Project Title: It’s a Mad, Mad World!

Project Idea: The Director of Marketing and Director of Manufacturing have approached your team of structural engineers with a “crisis situation.” A highly profitable sales tool is in danger of being banned from aircraft. In previous years, members of tour groups became confused with the logic of flight paths that were depicted on the two-dimensional maps distributed by the tour leaders of Transatlantic Tours. In response, a highly successful three-dimensional globe was designed that allowed tour participants to understand and follow the flight path of their tours. Currently Transatlantic Tours has been receiving complaints from airlines regarding globes being dropped in the aircraft during the flights and the resulting safety concerns. Your team has been charged with redesigning the souvenir to remedy this situation in order to prevent a profitable sales tool from being banned by the airplanes. The marketing director suggests a truncated sphere that will sit safely and snugly in the cup holder of the pull-down tray. Create a presentation of your plans for a replacement souvenir to the marketing and manufacturing departments. As always, the manufacturing department requires a cost-effective design, i.e., one that requires a minimum amount of material. Provide and justify diagrams and formulas so that the manufacturing department can implement the design for any flight path. (See Notes to Teacher).

Entry Event:Arrange the classroom to appear as an office of an engineering design firm. Post designs of a variety of appliances and equipment. Have the students sit around “work tables” and announce the imminent arrival of the Marketing and Manufacturing Directors. The directors arrive to request your team’s solution to a “crisis situation.” They charge your team with redesigning a souvenir to prevent it from being banned on airplanes. The marketing director needs your team to create plans for a souvenir globe that will sit safely and snugly in the cup holder of the pull-down tray and to present your design specifications to the Marketing Department. As the conversation progresses, provide students with two-dimensional maps of the world and ask them to determine a possible route between the two cities. Provide student teams with globes and assign the same task. Ask the students to compare the routes, determining if the distances appear the same and if not, which route appears to be the shortest. To highlight the difference between routes determined on a world map and determined on a globe, suggest that students complete the same task charting a route between New York and Bombay, India (now Mumbai, India). Globe applets http://nlvm.usu.edu/en/nav/frames_asid_308_g_3_t_3.html?from=topic_t_3.html and http://www.joelduffin.com/opensource/globe/) provide a visual aid for illustrating the calculation of Great Circle paths and comparing them to two-dimensional map distance.

West Virginia College- and Career Readiness Standards:Objectives Directly Taught or Learned Through

Inquiry/DiscoveryEvidence of Student Mastery of Content

Habits of Mind1. 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.48Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

Successful completion of teacher-determined criteria on Pythagorean Theorem Application with Rubric

Successful completion of teacher-determined criteria on Clinometer Activity with Rubric

Successful completion of teacher-determined criteria on Sample Truncated Globes with Rubric

Successful completion of teacher-determined criteria on Project Scenario Presentation with Rubric

M.2HS.49Explain and use the relationship between the sine and cosine of complementary angles.

Successful completion of teacher-determined criteria on Pythagorean Theorem Application with Rubric

Successful completion of teacher-determined criteria on Clinometer Activity with Rubric

Successful completion of teacher-determined criteria on Sample Truncated Globes with Rubric

Successful completion of teacher-determined criteria on Project Scenario Presentation with Rubric

M.2HS.50Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

Successful completion of teacher-determined criteria on Pythagorean Theorem Application with Rubric

Successful completion of teacher-determined criteria on Clinometer Activity with Rubric

Successful completion of teacher-determined criteria on Sample Truncated Globes with Rubric

Successful completion of teacher-determined criteria on Project Scenario Presentation with Rubric

Performance Objectives:Know

Data can be collected, organized and analyzedData can be presented numerically, analytically, graphically, and verballyTrigonometric ratios, including sine, cosine and tangentPythagorean Theorem and related vocabulary Circle structures and vocabulary: center, radius, diameter, circumference, central angle, arc, arc measure, arc lengthThe relationships among radius, diameter, circumference, angles and their subtended arcs

