Linear Functions

Students will work in pairs to understand linear_functions1_launch task and determine how to model the area (MP1, MP6).  Leave this very open-ended for students and try not to guide them in any particular direction.  Just have them come up with different combinations that will result in a given area.  

Some students, based on previous work, will start to make ordered pairs out of their list.  Don't worry if not all do, this will come out in the discussion.  Make note of which students are making an organized list compared to those that are just picking random values of x.  Also pay attention to the values that students are choosing (all whole numbers or rational numbers as well?)

During the discussion, you will want to call on students (in a strategic way) to share their thinking. Have those that tried a more random approach go first and then compare that approach to a more systemic approach.  Lastly, have students that organized their answers into ordered pairs share why they did so.  This will lead into having students graph their results.

Have graph paper available for each pair of students and have them graph their answers.  Pay particular attention to how students are labeling their axes to see if they are understanding the independent/dependent variable concept.  If this comes up in conversation you can help students determine why the side length should be on the x-axis and the area should be on the y-axis.

When leading this discussion with students, let them do most of the talking.  Have students do a turn-and-talk to discuss how they would write a function to model this situation (MP3, MP4).  As students share out their ideas, try to have each subsequent answer build on the previous as opposed to each student just sharing their own ideas.  This will keep the conversation focused and heading in the right direction and allow the students to do most of the thinking.  

(Answer A(x) = 3x)

On the second page, you are going to guide students to an understanding of the relationship between the graph of a function and the equation for that function (MP4). Have students answer the first question with their partner and come up with an answer (10.5 sq. units).  They should be able to determine this value using their graph.  Then have them draw a comparison with the equation for the function A(x) = 3x and how a value of x = 3.5 will give them the same answer.  Let students work with these two ideas in their partnerships before sharing out with the whole class (MP3).  Once students understand the connection with this value, help them to see that any point on the graph of the function will also make the equation true (MP2).

Explain to students that this investigation explores the use of functions to model the perimeter of a rectangle. linear_functions1_investigation.pdf follows the same format as the investigation of area in the Launch.  

You could just as easily post the following and not have to show all of the slides:

(1) What function could be used to model this situation?

(2) How can you represent this situation using a model or graph?

(3) If the length of the rectangle is 3.5 units (or any other value you choose), what is the perimeter based on the graph/based on the function?

(4) Show that this will work for other values as well

While students are working with partners, move from group to group to listen to student discourse and collect some formative data on student understandings.  It always helps to have a class roster when doing this so that you can make notes by individual student names.  

The final set of questions on the last slide will serve as the closure and will get students thinking more deeply about the content.

Question 1: Will help students start to compare the two functions and their output values based on certain input values.  As an extention to this question, you could also ask students what the perimeter of the rectangle is when the value of x is zero.  The point (0,6) is on the graph but does this make sense in the context of the question?  This could tie back to domain and range if those were words that students posted at the beginning of the lesson.

Question 2: This question encourages students to start thinking about systems of equations and the fact that there is an x-value where both functions will have the same numerical value.  In this case, we would not graph both of these functions on the same axes because the dependent variables represent different quantities.

Question 3: This last question is designed to get students thinking about the slope of the lines and rate of change.  You could add an extention to this question as well: The two functions have the same output value when x = 6.  For x values greater than 6, will the value of the perimeter ever be greater than the value of the area?  Why?

This unit is going to be all about students interacting with the concepts of linear and exponential multiple times.  Not all students will walk away from this first lesson with the same depth of understanding and that is fine.  Over the course of the unit their understanding of the main concepts will be deepend with each subsequent experience.