While I circulate around the room to check homework with the homework rubric at the start of today's lesson, students answer some TI NSpire Navigator questions about end behavior: Warm-up Polynomial End Behavior is a set of five questions that assess students' understanding of end behavior and the formal notation used to express it.
The homework packet (see WS Review and Polynomial End Behavior) from the previous lesson was substantial, so I leave a little more time than usual for going over this assignment.
After the warm-up, my students work on Polynomial Graph Matching. I arrange students in pairs for this activity, choosing partners based on ability. For each partnership, I'd like to have one student who is strong with graphing and one who is strong in algebraic manipulation. Both skills are required for success in this activity.
Polynomial Graph Matching is a set of 20 cards with algebraic and graphical representations of polynomial functions. I included only algebraic functions in factored form to make it easier for my students to connect the graphs to the functions. The first time I used this activity, I printed out the cards on card stock and cut apart the cards. I have a set of storage drawers in my closet labeled with each unit I teach, so after I make them the first time I collect then and put them in the appropriate drawer for the next time I teach the unit. The last page of Polynomial Graph Matching is a record sheet. I print this out on plain paper and provide one sheet to each pair.
Up to this point, we have not yet explicitly discussed the relationship between the zeros of a polynomial and the x-intercepts of its graph so the matching activity will require students to think through and discuss this relationship [MP3]. While my students work I will use the 3 Cup System to provide support where needed without being "too helpful" [MP1]. I make written notes on which students seem to understand end behavior and which students are able to connect the linear factors with the roots of the equation. When students have matched up the nine pairs, they fill in the record sheet and turn it in to me. I remind them to leave the matched cards in front of them for our discussion.
By lining up all the record sheets in front of me, I can quickly see which of the graphs were hard and easy for my students. We go through each of the matches, discussing how we know which algebraic function pairs with each graph [MP2, MP3].
The Common Core places increased emphasis on student understanding of function behavior, especially when it relates to interpreting graphs, tables, and symbolic representations. There are five separate high school standards that directly address interpreting the behavior of polynomial functions (HSA.APR.B.3, HSA.SSE.B.3, HSF.IF.B4, HSF.IF.C.7, HSF.IF.C.8). Can the students recognize roots, intercepts, and end behaviors on a graph? Can they determine those same values from a factored polynomial function? Can they manipulate a polynomial in standard form so that it is factored?
This Polynomial Graph Matching Activity is an entry point for this important content. Ten polynomial functions are presented in their factored form and compared to graphs in a one-to-one relationship. This allows students to apply what they know about end behavior, and then make and test predictions about intercepts. From this foundation they can begin to generalize their understanding and hopefully see the benefit of manipulating polynomial expressions from their standard form to a factored form.
After reviewing the correct pairs for the Polynomial Graph Matching Activity, we discuss how the process of finding the x-intercepts of a polynomial relates to the Fundamental Theorem of Algebra and the Remainder Theorem. It's important to me that my students to not think of Algebra 2 as a long list of unrelated topics, so I am careful to give them time to connect new knowledge to what we have already learned. I will structure this discussion so that students can make sense of the following concepts:
This last point leads to a discussion of multiplicity, which will be a new concept for my students. Using examples of factored polynomial functions like
After these final polynomial graph concepts have been explored, we take notes on the most efficient way of producing a good sketch of a polynomial graph. I advise my students to use a process something like this:
As an exit ticket, I ask students to produce a sketch of two polynomial functions using the strategy presented in the notes. Exit Ticket-Graphs of Polynomials has two functions for students to graph:
I want students to reflect on the differences between even and odd functions.
For homework, student will work on 16 questions related to the graphs of polynomial functions in WS Graphing Polynomials. These exercises focus on my students' ability to connect the algebraic form of a polynomial function to the graph of the function [MP2].