DoImplement a method to collect, organize, and analyze related data

Present the analysis of the related data numerically, analytically, graphically and verballyUse the ratios of similar triangles (trigonometric functions) to find unknown side lengths and angle measure Use the Pythagorean Theorem to determine unknown side lengthsCreate and justify relationships between points, lines (including parallel and perpendicular), and planes in a spherical geometryCreate a formula for determining the height of a truncated sphere given the circumference of its base.Create and justify relationships between parallel lines in a spherical geometryCreate and justify relationships between perpendicular lines in a spherical geometry

Driving Question:How does an understanding of geometry impact the design requirements of scale model souvenir that depicts a flight path?

Assessment Plan:Major Group Products Geometries Comparison

Create a presentation of the relationships between points, lines (including parallel and perpendicular), and planes in a non-Euclidean geometry. Compare the characteristics of points, lines (including parallel and perpendicular), and planes in the non-Euclidean geometry and Euclidean plane geometry. Examine at least three theorems or postulates of Euclidean plane geometry and test their validity in the non-Euclidean geometry. Include a clear description of the process you followed as you explored the theorems and your conclusions. Provide diagrams and/or models to support your conclusions. (The content addressed in this optional assessment exceeds the scope of the Content Standards.) Pythagorean Theorem DemonstrationResearch and present a proof of the Pythagorean Theorem. Present a clear explanation of the proof that is supported with diagrams and/or models.

 Pythagorean Theorem Application You are an engineer for a company that manufactures large screen televisions. Company research has determined that the optimum dimensions for television screens approximate the Golden Ratio. The Sales Department needs to be able to provide customers with the dimensions of televisions based on its size. Provide the Sales Department with a formula to determine screen dimensions given its size (a television’s size is its diagonal measure). Include diagrams, examples and mathematical justifications in your report. Clinometer Activity Construct and use a clinometer to measure the angle of elevation of several inaccessible objects outside of the school; create a report that demonstrates and explains the use of trig ratios to determine the heights of these objects. Sample Truncated Globes The manufacturing division needs to plan the construction phase of the truncated globe souvenirs. It has requested that your engineering division construct sample models of two different truncated globes. The models must adhere to company standards

mandating that they be as economical as possible, i.e. use a minimal amount of material. The base of the scale models must also fit into the cup holder of the airline’s pull-down tray. Create a report to explain the method your engineering team used to determine the dimensions of the truncated globes. Include diagrams and a mathematical justification for each of the models. (See Notes to Teacher) Project ScenarioThe Director of Marketing and Director of Manufacturing have approached your team of structural engineers with a “crisis situation.” A highly profitable sales tool is in danger of being banned from aircraft. In previous years, members of tour groups became confused with the logic of flight paths that were depicted on the two-dimensional maps distributed by the tour leaders of Transatlantic Tours. In response, a highly successful three-dimensional globe was designed that allowed tour participants to understand and follow the flight path of their tours. Currently Transatlantic Tours has been receiving complaints from airlines regarding globes being dropped in the aircraft during the flights and the resulting safety concerns. Your team has been charged with redesigning the souvenir to remedy this situation in order to prevent a profitable sales tool from being banned by the airplanes. The marketing director suggests a truncated sphere that will sit safely and snugly in the cup holder of the pull-down tray. Create a presentation of your plans for a replacement souvenir to the marketing and manufacturing departments. As always, the manufacturing department requires a cost-effective design, i.e., one that requires a minimum amount of material. Provide and justify diagrams and formulas so that the manufacturing department can implement the design for any flight path. (See Notes to Teacher)

Major Individual Projects

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

CollaborationCollaboration Rubric

Written Communication

Critical Thinking & Problem Solving

Content KnowledgeGeometries Comparison with Rubric

Pythagorean Theorem Demonstration with Rubric

Pythagorean Theorem Application with Rubric

Sample Truncated Globes with Rubric

Project Scenario Presentation with Rubric

Oral Communication OtherOther classroom assessments for learning:

