Grade 8 PARCC math: parameterized equations

The following fill-in-the-number 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:

Both of the equations shown have the same solution when solved for x.

$\frac{1}{2}(x+6) = 2x \textrm{ and } 5-2x=7x-h$

What is the value of h?

Answer and references

Correct answer: 13.

Common Core Math Content 8th grade, Expressions & Equations:

Solving linear equations in one variable, even if the solution has a variable in it, here h, isn’t technically explicit in the Common Core standards in eighth grade or earlier, but the exact standards in the 8.EE Common Core aren’t really the point of this problem.

We note that an eighth-grade math standard also requires students to analyze and solve pairs of simultaneous linear equations, and that standard is possibly closer to the technical skills in this math problem. We have two simultaneous equations that have one solution, but one of those equations uses only one variable.

In other words, the problem is sort of “in between” the standards, and that’s fine, because it tests students ability to reason as much as it tests their ability to solve a system of equations. So, emphasis here is on the word “analyze” in the standards, less on the “solve” part.

We find, on the Common Core document (page 52), that eighth graders are required to reason about expressions and equations, including modeling an association in bivariate data with a linear equation, and solving linear equations and systems of linear equations. … “Students use linear equations and systems of linear equations to represent, analyze, and solve a variety of problems,” the standards document says.

So by giving the condition that both equations have the same value of x, students must show reasoning in developing a solution strategy where they find x in the equation that lets them do that without interference from h or any other unknowns, and then plug that value for x into the other equation to find the second unknown. In order to do this, they must remember that x has the same value in both equations, which we told them.

Solution strategy (there are others)

Find x from the equation without h, and plug it in to find h in the second equation.

In solving the first equation for x, we get

$\frac{1}{2}(x+6) = 2x$

$x+6 = (2)(2x)$

$6 = 3x$

$2 = x$

Now that we know x is 2, given that it’s the same for both equations, substitute that value for x in the second equation and solve it for h.

$5-2x=7x-h$

$5-2(2)=7(2)-h$

$5-4=14-h$

$1-14=-h$

$h=13$

Analysis of this question and online accessibility

Fill-in-the-number questions prevent students from entering words and may even prevent them from typing a decimal point, since an answer is a whole number. I can’t be sure what constraints technology might have placed on students’ ability to enter a decimal point.

The question format prevents guessing, since no options are presented, and the student must enter a number from scratch.

The question is accessible for students on any device they may use or on paper. However, because paper test-takers would be required to respond to a multiple-choice format for this question, validity, reliability, and fairness measures may differ among the various delivery modes. Students who don’t know how to do the problem as required in the Common Core might be able to guess the correct answer, from the four options on a paper-based test, whereas students answering the question online won’t have those pre-formatted answers available at test time and won’t be able to plug in values for h in a guess-and-check manner to see if they work.

Systems of linear equations from the Utah State Office of Education
Solving system of linear equations on Purple Math

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

Purpose of this series

To help eighth graders 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 eighth-grade math problem PARCC released. The series can be found here.