Algebra II : Unit #2 - Modeling with Functions : Lesson #20

Absolute Value Functions: Transformations Day 1

Objective: Students will be able to graph transformations of absolute value functions.
Standards: HSF-IF.C.7b HSF-BF.B.3 MP2 MP3 MP4 MP5 MP6 MP7
Subject(s): Math
60 minutes
1 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- Absolute Value Functions Day 1 asks students to create their own questions from a scenario. 

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

2 Introduction to Absolute Value Functions - 7 minutes

This is the first day of a two-day lesson.   I have included the entire (2-day) PowerPoint, (with two separate exit tickets) as an option if the first day goes faster than expected. This way you can simply continue to as far as your class gets.

This lesson begins with a simple scenario.  The average monthly temperature for January in a northern Canadian city is 0 degrees Fahrenheit.  I ask the students to create a chart that shows the difference between a set of daily temperatures and the average and then graph this information.   We discuss the function that models this distance (Math Practice 4).  The purpose of this short activity is to allow the students to recall their conceptual knowledge of absolute value and connect it to its graph on a coordinate plane.  

This is an appropriate time to discuss the shape of the absolute value graph.  Why does it have a “v” shape?  How does it relate to f(x)=x?  I ask the class some leading questions to get a class discussion.

3 Translations of Absolute Value Functions - 15 minutes

Today’s lesson is my students’ first experience with transformation of functions.  We revisit this concept repeatedly in subsequent units for each separate parent function.  We begin with first vertical and then horizontal Translations.  My students use graphing calculators to explore each one (Math Practice 5) and then write a statement generalizing how each affects the parent function (Math Practice 7).  We then discuss these statements as a class.  For horizontal translations, I ask my students why these move opposite to the sign in the equation.  This is a good place to use a think-pair-share.  It may be a good idea here to rewrite the function like:  f(x) – c = abs(x – b).   This may help the students reason out why the horizontal translation goes opposite while the vertical translation doesn't seem to. 

Please see the PowerPoint for detailed presentation notes.

4 Reflections, Stretches, and Shrinks of Absolute Value Functions - 15 minutes

My students perform a similar procedure as in the last section for Reflections, Stretches, and Shrinks.  We start with the reflection f(x) = -abs(x).  I have also included the function f(x) = abs(-x).  I ask them to predict the transformation given the previous reflection.  Some may be surprised at the result.  This is a great place to do a think-pair-share on why the graph didn’t seem to change (Math Practice 2). 

Now the students look at stretches.  This should be run similarly to the translations.  The only difference is that the students need both a general and specific knowledge of what is happening.  Generally, they should know that the bigger the number, the narrower the graph.  More specifically, they should know that when something is multiplied by a four, it is four times taller.  I have them investigate and then write a general statement about this. 

Once they have a decent idea of stretches, I introduce shrinks.  This throws a kink into the statement they just wrote.  This is a great example of the need for precision (Math Practice 6).  I have them rewrite their generalization with this new information.  At this point, I ask for a volunteer to share theirs as a working model.   The students analyze this definition for completeness.  If the students are feeling lost, I ask a guiding question like: “Does this clear about what happens with both numbers bigger than one and numbers smaller than one?”, or “Is it enough to say a number smaller than one?”

Writing To Reinforce New Concepts
Writing Across the Disciplines

Engaging students in a lesson is fundamental in a Common Core classroom.  Gone are the days of mindlessly copying down notes taken from a text book.  I tell my students that my lessons are an experience.  With this in mind, I very rarely pre-write a definition or description of a mathematical concept into my PowerPoints.  This is something that is developed in class.  Often, we will talk about the concept, highlight key terminology, and then each student will write their own description.  This is crucial to their understanding.  Here is an example from today’s Notes on transformations.  One drawback comes from students whose summaries are not up to par due to misunderstanding or lack of effort.  I use several strategies to combat this.  I walk around and do a visual check during class where I can encourage and correct their efforts.  I often have students share their summaries with their partner which adds a layer of accountability and error detection.  Finally, I regularly check notes for points which ensures that they are putting forth a decent effort.  

5 Exit Ticket - 3 minutes

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

This Exit Ticket asks students to write and graph a function that uses two of the transformations that they have learned.