Quizzes/ tests Practice presentationsProject Scenario Presentation

(Check all that apply) with RubricSelf-evaluationKnowledge Rating Scale

Daily Individual Learning Log with Rubric

Weekly Group Learning Log with Rubric

Final Project Evaluation – Team

Notes

Peer evaluationWeekly Group Learning Log with Rubric

Final Project Evaluation – Team

Checklists/observations

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

Survey Focus GroupDiscussion Task Management Chart

Daily Individual Learning Log with Rubric

Weekly Group Learning Log with Rubric

Journal Writing/ Learning LogDaily Individual Learning Log with Rubric

Weekly Group Learning Log with Rubric

Other

Map the Product: Project StoryboardProduct: Project Scenario PresentationKnowledge and Skills Needed Already Have

LearnedTaught Before the Project

Taught During the Project

1. Presentation skills   x   2. Research skills   x  3. Collaboration skills   x  4. Ability to write and solve proportions  x    5. Ability to find squares and square roots  x    6. Ability to simplify radicals    x  7. Ability to identify congruent triangles: SSS, SAS, ASA, SAA    x  

8. Ability to apply the Pythagorean Theorem  x   x 9. Ability to find and apply trigonometric ratios     x 10. Ability to generalize results to make a conclusion     x 

11. Ability to compare hypothesis to conclusion     x 12. Ability to analyze patterns and create an algebraic representation to model the pattern    x  

Resources:School-based IndividualsForeign Language Teacher (especially if teacher travels abroad with student groups)TechnologyGraphing Calculators; ComputersCommunityDirector of Marketing and Director of Manufacturing for a local business (or individuals to assume these roles)MaterialsSuch as Styrofoam balls and tools for cutting tools Styrofoam balls when creating sample truncated globes; globes, rulers, protractors, compassesWebsiteshttp://nlvm.usu.edu/en/nav/frames_asid_308_g_3_t_3.html?from=topic_t_3.html and http://www.joelduffin.com/opensource/globe/) provide a visual aid for illustrating the calculation of Great Circle paths and comparing them to two-dimensional map distance.

Manage the Process:Before the Project BeginsCompile materials (including globes, Styrofoam balls, cutting tools, compasses, etc.) in a clearly identified classroom location. Display travel posters throughout the classroom.  See Project Storyboard. Prepare a packet of task management materials for student use and locate these in a clearly identified place. Include organizational materials such as: task checklist, project-related rubrics, student weekly planning sheets, student planning briefs, student learning logs, student investigation briefs, student project briefs, student presentation briefs, student research logs, project milestones records, etc. (The Buck Institute for Education’s Project-Based Learning Handbook can serve as a valuable resource in developing these materials.)

 Create students teams of three for the project. The teams can be created based on student interests or learning styles. This grouping will allow students to work, discuss, and discover with other students and with the teacher. If it is necessary to create a team of four students, divide the responsibilities of the design engineer and computer engineer among three students. Distribute Team Roles and allow the students to decide the role of each. Each team of students develops a team contract to be signed by each team member in which they agree to assume the roles and responsibilities necessary to complete the project. If this is a new concept for students, a class discussion and an Internet search of sample contracts can focus student thought.

 Launch the ProjectArrange the classroom to appear as an office of an engineering design firm. Post designs of a variety of appliances and equipment. Have the students sit around “work tables” and announce the imminent arrival of the Marketing and Manufacturing Directors.  The directors arrive to request your team’s solution to a “crisis situation.” They charge your team with redesigning a souvenir to prevent it from being banned on airplanes. The marketing director needs your team to create plans for a souvenir globe that will sit safely and snugly in the cup holder of the pull-down tray and to present your design specifications to the Marketing Department.  As the conversation progresses, provide students with two-dimensional maps of the world and ask them to determine a possible route between the two cities. Provide student teams with globes and assign the same task. Ask the students to compare the routes, determining if the distances appear the same and if not, which route appears to be the shortest.

