# Step Functions

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## Objective

Students will be able to write and graph step functions

#### Big Idea

Use real world settings to distinguish between step functions and other general piece wise functions. Can you make the call?

## Warm up and Homework Check

10 minutes

I include Warm ups with a Rubric as part of my daily routine. My goal is to allow students to work on Math Practice 3 each day. Grouping students into homogeneous pairs provides an opportunity for appropriately differentiated math conversations. This lesson’s Warm Up- Step Functions asks students to write a piecewise function given a scenario.

I also use this time to correct and record the previous day's Homework.

## Introduction to Step Functions

15 minutes

This lesson begins with a simple step function scenario:  AT&T offers a 10¢ per minute cell phone plan.  I ask the students to both graph and write an equation for this situation (Math Practice 4).  Most students will want to do f(x)=0.10x along with the corresponding graph.  Once they have had some time to work, I put this situation on the board and tell them that there is a problem with this model.  I then ask them to do a think-pair-share to figure out the problem.  It is important to clarify at this point that talking for 2 minutes and 1 second will be counted as 3 minutes.  I have them redraw this graph up through five minutes including the proper open and closed end points.

Now that they have a graph of this situation, I show how to write it as a floor function.  We also talk about ceiling functions and the difference between the two.

I then give them another scenario to model with a function and a graph: The seniors are going on a field trip.  Each bus has a capacity of 60 students.  This will help solidify this new concept.  I also have them find the domain, range and intercepts for this function.  It is good to compare the graph for this specific scenario to the general graph of the function.

## Is this a Step Function?

8 minutes

The next scenario isn’t really a step function although it does share similarities with one.

I took my dog to the groomer to get a bath.  Here is the pricing:
0-30 pounds  \$15
31-70 pounds \$25
> 70 pounds \$35

I give this to my students without a warning that it is different.  Once they have had a chance to work on it a bit, there is a good opportunity for some discussions about the similarities and differences between this and a step function (Math Practice 7).  They will then write a piecewise function and graph this scenario.

## Write your own Step Function

15 minutes

The last activity in this lesson begins with a brainstorming session on situations that can be modeled as step functions.  Students will come up with examples as pairs and then we will make a list as a class.  Each student will then create their own step function scenario which they will switch with their partner to graph and write a function (Math Practice 2).   This activity will help students deepen their understanding of step functions vs. other piecewise functions.

## Exit Ticket

2 minutes

I use an exit ticket each day as a quick formative assessment to judge the success of the lesson.

Today’s Exit Ticket asks student to write a step function from a real life scenario.

## Homework

This short assignment gives the students 4 scenarios that are either step or piecewise functions.  Students need to determine whether they are step or piecewise (Math Practice 7), write a function to match the scenario and graph it on a coordinate plane.