f (n) = 6 + 12n. 

Now we move on to the geometric sequences. This time, the trick is that the common ratio must be the base of the exponential function -- that is, f (1)r^(n - 1). In each case the first term is 1, so all that remains is r^(n - 1). Therefore the third sequence is f (n) = 6^(n - 1) while the last is f (n) = 12^(n - 1).


And as for the girl and the guy from the Pre-Calc class, neither one answers this question -- since for some reason, it didn't print on the SBAC Practice packet! At least this packet is much better than what I had at the old charter school two years ago -- where none of the SBAC Practice problems could print.

SBAC Practice Exam Question 13

Common Core Standard:

Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

SBAC Practice Exam Question 14

Common Core Standard:

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

Commentary: In the U of Chicago Algebra I text, the first lesson in which multi-step equations with both the Distributive Property and variables on both sides is Lesson 6-8, "Why the Distributive Property Is So Named." Arithmetic sequences appear in Lesson 6-4, "Repeated Addition and Subtraction," though they aren't called this. Meanwhile, although exponential functions are in Lesson 9-2, "Exponential Growth," geometric sequences don't appear at all.