| Title | This Pool Is Too Cool | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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| Creator: | Reger, Neil nreger@access.k12.wv.us | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Source: | TLI 2008 Staff PBL Plans | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Contributing Authors: | Traci Knight, Jerry Pomeroy, Neil Reger | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Project Idea: | You are engineers working for Hi-Tech Pools, Inc. The company is designing a rubberized mat that surrounds the border of a rectangular pool, to assist in heating the pool. In order for the mat to work properly, the mat must be of uniform width around the entire pool, and the area of the mat must be the same as the area of the pool’s surface. Prepare a presentation to the design branch of Hi-Tech Pools, detailing the derivation of a formula, accompanied by a drawing to find the width of the mat, if you are given the length and width of any rectangular pool. In addition to presenting facts about the rubberized mat, include a design of an alternate heating system that would be more appealing to the consumer. Support all reasoning through sound mathematical evidence. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Entry Event: | Invite a representative from a local in-ground pool company to initiate student interest by sharing his knowledge about pool heating systems. Provide the representative with the following ideas to guide his presentation. Use Google Earth and “Fly To” Los Vegas, Nevada. Observe the number of swimming pools. Discuss the amount of energy being used to heat the pools and the different types of heating systems. Compare the amount of energy used to heat a pool in Los Vegas to a pool in your area. Open the discussion about why you would want to heat a swimming pool. Look at http://www.ehow.com/how_2247574_reduce-swimming-pool-heating-costs.html and discuss how to reduce swimming pool heating costs. Discuss ideas of passive solar heating solutions. |
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| Power Standard: | Students select and use a variety of problem solving strategies, both algebraic and graphical, with and without technology, to analyze quadratic equations over real numbers, communicate results and justify solutions in a clear and concise manner when solving real-world problems. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Content Standards & Objectives: |
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| Performance Objectives: | Know: One form of a polynomial expression may be more useful than another Various methods of factoring higher order polynomials An nth degree polynomial can be written as n linear factors The zeroes of a polynomial equation are the x-intercepts of the graph The connection between the factored form of a polynomial and the graphical representation The relationships among the graph, equation, factors and zeroes of a quadratic function Various methods of solving quadratic equations The quadratic formula The discriminate can be used to find the number and nature of the roots Some real life situations that exhibit characteristics of change can be modeled by a quadratic equation
Do: Transform polynomial expressions Use the factored form of higher order polynomials to find the graphical representation and vice versa Apply the techniques of factoring, completing the square, and the quadratic formula to solve quadratic equations Derive the quadratic formula, develop and solve quadratic equations, understand multiple representations of quadratic equations Identify the maxima and minima of a quadratic equation Use words, graphs, tables, and equations to generate solutions to practical problems Extend the nature of collected, discrete data to that of a continuous function that describes the known data set Generalize the results to make a conclusion Compare the hypothesis and the conclusion |
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| Driving Question: | How can mathematics be used to determine the design requirements of an efficiently heated pool? |
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| Assessment Plan: | Construction of a Scale Model Product: It has been several days since your engineering division at Hi-Tech Pools, Inc. received the matting project. The head engineer reports to your group that the President of Hi-Tech Pools, Inc. is skeptical of your project. He does not believe that it is possible to construct a mat of uniform width around a rectangular pool, so that the area of the mat is the same as the area of the pool’s surface. The Chief Engineer is requesting that your engineering division construct a two-dimensional scale drawing, using rational number measurements that represent an example of a pool with a mat meeting these specifications. The scale drawing must be constructed on cardboard, poster board, or cardstock paper. After completing the scale drawing, cut the mat into pieces in such a way that the pieces can be laid on top of the pool’s surface to prove that the areas are the same. Prepare a short presentation for the Chief Engineer that would simulate his presentation to the President of Hi-Tech Pools, Inc. Include a numerical justification of your solution and the method that you used to determine the dimensions of the mat and pool.
