Project Based LearningImagine That!

Math II

Project Title: Imagine That!

Project Idea: You have been selected for a summer internship with the West Virginia Department of Energy. Your supervisor informs you that you will be working in teams to analyze the energy efficiency of several living complexes that are being developed in the area. Each team will be assigned a different housing unit to evaluate. You are expected to design a working circuit that represents the electrical flow of the unit, determine the impedance and graph the projected monthly electrical expenses. You must then compare your findings with that of another team and make a recommendation as to the most cost efficient housing unit. Remember, for this project you are the electrical engineering analyst for the state and your final project must provide potential residents a valuable resource to assist them in determining the most cost efficient units.

Entry Event: Invite a representative from the local power company to visit giving the students their congratulations letter (Power Congrats) and folder welcoming them to the WV Department of Energy’s summer internship program. Provide the guest with the following ideas to help facilitate the presentation:

Discuss with the students how the power company determines the electrical costs for each house. Show the students how power usage readings are taken. Allow the students to take their own power readings. Address the energy consumption of various common household appliances and the push to increase energy

efficiency. Explain that he will return to hear their recommendations.

West Virginia College- and Career Readiness Standards:

Objectives Directly Taught or Learned Through Inquiry/Discovery

Evidence of Student Mastery of Content

Mathematical Practices1. Make sense of problems and persevere in

solving them.2. Reason abstractly and quantitatively.3. Construct viable arguments and critique the

reasoning of others.4. Model with mathematics.5. Use appropriate tools strategically.6. Attend to precision.7. Look for and make use of structure.8. Look for and express regularity in repeated

reasoning Look for and express regularity in repeated reasoning

Proficient students clarify the meaning of real world problems and identify entry points to their solution. They choose appropriate tools and make sense of quantities and relationships in problem situations. Students use assumptions and previously-established results to construct arguments and explore them. They justify conclusions, communicate using clear definitions, and respond to arguments, deciding if the arguments make sense. They ask clarifying questions. Students reflect on solutions to decide if outcomes make sense. They discern a pattern or structure and notice if calculations are repeated, while looking for both general methods and shortcuts. As they monitor and evaluate their progress, they will change course if necessary.

M.2HS.4Know there is a complex number i such that i2 = -1, and every complex number has the form a + bi with a and b real.

Power Learning LogHousing Circuit DiagramPower CollaborationCircuit Construction located www.physicslessons.com/exp22b.htm

M.2HS.5Use the relation i2 = –1 and the commutative, associative and distributive properties to add, subtract and multiply complex numbers. Instructional Note: Limit to multiplications that involve i2 as the highest power of i.

Power Learning LogHousing Circuit DiagramComplex Operations ExercisePower PresentationComplex Comparison

Performance Objectives:

KnowRecognize complex numbers in the form of a + bi

Add complex numbersMultiply complex numbersDivide complex numbersSubtract complex numbersSimplify complex numbersCreate data tablesDevelop equations to compare data

DoDevelop equations for data comparisonsAnalyze solutions determined by comparing dataMake recommendations based on analyzed dataMake informed choices among available advanced technology systems, resources and servicesfor completing curriculum assignments and projectsUse multiple electronic sources of information and multiple technology tools and resources toolsCollaborate with others, to formulate a hypothesis, to solve problems, make decisions, and present and justify the solutionsUse advanced features and utilities of spreadsheet software to perform calculations and organize, analyze and report data

Driving Question: How are imaginary numbers used to light up your house?

Assessment Plan:

Major Group ProductsStudents will construct a parallel and series circuit.Students will be assigned varying impedances for their circuit. (note that though electrical engineers use a + bj for complex numbers, students will receive impedance amounts in a + bi.Students will develop a linear equation to represent the electrical costs for their assigned housing units.Students will compare their findings with those of the other teams to determine which unit is the most electrically cost efficient.Students will prepare a power point presentation, including table and graphs that justify their recommendation for the most cost efficient unit.

Major Individual ProjectsStudents will work individually on skills prior to beginning the circuit diagram.Students will complete the online circuit training.Students will maintain a Daily Learning Log.Students will complete the Complex Operations Exercise.Students will complete a self-reflection of the project Power Self Reflection.

Assessment and Reflection:

Rubric(s) I will use: (Check all that apply.)