 To highlight the difference between routes determined on a world map and determined on a globe, suggest that students complete the same task charting a route between New York and Bombay, India (now Mumbai, India). Globe applets http://nlvm.usu.edu/en/nav/frames_asid_308_g_3_t_3.html?

from=topic_t_3.html and http://www.joelduffin.com/opensource/globe/) provide a visual aid for illustrating the calculation of Great Circle paths and comparing them to two-dimensional map distance.

 Project Scenario: The Director of Marketing and Director of Manufacturing have approached your team of structural engineers with a “crisis situation.” A highly profitable sales tool is in danger of being banned from aircraft. In previous years, members of tour groups became confused with the logic of flight paths that were depicted on the two-dimensional maps distributed by the tour leaders of Transatlantic Tours. In response, a highly successful three-dimensional globe was designed that allowed tour participants to understand and follow the flight path of their tours. Currently Transatlantic Tours has been receiving complaints from airlines regarding globes being dropped in the aircraft during the flights and the resulting safety concerns. Your team has been charged with redesigning the souvenir to remedy this situation in order to prevent a profitable sales tool from being banned by the airplanes. The marketing director suggests a truncated sphere that will sit safely and snugly in the cup holder of the pull-down tray. Create a presentation of your plans for a replacement souvenir to the marketing and manufacturing departments. As always, the manufacturing department requires a cost-effective design, i.e., one that requires a minimum amount of material. Provide and justify diagrams and formulas so that the manufacturing department can implement the design for any flight path. (See Notes to Teacher)

 Distribute Vocabulary Knowledge Rating Scale, a document designed to assist with unit vocabulary. Distribute the Weekly Learning Log – Group, and the Daily Learning Log – Individual, documents designed to organize and chronicle student project progress. In these documents, student teams record and self-assess their progress and record future plans. These documents provide the teacher a quick overview of both the group dynamics and the project progress. They also serve as a daily journal entry where students individually provide a status report and can also respond to posted prompts.

 Ask students to compile all artifacts (including the Vocabulary Rating Scale, Daily Learning Log – Group, and Daily Learning Log - Individual) in either a Personal Portfolio or in a Team Portfolio.

 Announce a deadline for the completion of the project and other intermediate “due dates.”

 Identify the location in the classroom where project-related documents and resources can be found. Note that included in these materials are the Know-Need to Know Chart, the Weekly Learning Log – Group, and the Daily Learning Log – Individual, Task Checklist, and all project-related rubrics.

 Provide students with globes and in a class discussion begin a class list of the essential design modifications. Allow this discussion to evolve into team discussions and the creation of a class Know–Need to Know Chart. Student teams create and update their own Know-Need to Know Chart that will also serve as an artifact of team process and progress. Encourage students to add any mathematical concepts that they wish they knew more about or wish they could expand, especially recording any obstacles or difficulties in their design modifications; students should update it often as they work on the project.

 Post the Driving Question: How does an understanding of geometry impact the design requirements of scale model souvenir of a flight path?

Explore non-Euclidean Spherical Geometry (The content addressed in this optional investigation exceeds the scope of the Content Standards.) Expand the maxim that “the shortest distance between two points is a straight line” to determine a definition of “line” on the globe. Discuss the concept of parallel and perpendicular lines on the globe and introduce the idea of non-Euclidean geometries. Student teams research Euclid’s Postulates, Euclidean plane geometry and non-Euclidean spherical geometry and in a classroom discussion, create a classroom chart that compares Euclidean plane geometry with non-Euclidean spherical geometry.

Distribute the Geometries Comparison and assign its due date.

Flatland the Movie can be used to prompt conversation comparing and contrasting Euclidean (two-dimensional geometry) and non-Euclidean (three-dimensional geometry). It relates the story of beings that live in two-dimensional land and are confronted with a three-dimensional being.  Explore the Pythagorean TheoremPrompt student recall of the Pythagorean Theorem, helping them to state the theorem using the terms hypotenuse and legs. Use interactive geometry software, on-line resources and/or classroom activities that ask students investigate the Pythagorean Theorem (cutting up one or both of the squares constructed on the legs of a right triangle so that they exactly cover the square constructed on the hypotenuse). Provide opportunities for students to apply their understanding of the Pythagorean Theorem. Student teams then research and present a proof of the Pythagorean Theorem (Pythagorean Proof Presentation).

