Real-World Applications of Parabolas

Objectives

In this unit, students explore real-world applications of parabolas. Students will:

· develop a conceptual understanding of each extracted piece of information from a quadratic function.

· visualize a given scenario and illustrate the function on paper, defining and explaining each attribute in the context of the problem.

· create real-world scenarios best modeled by quadratic functions.

· classify a real-world scenario according to its function and argue the reasoning for the use of such a model.

Essential Questions

· How can a given function be represented? What connections can be made between the various representations?

· Which function best models a given real-world scenario? What does the function look like in the real-world?

· How can an equation, table, and graph be used to analyze the rate of change and other applicable information, related to a real-world problem and the representative function?

Vocabulary

Quadratic Function: A function in the form of y=ax2+c, or y=ax2+bx+c, where a ≠ 0.
Parabola: The shape of a quadratic function. [IS.1 - Preparation]

Duration

90–120 minutes/1–2 class periods [IS.2 - All Students]

Prerequisite Skills

Prerequisite Skills haven't been entered into the lesson plan.

Materials

· graph paper [IS.3 - All Students]

· basketball (if desired)

· Lesson 3 Exit Ticket (M-A2-7-3_Lesson 3 Exit Ticket.docx) [IS.4 - Struggling Learners]

Related Unit and Lesson Plans

Related Materials & Resources

The possible inclusion of commercial websites below is not an implied endorsement of their products, which are not free, and are not required for this lesson plan.

· http://www.mathwarehouse.com/geometry/parabola/real-world-application.php

Formative Assessment

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· Observe work on independent and collaborative activities by examining selected students’ individual work. Ask questions that require students to explain the reasoning they used to complete each task. The quality of responses will reflect the degree of engagement with the objectives of each part of the lesson. [IS.12 - Struggling Learners]

· Observe student performance on the cumulative PowerPoint activity in the Lesson 3 Exit Ticket (M-A2-7-3_Lesson 3 Exit Ticket.docx), as it reflects what each presenter can teach and therefore understands. [IS.13 - All Students]

Suggested Instructional Supports

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Active Engagement

W:	The entire lesson involves active engagement and exploration, without any explicit instruction from the teacher. Students are encouraged to visualize, problem-solve, and make connections. Students are offered opportunities to sketch out ideas and argue explanations.
H:	The openness of the lesson and autonomy given to students hooks them. The engaging and thought-provoking activities hold students’ attention. How many people have actually thought of going down a slide as a parabolic function?
E:	The lesson is divided into two parts, with Part 1 serving as the initial exploration and inquiry into quadratic functions in the real world. Part 2 provides higher-level thinking, whereby students must determine two examples from two separate arenas, as well as determine a belief and argue on its behalf.
R:	Each activity, as well as the cumulative PowerPoint review assignment, prompts students to reflect, revisit, revise, and rethink. Students are put in charge of their own learning of quadratics in the real world.
E:	The collaborative work, including the cumulative PowerPoint, will provide ample opportunities for students to evaluate their understanding.
T:	With an even split of independent activities and collaborative activities, with discussion time provided at the close of each, all students are offered support to learn, reflect, and engage in the learning of quadratic functions in the real world.
O:	The lesson is organized in a completely open manner for the structured and loosely structured activities/explorations. Abstract thinking is the focus and goal of the lesson.

IS.1 - Preparation

Consider reviewing and/or reteaching (possibly through the use of a graphic organizer (e.g. a Frayer Model, Verbal Visual Word Association, Concept Circles)) all of the vocabulary needed for this lesson. This could include: altitude, vertex, domain, range, x-intercept, y-intercept, transformations, and function.

IS.2 - All Students

Consider preteaching the concepts critical to this lesson, including the use of hands-on materials. Throughout the lesson (based upon the results of formative assessment), consider the pacing to be flexible to the needs of the students. Also consider the need for reteaching and/or review both during and after the lesson as necessary.

IS.3 - All Students

Consider providing students with various graphs of parabolas with notes explaining how the characteristics of the graph connect with quadratic functions.

IS.4 - Struggling Learners

Consider having struggling students preview the topic of parabolas and quadratic functions on www.khanacademy.org. Here you can search for lessons provided by the teacher. You may also consider having the student view the Khan Academy to review for homework.

IS.5 - Struggling Learners

Consider encouraging struggling students to use a graphing calculator for this activity

IS.6 - All Students

Consider, based on student needs, providing students with a completed graph of the path of the jet so that students may analyze in order to answer questions 1-5. Once students understand the concepts involved in interpreting graphs of quadratic functions, then focus on instruction related to sketching graphs.