CollaborationPower Collaboration

X Written Communication

Critical Thinking & Problem SolvingPower Thinking and Reasoning

X Content Knowledge

Oral Communication OtherOther classroom assessments for learning: (Check all that apply)

Quizzes/ tests Practice presentationsPower Presentation

X

Self-evaluation NotesPeer evaluation Checklists/observations

Power ObservationPower Phase

X

Online tests and exams Concept mapsReflections: (Check all that apply)

Survey Focus GroupDiscussion Task Management ChartJournal Writing/ Learning LogPower Learning Log

X OtherPower Learning Log

X

Map the Project: See Imagine That! Storyboard

Product: Complex Operations Exercise

Knowledge and Skills Needed Already Have Learned

Taught Before the Project

Taught During the Project

1.  Know how to use distributive property of multiplication X2. Ability to add complex numbers X3. Ability to subtract complex numbers X4. Ability to multiply complex numbers X5. Ability to divide complex numbers X6. Ability to calculate the powers of i X7.  Find the conjugate of an expression X

Resources:

School-based Individuals:Maintenance Workers (provide access to areas where students can take various voltage readings with power company representative)

Technology:Graphing Calculators Spreadsheet Software Word Processing Software Computer Lab Digital Cameras Flash Drive Microsoft Excel and Power Point

Community: Representative from local power company

Materials:Folders Copies of Learning Log sheets Graph paper Easel paper

Websites:Circuit definitions and symbols:http://www.intmath.com/complex-numbers/8-ac-circuit-definitions.php

Circuit construction:www.physicslessons.com/exp22b.htm

Group Contracts:www.novelapproachpbl.com/ProjectManagementTools.htm

Complex Numbers - www.webmath.comInteractive Math - www.intmath.com/Complex-numbers/imaginary-numbers-intro.php

Manage the Process:

Before the Project BeginsCompile materials in a clearly identified classroom location to be used as a resource center (include a list of website resources and how they can be utilized). Display several circuit drawings posted throughout the room.Contact the local power company and make arrangements for a representative to visit the classroom. Explain the project and encourage them to bring equipment that will allow the students to take different voltage readings from common classroom items.

Prepare welcome folders for each student. Each folder should contain a welcome letter, the project scenario and requirements, a copy of all grading rubrics, a website resource list, several copies of the Learning Log sheets, and a copy of their specific housing unit diagram. Create student task forces of two for Phase I of the project. Teams can be created based on student interests or learning styles. The grouping will allow students to work, discuss and discover with other students and the teacher. (note that once students advance to Phase II of the project, teams will be combined making a task force of four students).  Launch the ProjectArrange the classroom so that students can work in teams of two. Have the representative from the power company congratulate the students on being accepted into the West Virginia Department of Energy summer internship program. If knowledgeable, have him discuss internships that are offered at the power company. Then encourage him to interact with the students allowing them to take various voltage readings of classroom items. Have him initiate a discussion on various household items and their energy consumption as well as the power company’s efforts to have residents be more energy efficient. As he completes his presentation, launch into a discussion of their upcoming internship. Pass out the welcome folders to each student in the classroom. Encourage the representative to stay while the project is being introduced and, if possible, have the representative return to see the presentations. After the students have reviewed their welcome packets, assign task partners. They should sit next to their task partner and review every aspect of their welcome packet with special emphasis on the Learning Log. They must complete a Learning Log entry daily and keep all their notes in their log book. These books will be turned in for a grade as part of the final presentation.  Creating at least four different housing unit diagrams will accommodate up to 24 students. During phase II, have the students read through their housing unit diagrams and begin completing their first Learning Log entry. They should have questions as basic as what the symbols represent. Have them go to the resource center to find websites that could help answer their questions. If necessary, help the teams develop a plan of attack to accomplish their task. They should focus on the task checklist for Phase I. Once they have identified the various symbols, they will need to understand how a circuit works. You may want to provide a handout of the various circuit symbols or post them at the resource center so students are not spending excess class time researching information that applies more toward engineering skills. They must go online to construct circuits that are in series and parallel. The purpose of the project is not to become experts at constructing a circuit, so it is only necessary that they complete the tutorial at http://www.intmath.com/complex-numbers/8-ac-circuit-definitions.php and gain a basic understanding so that they can complete their project. They should now be able to identify whether their household unit’s circuits are parallel or series. They should be able to accurately label the diagram from the information given. If not, they must research the various electrical circuit symbols until they can label the diagram. As part of differentiated instruction, you may need to assist some students when labeling the diagram to identify the complex components. Unless they have already been exposed to complex numbers, you will need to instruct them on the basic operations of complex numbers. They will need to use this information to calculate the total resistance of their assigned housing unit. This is a good opportunity to confirm each task force understands how to find the total resistance. (note: if students try to get help online, many sites will tell them to convert their findings to polar coordinates. This is not necessary, and by not converting the coordinates, they will be forced to utilize the powers of ‘i’). All the housing units are operating on 120 volts. It is helpful to keep practice examples of calculating the total resistance at the resource center (with the answers) so they can practice their calculations. Each housing unit is operating on 120 volts and some have monthly service charges (differentiated instruction). Students will need to understand that electrical engineers substitute the letter ‘j’ for ‘i’ , so they may find information online that shows the resistance in ohms as a + bj, but that is the same as a + bi. They should also recognize the difference when an inductor (a + bi) is included as opposed to a capacitor (a – bi).  This is an excellent opportunity for differentiated instruction. The goal of the project is not that they gain electrical engineering skills, but to apply aspects of complex numbers to the real world. If you have students that are unable to grasp the concept of the circuit in a timely manner, have available the completed diagram and formulas so they can work on the math and not be detained by the engineering aspects of the project. They then need to take their findings and create a linear equation that represents electrical cost compared to energy use. Several of the housing units have additional service costs that will need to be included into their equations. You may need to include a step by step process for developing the equation to assist some students.  

