Are Absolute Value Functions Linear?

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SWBAT qualitatively understand the relationship between a linear and an absolute value function.

Big Idea

Is an Absolute Value function linear or non-linear? Students explore and explain their reasoning.

Launch: Applying Prior Knowledge

10 minutes

I begin the lesson refreshing the concept of absolute values by handing out Entrance Slips (Launch Entrance Slip.docx). This task works well in a Think, Pair, Share style, using one slip per pair. 

My purpose here is to tap into student's prior knowledge on absolute values and/or add to the ideas they already have in their minds. I want students to realize one important application of absolute values which is to express distance, but I don't emphasize the textbook definition, "distance from zero". (See my reflection on "Defining Absolute Value as distance from zero" 

As students answer the questions I stroll through listening in on their conversations and their thinking, especially with questions 5 and 6. If a students writes a negative number for any of the other questions, I may ask whether it makes sence to say, for example, that I live -5 blocks away from school. Students quickly pick up on this and understand that distance must always be expressed with positive numbers. 


New Info

10 minutes

Students will have covered the Functions unit by now and may have good notion of domain and range. Yet I don't want to assume too much, so I poke into previous knowledge asking questions about domain and range and listening to their responses. I make sure students recall that domain is the set of all possible values for the independent variable or input (Students will say all values of x), and the range, all possible values of the dependent variable, or output. Using a few examples on the board and quizzing the class for a couple of minutes is always good to get a sense of how much they have retained and maybe clear up any incorrect ideas.

Here are two video files that I may use to share with students that may need some extra help with domain and range. Students can watch these short vids before or during the activity. 

domian and range 1

domain range 2

My purpose in this section is for students to determine the domain and range of the linear function y = x, and y = lxl without graphing or using calculators.  I write these two equations on the board and ask the same pairs of students to take a few minutes to analyze and state similarities and differences between domain and range of both functions. I expect students to realize that the domains are the same but the ranges aren't. This should be a clue to whether absolute functions are linear or non-linear. Students will confirm this in the activity. 

Activity: Explore

25 minutes

I like to keep the same pairs of students yet hand each student an Activity Sheet. Activity Sheet.docx

Students are asked to graph y = x and y = lxl at the start of the activity sheet. Many students will not use negative domain values when graphing y = lxl and will obtain only have of the graph and assume that the functions is linear. I walk around expecting this to see this error. This happens because some students are used to using only positive integers like 1 through 5 as the x values. I try and make these students connect our previous conclusions about the domain and range of both functions (New Info section), to their graphs. If they still don't realize their mistake, I simply ask that they include a negative domain value in their t-chart. They will then see that the graph takes a turn. 

At the end of the activity sheet students should have determined that Absolute Value functions are non-linear, based on the range differences and the meaning of linear functions. 


Closure: Now I Know

10 minutes

Once again, after new learning occurs, I want students to reflect on or summarize what they know and didn't know before, with respect to the lesson objectives. (See my reflection on how important this is for ELL students)

I end by writing on the board in big letters...."Now I know...." I then call on students to complete the phrase stating one thing they've learned. I call on different students and I write what they respond clearly on the board. Some statements are...

  • Now I know that absolute value functions are non-linear. 
  • Now I know that the domain of both linear and absolute value functions is all real numbers, but their ranges are different.
  • Now I know that the range of absolute value functions are positive intergers. (This one I use to start a discussion. I start by asking,,,,Is this always true?)
  • Now I know that the graph of an absolute value function is V shaped and not a line.