Simplifying Rational Expressions Day 1

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SWBAT multiply rational expressions and simplify the resulting expression.

Big Idea

Polynomials are like the integers; rational expressions are like rational numbers. Multiplication of fractions makes multiplying rational expressions easy to understand.

Opening Conversation

10 minutes

The opening for this class sets the stage by calling to mind the notion of cancellation and then dealing with a common misconception.  Essentially, the conversation is based on three questions included in the notes for the opening conversation.  I would suggest writing the questions on the whiteboard one at a time and having students think-pair-share to jump start the discussion.  Each one should be handled briefly since these ideas are not new, just in need of refreshment.

With the first question, in particular, it is important to ask students how they know that the simplification is incorrect.  A simple way to confirm this is by "plugging in" some value for x that does not yield 3/2 as a result.

Some Elementary Rational Expressions

10 minutes

Hand out Simplifying Rational Expressions.

In the first 10 minutes, students should work individually to complete only the first four problems.  These put before the student the four basic cases that will arise again and again as we move forward.  Essentially, these are the four permutations of (x +/- n) divided by (x +/- n), and it's vital that students understand when these two binomials may be cancelled and when they cannot.  In fact, it might be a good idea to present these problems as a separate worksheet or simply to put them on the board before handing out the rest.  As students finish these problems check for understanding.

You may need to have some preliminary classroom conversations if you feel these first problems might be too challenging for your students or to just refresh their reasoning skills about ratios, opposites, and the commutative property.

Simplifying More Complex Expressions

25 minutes

Working independently, but verifying their results with their classmates, students should aim to complete the rest of the problems on the handout by the end of class.  Any remaining problems will become homework.  Emphasize to students to importance of considering the structure of the equation and making use of it for simplification. (MP 7)

Some structural elements to keep in mind:

  • recognize a difference of two squares & factor it
  • recall the algorithm for dividing a fraction by a fraction
  • factor (-1) out of (x - n) to obtain (n - x).

Finally, the graph is intended to keep the concepts from the previous lessons fresh in the students' minds, as well as to give some greater depth to these abstract expressions.  Students should be able to identify the discontinuity at x = -2, and the graph should clearly exhibit the asymptotic behavior of the function (both vertically & horizontally).  It is not important at this time for students to consider the discontinuity that was removed during the simplification process.