A Speedy Makeover for the Intermediate Value Theorem

As a college algebra teacher, I was not satisfied with the way I presented the intermediate value theorem last trimester.  I felt the lesson was somewhat isolated from other concepts we had studied and definitely was disconnected from the real world.  My approach lacked a hook and was laddened with procedure.   Committed to teaching the concept better this trimester, I recorded the following video while (someone else) was driving:

I know, not a high quality masterpiece, but I think I captured what I needed to illustrate the theorem.

I ask the students to draw a graph of the speed of the car with respect to time.  After playing the video a number of times, I had them share their graphs with their seat partner.  As I circulated the room, I noticed their results fell into one of these three categories:

Graph A

Graph A


Graph B

Graph B


Graph C

Graph C

After examining the options, I had them choose which graph they felt represented the situation most accurately.  Spoiler Alert:  The overwhelming majority of them chose Graph B.  Their reasoning:  it’s unclear what happened to the speed between seconds 10 and 15 therefore, there should be a space in the graph.  Those vying for Graph C cleverly argued that there was no audible “revving of the engine,” indicating that the car continued to slow.  Others supporting C claimed that even though we could not see the speed, they know how a speedometer works and can make a reasonable assumption about what happened in that time frame.

Enter this student’s graph and the Intermediate Value Theorem (trumpets):

IVT graph

I liked this students “shading” through unknown speed region, so I projected it for everyone to discuss.  They were able to determine the value of the function at ten seconds, f(10), was approximately 45 miles per hour and the value of the function at fifteen seconds, f(15), was approximately 35 miles per hour.  They also knew that the car must have reached 40 miles per hour sometime in between 10 and 15 seconds.  “How do you know that?” I pryed.  Gem response of the day:  “Well, speed is continuous and I can’t go from 45 mph to 35 mph without going through 44, 43, 42, 41, 40 mph, and so on.”  Bingo.  Intermediate Value Theorem.  No boring procedural explanation necessary.

We applied this “new” knowledge to a polynomial function so that they could get a handle on some of the algebra and notation used.   And as a bonus, they also seemed to grasp that this theorem does not only apply to crossing the x-axis, a common misconception students had last trimester.

Moving forward, I’ll definitely work on creating a better video!

Diagnostic Questions – A Tribute

The brilliant Diagnostic Questions website has been live for a few months now.  After using it once, I’ve completely convinced of its profound positive potential in my classroom.  Craig Barton and Simon Woodhead really have outdone themselves in creating a database of diagnostic questions.  If you have struggled with anticipating student responses or identifying sources of errors, this resource is a total winner.  The site allows teachers to quickly create quizzes that identify student misconceptions.  For example, here is a question from the ‘Probability – Experimental’ section:


All of these answers are carefully crafted so that the teacher can see what students aren’t grasping. Here is an example of student work:


Now I’m able to see how this student got 13, rather than just marking it wrong and moving on.  Obviously this isn’t a new phenomenon to have students explain their answers to multiple choice questions.  However, two key features of this website make it noteworthy above other sites that feature multiple choice questions:

1.  The deliberate multiple choice answers chosen to manifest misconceptions

2.  The easy-to-use format of the site allowing teachers to quickly create, administer, and grade these quizzes.

Thank you, Mr. Barton and Mr. Woodhead for this tremendously helpful tool.  I look forward to contributing questions to your database in the near future.

Surgery for Function Operations

My college algebra course boasts one of the driest textbooks on the planet. It’s one of those versions that has exercises from 1 to 99 for each section…brutal.   Can you relate?
The topics for college algebra are very standard and cover little more than what students should have encountered recently in their algebra 2 course. I therefore decided that this class would lend itself quite nicely testing out the theory that a high-level, rich question questioning can be facilitated from a traditional, drill-and-kill style textbook.

Previously, I recall that Operations on Functions was a particularly awful topic for both me and my students.  The textbook presents this concept in exactly the way you might think:

f(x) = [expression involving x]  and g(x) = [similar expression involving x]

Find f(x) + g(x), f(x) – g(x), f(g(x), f(x) *g(x), f(x)/g(x)…f(snoozefest)…you get the point.  It’s boring, they’ve done it before, and there’s not much high-level thinking involved.

Fortunately, it’s fixable by asking new questions from the same problems.  For example, have students choose a pair of functions from the book.  We have 99 choices after all!  For example, something quadratic and something linear,  like f(x) = x^2 + 1 and g(x) = 2x+4.

Here come the questions:

  • Which of these function operations are commutative and which are not?  How do you know this?
  • Does this work for all functions, or just the ones that you chose?
  • For what values of x are the non-commutative function operations equal?
  • What do you notice about those values of x for the different operations?
  • Can you prove any of your results?
  • How do the graphs of these new functions compare to the original graphs?

Compositions of functions are the most fun!  Here come some more:

  • For which values of x is f(g(x)) > g(f(x)) for your specific functions?
  •  Do your results hold true if both functions are quadratic?
  • Both linear?
  • How are the graphs of f(g(x)) and g(f(x)) related to both f(x) and g(x)?
  • Don’t forget about f(f(x)) or g(g(x))! How do those relate to our original functions?
  • What about g(g(g(x))) and g(g(g(g(x))))?
  • What do you notice happening each time we compose the function with itself again?
  • Can you generalize your conclusions based on the number of compositions and tell me what g(g(g…g(x)…)) would look like?
  • What do you notice about each of these compositions?
  • What do you notice about their graphs?

A personal favorite of mine is:  If 4x^2 + 16x + 17  =  f(g(x)), what could f(x) and g(x) have been?  This works really well with whiteboards and partners.

I might have students throw out any questions that they find interesting.  In fact, I’ll bet we can come up with at least 99 questions more intriguing than the ones given in the textbook.  Then let them choose which one(s) pique their curiosity.   Now hopefully we’ve taken the time that they would have spend doing 1-99 from a book and turned it into time better spent.