Demonstrating and Applying the Derived Formula: In only a few days, your engineering team will make their presentation to the design branch of Hi-Tech Pools, detailing the derivation of a formula, accompanied by a graphical representation to find the width of the mat, if you are given the length and width of any rectangular pool. The Chief Engineer is requesting from each of you, a persuasive essay that demonstrates the use of your formula, shows a drawing for your Scale Model Product, includes the widths of mats for a minimum of two different size pools, shows a drawing of your team’s alternate heating system and justifies why your team’s alternate heating system would be more appealing to the consumer.
Project Scenario: You are engineers working for Hi-Tech Pools, Inc. The company is designing a rubberized mat that surrounds the border of a rectangular pool, to assist in heating the pool. In order for the mat to work properly, the mat must be of uniform width around the entire pool, and the area of the mat must be the same as the area of the pool’s surface. Prepare a presentation to the design branch of Hi-Tech Pools, detailing the derivation of a formula, accompanied by a drawing to find the width of the mat, if you are given the length and width of any rectangular pool. In addition to presenting facts about the rubberized mat, include a design of an alternate heating system that would be more appealing to the consumer. Support all reasoning through sound mathematical evidence.
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| Assessment and Reflection: |
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| Map The Product: | Map the Project Storyboard
Product: Project Scenario
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| Resources: | No Data Entered | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Manage the Process: | Before the project begins: Divide students into teams of 3 or 4 students for major group products and projects.
Prepare a Resource/Learning Center for differentiating and tiering. Include the following possible tips or hints (mathematical knowledge students will need to know to complete this project) in the project Resource/Learning Center:
As a homework assignment at the end of each day, each student will use a word processor to keep a daily writing journal that includes accomplishments and a reflection of lessons learned. All entries will be in complete sentences.
Launch the Project. Driving Question: How can mathematics be used to determine the design requirements of an efficiently heated pool?
Entry Event: Invite a representative from a local in-ground pool company to initiate student interest by sharing his knowledge about pool heating systems. Provide the representative with the following ideas to guide his presentation. Use Google Earth and “Fly To” Los Vegas, Nevada. Observe the number of swimming pools. Discuss the amount of energy being used to heat the pools and the different types of heating systems. Compare the amount of energy used to heat a pool in Los Vegas to a pool in your area. Open the discussion about why you would want to heat a swimming pool. Look at http://www.ehow.com/how_2247574_reduce-swimming-pool-heating-costs.html and discuss how to reduce swimming pool heating costs. Discuss ideas of passive solar heating solutions.
Distribute the Project Scenario to each student: You are engineers working for Hi-Tech Pools, Inc. The company is designing a rubberized mat that surrounds the border of a rectangular pool, to assist in heating the pool. In order for the mat to work properly, the mat must be of uniform width around the entire pool, and the area of the mat must be the same as the area of the pool’s surface. Prepare a presentation to the design branch of Hi-Tech Pools, detailing the derivation of a formula, accompanied by a graphical representation to find the width of the mat, if you are given the length and width of any rectangular pool. In addition to presenting facts about the rubberized mat, include a design of an alternate heating system that would be more appealing to the consumer. Support all reasoning through sound mathematical evidence.
Distribute Know/Need to Know Log to individual teams to be used as periodic formative assessment.
Distribute Hi-Tech Pools Team Roles descriptions to each student. For groups of 4 students, two of the students can share the responsibilities of Design Engineer or Research Engineer. As an assignment, each team submits a Team Contract. Examples and ideas for writing contracts can be found at http://www.google.com and searching “employment agreement contracts.”
Distribute the Hi-Tech Pools Checklist to each team.
Distribute Hi-Tech Pools Adapted Knowledge Rating Scale Vocabulary Development to each student. Use as formative assessment. Students update the vocabulary development throughout the PBL experience.
Use a word processor to begin a daily writing journal that reflects on a summary of the lessons learned.
Explore Linear, Quadratic and Cubic Equations. Essential Question: What makes a set of data linear, quadratic, or cubic?
Begin with a formative assessment of degree of polynomials: Project and discuss http://www.univie.ac.at/future.media/moe/tests/var/polynome.html.
Students will work in pairs on Identifying Linear, Quadratic and Cubic Functions Activity. This activity helps students identify data sets as linear, quadratic and cubic by using first, second and third differences. Before given this activity, students will need to know how to enter an equation, graph an equation, use the list function, use regression, enter the regression equation and graph the regression equation on a graphing calculator. Directions are included in the activity.