Distribute Pythagorean Theorem Application with Rubric. 

Differentiated Assignment: Challenge students to demonstrate/prove the Pythagorean Theorem by constructing a shape other than a square on the legs and hypotenuse of a right triangle—for example a semicircle, an equilateral triangle, a regular pentagon, a regular hexagon, etc.  Explore Similar Triangles, Corresponding Parts of Similar Triangles, and Trigonometric Ratios Ask students to construct a right triangle in which the hypotenuse is twice the length of one of its legs; students now construct a second triangle with the same attributes. Students measure the length of the three sides and record them in a posted classroom chart or Excel worksheet. Students then create the ratios and their decimal equivalents that relate the longer leg/hypotenuse, the shorter leg/hypotenuse, the longer leg/shorter leg, and the shorter leg/longer leg. As student teams compare the class ratios, they should notice that the ratios comparing the corresponding sides are equivalent. Ask students to measure the angles of the triangles, determining that the triangles are similar. Introduce the idea of naming the equivalent ratios that compare the sides of these similar triangles—sine, cosine, and tangent. Find sin 30 and relate it to the previously created ratios and their decimal equivalents; comment on and discuss that this is equivalent to the ratio of shorter leg/hypotenuse; notice that the sin 30 can be identified as the ratio comparing “the side opposite the 30 degree angle” and the hypotenuse. Similarly determine and identify the remaining ratios (cos 30, tan 30, sin 60, cos 60, tan 60, sin 90, cos 90, tan 90) and relate them to definitions of these trig ratios. Ask students to create a Quick Fold booklet, recording and illustrating one ratio on each page. Add other examples of each ratio, eg., tan 45 = 1.  Use interactive geometry software, on-line resources and/or classroom activities to explore similar triangles and trigonometric rations. Provide students with teacher-created skill practice.  Distribute Clinometer Activity (http://mathcentral.uregina.ca/RR/database/RR.09.97/bracken1.html#pyth). After constructing an angle-measuring device called a clinometer, student teams use it to measure the angle of elevation for several inaccessible objects outside of the school and then use trig ratios to determine the heights of these objects. The Marketing Director returns to request that students construct sample truncated globes for the manufacturing division. Distribute Sample Truncated Globes. Explore Special Right TrianglesUse geometry software and/or constructions to allow students to explore the relationship between the measure of the legs of an isosceles right triangle and the measure of its hypotenuse (45-45-90 triangle). Ask students to use the Pythagorean Theorem to find an algebraic relationship between these lengths. Use geometry software and/or constructions (construct an equilateral triangle with an altitude from one vertex to an opposite leg) to explore the measure of a 30-60-90 triangle. Ask students to use the Pythagorean Theorem to determine an algebraic relationship between the length of the legs and the length of its hypotenuse.

 Preparation for and Presentation of Proposals Discuss the completed rubrics for the Pythagorean Proof Presentation, Sample Truncated Globes and Clinometer Activity; provide design teams opportunities to revise and practice their presentations through this final week of the unit project. The unit concludes with team presentations of their Design Revisions to the Marketing Department.

 NOTE: Student products from this project can be used when analyzing circles. For example, the determined dimensions of the truncated sphere provides an example of the theorem: If two REVISION chords REVISION intersect in a circle, then the products of the measures of the segments of the chords are equal.

Project Evaluation:Distribute Collaboration Rubric to each student.Distribute the student reflection (Final Project Evaluation - Team) of their individual and team collaborative success during the project.Distribute the student reflection (Final Project Evaluation) of the effectiveness of the project design.

