The previous lesson (Part 1) closes with an Exit Ticket where students indicate a functions scenario they understood well, and one they did not understand clearly. My experience is that many students understand the username-password idea easily. This is of no surprise. For ideas not well understood, some students wrote of scenario 3 (see 3 Scenarios), the input values being restricted where Andres listed side lengths in the "x" column of his table. Some students may even write that they are still unclear about what a function is, in general. This is of no surprise either.
I begin part 2 of the lesson addressing the issues written on these Exit Slips.
Example: In response to scenario 3 in yesterday's lesson, I project the image on the board again, and I ask students if they realize that Andres could have used any positive number for the length of a side of the square. So tell the class to state 3 more input values with their corresponding output values. Once they nominate these, I ask if the relation still represents a function. I then suggest a negative value, so they can give me the area, or output value. Say -4. The area would be 16, and I try to make them realize that the relation is still a function, but.....can -4 be an imput value according to the problem? So despite the relation being a function, there are values I cannot use because of the particular situation. These values are the restricted values.
After addressing the parts where students were confused, and answering questions to their satisfaction, I proceed with part 2.
To launch the lesson I project RELATIONS_FUNCTIONS for the entire class to view. After a few minutes, I begin to call on individual students to state why they believe the given relation is, or is not, a function.
My students usually don't have any difficulty with table samples. I repeat the idea of username-password in this activity, so they could make connections with an idea they are familiar with. Some difficulty may come up with the second of the mapping examples. Yet again, I refer to email accounts and ask if it is possible for many of us here in the room, to have the same password, which would be the output, while having different user names. Students quickly realize that yes, this relation is a function.
With the ordered pair example, students should realize that the relation is a function, and that in order to make it a Non-function, they can simply repeat any of the input values, keeping the same output values.
I introduce two new terms to the class. They are Domain and Range. I tell my students that the Domain is the set of all possible values of the input, which are usually the x values, and the Range is all possible values for the output, usually represented by y.
I like to refer back to Scenario_3 of the previous lesson, Andres's list of side lengths and areas. I ask students to state all the possible values for a side length. I get various values in response. Then I ask students to provide an expression that includes all of the possible lengths, including those mentioned. Then I ask: Are 0, and negative values in the domain?
A Common mistake that my students make is to state the domain as x > or = 1. I quickly ask if 0.5 or 0.6 can be used as an x value, to which they quickly realize why x > 0 is in fact the domain. To conclude, I call on volunteers to provide an expression representing all of the possible values for y, the output set (area), of scenario 3.
The task that I pose for my students during the application section is to think of three relations:
I project the launch resource (RELATIONS_FUNCTIONS) for my students to use as a reference. I encourage my students to create one of the three relations so that it is a relation, but not a function. For each example, I ask students to state the domain and range values of the relation. I find this task works well with my students working in pairs.
Once students are done, I call on random groups to share their work, making sure that all three forms are exhibited. I also make sure that Non-functions for all 3 forms are shared with the whole class. The document camera comes in handy here so students could come up and project their examples, and explain why or why not each relation is a function.