The following 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:
Two real numbers are defined as:
- a = 0.444444444444 . . .
- b = 0.354355435554 . . .
Determine whether each number is rational or irrational. Is the product of a and b rational or irrational?
Justify your answers.
Enter your answers and your justifications in the space provided.
Resources for further study
The Oswego City Schools in New York developed the Regents Exam prep center, courtesy of Lisa Schultzkie. It has a very nice page explaining the difference between rational and irrational numbers, here.
The Khan Academy, developed by Sal Khan, an engineer who has created a library of thousands of video lessons, has a landing page dealing with rational and irrational numbers. The site features more than a dozen lessons on the subject, including some interesting algebraic proofs.
Purple Math, developed by Elizabeth Stapel, a math teacher from the St Louis area, has a page or two, starting here, to explain the different types of numbers, including a lesson about differentiating rational and irrational numbers.
Analysis of this question and online accessibility
The question measures knowledge of the Common Core standard it purports to measure and tests students’ ability to explain why the product of an irrational number and a nonzero rational number is an irrational number, as well as identifying decimal representations of numbers as rational or irrational.
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 and may not be able to achieve full credit (3 points) for this question if they are too unfamiliar with the operation of the tool in the online test delivery system.
No special accommodation challenges can be identified with this question, so the question is considered fair.
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.