Composition of Functions- Open Middle Problem

Directions: Using the digits 1 to 9, fill in the red and blue boxes to create two functions f(x) and g(x) such that f °g exists. You can only use a number once per red box and once per blue box.


Can you create two functions f(x) and g(x) such that g°f exists?

Can you create two functions f(x) and g(x) such that  f °g and g°f exists?



How I use Open Middle Problems in the Classroom

I love talking about OM problems, and when I do I always send the message that OM problems can fit into almost any aspect of your lesson- that you can decide when or where. I always assumed that people understood what I meant- now I realize I was being that teacher! I have had many people come to me after talks saying how they would like to know how I use it within my classroom. Before I forget to address this and give another presentation, I want to have an outline in my blog.

My daily routine typically looks like this: Bellwork -> Lesson ->Exit Reflection.

OK, this looks pretty simplistic, but I try to keep it that way. When I first started teaching, my school implemented the Developmental Design for Middle School and in their training, they had a lesson template that really connected to me for how I wanted my day to progress. I have been using the Origins Lesson Plan Template ever since (and each new Administrator asks me about it- but has always seen the value of it to me and my classroom). It looks like this:


As you can see, even though I go through 3 transitions in my classroom, how they look and feel can vary. It gives my students the safety of knowing what the daily routine is, but give them the variety to keep things fresh, new, engaging.  So my

Bellwork -> Lesson -> Exit Reflection  translates to:

Plan & Prepare -> Options & Work -> Exhibit & Reflect on the lesson plan.

Now, in my talks I say that OM problems can be used at any point in this, so let’s look at how that could happen.

1)   Bellwork OR Plan & Prepare

Everything I use in class has a purpose, it’s never time-filler.  As such, I make sure to pick bellwork that will get students thinking of the upcoming lesson OR use concepts we talked about yesterday so that we can build upon them in the lesson for today.  So one OM problem I like using for bellwork when we talk about functions in 8th grade is this:


I like this problem because even though it seems too open-ended, it can generate a lot of good discussions.  Unlike a lot of other OM problems, it doesn’t provide students with a “fill in the box” format but it allows students to generate as many (or as little) points in the table as they want. I typically have students write their tables on the board and as a class we discuss whether they are a function or not.  There are times I have DESMOS on the SMARTBoard and I enter the table when we are unsure. In 8th grade my students have an idea of what functions are or are not, and this sparks a lot of prior understanding and learning which primes them for the lesson.

2)  Lesson OR Options & Work

I will keep with the functions theme through this example, but I will not use an OM problem more than once in any part of my day. It’s good to use them to challenge students, but if I am throwing too many/too often at students it looses its effectiveness.  Let’s say I get done with a great lesson on functions and want to assign practice problems for students to explore, solidify and demonstrate their understanding. I do not believe in the “Do 1-60 odds” philosophy of homework, I did not benefit from this practice as a student and do not believe in it as a teacher. I do believe in providing students with a homework assignment that is manageable and have them thinking of math outside the classroom for 15-20 minutes of their day.  As such, I like OM problems for this reason. Consider this problem:


Consider working on this problem as a 8th grade student.  Will they get a lot of practice? How many problems were assigned? Do they know what rate of change is and how it effects the points that lie on the line?  Will they have understanding of a function if they solve this problem? How can they demonstrate this? I believe they will have a solid understanding of functions and are ready to continue their learning of functions.

3)  Exit Reflection OR Exibit & Reflect

I’ve had a great lesson on linear functions, and I am positive all of my students understand what we covered. How can I be sure? Well I give them an Exit Ticket OM problem to check that understanding and provide data for tomorrow’s lesson. remember when I had the OM bellwork to tell me what a function wasn’t? Well I could give them that problem again with a few more constraints (limited number set, minimum number of points, etc) or I could give them something like this:


I would like you to reflect on this. If your students can correctly answer this, was it a good learning experience?

Those are they ways OM problems would appear in my classroom, implemented in any of the 3 transitions of my classroom. I hope this helps you envision how you can use OM in your classroom to make homework problems more challenging and interesting.


This IS a Test

This IS a test, it’s a test that has been a long time coming for me, but one I have prolonged as much as possible.  The title of the blog was supposed to catch your eye, make you think of a man I am truly honored to call a friend, José Vilson. I wanted the name to make you think of his book, and the many issues we have in classrooms all over America. I want you to also think about those same issues that lie outside the classroom, that permeate every second of every day. That issue is the one of race.