Key Words: Trigonometric ratios, Pythagorean Theorem, circles, arc measure, arc length

Project StoryboardDay 0:

Before the Project Begins

Create classroom center for needed materials, related information and organizational aides, prerequisite skills

Determine due dates for individual and group assessment products

Divide the students into teams

Days 1 – 2: Launch the Project

Entry Event 

Know/Need to Know Chart

Team contracts 

Brainstorm/Investigate Design Options/Report out

 

Days 3 - 4:

Teach/Investigate/Assess

Explore Spherical Geometry

Days 5 - 9:

Teach/Investigate/Assess

Explore the Pythagorean Theorem

Non-Euclidean Geometry Presentations

Days 10 - 14:

Teach/Investigate/Assess

Explore/Review Similar Triangles and Corresponding Parts of Similar Triangles

Explore Trigonometric Ratios

Days 15 – 16:

Teach/Investigate/Assess

Explore Special Right Triangles

Days 17 – 19:

Project Scenario

Outline

Oral presentation practice

Days 20 – 22:

Project Scenario

Final presentations

Day 23:

Reflect

Final Evaluation (Self/Team)

Final Project Evaluation

Project Presentation Rubric

4 3 2 1

Organization

15%

We present our data and plan in a clear and logical sequence that is complete and easy to follow.

We present our data and plan in a logical sequence that is complete and relatively easy to follow.

We present our data and plan in a manner that is complete, but its disorganization often makes it difficult to follow.

Our presentation is incomplete and/or disorganized.

Subject Knowledge

30%

We demonstrate our understanding of the mathematical concepts related to the project with explanations that are clear, thorough, and mathematically correct.

We demonstrate our understanding of the mathematical concepts related to the project with explanations that are mathematically correct.

We demonstrate our understanding of the mathematical concepts related to the project with explanations that are primarily mathematically correct.

We are unable to adequately answer questions related to the mathematical concepts of the project.

Public Speaking

15%

We speak so that our presentation can clearly be heard; we use proper grammar and correct pronunciation; we appropriately use mathematical vocabulary to demonstrate an understanding of the terms.

We speak so that our presentation can generally be heard; we generally use proper grammar and correct pronunciation; we use mathematical terms appropriately.

We speak softly so that our presentation is difficult to hear; at times, our use of grammar and pronunciation detracts from the presentation; we use mathematical terms appropriately, but infrequently.

We speak softly so that our presentation is difficult to hear; our use of grammar and pronunciation detracts from the presentation; we seldom use mathematical terms, or use them inappropriately.

Group Participati

on15%

Each member of our group participated with relatively equivalent roles.

Each member of our group participated, but our roles were not equivalent.

Each member of our group participated, but not all spoke.

Not all members of our group participated in the presentation.

Graphics and/or Display

Materials20%

Our graphics and/or display materials are designed to explain and support our presentation.

Our graphics and/or display materials are related to our presentation.

Our graphics and/or display materials often distract from our presentation.

Our graphics and/or display materials are unrelated to our presentation or none are provided.

Developed using Rubistar (rubistar.4teachers.org)

Team Roles:

Project ManagerResponsibilities: LEL

Maintain the team schedule Take the lead on written work, including the community survey, proposals

and group presentations Track team progress Ensure completion of team products Recognize and use the strengths of other members of the team Check up on progress Deal with schedule changes or setbacks in the research, design and creation

process Assist the design engineer and computer engineer to ensure successful

completion of project tasks

Design EngineerResponsibilities:

Conceptualize and think though alternate recreation area design ideas and solve related problems

Lead in the development of design of all models, including sample truncated globes

Analyze the design to ensure accuracy Base decisions on sound mathematical reasoning Analyze ideas with experimentation and testing Think “outside the box” for a better, cheaper, more efficient, or elegant way

to accomplish a task or solve a problem Assist the project manager with scheduling, writing, and organization Help the computer engineer develop a sound research plan and produce

reports

Computer EngineerResponsibilities:

Lead in the creation of the community survey and design a method for organizing, interpreting, and presenting the results

Lead and organize research on non-Euclidean geometries and the Pythagorean Theorem

