The following multiple-droplist 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:
Select the values and signs from the drop-down menus that correctly complete the solution by factoring.
(x _(A)_ _(B)_) (x _(C)_ _(D)_)
x = _(E)_; x = _(F)_
Choices for A and C: + or –
Choices for B and D: 1, 2, 3, or 4
Choices for E: –4, –3, 1, or 2
Choices for F: –2, –1, 3, or 4
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 x2 – 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 l must be factors of a and m and n must be factors of 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 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 drop-down menus in that the choice the student wishes to select, incorrectly, may not be among the options. Not all possibilities can be made available to students with a question format like this, and not all options for the solutions go with every possible option for the factoring.
No special accommodation challenges can be identified with this question, so the question is considered fair.
What would be a better way to ask this question on a test? Can you identify a context the problem might be useful for?
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.