IS.7 - Struggling Learners

It may be beneficial to complete this activity in groups (small or large) so that struggling learners may communicate with others and benefit from the thinking processes that are shared. Be sure that conversations are facilitated within the group(s) so that student thinking is shared.

IS.8 - All Students

Consider a brief review of rate of change (from their understanding of linear functions) prior to this activity. Also consider doing one example together as a group in order to explain the vocabulary, concepts and thought process involved in solving the example. Using formative assessment, determine if it is necessary to do an example together with the groups before allowing them to complete the activity in their individual groups.

IS.9 - All Students

Consider providing a picture of the slide that is to be considered for this activity. Remember that students may have differing prior knowledge related to slides.

IS.10 - Struggling Learners

Consider providing review/direct instruction on the concept of transformations before having the struggling students complete this activity.

IS.11 - All Students

Consider viewing “Supporting Students Through a Wrap-Around Instructional Plan” (Marjory Montague) on www.naset.org.

IS.12 - Struggling Learners

Prior to teaching this lesson, consider the prior knowledge of struggling students as well as misconceptions that are likely to surface. Have a game plan for how to correct misconceptions and connect current concepts to prior knowledge.

IS.13 - All Students

Consider viewing the publication, Teachers’Desk Reference: Essential Practices for Effective Mathematics Instruction in order to review the sections on formative assessment as well as assessing and advancing questions. This publication can be found at: http://www.pattan.net/category/Resources/PaTTAN%20Publications/Browse/Single/?id=4e1f51d3150ba09c384e0000

Instructional Procedures

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Part 1

Activity 1  [IS.5 - Struggling Learners] [IS.6 - All Students]

The path of a fighter jet is represented by the function h = –15m²+ 60,000. Sketch the graph of the path of the jet and answer the following questions:

1.      What is the approximate height after 28 minutes? 54 minutes?

2.      How many minutes does it take for the jet to reach an altitude of 46,000 feet?

3.      What is the vertex, and what does it mean in the context of the problem?

4.      If another fighter jet, represented by the function h = –17m²+ 56,000, sets out at the same time for the same destination, which fighter jet will make the quickest trip? Why?

5.      What is the domain and range of each jet path? Explain the meaning in everyday terms, relating it directly to the context of the problem.

Activity 2

A tennis ball is thrown into a trash can. Draw the path of the ball, from the moment it leaves the thrower’s hands. On the graph, representing the path of the ball, label and describe each part of the process, as time continues. Provide information, regarding the vertex, domain, and range. Describe and explain in terms of the problem. What are the x-and y-intercepts? What do they mean in the context of the problem? [IS.7 - All Students]

Students will compare graphs and discuss possible reasons for differences and similarities. Students should discuss the idea of rate of change with quadratic functions. What kind of rate of change do we have here? (Encourage students to use an actual tennis ball to investigate the function.)

Part 2

Activity 3

Divide students into groups of three to four. Create two real-world scenarios of quadratic functions; model with tables and graphs; and pose and answer at least three appropriate questions for each. Create one problem from the science realm and one from the business arena. Create a visual to present to the class. [IS.8 - All Students]

Activity 4

Monique states that a person traveling down a slide represents a parabolic function. Determine if you concur or dissent. Provide supporting evidence, including representations, examples, and explanations. [IS.9 - All Students]

Activity 5: Performance

Divide students into groups of three or four. Students will create an animated PowerPoint presentation with a twofold purpose. Purpose 1 is to illustrate transformations of functions. Transformations should be supported and explained by audio, while animation is used to show the different transformed functions. Purpose 2 is to illustrate a real-world quadratic function in action. The PowerPoint will pose a problem; simulate a real-world occurrence; represent the function in equation, tabular, and graphical forms; and pose and respond to three or four questions.  [IS.10 - Struggling Learners] [IS.11 - All Students]

Extension:

·         Have students provide examples of related absolute value and quadratic functions. Students should illustrate each with appropriate representations, while explaining the reasons behind the choice in function for each example. In other words, why was an absolute value function used to model this piece of the example? Could we use a quadratic function? If we used a quadratic function, what would change regarding our interpretations? (and vice versa) Encourage students to delve deep into their understanding of connections of absolute value functions and quadratic functions with the real world. The intent is to see if they can make connections between the two types of functions, thus exhibiting a very high level of conceptual understanding related to these ideas.

Real-World Applications of Parabolas

Lesson Plan