From the 8th grade Pennsylvania System of School Assessment (PSSA) Mathematics Item and Scoring Sampler for Grade 8:

**Extra Credit:**

**1. In what sense is this a complete explanation?**

*Hint:*Consider the terms "even," "exact value," and "never stops." Consider as well the difference between answering a question and rephrasing the question as a statement.

2. Discuss the human element involved in scoring responses to open-ended questions on tests taken by millions of students.

3. Discuss the human element involved in scoring the verbal explanations submitted by tens of thousands of English-impaired students.

## 2 comments:

I consider the alleged complete answer to be completely non-responsive to the question and would mark it as such were I grading this test. As I read it, the question is asking the student to prove that the square root of the sum of a perfect square and 1 can never be rational. The 'answer' does not do this, but rather, as you point out, merely asserts that all the hypotenuses in the table are irrational, without demonstrating or explaining why this must be so.

A responsive answer would have to demonstrate 1) that none of the hypotenuses outlined in the problem can be perfect squares and 2) that the square roots of integers that are not perfect squares must be irrational.

I am currently homeschooling my 6th grade daughter using Art of Problem Solving Prealgebra and we have covered irrational numbers, square roots, and properties of right triangles. She might quickly be able to perceive that the hypotenuses can not be perfect squares (I would not be surprised to see such a question in AoPS), but to put that insight into clear explanatory sentences would be time-consuming and she would not finish the rest of the test.

Moreover, although my daughter might be able to follow a proof that the square roots of non-perfect squares must be irrational, I would have no expectation that she could develop such a proof on her own. It would be a truly exceptional 8th grader who could come up with such a proof while taking a test and still have time to answer any other questions.

I would be curious to know how the graders for the test would rate an answer saying that all the hypotenuses in the table must be irrational since the square of the given hypotenuses can never be a perfect square, and that the square root of integers that are not non-perfect squares are irrational.

I agree. An actual proof of this would be hard. The question also isn't asking about whether the answer is ever a whole number, but a "rational" number. So you'd have to prove that the sqrt(n^2 + 1) can never be expressed as a fraction, which is a whole different level of complexity.

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