Think-Pair-Share discussion questions after the activity should include the following: What do common differences represent? What would happen if you continue to find common differences after you have constant differences? What are the regression equations for the following tables? Graph each regression. x | f(x) x | g(x) 1 | 6 1 | -1 2 | 11 2 | 13 3 | 18 3 | 51 4 | 27 4 | 125 5 | 38 5 | 247 6 | 51 6 | 429 7 | 66 7 | 683
Explore Quadratic Data. Essential Question: What makes a set of data quadratic?
Identifying Linear, Quadratic and Cubic Functions Activity Assessment
Use one of the following activities to provide opportunities for students to generate data that models the height of an object falling due to the force of gravity. Students will work in teams to acquire data using motion detectors. They will analyze the data to find a function. http://www.dlt.ncssm.edu/algebra/HTML/12.htm Directions are included in the activity.
Think-Pair-Share discussion questions after the activity should include the following: What methods can you use to solve three equations with three variables? What function represents the height of a free-falling object with respect to time? What is the name of the graph of a free-falling object with respect to time? What equation do you use to find average speed? Is this function linear, quadratic or cubic?
Explore Zeroes and x-intercepts. Essential Question: How are the graph, equation, factors and zeroes of a quadratic function related?
Factoring Quadratics Investigation using Algebra Tiles: Students will work in teams to investigate factoring quadratics using algebra tiles (http://nlvm.usu.edu/en/nav/frames_asid_189_g_1_t_2.html?open=activities). Student Directions: On your own paper, write the title, "Factoring Quadratics Investigation using Algebra Tiles," and your team’s name. Write the activity name. Follow the directions for each activity. Copy and answer all questions. Use a straight edge to copy the algebra tiles solutions to your paper. For each investigation, write the expanded polynomial and its equivalent factored form.
See a video documentary of this project at http://wvde.state.wv.us/professional-development/model-classrooms/videos/?vid=neilreger.flv. Factoring Quadratics Practice using Algebra Tiles: Students will work in teams to practice using algebra tiles to factor quadratics (http://strader.cehd.tamu.edu/Mathematics/Algebra/AlgebraTiles/AlgebraTiles2.html). Since the problems are randomly generated, this same site can be used for assessment. Student Directions: On your own paper, write the title, "Factoring Quadratics Practice using Algebra Tiles," and your team’s name. Write the quadratic. Write the equivalent factored form. Use a straight edge to copy the algebra tiles solutions to your paper. Do the first 10 investigations.
Solving Quadratics Equations by Factoring: Students will work in teams to practice solving quadratic equations by factoring (http://www.mathmax.com/develmath/chapter/ch_ep/SEC11_7_EP47.html). Student Directions: On your own paper, write the title, "Solving Quadratics Equations by Factoring," and your team’s name. Recopy each problem and show all work to find the solution. Answers: http://www.mathmax.com/develmath/chapter/ch_ep/EP47_ANS.html
Think-Pair-Share discussion questions after this activity should include the following: Why do you need to set the quadratic equation equal to zero before you factor? What is the difference between factors, zeroes, roots, solutions and x-intercepts?
Construction of a Scale Model Product Challenge Essential Question: Why is the use of a scale model important in mathematics?
Distribute Construction of a Scale Model Product challenge to each team.
Additional Entry Point for Differentiation: Use http://nlvm.usu.edu/en/nav/frames_asid_189_g_1_t_2.html?open=activities and build the scale drawing out of Algebra Tiles. Start in the center frame of the screen and build the scale pool with one xy-tile. Build the scale mat around the pool using x-tiles, y-tiles and 1-tiles. Check the scale pool and mat to see if it meets the design requirements. Move the mat pieces on top of the pool. Move the sliders until the scale pool and mat meets the design requirements. Determine the possible measurements of the pool.
Explore the Vertex Form. Essential Question: Why is one form of a polynomial expression more useful than another?