Students should complete the Phase I portion of their task checklists before being assigned new task force teams for Phase II. Phase II focuses on analyzing results, comparing findings and making a recommendation. If the team cannot explain their graphs from Phase I, they will have difficulty justifying their recommendations in Phase II. For differentiated instruction, you may need to keep a more detailed step by step process that compares results from Phase I and applies them in Phase II. It may be necessary to address the importance of the line intersection and what that indicates.  Students should present their findings in a professional power point presentation. However, if the computers are unavailable, they can present their finding on an easel pad.  Circuit ConstructionWhen the students work through the physics lesson located at the website www.physicslessons.com/exp22b.htm they will be given the opportunity to construct both parallel and series circuits. Though in their assigned housing units, the circuit has already been completed, they need to understand how the resistors, capacitors, etc impact the circuit. They should turn on the pop up option in the program to provide additional information about the various items available for the circuit. They should tap on “the hand” and complete each aspect of the tutorial. They should record their notes in their Learning Log. Your notes should explain what you learned and include diagrams when applicable.   Housing Circuit DiagramYou will provide the students in their initial welcome packet folder a detailed housing unit. Make sure you have put the same diagram in both partner’s packets. You may want to label the folders before handing them out to ensure that everyone is receiving the intended housing unit diagram. The diagram will have all the necessary information they will need to calculate the impedance for Phase I. Students must complete each task in the project to understand how to apply the data. The bottom of the diagram will also give them all necessary information to develop their linear equation.   Complex Operations Exercise Depending on the prior knowledge of your class, you may need to teach the basic operations of complex numbers as well as how to apply the powers of ‘i’. When students complete the exercises in each of the operations, have them include them in their log books for future reference. They will need to apply these skills when calculating the total resistance.  Complex ComparisonIf students have access to Microsoft Office software, you will want to include a help site for creating graphs and charts to assist students who do not have experience with developing presentations. Students should include in their log books the steps they took on the graphing calculator to determine the intersections of the two lines, etc.

Project Evaluation: Distribute the Power Self Reflection to each student

Key Words: complex numbercomplex number systemimaginary numbercircuitsconjugate

Congratulations. You have been selected for an internship with the West Virginia Department of Energy. This opportunity will provide you a better understanding of our state’s household energy consumption. You will be assigned to a task force evaluating the probable electrical energy consumption of several new housing units being developed in the area. Your team is expected to construct a circuit model of the electrical flow in the unit to determine the impedance of the circuit. You must then create a graph that exhibits the estimated monthly energy costs for the unit.

Once you have completed your circuit model, you will compare your results with the results of the other task forces, and make a recommendation as to the most electrically cost efficient facility. Your recommendation must be professionally prepared in a power point presentation, showing cost comparison data charts and graphs as well as a written explanation for your recommendation. Your presentation will be reviewed by electrical engineers as well as prospective residents, so ensure you are prepared to answer questions.

This will be an exciting summer and we look forward to having you join us.

Sincerely,

Redi Kilowatt

Project Scenario

You have been selected for a summer internship with the WV Department of Energy. Your welcome folder contains the information you will need for the program. Make sure you read through the entire packet before beginning and ensure that you have all required documents. You will be assigned a task force partner once everyone has had a chance to review their packets.

Your summer internship responsibilities will occur in two phases. Phase I of your project requires your task force to review the electrical circuitry of a housing unit and determine the total impedance (resistance). You must then develop a linear equation that relates power usage to monthly electrical cost. Once you have completed the Phase I requirements, you will review your findings with your project leader.

During Phase II, multiple task forces will be combined. You will compare and contrast the electrical costs of each unit. Your task force must then make a recommendation as to which unit is the most cost efficient.

You are expected to present your findings in a professional presentation that utilizes data charts and graphs that support your recommendation. The Department of Energy requires all task force members to maintain and update a daily log book, which must include any notes taken during the day. Your log book must be submitted with your final project.

You will have access to Microsoft Word, Excel and Power Point, graphing calculators and digital cameras for use during your presentation. The Department of Energy’s resource center contains additional information on circuit symbols, websites and practice problems dealing with complex numbers that may assist you in completing your task.

This project is expected to challenge you as individuals and teams, so work hard and have fun.