Simplifying Rational Expressions, Day 2

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SWBAT simplify rational algebraic expressions and products of these expressions. SWBAT graph rational algebraic equations and identify points of discontinuity.

Big Idea

Simplifying rational expressions isn't complicated until zero gets involved!

Looking Back: What did you notice?

10 minutes

Discussing the problems from the previous lesson, ask students what they noticed as they worked to simplify the given rational expressions (see video).  For this discussion, I'll act as a scribe and interpreter at the board, taking student input, asking whether others noticed the same things, and helping students formulate their thoughts in mathematical language.  In the end, I'd like them to produce a list like this one.  I've gone into a little more detail about this discussion in this video.

Also, ask students to explain why the graph was discontinuous at x = -2.  They should be able to explain that at this point the denominator equals zero and division by zero is undefined.  Also, ensure that students recognize the asymptotic behavior of the graph.

Simplifying more complex expressions

10 minutes

At this point, I hand out Simplifying Rational Expressions 2.

Working individually at first, as usual, students should aim to complete the first problem entirely on their own.  The directions may need some clarification: since factoring a cubic polynomial is no simple matter, I suggest making use of the Factor Theorem.  Students should first factor the two quadratic denominators, then use long division or synthetic substitution to determine whether any of these factors are common to the numerators.  If some students are just able to correctly "see" the factors of the cubic polynomial, I'm fine with that.

Once the majority of the class has completed the first problem, ask one of the students to write his or her simplification on the board for all to see.  If everyone agrees, great; if not, ask that student and others to explain their thinking so that the truth will come out in the end.

Simplification & Discontinuity

25 minutes

For the remainder of class, I allow the students to work together in groups of no more than three students.  Their aim is to complete the remaining problems on the handout; any unfinished problems will become homework.

In the second section of the worksheet, I've deliberately asked students to check their simplification with x = 5 because they will find that in two of the four cases they do not really have equivalent forms.  In one case, the original equation is undefined while the simplified form is not.  In the other case, both forms are undefined for x = 5.  (This of course leads to the question of whether or not two undefined terms are equal.)

I included these questions to give students an important opportunity to make arguments and critique one another's reasoning.  As a class, we will have to answer the question of how a valid simplification process can change the value of the expression.

Finally, to see how the simplification process affects the value of the expression for all values of x, students are asked to graph both an expression and its simplified form.  They should see that the two expressions have a different value at only one point; this should help to restore confidence in the validity of the simplification process.