Detailed Descriptions of our Puzzling Polynomials
Lesson 4 of 15
Objective: SWBAT describe polynomial functions using accurate terminology, identify the number of turning points and state whether the functions are even or odd.
Have students answer the built in clicker questions on pages 2-4 of the Flipchart - Describing Polynomials to assess students intuitive understandings about increasing, decreasing, and constant functions. I am predicting that students already have an intuitive understanding about anything increasing or decreasing should make these go smoothly. Students may need to be reminded to “read” the graphs from left to right just like we read a book.
Students will first work through just part 1 of the Student Notes - describing polynomials to have a little more practice identifying increasing and decreasing parts of a graph. I am going to allow calculator use to sketch this function as my goal in this part of the activity is not whether or not they can make an accurate graph by hand, but whether they can identify increasing and decreasing parts of a graph.
In part a, the question does not specifically address how students should ‘indicate the regions.’ And I think this is the great part about this question. I am curious to see how students go about showing these regions. Any method at this point would be fine. And this really is the point, which leads us into the next section.
Next present page 6 of the flipchart to students and have them add the formal definition of increasing functions to their Personal Dictionaries (or maybe just write a statement about what increasing or decreasing means to them… that might be even better!).
Since this mathematical definition is pretty complex for students (especially in comparison to their intuitive understanding of increasing) it is a great opportunity to have them break apart what the math words and symbols are communicating. On page 7 of the Flipchart - Describing Polynomilas (p.6-11) there are some discussion questions to prompt the class to think about what this formal definition means. I don’t intend on even showing pg. 7 to my classes, but instead just using these as prompts to break down the definition while I am displaying the definition to the class. So I will print this page ahead of time and use the questions/prompts to get the students talking about the definition.
While students are communicating about this definition it is a great time for students to hone in on their skills at Mathematical Practice 3: Construct viable arguments and critique reasoning of others. Present pages 8 and 9 and have students add these definitions to their dictionaries.
Model the example problem for students on page 10. During this time, I plan to introduce interval notation as the uniform way we will notate the intervals on which a function is increasing and decreasing. Now students should complete part 2 on the describing polynomials notes. Toward the end of this allotted time, be sure to take questions from students over these practice problems.
To assess students understanding of even and odd functions and to help students summarize their learning today, have students answer the 5 questions on pages 14-18 of Flipchart - Describing Polynomilas (p.14-18).