# Increasing, Decreasing, or Constant

In algebra 2 I am tasked with teaching students to write the interval on which a function is increasing, decreasing, or constant.  The idea of what a function is doing is not difficult to determine, but the subtleties of conveying that information can sometimes be a challenge.  Students often struggle to write the interval in terms of the x-values only, and often cannot remember whether to use open or closed brackets.  Here are some ways I have tried to teach what brackets to use over the years.  Although i’ll admit that this concept is a rather small component of algebra 2, I have found that I am faced with these three possible strategies often when teaching new concepts.

Option 1: Just tell them what to do

It’s easy enough to just tell students to use open brackets every time.  Most students like this and they feel like they learned something.  They can just tell themselves, “whenever I am writing if a function is increasing, decreasing, or constant, I should use open brackets.”  Students see success on assessments as a result, but rarely understand why they are doing what they are doing.  As I improve my repertoire of teaching tools and strategies, I find that I use this method less and less, only occasionally reverting to it out of desperation or an extreme shortage of time.

Option 2: Use the definition

I’ve used this strategy before with some success.  I give students the textbook’s definition of increasing, decreasing, and constant and ask them to decipher it in groups.

Definition: a function $f$ is said to be increasing on an open interval I, if for all $a$ and $b$ in that interval, $a < b$ implies $f(a) < f(b)$.

This certainly teaches students an important skill: to read and make sense of definitions in a book.  In order for students to be self-sufficient, I believe it is essential for them to be able to learn concepts without me explicitly teaching them.  However, assuming students can decompose this definition into plain English, I would expect that only about 20% of students would truly understand the mathematical meaning of “increasing” and none would understand why the book only talks about open intervals.

Option 3: Connect it to the “real world”

I used this strategy recently and was quite satisfied with the results.  I began by walking forwards and asked students, “am I walking forwards or backwards.”  All students agreed that I was walking forwards.  I then walked backwards and asked the same question.  Again, students agreed that I was walking backwards.  Finally I had one student take a picture of me at some point as I alternated walking forwards and backwards.  I put the picture under the document camera and asked the same question.

Walking forwards or backwards?

All of the sudden there was some disagreement.  Students quickly concluded that they can’t tell if I’m walking forwards or backwards from a single picture and need to know what happened before and after the photo.  I then asked them if a single point on a graph is increasing, decreasing or constant.  “We need more information” they all agreed.  The truth is, we don’t ever know whether something is increasing, decreasing, or constant without knowing where the points near it are.  When I asked the class to write a sentence for why we use open brackets at the end of class, every student gave a (more or less) adequate response.

Now of course these three strategies are not the only three.  Of course I should talk about the definitions and teach students to use their book.  I may even revert to telling an individual student to only use open brackets.  As any teacher knows, no single strategy is the fix-all for education, but instead, it’s a matter of matching one’s teaching methods with the complexities of the students, the content and the thousand other factors that go into day-to-day teaching.