The following constructed-response 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 (#29):
A set of points in the xy-coordinate plane meets two conditions, as described.
- Condition 1: The y-coordinate is positive.
- Condition 2: The sum of the coordinates is greater than –2.
Create a system of inequalities described by the two conditions.
Enter your answer in the space provided.
A graph of the solution of the system includes points in different quadrants of the xy-coordinate plane. Explain why all points in the first quadrant are part of the solution.
Enter your answer in the space provided.
Answer and references
In the first quadrant, all x-coordinates are positive and all y-coordinates are positive, so the first condition is satisfied. Adding two positive numbers always results in a number greater than –2, which means the second condition is satisfied.
Common Core Math, High School Algebra, Reasoning with Equations & Inequalities
The question requires students to base explanations on the principle that the graph of an inequality in two variables is the set of all its solutions plotted in the coordinate plane. The standard is found in the Common Core, Math.HSA.REI.D, which requires students to “represent and solve equations and inequalities graphically.” The reference to the first quadrant in Part B of the problem, where two of the three points are earned, ties the question to the graphical representation of an inequality, as required by the Common Core.
Specifically, standard HSA.REI.D.12 requires students to be able 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.”
The problem references Math Practice 3, which requires high school students, in part, to “justify their conclusions, communicate them to others, respond to the arguments of others, … reason inductively about data, [and] make plausible arguments that take into account the context from which the data arose.”
Solution strategy (there are others)
Represent the conditions using a system of 2 inequalities in 2 unknowns.
First, you have to know that the first quadrant is the upper right quadrant of the coordinate grid, where both the x- and y-coordinates are positive. For an explanation of the different quadrants, see this explanation on Wolfram MathWorld.
Plot of y
> 0 (yellow) and x
> –2 (blue) on the xy
Then, realize that having a y-coordinate that’s positive (condition 1) means all values on the coordinate plane above the x axis will be true, as shown in yellow on the graph above. Next, realize that the (non-inclusive) boundary for the second condition will be a line:
That line, which would be better represented as a dotted line, since the points on the line itself aren’t part of the solution, is shown in blue on the coordinate plane above, and the region above it is shaded blue. Every point in the blue-shaded region satisfies the second condition in the problem.
I have shaded the region of the graph that satisfies both condition 1 and condition 2 in green, since that’s the color that blue and yellow make when they’re combined. Note that the green region is part of the solution for both condition 1 and condition 2. That is, the green half-plane is both blue and yellow, shown in green.
Part B asks you to explain how you know all the points in the first quadrant will be part of the solution. They will, because all points in the first quadrant have a positive x-coordinate and a positive y-coordinate. When you add two positive numbers, you will always get a positive number, and any positive number is greater than –2, which means the second condition will be satisfied by any point in the first quadrant. The first condition is satisfied for the same reason: all points in the first quadrant have a y-coordinate that’s greater than 0.
Analysis of this question and online accessibility
The question is valid in that it tests students’ ability to explain their reasons for what they know about the solution to inequalities on a coordinate plane.
The question can be delivered online or on paper. However, because students can’t draw using the online tool and will be forced to use words, those who take the test online may be at a disadvantage with this question if they are accustomed to solving or explaining their solution to problems like this using diagrams or graphs. This solution strategy isn’t the only one available, though, but the PARCC test, with the “equation editor” tool, unnaturally forces students’ hand: they must explain their solution in paragraph form, not by using a graph in their explanation, which is also a valid way of explaining a mathematical scenario, if they are taking the test online.
This means validity and reliability measures for the item may suffer from the different test-taking environment, in terms of responding to the question, experienced by students who take the test on a keyboard and those who take it freehand.
No special accommodation challenges can be identified with this question, so the question is considered fair.
Resources for further study
Purple Math, developed by Elizabeth Stapel, a math teacher from the St Louis area, has developed several pages dealing with finding the solution to a system of two inequalities in two unknowns using 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 several videos about working with inequalities on the coordinate plane, 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.