To provide students additional exposure to the definition of a function, I include three representations in today's warm-up and ask if they are functions. I intentionally included a table that at first glance seems to meet the criteria for a function, but upon closer inspection it has two outputs for the same inputs. I graph these data points and remind students of the vertical line test to verify that this table does not represent a function.
The graph I provided is non-linear which my students have had little exposure. I ask if that graph passes the vertical line test and the students (after some discussion about how the sides of the graph approach a vertical line but are not vertical) confirm that it does pass the test and is then, a function.
Finally, I provide the students an equation. I ask them what strategy we should use to determine if it is a function. One student suggests we graph it since it is in slope-intercept form, so I ask him to come to the board to do it. The other students in the class verify his work, so I ask if it, too, passes the vertical line test. He reports that it does and the class concurs with a consensus of 'thumbs-up' signals.
For today's work time, I have selected a performance task for pairs (or trios, when needed) called Charity Walk-A-Thon. I have borrowed this task from Anneberg's Missing Link Curriculum Series ( http://www.learner.org/workshops/missinglink/workshops/patterns_functions.html ). It is a wonderful task that requires students to use their previously acquired representation skills to justify their selection of the sponsor pledge amount they think would bring in the most money for their fundraiser.
Student understanding of this task is built upon our previous work with the Rule of 5 (story, picture, table, rule/equation, and graph)that we have used on assignments.
I introduce the task by giving each student pair a copy of the task. I then show it on smartboard and read through the assignment. I ask for any clarifying questions and then set the timer for 35 minutes. Once students begin working, I circulate the room providing guidance through questioning like: "What will your tables look like for this assignment?" and "What scale are you going to use on your graph?" This keeps students moving toward task completion.
One common difficulty students encounter with this task is representing the d = $.50m + $5 on the graph. Typically, they have scaled their graph by either one or two, so the half throws them off. To get them on track, I typically ask, "Where would the point (1,$5.50) be on the graph?" which is usually all the help they need to figure it out for themselves.
As student pairs complete the assignment, I asked them to turn their papers over to show me they are done. When the timer sounds after 35 minutes, I gather their attention for closure.
For closure, I ask students to look at the recommendation they made in their work and report to the corresponding corner of the room (where I have hung posters with the names Leanne, Gilberto, and Alana). I count down from 10 to expedite the process. Once students are in their respective corners, I instruct them to work with the other teams in their corner to come up with a convincing argument for their suggested pledge amount. I give them only 2 minutes to do this. When the timer sounds, I ask which corner would like to begin and then follow with the other two groups. I end by allowing students to change their minds and move to the corner they believe has the best recommendation.