Identifying Functions and Providing Rationale
Lesson 3 of 13
Objective: SWBAT identify if a relation is a function or not, and explain that there must be exactly one output for each input, and apply the Vertical Line Test for proof.
The purpose of this lesson is for students to determine functions from different representations, and provide proof and reasoning. It builds from the previous lesson in this unit about relationships between two variables, and determining if a situation represents a function or not. It accesses students prior knowledge about the school's snack machine which makes the content relevant to them.
This Warm Up is intended to take about 10 minutes for the students to complete and for me to review with the class. While students are working on the Warm Up, I walk around to observe each student's response(s) and method(s). I purposely look for students that have different methods to share as we review the Warm Up as a class. Students share verbally, as well as by using their individual white boards. The video below shows an example that a student might give.
After reviewing the Warm Up, I hand students a Frayer Model for students to complete as preparation for today's Guided Practice. The Frayer model includes the following four parts:
- The definition of a function
- A real world example of a function
- Examples of different representations of functions
- Non-examples of functions
This Frayer Model takes about 10 minutes of class time. The first five minutes are a time for students to write down every thing they know at this point in the lesson.
I expect my students to complete the Frayer Model using their own words or illustrations. After students complete what they know, I will have them begin the Guided Practice. After the Guided Practice, students fill in any more examples or parts of the Frayer Model that they have not completed. Sometimes, I check the students' notes as we discuss the Frayer Model. An example of the Frayer Model is in the video below.
After the lesson, the Frayer Model is to be kept in the student's notebook for reference. For more information about creating and using Frayer Models, you might enjoy the resources at www.worksheetworks.com.
Today's Guided Practice is meant to provide students with examples of the different representations of functions. I allow students about three minutes to complete Problems 1 and 2 by plotting the points. Then, students plot the points from one and two in the next two graphs. Students are to prove if the relation of each example is a function or not using the vertical line test.
After reviewing page one, I provide students about four minutes to complete page two. Page two of the Guided Notes has three examples of mapping. I use different types of mapping diagrams to increase student awareness of the different possibilities. The students have to determine if each relation is a function, and again prove that it is not a function by applying the vertical line test. Students not only have to apply the vertical line test when graphing, but circle one location on the graph where the function fails if it is not a function.
One common mistake of my students is to count the x or y axis as another location where the vertical line crosses. I want to be clear to the students that the vertical line must cross the graph itself at more than one location to fail the vertical line test.
I again ask the class why the vertical line test fails. Students should be making the connection that if a relation or graph fails the vertical line test it is because an input has more than one output. This reminds students of the definition of a function that each input can only have exactly one output.
Students then complete the third and final page of the notes. On this page students are given a set of ordered pairs, tables, and graphs to determine if each set of numbers create a function. Students are instructed to explain and to show reasoning. When students have completed the Guided Practice, I allow them a few minutes to add to the Frayer Model if necessary. If students have completed the Frayer Model, then they may begin working on the Exit Slip.
The Exit ticket is at the end of the lesson, for the students to hand in before they leave. I hand the students the Exit Slip about five minutes before the class is over.
I want students to be able to explain the connections between the different representations we used in todays and yesterdays lessons. Students should be able to identify a function, and explain using reasoning and providing evidence of why a relation is or is not a function. Students should understand by the end of this lesson that a relation is a set of ordered pairs or a relationship between two variables. A special kind of relationship is called a function. A relation is a function when each input in the domain has exactly one output. When an input has more than one output, it is not a function, and will also fail the Vertical Line Test. It will fail the Vertical Line Test when a relation has two or more outputs for the same input because the vertical line crosses more than one point on the graph.
Students should continue to understand that if the two variables in a function have one variable that is determined by another variable, then an equation can be written for the function. This equation models the function by representing all the values possible in the domain and range. However, students should also realize that the domain or range may be restricted depending on the situation.