Finding Average Rate of Change of Polynomial and Non-Polynomial Functions

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Objective

SWBAT find average rate of a Polynomial and non-Polynomial Functions using tables, as well as a secant line intersecting a graph.

Big Idea

For students to realize that finding the average rate of change between points on a function is equivalent to finding the slope of the secant line.

Warm Up

10 minutes

The purpose of this lesson is for students to find the average rate of change between an interval in the domain given a function in different representations.  I expect the students to make the connection between finding the average rate of change and the students' prior knowledge of using the slope formula.  

In this Warm Up, I provide students with the opportunity to find the average rate of change from a table, a graph, and a given function.  I expect the Warm Up to take about 15 minutes for the class to complete and for me to review with the class.  This is to prepare students for the Partner Work where students continue to work on finding the Average Rate of Change from different representations.  

Problems one through eight of this Warm Up I used from the following website:

http://www.wccusd.net/cms/lib03/CA01001466/Centricity/domain/60/lessons/algebra%20i%20lessons/AverageRateOfChange10062013v3.pdf.  (last accessed 6-07-15)

When reviewing the Warm Up, I want students to understand the definition of a secant line and how finding the slope of the line is used to find the average rate of change between two points on a function.  I demonstrate reviewing the Warm Up with the students below.

 

Partner Work

30 minutes

After reviewing the Warm Up with students, I allow students to keep the Warm Up to access during the Independent Practice.  In the Warm Up, I provide how to find the Average Rate of Change of a Function from different representations.  These representations include from a table, graph, and an equation. Students comprehend quickly when making the connection of finding the Average Rate of Change of a function to their prior knowledge of the slope formula.  Therefore, I move forward in this lesson and assign them this online worksheet for students to work individually.  Students may discuss the worksheet with their table partner as they are working.  I instruct students to work the worksheet on their own paper and to show their work.  I also instruct them to work problems one, five, and seven first, for us to check in about five minutes.  This provides feedback to students about the understanding and accuracy of their work before completing the worksheet.

(online worksheet last accessed 5-21-15)

After reviewing problems one, five, and seven, I let students resume working on their own.  I also make a note on the board that number 14 is not to be a linear function.  Students may be assigned less problems, even or odd if necessary.  I assign the remaining problems that students do not complete as homework.  I feel these problems are important for students to understand.  These questions are given on the PARCC exam, and also relate to finding Average Rate of Change in a later Calculus Course.  I have students hand in their work from the worksheet the following day for me to check.

Exit Slip

10 minutes

With about 10 minutes left in the period, I hand each student an Exit Slip.  In this Exit Slip, students are to find the average rate of change between the second and the fifth year of the $10,000 deposit. Students are provided with the Exponential Function modeling interest compounded annually.  

Students are to use the given function to find the average rate of change in the interval x=2 to x=5, with x representing the number of years since the initial deposit. I wanted to provide an application problem at the end of the lesson to apply their knowledge on finding the average rate of change of a function.  This problem also models problems similar to the PARCC Exam.

Most of the students substituted two and five for x to get the output values of $10,816 and $12,166.53 respectively.  Then divided the difference of the output values by 3 years to find an average rate of change of 450.18 dollars per year.  Some of the students created a table of values for years two to five years and then found the average rate of change.