The following inline choice question, explained here in hopes of helping eighth-grade students and their parents in Maryland and Illinois prepare for the PARCC test near the end of this school year, appears on the released version of PARCC’s Spring 2016 test for grade eight math:
Information about two linear functions is shown.
Select from the drop-down menus to correctly complete each sentence.
The average rate of change of function P is __A__ the average rate of change of function Q.
The y-intercept of function P is __B__ the y-intercept of function Q.
Choices for both _A_ and _B_:
• less than • equal to • greater than
Find the slope and intercept of function P and function Q.
Function P is defined by a rule. We can translate this rule directly into an equation that describes a linear function:
Since that is in the slope-intercept form for the equation of a line, y = mx + b, the slope is 2, which also represents the rate of change, and the y-intercept is 3.
For function Q, we have to compute the rate of change, which is given by
The change in y, 7.5 – (–4.5), is 12, and the change in x, 5 – (–3), is 8, giving us a rate of change of 1.5.
Where does function Q cross the y-axis? For this, we can plug in one of the points that we know is on the line for function Q and find the y-intercept:
Analysis of this question and online accessibility
Eighth-grade math students have to grasp the concept of a function as a rule that assigns to each input exactly one output. Functions describe situations where one quantity determines another. This skill and the understanding of the underlying notion of a function is mostly new in eighth grade, even though this understanding has significant prerequisite understanding that is part of the sixth- and seventh-grade Common Core standards.
In eighth grade, students learn to translate among representations and partial representations of functions, noting that rule-based, tabular, and graphical representations may be partial representations. They should be able describe how aspects of the function are reflected in the different representations.
The question can be delivered online and on paper-based tests, and the validity, reliability, and fairness measures should not differ significantly among the various delivery modes.
The use of the adjective “average” to describe the rate of change of a function that a test-taker knows is linear (because the problem identifies both P and Q as linear functions) is at best redundant and at worst misleading to eighth-graders who may incorrectly think, for a moment, that the rate of change for a linear function itself is subject to change and an “average” value must somehow be computed. The word isn’t mathematically incorrect, however, and its use simply makes the problem less direct, not invalid.
Still, the use of the word average does force our analysis to wonder whether this word is a Freudian-like slip or a canary in the mine that reveals for us all an understanding of linear functions on the part of the people who developed this test question that is essentially below the eighth-grade level. If, on the other hand, the word is used intentionally, it is misleading.
- Khan Academy finds “average” rate of change for linear and nonlinear functions
No special accommodation challenges can be identified with this question, so the question is considered fair.
