Fantastic Radicals and Where to Find Them

September 13, 2016

Radicals came up this week in a high school math PLC (that is Professional Learning Community). They were trying to figure out when students would be learning how to simplify radicals. For example:

Prior to common core, these guys had a nice chunk of a chapter dedicated to them. First we would remember perfect squares, then depending on the conceptual focus of the text, we would first simplify or learn to multiply and divide, then vice versa. Finally we would add and subtract thus using the simplifying, and finally rationalize the denominator. In every day terms, this means "in a fraction, you cannot have a radical on the bottom". Why you ask? Because it isn't truly simplified. And that was that. 

Now, if you are ANYTHING like the students I taught you are thinking to yourself...why? Why do I care what a radical is in the first place? Even IF I cared, why would I simplify, or perform operations on said useless giant check-marks. And if THAT interested you, I bet the selling point of rationalizing with "because I said so" surely made you cringe. 

Most of us grew up learning math this way. Why am I learning this? Because it is on the test, because it is your job, BECAUSE I SAID SO! Ugh. I will even admit to uttering these phrases (except the last. I only use that with my 3 year old daughter who incessantly asks why without actually listening to my well thought out responses and thus degrading the quality of my responses.)

Fast forward to 2010. Common Core Standards come out and LOW AND BEHOLD, no simplifying radicals to be found! And it most certainly isn't broadcasted. It used to be taught somewhere between 8th and 9th grade. In Boise, we discovered it when we realized some of our 9th graders couldn't do some stand by Pythagorean Theorem problems. At that time, we didn't have a textbook, so we assumed it was an implied prerequisite. Then we got a textbook (CPM) and after searching HIGH AND LOW, we found it as an insignificant sentence at the bottom of a page of notes. Frankly, it seemed like it was only put there so the textbook could use their old answer keys with simplified radicals. Now with my current textbook, it seems to be the same situation. This book has a grand total of 4 sentences with 2 "properties" boxes between 2 separate lessons. Again, not the focus of the lesson. This time, it seems it has been included not only to use simplified radicals in the answer key, but to pacify educators. 

Cut to today where I am watching the emails fly back and forth between teachers. When should we teach this. OBVIOUSLY this was an oversight of the common core, THIS IS A VITAL SKILL. Lets take our short time and teach it. So I posed the same question I pose here:

Why do we need to know how to simplify radicals?

I will admit, my research is minimal. What I have found all focuses on very shallow answers. What I was told (professor from BSU) is that prior to calculators, being able to manipulate irrational numbers was inaccurate. We could only estimate roots or calculate them by hand (I had a student who wants me to learn this archaic skill and teach it that way!). The ability to simplify and perform operations on radicals (add, multiply, etc.) by hand enabled mathematicians to continue their calculations without losing precision. Furthermore, estimating and calculating when denominators that were irrational (radicals) created more work.

Now we have calculators and can maintain precision without the need for simplifying radicals. So, should this skill disappear with the need for an abacus? Or is there some legitimate purpose and the standards simply missed something? And if they are an outdated skill, how do we convince textbook companies and high school math teachers to change this widely held belief?