N-RN.1 (Real Number System) - The Meaning of Rational Exponents

While I'm trying to update every day in August, I might as well start taking a closer look at the Common Core standards, which have now been in place for one year in high school Algebra 1 classes in New York. The first standard I find is N-RN.1. The first N stands for Number and Quantity. Unfortunately, the RN stands for Real Number System and not Registered Nurse because the latter would be helpful when you got sick of all this! I could have helped that telling that joke -- I chose to tell it any way, if only because I had to look up what the letters meant, particularly the first "N".

The entire standard reads as follows: Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.

It then has two components, one for Algebra 1, the other for Algebra 2. The Algebra 1 component is Evaluate integers raised to rational exponents (Algebra 1 - V.9). Don't ask me about the "V.9", I've done enough searching for this column.

I used to start teaching each year with Order of Operations, something that all the students should have seen before, and yet seemed to forget about. They know an acronym, such as PEMDAS, but don't know what it means. Oh, they know what the six letters mean, but they don't get the concept. And even when they can explain the concept, when push comes to shove and the pressures on (and they're taking a quiz), you find them calculating from left to right as if they hadn't learned anything. I had to change that when we started welcoming calculators into the lesson (even before we started requiring them). The calculators were down the work for them, so they didn't have to learn it, right? Wrong. I just adapted the problems. I started added more operations within fractions and adding exponents, forcing them to pay attention to what they put into their calculator. For instance, you need explicit parentheses to group things in a calculator because the numerator and denominator are implicitly grouped.

But Common Core changes that. They have to learn it much earlier, so they're ready for Algebra by the time they get to high school. Yeah, right. I'm still doing it. But wait, there's more.

After parentheses, come the exponents. Some students know the concept of exponents, and some just press buttons on the calculator. They know (or they'll learn!) that they are multiplying the factor some number of times. What they haven't seen before is a fraction as an exponent. What do you do when you see a fraction? Hide under the desk, usually. Wait for it to be over. I'm not exaggerating much.

First, I have to teach the concept that exponent 1/2 means take the square root, as stated in the gem. And then see how well they know they're square roots, either with or without a calculator. (I try to get them to learn up to 256, with some success.)

Second, I have to teach them that exponent 1/3 means take the cube root. This usually entails explaining what a cube root is, and where to find it on the calculator. Maybe reviewing what "cubing" means, and possibly through in Volume = length X width X height somewhere, just so I can spiral back to it when that comes around again. Depending on the results, I could try to conquer the concepts of fourth and fifth roots. Seriously, a practice webpage asked me (729)^(1/6).

At this point, I'm asking: are they getting it, or are they pushing buttons? I don't mind the button pushing if they understand the concept, because then they'll start recognizing patterns and will be smarter about their button pushing.

Okay, so the next step is the real doozy: explaining exponents of 2/3 or 3/5. The students have to deconstruct the fraction. That is, they need to know that 2/3 = (2)X(1/3), so they need both to square it and then take the cube root. And then I'll suggest that you take the cube root first, so that they're dealing with smaller numbers. Sometimes this makes sense to them.

Finally, there's the kicker: improper fractions. If fractions are Dr. Frankenstein, improper fractions are his monster. They're a whole new level of scary, and the first thing they want to do is turn them into mixed numbers. Or decimals. No! Wait! Stop!

Taking the (5/2) power of 16 is as simple as taking the fifth power of the square root of 16. Okay, read that again like you're a ninth grader.

No, it's not that difficult to do once the concept is learned, but it's something they haven't seen before, and they're learning it earlier. I can't remember exactly when the first time I was required to find a third or fourth root. I was probably using logs to do it. But then, I didn't have the calculators that they have today on their phones. They'll have the answers at their fingertips, but I'll make them show the work so I know they get the idea behind it.