Why is “Simplify” So Damn Complicated?

Making my classroom rounds this week, I came across a class reviewing concepts for the upcoming Pennsylvania Keystone Exams in Alegebra 1.  The PA Department of Education provides an eligible contect document with sample items on its website, and the class was working on the following question:

Pretty standard problem.  Factor the numerator and denominator, cancel common factors, and you’re home.  But this class was struggling with the factoring review, so I stepped in with a different approach.  How about taking the given expression, and using a graphing calculator to evaluate it?  Sadly, the class was not familiar with the Table on their TI-84’s, but understood what it did right away:

Some nice discussions emerge here.  What’s with that “error”?  Is our calculator broken?  And some evidence over this function’s behavior emerges.  Note the slowly increasing values of y.

But how does this help us with the question at hand?  A number of students recognized that the correct answer would be the expression which had the same Y-values.  In essence, simplfying produces a different-looking expression with the same outputs as the original.  So, let’s try the answer choices.  Here’s A:

No dice.  Values are much different.  And a fantastic opportunity to discuss the difference between an output of zero, and an undefined output.  But eventually we get to D, and can check the tables:

Looks pretty good, butttt……..what’s with the errors?  And they seem different for some inputs.  But now we can review and discuss domain, and look at those pesky domain restrictions in a new light.

So, am I a bad person for bypassing the factoring review, and encouraging calculator use?  After the discussion, I reminded the class that factoring is a skill they need to have in their toolbox, but the alternate discussion of equivalent forms and assessing values was also worthwhile.  I feel good.

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This classroom visit got me thinking about the nature of the word “simplify” in math class.  How often do we ask students to “simplify” in math class, and in what contexts?

Sometimes we want to simplify an expression:

Or maybe we want to simplify a rational expression:

Or perhaps we want so simplify a radical expression:

And make sure you simplify when there is a radical in the denominator (unless you are taking AP Calc, in which case we don’t care about such silliness)

For different situations, we have subtle differences in what it means to simplify, but is there a common goal of simplifying?  Is it just to make things look pretty?    And is a simplified expression always the most useful?  When is it not?

I’m curious if anyone has a short and snappy answer to “what does it mean to simplify an expression?”.  I invite you to participate and contribute your response on Todays Meet (click to participate).  If you have never used Today’s Meet, it is a nice, free way to gather responses.  Simply provide the link and start a conversation!  Feel free to share the link with your students a “bell ringer” activity.  If we get some responses, I’ll make a later blog post about them.