Introducing Functions

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SWBAT identify the independent variable, the dependent variable, input, output, domain, and a range of a relation and analyze the relation to determine a function.

Big Idea

To use the concrete example of (pieces of furniture, room location) to develop deeper student understanding of a function that leads to quantitative examples.

Warm Up

15 minutes

My goal in this lesson is for students to get a deeper understanding of functions by moving from categorical data to quantitative data. Today's Warm Up takes about 15 minutes and leads into the PowerPoint examples.  For the Warm Up, I have students compare the relationship between pieces of furniture and room locations in a house.  

To begin, I give students about three minutes to write down their ideas about the relation between pieces of furniture and room locations in a house.  After the three minutes, I call on students to share out their responses.  I first list their ideas in pairs, close together, horizontally across the board as shown in the example date below:  

bed bedroom  table dining room  table living room   toilet bathroom  couch living room   refrigerator kitchen

I purposely collect answers from students that are in the different situations of a relation. I choose students that give me two inputs that are the same, and students that give me two outputs that are the same. 

Next, I place parentheses around the pairs, and a comma in between the words.

 (bed,bedroom)  (table,dining room) (table,living room) (toilet,bathroom) (couch,living room) (refrigerator, kitchen)

The students immediately recognize the relationships as a set of ordered pairs which I define as a relation.  We discuss that a relationship has at least two items being compared to form a relationship.  From this starting point we work towards the definition of a function.

I begin asking the students what they remember about functions.  If the students do not remember what a function is, I explain that a mathematical function is a special kind of relation. A function describes a relationship in which each input maps to exactly one output value.  It can also be worded that a function has exactly one output value to each input value.  I use this definition of a function to transition to the Guided Practice section of the lesson.




Guided Practice

20 minutes

After introducing my students to the definition of a function, I begin a more formal presentation using these Power Point slides.  I consider this presentation to be Guided Practice.  As we proceed through the presentation, I will be questioning my students. Today, I plan to ask questions at the beginning of our discussion of each slide, to assess their prior knowledge. I will also give the students time to think with their elbow partner before sharing out. When necessary, I will lead using the information that I want students to think about or record in their nores.

We begin our Guided Practice by working through the following slides below:

  1. Definition of a Function
  2. Domain and Range
  3. Two Output values that are the same
  4. One to One
  5. Who has a Job?
  6. Review of Vocabulary
  7. Exit Slip- students complete on their own at the end of the lesson.

I instruct students to find the examples of each from the set of ordered pairs or relation that we stated in the Warm Up.  I refer them back to the list below.

(bed,bedroom)  (table,dining room) (table,living room) (toilet,bathroom) (couch,living room) (refrigerator, kitchen)

I begin on slide one above by questioning students about what they remember about a function. Together we come up with the definition that each input has to map to exactly one output.  I demonstrate introducing the definition of a function, Domain and Range, and the Vertical Line Test which I show in the video below.


By slide three, students are able to work with their elbow partner to show two output values that are the same. It is easy to see by mapping that the relation of (table, living room) and (couch, living room) is a function.  Each input maps to exactly one output.  I have students share out after giving them about two minutes to work on this example, and demonstrate if two output values the same still determine a function.  I use student examples and demonstrate with mapping and the coordinate plane.  On the coordinate plane, the points are horizontal and therefore passes the vertical line test.

In the fourth slide above, I explain to students that if I switch the input and the output of an ordered pair, and it also maps to exactly one value of the range, then it is a one to one function. In our examples we discuss the possibility of the ordered pairs (toilet, bathroom) and (refrigerator, kitchen) being one to one.  Even though we did not list it, some people have refrigerators in the garage or other places in the house.  So for our example, we give (toilet, bathroom) as our example.  The bathroom should be the only room that maps to the toilet also.   

On the fifth slide listed above, I ask if any students have a job, and if they do not mind telling us how much they get paid. I use this example to introduce the independent and dependent variable. Students begin responding that one hour is $7.25, two hours is $14.50, and so on until we have a table of values. I graph a few points on the coordinate plane and again we apply the vertical line test to determine that this relation is a function.  

Finally, I give students a few minutes individually to write down all the names we have used today for x and y.  Then I instruct them to share with their elbow partner.  I randomly call on pairs to share out, and complete the t-table of the different names.  Students respond with answers to the questions on the Review slide as well before completing the Exit Slip.










Exit Ticket

10 minutes

I hand students today's Exit Slip with about five minutes left in class.  I expect that my students have a strong understanding for relations, domain, range, mapping and determining a function at this point in the lesson.  Therefore, I do not expect this Exit Slip to take long.  However, I am aware that this lesson has a significant load of vocabulary for functions. I want to read what my students write, and I want to give them adequate time to use their own ideas and try to employ precise mathematical language.

If necessary, I will assign the Exit Slip as homework if time does not permit for students to complete it in class.  I will use the Exit Slip as a quick formative assessment to check for student understanding of a function.

One common mistake for students to make is confusing the fact that if a relation is not a function, the solution set for the domain and range may still be identified.  I will be checking for this and other common mistakes of students, in order to clear up these misunderstandings in future lessons.