GJordansays: April 16, 2012 at 12:04 pm
Hi Bill,Thanks for the opportunity to ask questions for you and the community. I looked for simplyfing radicals as an individual learning standard and was unable to find it. Is this purposeful have I overlooked this skill? 8.EE is close and so is N.RN.2 . Is this like the above conversation about simplifying fractions?

April 16, 2012 at 12:16 pm
You’ve got the right standards there, particularly N-RN.2. I would also include A-SSE.3. Students should be able to rewrite √12as 2√3and vice versa, but neither of these is simpler than the other. The emphasis in the standards is on transforming expressions into different forms for a particular purpose, as described in A-SSE.3. So yes, it’s similar to the conversation about fractions. The word “simplify” does not occur in the standards (except in one grade level introduction, which was an editing error).
Lisa says: April 3, 2012 at 11:35 am
I am looking for some guidance regarding what the expectation is for N-RN.3. It says to “explain why the sum or product of … is rational; …..that the sum of a rational number and an irrational number is irrational; …” How much is expected at this level? If a student is given an item for this standard will the student response include much more than the definition as a way of explaining?

Bill McCallumsays: April 5, 2012 at 6:59 am
It’s a good question, Lisa. Because of the work of Deborah Ball and others, we have a good idea of what reasoning and proof can look like in elementary grades: students can explain why the sum of two odd numbers is even, for example, using visual representations of odd and even. In high school, we see geometry as a place where students learn to produce mathematical proofs. But Middle school has been a bit of wasteland for reasoning and proof. This standard provides an opportunity for that. One way of presenting Farshid’s argument to students might be to make the explicit connection with earlier understanding of the relationship between addition and subtraction, so that students can see that rational + irrational = rational would be the same as irrational = rational – rational, an impossibility. By the same token, rational times irrational = rational would be the same as irrational = rational/rational, also an impossibility. Then perhaps you could ask “by the way, how do we know that rational plus rational = rational?” This could be an opportunity to see the formula for fraction addition as not just a computational device, but as a fact about the system of rational numbers (that it is closed under addition).
There’s a danger that assessment will drive all this away, of course, attempting to reduce this standard to some mindless exercises; we have to resist that.

lmhenry9 says:

April 5, 2012 at 8:32 pm
I teach HS Math (specifically Algebra 2). How do you envision how math class would be taught with the Common Core Standards? I think many teachers teach math in a fairly “traditional” way – instructing students on how to do (whatever) and then assign problems to be completed. How is our “mode of business,” if you will, going to change?

April 13, 2012 at 10:19 am
Dear Lisa, I don’t see the standards as dictating any particular teaching method, but rather setting goals for student understanding. Different people have different ideas about what is the best method for achieving that understanding. That said, I think it’s pretty clear that classrooms implementing the standards should have some way of fostering understanding and reasoning, and classrooms where students are just sitting and listening are unlikely to achieve that.

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http://commoncoretools.me/2012/04/02/general-questions-about-the-standards/April 16, 2012 at 12:04 pm

Hi Bill,Thanks for the opportunity to ask questions for you and the community. I looked for simplyfing radicals as an individual learning standard and was unable to find it. Is this purposeful have I overlooked this skill? 8.EE is close and so is N.RN.2 . Is this like the above conversation about simplifying fractions?

- Bill McCallum says:

April 16, 2012 at 12:16 pmYou’ve got the right standards there, particularly N-RN.2. I would also include A-SSE.3. Students should be able to rewrite √12as 2√3and vice versa, but neither of these is simpler than the other. The emphasis in the standards is on transforming expressions into different forms for a particular purpose, as described in A-SSE.3. So yes, it’s similar to the conversation about fractions. The word “simplify” does not occur in the standards (except in one grade level introduction, which was an editing error).

Lisa says:

April 3, 2012 at 11:35 am

I am looking for some guidance regarding what the expectation is for N-RN.3. It says to “explain why the sum or product of … is rational; …..that the sum of a rational number and an irrational number is irrational; …” How much is expected at this level? If a student is given an item for this standard will the student response include much more than the definition as a way of explaining?

Bill McCallum says:

April 5, 2012 at 6:59 am

It’s a good question, Lisa. Because of the work of Deborah Ball and others, we have a good idea of what reasoning and proof can look like in elementary grades: students can explain why the sum of two odd numbers is even, for example, using visual representations of odd and even. In high school, we see geometry as a place where students learn to produce mathematical proofs. But Middle school has been a bit of wasteland for reasoning and proof. This standard provides an opportunity for that. One way of presenting Farshid’s argument to students might be to make the explicit connection with earlier understanding of the relationship between addition and subtraction, so that students can see that rational + irrational = rational would be the same as irrational = rational – rational, an impossibility. By the same token, rational times irrational = rational would be the same as irrational = rational/rational, also an impossibility. Then perhaps you could ask “by the way, how do we know that rational plus rational = rational?” This could be an opportunity to see the formula for fraction addition as not just a computational device, but as a fact about the system of rational numbers (that it is closed under addition).

There’s a danger that assessment will drive all this away, of course, attempting to reduce this standard to some mindless exercises; we have to resist that.

- lmhenry9 says:

April 5, 2012 at 8:32 pmI teach HS Math (specifically Algebra 2). How do you envision how math class would be taught with the Common Core Standards? I think many teachers teach math in a fairly “traditional” way – instructing students on how to do (whatever) and then assign problems to be completed. How is our “mode of business,” if you will, going to change?

- Bill McCallum says:

April 13, 2012 at 10:19 amDear Lisa, I don’t see the standards as dictating any particular teaching method, but rather setting goals for student understanding. Different people have different ideas about what is the best method for achieving that understanding. That said, I think it’s pretty clear that classrooms implementing the standards should have some way of fostering understanding and reasoning, and classrooms where students are just sitting and listening are unlikely to achieve that.

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