## [Testing Testing] SBAC Grade 11 Practice Test #7

We’re starting a new category of post here called “Testing Testing”. Each post will analyze a released test item. While this would normally focus just on the mathematics of the problem, the SBAC and PARCC exams are also promising new formats and interfaces for testing, which can potentially have effects on how problems are posed, read, answered, and scored. Writing assessment items for these new interfaces will be especially challenging, and it may take more revisions than usual to build a high-quality test item.

Recently, SBAC (short for Smarter Balanced Assessment Consortium) released a set of Practice Tests, giving everyone the chance to study the testing environment and suggest improvements. I took the 11th grade mathematics exam, with 23 problems, trying to get a sense of the quality of the mathematics and the quality of the interface. Several posts will talk about items from the SBAC Practice Test, along with suggestions to SBAC on how to improve both the mathematics of the exam and the interface students will use when taking it.

Here’s Problem 7 from the SBAC Grade 11 Math Practice Test.

At first glance, this problem looks pretty innocent, and has a useful testing target, specifically HSA-REI.D.11, finding approximate solutions to f(x) = g(x) when f and g are given by tables. The test designers were even careful to use function types prescribed by the standard: there’s a quadratic rule matching f and an absolute value rule matching g. This could have been a good item, expecting students to interpret the table in a meaningful way.

However, there are two serious errors in the problem. First, the problem is never stated. The problem asks to “show the interval for x in which the solution to this system of equations lies”. But there is no system of equations. There’s not even an equation at all, just a table. To get this question right, the student must figure out the intended question, the interval for x that contains the solution to f(x) = g(x). Note that this still isn’t a system of equations, just a single equation with a single variable.

Second, and even more serious, there is more than one correct answer to the problem. The only thing we are told about functions f and g is that they are continuous. We are not told that they are quadratic, absolute value, or otherwise. We cannot assume that the functions do not intersect in other intervals: in fact, they can easily intersect. These two functions match the given tables and intersect many more times:

$f(x) = x^2 - 5 + 100 \sin (2 \pi x) \text{ and } g(x) = |120-17x|$

Because f and g are continuous, we know they must intersect somewhere between x=5 and x=6, but we do not know for certain that they will not intersect elsewhere.

These issues are very serious and require correction. There are other minor issues: “f(x)” isn’t a function, it’s an output of the function f (see HSF-IF.A.1). Functions don’t have points, they have ordered pairs; graphs of functions have points. Neither of these minor issues spells doom for the problem, but these details can be corrected.

On the plus side, the interface for the problem is clean and the task for the student is clear (“Click the number line”). Clicking marks a unit interval in red, and clicking anywhere else changes the interval to reflect the new click.

This problem can be repaired simply by changing the text that appears on the left: “The table shows several inputs and outputs for continuous functions f and g. [table] The graphs of f and g intersect exactly once. Click the number line to show the unit interval for x in which the solution to the equation f(x) = g(x) must lie.”

I encourage you to take the SBAC Practice Test. I’ll have more to say about other problems. My hope is that through the work of the community we can improve these exams while they are still in their pilot stage.

## We Got A Problem #12: Tile Shufflin’

Online word games, such as Scrabble, now let you “shuffle” your tiles with the touch of a button.  As far as I can tell, when you click the “shuffle” button the 7 tiles are randomly rearranged into any of the 7! = 5040 different orderings.

And there’s an animation of the tiles moving.  Well… sort of.  Because, sometimes one or more of the tiles doesn’t move, because they are in the same position before and after the shuffling.

How does the probability of having one stuck tile compare to the probability of having no stuck tiles? How does the probability of having one stuck tile compare to the probability of having two stuck tiles? Three? Four?

A Scrabble variant called Lexulous gives players eight tiles instead of seven.  What happens then?

What is the approximate probability of having no stuck tiles?

## The Gambling Machine

The NY Times “Numberplay” blog recently posted an interesting question called “The Gambling Machine”.  The problem was based on a problem found in the NCTM 2011 slides from “Mathematics of Game Shows”, from the PCMI 2007 materials, and originally from the (thankfully cancelled) game show National Bingo Night.