Reflection: Intervention and Extension Simplifying Rational Expressions, Day 2 - Section 3: Simplification & Discontinuity

 

Order of operations became an issue today.  When faced with something like

many students wanted to cancel out the two c’s before doing anything else.  They recognized that the b’s couldn’t cancel because the first fraction was divided by the second, but they simply saw multiplication between the 2nd and 3rd fractions and wanted to do it first.  As one student explained, “Isn’t it true that you always do multiplication first in the order of operations?”

To clear this up, I took two approaches.  The first was conceptual.  I posed the problem 5 – 3 + 4 to the class, and everyone immediately said the value was 6.  “Well,” I said, “couldn’t I add 3 and 4 first, then do the subtraction, getting a value of -2?  What’s wrong with that?”  The students all agreed that the expression should have just one value (so it can’t be both 6 and -2), but they weren’t sure what was wrong with my approach.  I briefly led them to see that the operation of subtraction is equivalent to addition with the additive inverse.  That is, 5 – 3 + 4 is the same as 5 + (-3) + 4.  From here, we can rearrange the problem however we like: 5+4+(-3), (-3) + 4 + 5, etc.  The thing to see is that the numbers are attached, in a sense, to their operations.  Move that 3 where you will, it must always be subtracted from the total. 

Well, something similar is true of our fractions.  You can turn the whole expression into multiplication by replacing with its equivalent, .  Or you can rearrange the expression however you like, but must always be thought of as a divisor.

The second approach was more procedural; the order of operations defines a hierarchy.  However, some operations have priority over others (multiplication before addition, for instance), but some are at equal levels in the hierarchy (multiplication & division, for instance).  For operations that are at equal levels, we work from left to right.  That's just the way it is.

  Order of Operations
  Intervention and Extension: Order of Operations
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Simplifying Rational Expressions, Day 2

Unit 7: Rational Functions
Lesson 6 of 17

Objective: 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!

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Subject(s):
Math, rational expressions, Algebra, Graphing (Algebra), asymptote, Function Operations and Inverses, Algebra, master teacher project, rational function, discontinuity
  45 minutes
rat func discontinuous 2
 
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