Students will work in teams to investigate and explore the vertex form. Use http://www.analyzemath.com/quadraticg/quadraticg.htm to explore the vertex form, y = a(x – h)2 + k, as a transformation of the quadratic function y = x2. Student Directions: On your own paper, write the title, "Explore the Vertex Form," and your team’s name. Write the activity name. Follow the directions for each activity. Copy and answer all questions in activities A, B & C.
Think-Pair-Share discussion questions should include the following: Why is it helpful to change a quadratic function to parabolic form? What is the difference in the graphs of parabolas when there are two zeroes, one zero and no zeroes? What is the difference in the general forms of quadratics when there are two zeroes, one zero and no zeroes?
Explore Completing the Square. Essential Question: Why are the coordinates of the vertex of a quadratic function important?
Students will work in teams to investigate and explore completing the square. Students will take notes from http://www.purplemath.com/modules/sqrvertx.htm as an introduction to the method of completing the square and converting quadratic functions into vertex form by completing the square. Algebra tiles can be used to model this process. Use http://www.sascurriculumpathways.com -> Mathematics -> Intermediate Algebra -> Quadratic Equations -> Web Lesson 497 to explore applications of completing the square. Student Directions: On your own paper, write the title, "Complete the Square," and your team’s name. Follow the directions. Submit your record of notes and the two tables for assessment.
Practice completing the square and converting from standard form to vertex form. Check by graphing both equations.
Use real-world examples from textbooks or Internet resources to identify the vertex of a quadratic function as relative extrema.
Think-Pair-Share discussion questions should include the following: Why would you need to find the coordinates of the vertex of a quadratic function? Is there another name for a quadratic function?
Explore the Discriminant. Essential Question: Why is it important to evaluate the discriminant?
Students will work in teams to investigate and explore the discriminant. Student Directions: On your own paper, write the title, "Explore the Discriminant," and your team’s name. Use the method of completing the square to derive the quadratic formula. Discuss the information gained from the different values of the discriminant, b2 – 4ac.
Explore the graph of a quadratic function when the value of the discriminant is less than zero.
Think-Pair-Share discussion questions should include the following: Why would you need to find the value of the discriminant? What are the situations for the different values of the discriminant? What do each of these situations tell you about the nature of the roots? How do the graphs of each of these situations differ?
Presentation of Construction of a Scale Model Product.
Demonstrating and Applying the Derived Formula Challenge. Essential Question: Why is the use of mathematical models important in mathematics?
Distribute Demonstrating and Applying the Derived Formula challenge to each team member.
For help with elements of a persuasive essay, visit http://www.studygs.net/wrtstr4.htm.
Explore Factoring Higher Order Polynomials. Essential Question: Why is factoring important?
Students will work in teams to investigate and explore factoring higher order polynomials. Take notes from http://www.purplemath.com/modules/specfact2.htm as an introduction to the method of factoring sum and difference of cubes. Introduce the different methods of factoring higher order polynomials including factoring by grouping, and sum and difference of two cubes. Take notes from http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut27_gcf.htm (Factoring a Polynomial with Four Terms by Grouping Section) as an introduction to the method of factoring a polynomial with four terms. Use Degree Factors Intercepts Zeros Activity to practice factoring higher order polynomials and analyze the relationships among the factored form, zeros, degree, intercepts and shape of the graph. Directions are included in the activity. Teams may want to use http://www.mathsisfun.com/graph/function-grapher.php, a full featured graphing utility that allows them to save their work as a website link.
Class Discussion: Display all conjectures that the teams generated from Degree Factors Intercepts Zeros Activity. Discuss the validity and justification of each.
Presentation of Project Scenario. Invite parents and members of the community to be present during the presentation.
Self/Team Evaluation. Distribute Hi-Tech Pools Self-Team Final Evaluation to each team member. Each team member completes a self- evaluation and evaluates all other members of the team.
Differentiation: Classroom format includes a mix of whole group, collaborative group, paired and individual activities. Quadratic functions are modeled in a wide variety of ways using physical and virtual manipulatives, graphing technology and Internet web sites. All explorations offer a variety of entry points. A Resource/Learning Center is provided that includes materials to meet the needs of all learners. Step-by-step instructions should be provided for the special needs student. |
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| Project Evaluation: | Distribute Hi-Tech Pools Final Evaluation to each team member. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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