#### The following fill-in-the-blank 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:

What are the solutions to the equation ?

To respond, fill in the blank boxes next to two “** x = __**” prompts in the equation editor.

## Resources for further study

**Purple Math**, developed by Elizabeth Stapel, a math teacher from the St Louis area, has a six-part series on solving quadratic equations by factoring, as I did above. The series starts here.

The **Khan Academy**, developed by Sal Khan, an engineer who has created a library of thousands of video lessons, has a series of video lessons that demonstrate how to factor quadratics, starting here. He starts by factoring x^{2} – 14x + 40 as (x–4)(x–10), even though the text on his site says the second factor is (x–1). Despite the minor error in marketing, which would be unforgivable if done on the PARCC test, the video’s fine. Mr Khan clearly understands completely how to factor quadratics, even if he typed it incorrectly on his site.

Chapter 4, Section 4.4, of the book * Algebra 2, Illinois edition* by

**Ron Larson**

*et al*deals with solving quadratic equations by factoring. Students are trained to spot the structure in polynomial expressions of the general form

where * k* and

*must be factors of*

**l***and*

**a***and*

**m***must be factors of*

**n***.*

**c**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 it purports to measure and tests students’ ability, by recognizing structure in an expression, to factor a quadratic equation to find the zeros. It is considered to have a median cognitive demand.

(The assertion in PARCC’s evidence statement that the math practices are assessed [*integrate mathematical practices by not indicating that the equation is quadratic*] is only weakly supported in that simply not telling the student the equation conforms to some vocabulary word [*quadratic*] fails to test the underlying mathematics. Furthermore, by not requiring or scoring students’ use of technology-based tools and other methods to check the solution, the PARCC item here fails to incorporate a significant part of Common Core Math Practice 5.)

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 with the equation editor if they don’t find the simple integer solutions in this case and try to type in a more complex albeit incorrect answer or answers. However, since the score on the item depends only on the correct answers, students aren’t penalized and don’t suffer from an inability to use the equation editor.

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

## Challenge

Explain why, using the form of a two-column mathematical proof, **the quadratic formula would yield exactly the same solutions as the factoring method** used above. For reference the quadratic formula is

where * a* is the coefficient of

*x*

^{2},

*is the coefficient of*

**b***x*, and

*is the constant in a quadratic equation of the form*

**c**## 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.