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N.RN.3 states “Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.” Without requiring a formal proof, what should we look for from Math II students in their explanations of these fairly sophisticated questions? The standard points to physical situations, which then logically seems to indicate for specific situations. Yet the standard itself seems to require a general explanation (proof?), but doesn’t actually use “any” or “all” as part of the prompts.
Does it help to look at p. 17 of the Number System & Number Progression posted here: http://ime.math.arizona.edu/progressions/. I see that the document label on that page is misleading, you need to click on the link that says “Draft 6–8 Progression on The Number System”.
I’m not sure how the standard points to physical situations, could you explain?
Thank you for pointing out the 6-8 NS Progression actually includes high school as well.
The document I am referring to is Appendix A. Whether looking at page 25, Algebra I/Unit 5 or page 63, Math II/Unit 1, the Clusters with Instructional Notes column to the of the standard states “Connect N.RN.3 to physical situations, e.g., finding the perimeter of a square of area 2.”
We have since moved beyond this standard having needed a solution a couple of weeks ago. We handled my posted question by asking students to either state that a particular property was true or if false, by offering a counter example. Our in-class conversations went much deeper than this and included some discussion of formal proofs for a few of the properties.
This was our first go at this particular standard as we are phasing in Math 1/2/3 and replacing the A1/G/A2 track we use to have. We may handle this differently next year based as we learn more about this particular standard over the next year.
I think Appendix A is a little of base here. There’s a discussion of this from about a year ago here.
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