Math = Love: Function/Not a Function Open Middle Problem

Thursday, June 2, 2016

Function/Not a Function Open Middle Problem

My first exposure to an open middle problem was at a CCSS workshop in 2013.

We were given the following problem (SOURCE - Page 20) to solve:

I enjoyed the challenge, but I don't think I ever used the problem in my classroom.

As I've been thinking about practice structures to use in my classroom next year, my mind keeps turning back to open middle problems.  Never heard of open middle problems?  Seriously, you need to follow that link!

Open Middle's website defines the problems as follows:

Of course, I read this description of "open middle" problems AFTER I took a stab at creating my own.  The problem I created (and am getting ready to share with you) has a "closed beginning," an "open middle" and an "open end" because there are multiple solutions possible.  I'm guessing this is still allowed because it seems like many of the problems on open allow for multiple solutions.

Okay.  Time to share the problem I created.

I wanted to create an activity to help my students practice determining if a relation is a function or not a function.  I also wanted my activity to perform double duty.  I wanted students to be able to use the exact same activity to form relations that were functions and that were not functions.

I ended up creating three x/y tables that have 9 missing values (3 missing values per table).
Students are given the integers between -4 and 4, inclusive, to place in the empty spots.

I designed these numbers so they could be laminated and easily cut apart.  I think printing them on different colored paper than the three tables would also be beneficial.

Students are given two separate challenges to complete.

Challenge 1: Place the integers between -4 and 4 into the empty spots in the tables to form three relations that are also fuctions.  Each number can only be used once.

Challenge 2: Place the integers between -4 and 4 into the empty spots in the tables to form three relations that are not functions.  Each number can only be used once.

Of course, I had to print out the activity and try it myself to make sure that there were possible solutions.  As I was completing the first challenge, I thought to myself "That was easy!  Perhaps too easy."  Then I looked at the third table and noticed I had created the ordered pair (4, 3).  I had messed up on my own puzzle!  So, you must be careful!

I'm super excited to try this activity out with students next year!

One of my motivations for making this activity is a certain EOI question released by the Oklahoma State Department of Education.  It trips my students up EVERY SINGLE YEAR.  Almost always, they are convinced that the problem is flawed and that none of the sets of data represent a function.  In this "open middle" problem, I am essentially having them create functions like choice c and non-funcitons like choices a, b, and d.

I have to admit that this activity did not turn out exactly as I had planned.  I took pictures of my planning scribbles to give you some insight into how this activity came to be.

Originally, I thought I would have a table, a set of ordered pairs, and a mapping diagram.

I also changed my mind about using the digits 0-9 and decided on -4 to 4.

After I created a set of ordered pairs, a table, and a mapping diagram, I began having second thoughts.  In order to make a mapping diagram that could be both a function and not a function, I would have to break the normal protocol for making such mapping diagrams.  This is exactly what I tell my students NOT to do.

Thus, I settled on just using tables.  I had to edit some of what I had already made so that all the tables had the same number of rows.

Overall, it was a fun and stimulating process to try and make a problem that could be solved multiple ways.  If you have any feedback about how to make this problem better, I'd love to hear it.  And, I must give a special shout-out to my husband for testing out the activity for me and listening to me as I tried to figure out what in the world I was trying to accomplish.  

You can download the files to use this activity in your own classroom here.


  1. I really like this activity! If I adjust your starting table of values just a bit, you might be able to ask students to create one linear function, one non-linear function and one non-function. I might give this a try. This fits very well with outcomes from our mathematics 10 course.

  2. Thanks for including your planning on the planning. It is rare to see the rough draft of an idea, but it is very cool to catch a glimpse of the creative process.