The following multi-part constructed response question, explained here in hopes of helping algebra students 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 2015 test in algebra 1, here:
List the steps to solve the equation x2 + 12x – 28 = 0 by completing the square, and give the solution or solutions.
Enter your work and your answers in the space provided.
Explain what value or values of c make the equation x2 + 12x + c = 0 have one and only one solution. Justify your answer.
Enter your answer and your justification in the space provided.
Resources for further study
Purple Math, developed by Elizabeth Stapel, a math teacher from the St Louis area, has a two-page series on completing the square. She outlines the entire process, step by step, for the general case and then gives a few specific examples to show how to find the zeros of a quadratic equation by completing the square. The pages start here.
Chapter 4, Section 4.7 of the book Algebra 2, Illinois edition by Ron Larson et al is entitled “Completing the Square” and it is useful indeed. But in Section 4.8, he identifies a key concept regarding the discriminant of a quadratic equation. The equation has exactly:
- 2 real solutions if the discriminant is positive
- 1 real solution if the discriminant is zero
- 2 imaginary solutions if the discriminant is negative
Complete reference: 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.
Analysis of this question and online accessibility
The question measures knowledge of the Common Core standard and math practices it purports to measure and tests students’ ability to solve a quadratic equation by completing the square and understand the meaning of the discriminant in the quadratic formula. It is considered to have a high cognitive demand.
The reliance on a single solution strategy is one of the flaws in the Common Core. Sure, completing the square is a useful way to solve quadratic equations, but it is one of many solution strategies, including the quadratic formula, graphing the function, factoring, and a few other techniques students can use. In fact, completing the square is, in some ways, just an algebraic solution with the realization that sometimes you have to take one step back before you can take two forward in solving a math problem. That’s a useful concept: if you can’t break through a wall, try to go around it.
Consider, for example, the sentence in Math Practice 3 that says, “Students learn to determine domains to which an argument applies.” By directing students to use the method of completing the square—undoubtedly an appropriate strategy for this particular quadratic equation, since it won’t factor neatly—the Common Core robs educators of their ability to assess whether a student is able to decide for himself which approach to use. That prior restraint, imposed by the Common Core and amplified on the test, violates the spirit of the Common Core. But in any event, the question is perfectly aligned to the content standard.
The question can be tested online and should yield results that are as valid and reliable as those obtained on paper. Students online may experience difficulties using the equation editor to show their math in determining the slope. We have previously described problems with the user interface of the equation editor tool (see here), and we will not repeat that argument here. However, we will use the opportunity to remind those of you taking the PARCC math test to spend a little extra time on problems with the equation editor to:
- Make sure you have typed in all the work necessary to make your case
- Transfer all your work from scratch paper to the computer so you can receive credit for it
No special accommodation challenges can be identified with this question, so the question is considered fair.
Identify one value of c in Part B for which the quadratic equation will have two imaginary solutions and plot the graph. How does the graph differ from the one you found in Part B.
Purpose of this series of posts
Voxitatis is developing blog posts that address every algebra 1 question released to the public by the Partnership for Assessment of Readiness for College and Careers, or PARCC, in order to help students prepare to take the test this spring.
Our total release will run from February 27 through March 15, with one or two questions discussed per day. Then we’ll move to geometry at the end of March, algebra 2 during the first half of April, and eighth grade during the last half of April.