Scaling Mount Grading

We spent all week talking about rational and irrational numbers. A lot of time went into explaining that rational numbers can be written as fractions with integer numerators and natural number denominators (I did not use that jargon -- I talked about "positive and negative whole numbers" and the like). We devoted nearly 50 minutes of class time (spread over two days) to showing that certain square roots are irrational. We did the standard proof by contradiction, and for each of the three iterations of it that I did in class (square root of 2, of 3, and of 10), I emphasized the importance of including all the words in the assumption and in the proof. I even said, "When you do the proof for square root of 7 in the homework, you would get it pefectly right if you took my proof about square root of 3 and copied it exactly, just changing all the 3s to 7s." Every time we talked about decimal expansions, I changed to red pen and wrote "can not trust calculator" and "calculator lies."

Today I graded their efforts to respond to questions about rational and irrational numbers.

Most of my students felt that 3 * sqrt(2) / 5 * sqrt(2) was irrational because of the sqrt(2). They didn't remember that you can "cancel" (hate that word: prefer "divide out") the sqrt(2) that appears in both the numerator and the denominator. More alarmingly, many of them decided that 3.14159 was irrational, citing only "it can't be written as a fraction," and forgetting that 314159/100000 is a perfectly legit fraction.

The proofs, though. Oh my. Some took my words to heart and mimicked my proofs, changing the numbers as appropriate. The other students, though? Including several who I knew were in class? Oh my.

I've scanned their responses, and we will be critiquing them as a class at the beginning of class on Monday morning.