Teaching Techniques to Tackle Some Sticky Topics in Algebra 2 Presenters Liz McClain Steve Rives

Some Sticky Topics in Algebra 2  Piecewise Functions  Function Characteristics : Positive and Negative or (non-negative, non-positive) Positive and Negative or (non-negative, non-positive)

What Does Research Say?  The function concept is one of the central concepts in all of mathematics (Knuth, 2000; Romberg, Carpernter, & Fennema, 1993; Yerushalmy & Schwartz, 1993).  Understanding multiple representations of functions and the ability to move between them is critical to mathematical development (Knuth, 2000; Rider, 2007).

Piecewise/Compound Functions  Functions with restricted domains  Piecewise Function Activity  Extension of the Piecewise Activity  Technology

Part 1: Functions with restricted domains  Have students make a table and graph the function

Discussion of just “part” of the graph.  On overhead, erase part of my graph.  Discuss with students: “How could we define the domain?” “How could we define the domain?” “ How do you think we could write the rule to this function?” “ How do you think we could write the rule to this function?”

 Repeat.  Take the complete graph and erase a different part of the graph.  Discuss: “How could we define the domain?” “How could we define the domain?” “How do you think we could write the rule to this function? “How do you think we could write the rule to this function?

Discuss restricted domains.  Students erase a piece of their graph.  Students write their function with this restricted domain.  With a neighbor, students check each other’s work.

Part 2: Piecewise Function / Compound Function Activity  Materials: Students will need transparency paper and marker  Set Up:  Divide the classroom.  Half of the students graph the linear function (A), (A),  Half of the students graph the quadratic function (B)

 Students make a table and graph the function on their paper.  Students copy their graph onto a transparency.  Students pair up with students from the other group.

 Students transpose their graphs together, by laying their transparencies onto one another’s, and answer the following questions.  Together, do these graphs describe a function? (Explain)  How could you describe the new function rule?

Discuss Piece-wise Functions  A few groups share their results with the class.  Students evaluate the piecewise function.  Real world situations that require piecewise functions.  Students’ ideas? cell phone plans, IRS tax rates, purchasing in bulk cell phone plans, IRS tax rates, purchasing in bulk

Part 3: Piecewise Function More Practice  If time allows and if needed, you can continue this activity.  Students pick a function from a collection of functions you have defined.  Students make a table and graph the function on their paper.  Students copy their graphs onto transparencies.  Students find a partner and check each other’s work.  Students transpose their graphs together and create a piecewise function.  Pair of students finds another pair of students and they check each other’s work.  Have students share with overhead transparency and written function.  Summarize characteristics of Compound Functions/ Piece-wise Functions.

Using Technology to Graph Piecewise Functions  Part 1:Graphing Calculator  Graph Procedure Press Mode. Select Func and the default settings. Press Y=. Turn off all functions and stat plots. Enter Y= function. Use the TEST menu operations to define the piecewise function. Choose a graph style.

 Try graphing this piecewise function:

Part 2 : Via the Internet   ory/shared/concord/resources_files/Pi ecewise.html ory/shared/concord/resources_files/Pi ecewise.html ory/shared/concord/resources_files/Pi ecewise.html