# Building Functions 1- Open Middle Problem

Directions: Using the digits 1-6, at most once each, fill in the boxes to make a true statement.

What would changing f(x) to ___ x – ___ do?

# 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.

# The Guitar Class: MAPs Activity- Cutting out what you don’t need

Once again I am using the Shell Center resources for class, this time looking at Modeling Situations with Linear Equations.  Once again, I am looking at the student handout and although it is designed as a pre-assessment, I am wondering why they are supplying students with so much information- time to get the scissors out….

The Guitar Class:

To be totally honest here, I’m scrapping pretty much the whole handout and start from scratch, here is what I will show students.

Students should come up with ideas about rent, how many students are in the class, and how much each student should pay for the class.  They should also sketch a graph starting at a negative balance (showing the rent for the space) and increasing with each student enrolled.  I will then ask them to write down answers to the following questions on their paper:

Teacher Questions:

What values for each factor did you assign?

How did that determine where to start your graph?

How did that effect your Axes? How did it effect your scale?

What do you notice about your graph?  (hopefully they say 2 things, it’s increasing at a steady rate per student, and that is crosses the X axis)

Students will have a varied set of values for each factor, and that is OK, their graphs will all have the same general look, and they should recognize that where it crosses the X axis is where the music teacher breaks even for cost.  I don’t want to give away anything with this question, it provides a great platform for students to provide ideas and values, as well as creating their own ideas on how to graph the scenario.  Looking at each individual’s work will give me instant information on how much the student comprehends the material and what I need to address.

As students are done with their graphs and our group discussion is when I would put back in the original scenario from their handout:

I do not think I will ask students to write an equation right away, giving students the proposed values instead. I would then ask these questions for students to answer on their paper:

Teacher Questions:

What is the situation the music teacher faces? (using these values)

What is the teacher’s balance at the beginning of the class?

How does each student effects that balance?

Write an equation representing this situation.

I also would not show them questions 3 or 4, I would instead ask verbally these questions, have students record their thoughts and discuss each one as a whole class at the end of the hour:

Teacher Questions:

How can we determine the teacher’s total cost.  Write that as an equation.

How can we determine the teacher’s total profit?  Write that as an equation.

What factors do we need to consider to determine the cost for each student?  Write that as an equation.

Are there values that are unreasonable to charge students for taking his class?  Explain.

I will be trying this out with my groups tomorrow.  I plan on engaging their creativity and mind- which will produce great ideas to discuss.  I realize that this is a pre-assessment for their unit, but students will provide me with enough information on where their skills lie with my student handout and follow up questions.  I will write a follow-up to see if I “Cut too Much.”

# Functions- Open Middle Problems

Using the Integers 0-9 (and only each Integer once), create a set of ordered pair that represents a Function.

f(x) = (    ,    ),(    ,    ),(    ,    ),(    ,    ),(    ,    )

Graph the Ordered Pairs

Using the Rational Numbers 0-9 (and only each once), create a set of ordered pair that does not represents a Function.

(    ,    ),(    ,    ),(    ,    ),(    ,    )

Graph the Ordered Pairs

# Functions

Construct an X and Y axis for the graph and create a list of ordered pair representing the graph.  Does this graph represent a Function?  Justify your answer.

If you believe it is a function, change one ordered pair so that it is not.

If you believe it is not a function, change one ordered pair so that it is.