## Reflection: Joy Simplifying Rational Expressions, Day 2 - Section 3: Simplification & Discontinuity

Some students got really excited when they made sense of the removable discontinuity in one of these functions.  First, they cancelled out the only common factor: (5 – x).  Then they tried to verify their simplified form by evaluating the original and simplified forms when x = 5.  A few minutes later they asked for help.

“What did we do wrong? We only cancelled one factor, but the two equations don’t match.  In the original form we get 0/0, but in the simplified form we get -56/25.  What are we doing wrong?”

I responded by asking them if they knew why they were getting 0/0 in the original form.

One girl answered, pointing to the soon-to-be-cancelled factor (5 - x), “Well, that’s easy, it’s because 5 – 5 = 0.”

“Right,” I said, “but that's precisely the factor that you cancelled, isn’t it?  How about this: to verify your simplified equation, why don’t you try evaluating it for some other value, like x = 2.”

A few minutes later, one of the girls stopped me again.  “What’s that thing called where the graph is broken and you have that vertical line? … An asymptote?  No, I’m thinking of the other thing, where there’s just the one point missing?”

I told her it was called a removable discontinuity and walked away.  Another minute or two went by and suddenly the two girls started shouting, “I get it! I get it! It makes sense!  That factor is what causes the point to disappear!”  It was really wonderful to see.  Now, how about the rest of the class...

Getting Excited!
Joy: Getting Excited!

# Simplifying Rational Expressions, Day 2

Unit 7: Rational Functions
Lesson 6 of 17

## Big Idea: Simplifying rational expressions isn't complicated until zero gets involved!

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Standards:
Subject(s):
Math, rational expressions, Algebra, Graphing (Algebra), asymptote, Function Operations and Inverses, Algebra, master teacher project, rational function, discontinuity
45 minutes

### Jacob Nazeck

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