Here is another installment in my series reviewing the NY State Regents exams in mathematics.
This is question 25 from the Common Core Algebra exam.
I’ve already complained about the contrived, artificial contexts for these questions (why not just ask “Is the sum of these two numbers rational or irrational?”), so I’ll ignore that for now. What’s worth discussing here is the following sample student response provided by the state.
So, why is the sum of a rational number and an irrational number irrational? Because the sum of a rational number and an irrational number is always irrational. This circular argument is offered as an example of a complete and correct response.
I’m not sure there’s a way to rewrite this question so that it admits a sensible answer. That’s probably a good indication that it shouldn’t be on a high-stakes test.
As I’ve argued time and again, questions on these exams should stand as examples of proper mathematics. But questions like this actually encourage bad habits in students, and teachers too, who are being told that this constitutes an appropriate response to this question. This is yet another example of the danger of simply tacking on “Justify your reasoning” to a high-stakes exam question.