Monday, October 18, 2021

Algebra 1 PARCC: graphing inequalities


The following multiple-choice question, explained here in hopes of helping algebra 1 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 algebra 1 (#5):

Which graph best represents the solution to this system of inequalities?

x + y > 3
2x - y \ge 1

Answer and references

Correct answer: D.

Common Core Math, High School Algebra, Reasoning with Equations & Inequalities

The high school algebra standards in the Common Core, including those under this subheading, require students to be able to “represent and solve equations and inequalities graphically.” In this case, a system of inequalities is presented (HSA.REI.D.12), and students are required to “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.”

Solution strategy (there are others)

Graph each inequality and find the intersection of the half-planes.

When you graph an inequality, half the coordinate plane is shaded, because it’s the solution. When the y value is greater than (or greater than or equal to) the quantity with x, the half-plane above the line will be in the solution set.

We can rewrite the first inequality in the slope-intercept form for a line as follows:

x + y > 3
y > -x + 3

That means the slope will be –1 (the coefficient of the x term above), and the y-intercept will be 3. Also, since it’s a strict inequality, not a “greater than or equal to” inequality, the line will be dashed, since the boundary is not included in the solution set.

Looking at the four graphs in (A) through (D), you can see that only (C) and (D) have a dashed line with a negative slope and a positive y-intercept. Since the shading is above the dashed line in both of those, they are both possibilities for the correct answer, given just the first inequality in the problem.

The second inequality can be converted to slope-intercept form as well:

2x - y \ge 1
-y \ge -2x + 1
y \le 2x - 1

This line has a positive slope, a negative y-intercept, and a solution set that would be shaded below the line, which should be a solid line, because it’s a “less than or equal to” type of inequality.

In (C), the shading is above the solid line, so that can’t be the correct answer, leaving (D).

Analysis of this question and online accessibility

The question is valid in that it tests students’ ability to graph the solution set to a system of two linear inequalities in two variables.

Multiple-choice questions like this one can be delivered easily online or on paper. Validity and reliability should be comparable between different formats.

No special accommodation challenges can be identified with this question, so the question is considered fair.

A minor or irrelevant editorial issue remains in the use of the emphasized word “best” in the question. This usage is typically reserved for questions that possibly have more than one correct answer but one choice that’s a better explanation or match. There is no way (A), (B), or (C) could be considered possibly correct, assuming the axes intersect at the point (0, 0). Although the word doesn’t make the question incorrect, its use in the question is redundant.

Resources for further study

Purple Math, developed by Elizabeth Stapel, a math teacher from the St Louis area, has developed a few web pages that describe how to graph the solution to a system of linear inequalities on the coordinate plane. Her lesson begins here.

In addition, Sal Khan, an engineer who developed the Khan Academy, a set of thousands of tutorial videos about math and a few other subjects, has created an entire series of video tutorials and practice problems that do an excellent job of teaching students how to graph systems of two-variable linear inequalities, beginning here.

Ron Larson, Laurie Boswell, Timothy D Kanold, Lee Stiff. Algebra 2, Illinois edition. Evanston, Ill.: McDougal Littell, a division of Houghton Mifflin Company, 2008. The book is used in several algebra classes taught in Illinois high schools.

Foerster, Paul A. Algebra and Trigonometry: Functions and Applications, revised edition. Addison-Wesley, 1980, 1984. The book is used in several algebra classes taught in middle and high schools in both Illinois and Maryland.

Purpose of this series

To help algebra 1 students and their parents prepare for the PARCC test in math, as administered in at least six states, or to just master content on that test, we provide an analysis of every algebra 1 math problem PARCC released in 2016. The series can be found here.

Paul Katula
Paul Katula is the executive editor of the Voxitatis Research Foundation, which publishes this blog. For more information, see the About page.

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