|New York State Regents High School Exam, 2010
Source: New York State Regents High School Examination, Integrated Algebra - June, 2010
20. When 36 is subtracted from the square of a number, the result is five times the number. What is the positive solution?
(1) 9 (3) 3
(2) 6 (4) 4
What Are The Flaws in This Question?
- Since the word "solution" must be referring to the solution of an algebraic equation, this question seems to be testing how to solve algebraic equations. But it is not testing algebra at all! A student needs only to plug in the numbers listed as the choices to see which one works.
- Use of the word "solution" is confusing anyway. It seems to come out of nowhere because the first sentence makes no reference to an algebraic equation to solve. Perhaps a student should be able to deduce that, but the language adds an unnecessary level of complexity.
- Using the word "five" instead of the number "5" also adds an unnecessary level of complexity. Students should not have to go through the extra conceptual step of converting a word to its numerical equivalent.
- If a student does, in fact, use algebra, then there are too many steps to perform and too many concepts being tested for a simple multiple-choice question, which is a type of question where students cannot get partial credit for correct work shown. In this case, a student first needs to convert the word problem to an algebraic equation. Then the student needs to rewrite the equation and factor it. Then the student needs to find the two solutions of the equation and select the positive one.
- The wrong answer choices should be plausible and reflect common mistakes. The choice "6" seems okay because it's the square root of 36, reflecting the mistake of using only the left hand side of the equation. The choice "4" is okay because it reflects the mistake of taking the negative solution and making it positive. But where did the "3" choice come from?
- Standard test question writing requires numerical choices to be listed in ascending or descending order so that there is no bias in the ordering. However, in this question, the choices are NOT in numerical order.
- Referring to 6. above, probably the reason the choices are not in numerical order is because ordering them might cause confusion due to having the labels (1), (2), (3), (4) instead of the standard A, B, C, D for the choices. If the choices had been in numerical order, then choice (3) would be 4, and choice (4) would be 3. The way to get around this whole problem is to use A, B, C, D so that numerical answers aren't confused with numerical labels.
How to Fix This Question:
One way to fix this question is to rewrite it as an open-response question instead of a multiple-choice question; i.e., break it up into multiple parts, have students show their work, and give partial credit for correct work shown. One part would ask students to write the word problem as an algebraic equation. (This part alone could be a multiple-choice question.) Another part would ask for the solutions. There would be no need to ask only for the positive solution.