It is an issue I have struggled with, one that I have not engaged directly in- because I am a white man of privilege.  As such, I have never been sure of exactly where or when I should insert my voice on this subject. As someone of privilege, I am aware of how what I say (or what I do not say) can influence the conservation on race.  It wasn’t until one Sunday night, when I was trapped in MSP airport for 3 hours waiting to catch my flight home to Bemidji, that I decided to reach out for help. I want to be part of this conversation, but I truly do not know where to start.  That is where this blog, and those that respond to it, can help.

I before I type these next few paragraphs, I want you all to know this- that I am scared.  I am scared to continue, scared to hit publish, scared to share these thoughts publicly on social media because they represent an internal struggle within me and my life’s journey. They represent a struggle I continue to attempt to balance every day when dealing with my students and in my life. I am scared of the weight of judgement that will be rained down upon me as a man, a husband, a father, a teacher of our youth. Although I have this fear, I am continuing to type this, and hopefully when I reach the end I can finally find strength to hit send.

I was born in Virginia, and as such, I received a very strong message about people of color as I grew up.  Children are very observant of what is happening around them, and adults in the community did a good job in demonstrating the beliefs of those with privilege.  The issue was this- I never was comfortable with the message I was receiving. I kept silent, hearing those words but never pushing back on them. As a youth in a military home, questioning adults was not an option.  When I was 3, we moved to Minnesota and I grew up on a farm in a white community. I went to a school that was predominantly white, even though we were located between 3 reservations. I was not experienced in the issue of race, and my only really experience with people of color was still from the experiences of my early youth.

I continued to struggle with that view until college.  In college I lived in a diverse community, I had many friends- and those friends included many people of color.  I started to push back against those early images I retained from my youth. While living on campus, within the dorms, I was able to push past those prejudices and focus on the people I was interacting with. I did not have to worry about judgement from others about my friends there, although they still suffered from judgement every day.

As a teacher, I started teaching in a public school located on a reservation. Throughout my 15 years of teaching I have taught in a school that was primarily students of color.  I try to be aware of my message and my voice, but coming from my background I am sure I can’t help but fall into those times of “well intentioned” teaching. I need someone to push back on what I say or do in the classroom, sometimes it comes from my students. I have lost count of how many times I have been called racist by a new student, and each time it happens I reflect on what actually happened in the classroom before I respond.  Many times they are saying that as a knee jerk reaction. “Johnny, I need you to continue your work on your assignment” is immediately followed by “You’re racist.” It’s the other times that concern me. It’s like being slapped in the face, and I always try to reflect on the conversation, body language, body positions in the room, and who I addressed first in a situation in the classroom. If I ever have a doubt that what I said or done was wrong, I immediately apologize, although by that time it is too late- the damage has been done.

There are times throughout the year where the class will talk about race, and how that has affected their learning.  It is quite an eye-opening conversation with every group I have. Although many of the issues remain constant, there is always a new insight that I take from each talk.  All of the talks will have a racist versus prejudiced thread, and many of my students do not know the difference between the two. Many students do not know how to articulate themselves beyond using the word racist, so we talk about ways to talk with a teacher about how they are feeling and what factors within the classroom cause those feelings.  Many of my students come from very diverse and challenging situations, they have a need to be heard and respected for who they are. They typically are not allowed this, but it’s one thing I try to address and acknowledge. They need a place they can feel safe, respected and valued for being themselves.

Like my students, I need a place where I can feel safe and respected for this conversation of race. I want to learn more about what I am doing right or wrong, or even what I am implying by not saying anything at all. The biggest reason why I have been struggling with this for so long is that I feel ashamed.  I am ashamed because I have not pushed back against what I know is wrong for so long, but I also feel shame and fear of how my family and friends will react to my seeking knowledge and how I will interact with the topic of race. It’s the latter that has chained me into inaction, and although I keep struggling against those constraints- they keep me silent, firmly entrapped.

I want to have a conversation about race: within me, my classroom, my community, my country. I know that conversation won’t be neat or easy, but I also want to be able to know that while emotions may run hot- that I am having it with people in a constructive way.  This IS a test, we all are being graded on this every day, and for me- I need a lot of review and chances to retest so I can get it right.