Dividing Radicals Made Easy Through the History of Rationalizing

This Warm up is intended to take about 20 minutes for the students to complete and for me to review with the class.  This lesson is based from the History of Mathematics and the story of rationalizing. When students enter the classroom:

• I have a square root  table posted at the front of the room.
• We discuss the perfect squares as I post them on the board.  I ask, what is the value of square root of two?  
• Square root of one is one
• Square root of four is two
• So square root of 2 and square root of 3 are between one and two
• Looking at the square root table, square root of two is approximately 1.414 that we will use for our first exercise today.

I tell students that the calculator is a relatively recent invention. Before its arrival students had to calculate the fraction "1 divided by the square root of two" by hand.  So, that is their first task today, to divide 1 by the square root of two. I tell them to approximate the square root of two with the rational number 1.414.  

1. Students write 1 divided by 1.414 on their individual white boards and begin working. This problem is very difficult for students.  They begin struggling of what to do with the decimal.  It is interesting to watch as I walk around the room and try to prompt them with questioning.
2. Finally some students realize to move the decimal in the divisor of 1.414 three places, and then have to figure out to move the decimal after 1 three places.  If they finally get to the problem they need, the student is having to divide 1000 by 1414.
3. Next, they must add zeros after the 1000.000 and begin dividing into 10000 first which is the tenths place.  

I did not have very many students get the correct answer, but it does help them to appreciate the complexity of the algorithm for dividing radicals (without a calculator).  I have one sample of student work in the resources (Note: not a correct answer).

I do not reveal the answer before we move on to work problems two and three. I do let the students  know that to get a common denominator, you multiply by 4/4 or 3/3 or 2/2 because this is equivalent to multiplying an expression by one, using the Identity Property. I introduce rationalizing on top of this foundation.

I begin our work on Problem 2 by saying, "Mathematicians began rationalizing to change the form of the fraction so that it is easier to divide by hand. This was important before the invention of the calculator." Then I ask, "What radical can we multiply by to multiply by a one?"  With some assistance, the students agree on square root of two divided by square root of two.  So now I ask the students to divide the square root of two divided by 2. So students write out 1.414 divided by 2, which is much easier for them.  Most students were successful at finding that the fraction was equal to .707.

This Warm Up activity takes time, but it helps students remember why to rationalize the denominator when it has a radical.  It is not mathematically incorrect to leave a radical in the denominator. But, there are operations where it is helpful to have the number written in this form. Since these operations were once common, the practice of rationalizing the denominator was standardized, although it is less necessary these days.  

After the Warm Up activity, for Independent Practice I give my students a Rationalizing Denominators Worksheet.  There are twenty-four problems on this practice, so it will take the students about 30 minutes to complete.

This worksheet also helps my students to review their perfect squares and the writing of ratios.  It is intended to reinforce the discussion of rationalizing the denominators of fractions to simplify radical expressions.  This always seems to cause the students difficulty, so I am hoping the history lesson helps them remember the not only the procedure, but why we are rationalizing.  I want the students to recognize that the form is being changed, but not the number value.

After the students have completed the practice, I have them self-grade their own papers using a pen or colored pencil.  No erasers are allowed at this time.  Then I have them hand the worksheet in so that I can check their progress.

This Exit slip only takes about 10 minutes for the students to complete.  I use it as a quick formative assessment to check student understanding on being able to not only rationalize the denominator, but explain the reasoning behind it.  

I show a student sample of the exit slip in the resource section.

This student rationalized the fraction correctly, and expressed reasoning of why to rationalize. Even though most of the students rationalized correctly, several of the students skip over the parts that I ask them to write about the math.  Purposely planning for writing and discussing math to occur on a daily basis helps students to know it is expected every day.