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:

Calc 1

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:

Calc 2

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:

Calc 3

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.

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.


Algebra Technology

Those Funky, Funky Exponential Functions!

A neat math discussion came from an unexpected place today when a teacher in my department sought me out with the TI Nspire of one of her students in hand. The student was in a pre-calc class where exponential functions were being examined, and attempted the graph the following:

What should we expect to see?  How does this graph behave?  Here is what my new Nspire app gave me, which matches what the student calculator showed.


There seems to be a little funkiness around the origin which confused the Nspire, but the bigger issue is that the meat and potatoes of the graph is just wrong.  This about these values of this function and you’ll see why:

This function be-bops around in a quite interesting manner, and the TI-84 shows the graph nicely, as individual dots.  After going through some usual diagnostics in my head, and the list of dumb things kids sometimes do to calculators which cause them to act funny, the problem seems to be with Nspire. But this got me thinking about this strange function, and it’s behavior.  What happens if x = 1/2?  2/3?  3/2?  What’s the domain of this function?  And how do some of my other online math tool friends handle this one?

Wolfram|Alpha is our first contestant.  Show me your stuff:


How cool!  And what a neat discussion of complex numbers, and an interesting overlap between real and complex parts.  Wondering if anyone has insight into the domain and range though.  Is it that this function has no domain?  Or is it that  the domain is simply too difficult to express nicely?

Next up is my old friend Desmos.  I know you won’t let me down.  First, entering the function, Desmos does nothing (trust me, no screencap…nothing happens).  But, activate the table and you can plot some points.  I also added a few of my own at the end of the table:


A good effort, but wish there was some indication of the graph’s behavior without the table.

Overall, this is a tricky little function with a lot to talk about.  Put it on the board for your classes and let them think about:

  • What rational values cause the function to be undefined over the reals?
  • What rational values cause the function to have negative value?  Positive value?

Then, the plot THICKENS!  Later in the day, I was showing of my Nspire app and the goofy function to some math friends at a meeting, when an English teacher collegue joined the fray.  After giving us the obligatory “what a bunch of geeks” staredown, she grabbed my iPad and gave a few finger swipes at the graphs, changing the window values and……


Holy crud!  How cool, yet….pretty much not useful at all!  During the day, I also sent a note out to TI about what I had found, and a response was given later in the day.  Thanks for getting back to me so quickly TI folk!


Exciting Action at T^3 – Saturday

Back for a second day at the Texas Instruments Teachers Teaching with Technology Conference, and hearing some great ideas from the sessions I have attended.  The highlight of my day today was a panel discussion of  the Nspire app for iPad, which I have just loaded.  The panel was faciltated by TI brand ambassador Dr. Mayim Bialik, who has been a presence throughout the conference on sessions regarding the new app.  Dr. Bialik was joined on the panel by teachers Sheri Abel and Stephanie Ogden, who piloted the Nspire app in their classrooms.

Stephanie summarized her feelings of the ipad app:  “This is what other instructional tools aspire to be”, while Sheri shared a level of engagement from her students she had not before experienced.  As one example, Sheri had students drag and re-drag axes and functions in order to develop a conceptual understanding of domain and range.  The ability to drag and discover has been the greatest source of positive discussion from the app this weekend, and is seen as a gamechanger.

ipad appWhile the app is an exciting tool for discovery, the panel also noted that ipads certainly cannot be used for standardized tests.  To me, it will be a tough decision for schools to make a decision between ipads and handheld: the expense is not insignificant.  Also, schools may need to make a choice between continuing a TI 84 culture, or transitioning to an Nspire culture.
Coming soon, I will take the app back to my schools and hope to work through a lesson or two with classes.  I will share my reflections in a later blog post, so stay tuned….
Earlier in the day, I was excited to experience first-hand the new color 84, which is due to be available in stores for back-to-school time.  The session, 84 Plus C in Secondary  Math in Preparation for AP Calculus, was facilitated by Fan Disher
Fan walked the group through a number of problems from AP Calculus which require graphs.  This was a great opportunity to test-drive the new features.  If you are a veteran 84 user, you will identify the commands and their locations, with some new bells and whistles.  Here are my favorite new features:


The opportunity to graph functions in color, and have a color background, makes it so much easier to identify and compare graphs.  Pictures can be added as backgrounds.  The calculator comes with 5 pictures, just begging for function-modeling, and your own photos can be added by using TI Connect.

No more AAA batteries!  water fountainIf your math department spends crazy money each year just on batteries, this is a huge improvement. Calculators can be recharged using cables, or a docking station.  In chatting with TI trainers, they tell me that a calculator will hold a change for about 20-30 hours, or about 2 weeks of occasional use by a student.

Me calc

Besides utilizing the new features of the 84C, this session also modeled composition of functions, and how to use the 84 to facilitate class discussion of domain and range of composites.  Problems from AP Calculus were used to look at graphs, shaded areas, function tables, and TI’s real-type features.  Great stuff I will take back to my colleagues.

On Friday, I had the opportunity to sit down with Mari and Dale from TI to discuss some of my concerns about the latest wave of TI products, in particular that there is so much new stuff coming out at the same time….so much so that it becomes tricky for teachers to select a product and standardize their building technologies.  Dale, who has been involved with TI for 16 years, did an excellent job of walking me through TI’s thought process.  With the 84 color, TI has acknowledged that, while the Nspire and its iPad app have improved educational capabilities, the is a large core audience which is attached to the 84.  Ignoring them or forcing them into the Nspire would not be wise, or very nice.  Also, the overall look of the 84 had not changed for some time, even with vast improvements, and decreased cost, of memory and color screens.  So, now we have the 84 C!  And its improved interface is worth considering for your classroom.

If you read my blog post a few weeks ago, where I broke up with TI, I need to give you a status update on our relationship.  I’m happy to say that we talked things out, and I will be happy to be in a relationship with TI for quite some time!