0 Reflections on a Future Lesson: Rationalizing the Denominator

I actually wrote this in response to the Week 1 topic of Walk the Plank Wednesdays.  I'm so excited about what I wrote that I decided to break my unintended blog hiatus with this post.  I actually have yet to read Teach Like A Pirate.  I'm hoping to get to it this weekend.  Most of my reading of late has been either Embedded Formative Assessment (amazing!) or non-school related reading. 

Note to self: attending back-to-back week-long math teacher conferences is inspiring and really, really draining.  I saw all kinds of really cool ideas that I want to blog about, but my motivation has been extremely lacking.  So, without further ado, I present: Why I'm Actually Excited About Teaching My Students to Rationalize the Denominator Next Year

Until I went to a Pre-AP Math Conference last week, I had no idea why it was customary to rationalize the denominator of a fraction. I never questioned the practice when I learned it in school. Last year, I simply taught my students that “mathematicians don’t like to have radicals in the denominator.” They complained and weren’t quite satisfied with that explanation.

Next year, I will introduce the concept of rationalizing the denominator by taking away my students’ calculators and providing them with square root charts like those found in old math textbooks. I will ask them to find the numerical approximation of 1/sqrt(2). After suffering through having to divide 1 by 1.414, I hope they will ask for a better way.

Then, we can talk about how a fraction remains equivalent as long as the numerator and denominator are multiplied by the same value. I will ask students what they think we should multiply the numerator and denominator by. Hopefully, someone will suggest that we multiply the numerator and denominator by sqrt(2). (Of course, I’m going to make my students try out all the other possibilities that are suggested, too. They are going to absolutely love me!) Then, we will use our chart to divide sqrt(2) by 2 or 1.414 by 2. This division problem is much, much nicer. Most students should be able to do this in their heads.

I can see myself bringing out the stopwatches and breaking the class into two groups. I will put an expression on the board that involves a radical in the denominator. The goal is to find the numerical approximation of the value. One group will solve the problem using the square root chart without rationalizing the denominator. The other group will solve the problem using the square root chart with rationalizing the denominator. What is the time difference? If you had a homework sheet of 20 problems, how much time would you save by rationalizing the denominator?

I want my students to see that rationalizing the denominator does have a purpose. Or, it did have a purpose in the days before calculators. Yes, in this current day of technology, we just type 1/sqrt(2) into our calculator and let it do the work for us. Hopefully, seeing the historical reason behind the mathematical process will help my students understand why we rationalize the denominator and the process will be more meaningful to them.

With Common Core and the increased emphasis on writing, I'm always on the lookout for creative ways to fit more meaningful writing into my math lessons.  Oklahoma's current Algebra 1 EOI features specific questions on rationalizing the denominator.  I think I will have students write a memo/letter/speech/etc to the State Department of Education expressing their toughts on whether students should or should not still be required to rationalize the denominator in light of technological gains.  I need to look into the Common Core Language Arts standards and pose this question in such a way that it matches what students will be required to write on their end of year test.