Performance+Task


 * Performance Task **

The purpose of this task is for students to explore the relationship between average and instantaneous velocity. The situation they will explore is that of the position and velocity of a book thrown into the air, and how the position and velocity change over time. The students will take the role of exploring the different representations for the position of the book, and how these representations can be used to explore the average and instantaneous velocity of the book. They will present their solutions to their classmates as a way of explaining their understanding of the key points of the lesson. In addition, they will develop their own problem in which they use a position function to model a real-world problem, and develop questions that they could present to their classmates to allow them to further explore the concepts of instantaneous and average velocity. Finally, they will reflect upon the task and explore their understanding of the problems and the various representations.
 * __Performance Task Summary__**


 * __Performance Task Worksheet__**

Students will be provided with an introduction to the task and the packet below. Students will have access to their Calculus textbook, notes, and classmates. They will have had some experience with rates of change before the activity. In addition, they will be able to use a graphing calculator for parts of the activity. Finally, the teacher will provide support, as needed, to help students during the task.
 * __Needed Material and Supporting Information__**

**Goal** The goal of this task is for students to explore the concepts of average and instantaneous velocity, through the use of the concept of the derivative as a rate of change. They will also explore different ways of representing these concepts – algebraically, graphically, numerically, and verbally.
 * __GRASPS Components__**

**Role** The role of the students will be to explore the problems presented in the task, create their own problem to present to classmates, and reflect upon their learning and understanding during the task.

**Audience** The audience for the second part of the task – in which the students will create their own application problem – will be the student’s classmates. Their goal is to create a problem that is clear for their classmates and allows them to explore the concepts more in depth.

**Situation** The scenario involves the students exploring and modeling the position of a ball thrown into the air, as a function of time. They will answer questions related to the average and instantaneous velocity of the object, and explore different representations of this. Finally, they will create a novel problem to model their own scenario involving free-falling objects.

**Product** The students will turn in their answers to the exploration questions in the first part of the task, as well as the problem they develop themselves and their answers to the reflection questions at the end. See the above task worksheet for more details.

**Standards and Criteria** The NCTM standards, Enduring Understandings, and Essential Questions are explained on different tabs on this site. In addition, the rubric is provided within the task worksheet that is provided to students.

**Explanation** This task provides students with an opportunity to explore the concepts of average and instantaneous velocity and the concept of the derivative as a rate of change. They will need to explain their solution process throughout the task, and also will need to explain their thinking and the different representations they used during the reflection part of the task. Finally, they will need to be able to clearly articulate the underlying concepts that they are exploring when they create their own problem for their classmates to solve.
 * __Facets of Understanding Addressed__**

**Interpretation** This task allows students to interpret the secant and tangent lines to a graph, and the concept of the derivative, as representing the average and instantaneous velocity of an object. They will also need to interpret the different representations of these concepts, including graphical, algebraic, numerical, and verbal. Finally, they will need to interpret how the mathematical concepts and functions they are exploring relate to the real-world situation the task is modeling.

**Application** Students must apply their knowledge of secant lines, tangent lines, and the derivative to the problem of finding the average and instantaneous velocity of an object. They will also need to apply the functions, graphs, and tables, as well as the results they get when analyzing these, to the real-world situation that they are modeling.

**Self-knowledge** Students will need to reflect upon their own learning about the concepts explored. They will need to think about which aspects of the problem they understood well and which aspects stood out to them as the most important. They will need to use their knowledge of their own learning to create a new problem to be solved.