See piecewise.pptx slide 1
This warm-up activity allows students to revisit their working definition of a function seeing that you cannot have two distinct y-values for one x-value. This understanding is crucial for the entire lesson because we will be using step functions as a foundation to piecewise functions. In a typical step function, the point at one end of the line segment is defined and the point on the other end is not defined (closed circle and open circle, respectively). We will use this same understanding in piecewise functions. However, for all of this to come together students must understand that they cannot have two y-values for the same x-value (aka, two closed circles on the same vertical line).
Have students work with their partner to determine if the first two relations are functions and use as much vocabulary as possible to describe why they are not functions. Then, to provide an exemplar, students should work with their partners to make a sketch of a function, paying particular attention to plotting only one y-value for each x-value.
For this task, I tell students to "make their function as creative as possible" as this will discourage many of the students from drawing a basic linear function. The more complicated the function they draw, the more they have to continue to revisit the definition to ensure it remains a function.
Students will work with their partner on this opening activity. The goal is that students arrive at an understanding of the need for step functions. The step function that students will use to model Presidential salaries can be constructed without any prior knowledge. However, students will need to use persistence in order to understand the situation fully (MP1).
As a wrap up to this opening, I like to have one or two groups share their graphs and allow the class to investigate their graph with one particular concept in mind, "did this group create a function?"
What to watch out for: I try not to provide specific guidance to groups but only ask them questions to keep them focused on how to model the given situation. That said, few pairs, if any, will construct a step function that has the open and closed circles (points that are defined and undefined) in the correct place. This is the point of the wrap up to this section: Through a class dialogue and some thoughtful questioning, encourage the class to come up with a way to turn this step relation into a step function.
Questions to consider asking:
(as always "think-pair-share" or turn and talk are good ways to slow down student thinking and allow for deeper processing of your questions)
Pick your one or two crucial questions so that students can process deeply (if you emphasize everything you emphasize nothing):
1) Did this pair create a function?
2) Can you explain why it is not a function?
3) Look at the year 1969, did the President make $100,000 and $200,000 at the same time?
4) What graphical symbol could we use to represent the domain going up to the year 1969 but not including 1969?
Extension: Allow students to consider question #5 from the launch activity: "Is the President well paid?"
Students may think about things like:
1) How high is the President's salary compared to the citizens they represent? What would you have to know to answer this question? Where could you find more data to answer this question?
2) Has the relationship of the President's salary to the citizens they represent always been about the same? What were salaries for other citizens like when Washington was President? What about during other years?
*The consumer price index (CPI) is a great way to answer question #2. Students can convert 1700's dollars into 2013 dollars to make a comparison.
Encourage students to use data when considering the question, "Is the President well paid?" Have students take a position but then back it up with data that supports their position (MP2, MP4).
In this ticket out the door students will be answering questions based on a version of the 3-2-1 strategy. On a half sheet of paper or in their math journals they should write the following: 1) Three facts from the lesson about step and piecewise functions. 2) Two things that you understand really well about step and piecewise functions. (3) One question that you still have.
This ticket out will give some good insight into student understanding of the lesson based on the sophistication of the student answers. The qusestions that the students ask will also help to drive the instruction in the next lesson in order to clear up any misconceptions.