#WTPW – Week 1 – Rationalizing the Denominator

Walk the Plank Wednesday – Week 1 

Welcome to Week 1 of Walk the Plank Wednesday!!!!  A place where teachers ask for help transforming a not so exciting lesson/topic into a PIRATE lesson.  Teachers walk the plank and hope other people can help get the lesson into something amazing that students will be  running to class for.

Week 1’s topic (submitted by @jkindred13 – Love you for being to first smart one to submit a topic) — Rationalizing the Denominator in Algebra II – <insert scary music now!!!>  

Please add your ideas, thoughts, suggestions, struggles, links, etc  in the comments section below.

Information below submitted by @jkindred13

How the topic is currently taught — 

1. Basics are all there. Standards identified, learning targets included.
2. Have some examples: I do, we do, you do format.
3. I usually show kids that we make a special one with the denominator and multiply the fraction by the “special one” This emphasizes we are not changing the value of the expression just changing the looks.
4. Practice

What are some things that are/aren’t working with this lesson? —

Well, mostly it’s boring, unnecessary, old-fashioned, has no real-life application and the teacher hates teaching it.

HELP A SISTER OUT!!!!!

Any additional information —

After reading #TLAP, I’ve been thinking about approaching it as “Beauty is in the eye of the beholder”. That is an expression with a radical in the denominator is not considered beautiful by some, but when we rationalize then we have beauty. Sort of the ugly duckling becomes a beautiful swan.

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Posted on July 1, 2013, in Alg II, Walk the Plank Wednesday and tagged , . Bookmark the permalink. 6 Comments.

  1. What if students took time to turn the ‘beautiful swan’ into an ‘ugly ducking’ first? De-rationalizing the denominator and making it as complex and ugly as possible? Then have a conversation about why we might want the original form, then look at the math to transform from one look to another – coconstruct what math they performed and see if the math applies to going the other way.

  2. What about looking at patterns? Show them several examples like 1/sqrt(3) = sqrt(3)/3, see if they can figure out how the transformation took place? Then give them like like 3/sqrt(5) and see if they can use the pattern they discovered to rationalize the fraction. Continue with more difficult denominators.

  3. I still haven’t read Teach Like A Pirate. I bought it months ago, but I keep getting distracted by other books. I’m going to try to get to it over this holiday weekend. Here are my ideas for now. They may change after I actually read the book. 🙂

    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, I am always looking for creative ways to fit more writing into my curriculum. Our current state standardized test for Algebra 1 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 thoughts 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 problem in a way that matches the essays students will be required to write on their end of year tests.

  4. Wow, @jkindred started you with a really tough one. 🙂 I admire your taking this on!

    I think a really big question for students (and teachers) is, “Why rationalize?” My cursory searches of a few websites (Dr. Math, New York Regents, several math forums), is that, essentially, rationalizing the denominator allows a standard form by which to check that students are arriving at the same answer. Yes, students get some cooperative benefit from this idea sometimes, but ultimately, saying, “It’s convenient,” is not a very good answer to give a student about WHY they must rationalize.

    But I think you can turn that idea around into a hook, too, and maybe work in some number sense at the same time. What about this: Let’s take 1/sqrt(2) and sqrt(2)/2. Give the students that sqrt(2) is approximately 1.414, then have the students estimate the values of the two fractions (I encourage you to do this sans calculators) to the nearest tenth. See which one they find easier, and then be sure to show them numerically (maybe with a calculator) that the two fractions are actually equal.

    Do it again for 1/sqrt(3) and sqrt(3)/3. Then, since you said it was Alg II, if it’s in your standard, try a conjugate like (1 – sqrt(2))/(1 + sqrt(2)), showing it is equal to (2*sqrt(2) – 3); I get the feeling they will take your word on this. That may establish some value to simplification in their heads, the idea that one of these expressions (in most cases) is easier to work with. Then continue on @druinok’s suggestion, asking them how the first couple could be “shown” mathematically. You might culminate with a competitive event (a “rationalization bowl” or a matching game).

    By the way — I would be concerned about giving students the idea that an unrationalized denominator is always worse/uglier than a rationalized one. Yes, if the fraction is 6*sqrt(5)/sqrt(3), I’m hard-pressed to think of a way in which its rationalized form, 2*sqrt(15), is inferior. However, if I’m doing a trig problem or an area problem where I’m going to square a result with a radical in it, there might not be a need to rationalize first (try squaring 1/sqrt(3) versus sqrt(3)/3). And in certain trig identities or in doing derivatives with the limit definition in calculus, there are certain times when rationalizing the numerator is actually the best choice.

    Ultimately, I feel that rationalization within a fraction is less a style issue and more a transforming skill that allows students to put things in a form that allows them to connect with a particular idea or a computational need. Getting students comfortable with rationalization and when it can be used to bridge gaps or enhance understandings (division of complex numbers, or even casual class discussions like, “Bobby says tan(pi/6) = 1/sqrt(3) but Helen says tan(pi/6) = sqrt(3)/3, who’s right?”) is the best thing you can do for them.

  5. These are great comments so far. Lots of things to think about. I decided to make an activity that would allow students to practice rationalizing denominators. I made a tarsia. Check out my blogpost for the scoop. http://summathmadness.wordpress.com/2013/07/03/wtpw-week-1-rationalizing-the-denominator/

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