Lead and organize research on topics related to the design Organize the results of the research into a report Create an effective presentation of the sample truncated globes to highlight

the cost efficiency of the design Assist the project manager with scheduling, writing, and organization Help the design engineer develop designs that are in accord with the space

and cost requirements Help the design engineer ensure the accuracy of the design

Notes to Teacher:

In both the Project Scenario and the Sample Truncated Globes, students are asked to truncate a globe with a flight plan mapped onto it. Students need also to devise a formula that will provide the height of the truncated globe whose base will fit snugly into a cup holder. One possible method is outlined below to aid in the understanding of the intent of this project. It is provided for clarification purposes only. Students should be encouraged to find their own solution paths.

Determine the arc length of the flight path. Using an approximate circumference of the earth, determine the arc measure

of the flight path. In a two-dimensional model of the original globe with the portion to be

truncated identified (i.e., the arc of the flight path), construct an altitude from the central angle to the chord formed by connecting the endpoints of the arc of the flight path. The created congruent right triangles can now be used to determine the distance from the vertex of the central angle to truncating chord.

In addition to the measure of the radius, the measures of the angles of the right triangle are now known. These measures and trig ratios will yield the distance from the vertex of the central angle to the truncating chord. Subtracting this distance from the radius of the earth results in the needed height of the truncated globe.

Trig ratios will also yield the length of the truncating chord. Proportions can now determine the measures needed for scale models with

the necessary chord length (i.e., able to fit snugly into cup holders. Determine a formula that will model the above procedure.

Vocabulary Knowledge Rating Scale

WordKnow

It Well

Have Seen or Heard of

It

Have No

ClueWhat It Means and/or Diagram

Euclidean geometrynon-Euclidean geometry

Pythagorean Theorem

hypotenuse

leg

Golden Ratio

sine

cosine

tangent

similar triangles

30-60-90 triangle

45-45-90 triangle

central angle

arc

arc measure

arc length

WEEKLY LEARNING LOG: GroupStudents names:

Project Name:

Date:

This week, we had the following goals for project work

1

2

3

4

5

This week we accomplished…

1

2

3

4

5

Our next steps are…

1

2

3

4

5

Our most important concerns, problems or questions are…

1

2

3

4

5

www.novelapproachpbl.com

DAILY LEARNING LOG: Individual

Student name:

Project Name:

Date:

Today I had the following goals for project work

1

2

3

4

5

Today I accomplished…

1

2

3

4

5

My next steps are…

1

2

3

4

5

My most important concerns, problems or questions are…

1

2

3

4

5

www.novelapproachpbl.com

Team: _________________________ Date: _____________________

Know Need to Know/Wish I Could

Task Checklist

Individual Daily Individual Learning Log

Vocabulary Knowledge Rating Scale

Geometries Comparison

Pythagorean Theorem Application

Linear programming practice problems

Peer evaluation

Student reflection report

Other artifacts

Group Team contract

Know – Need to Know/Wish I Could Chart

Weekly Team Learning Log

Pythagorean Theorem Demonstration

Clinometer Application

Sample Truncated Globes

Project Scenario Presentation

Team notes

Final Evaluation

Final Project Evaluation

Other artifacts

Geometries ComparisonCreate a presentation of the relationships between points, lines (including parallel and perpendicular), and planes in a non-Euclidean geometry. Compare the characteristics of points, lines (including parallel and perpendicular), and planes in the non-Euclidean geometry and Euclidean plane geometry. Examine at least three theorems or postulates of Euclidean plane geometry and test their validity in the non-Euclidean geometry. Include a clear description of the process you followed as you explored the theorems and your conclusions. Provide diagrams and/or models to support your conclusions. Rubric

1 I define points, lines (including parallel and perpendicular), and planes in a non-Euclidean geometry.

___ /10

2 I accurately compare the characteristics of points, lines (including parallel and perpendicular), and planes in the non-Euclidean and Euclidean geometry.

___ /10

3 I accurately detail at least three theorems or postulates of Euclidean geometry.

___ / 5

4 I accurately and clearly describe the process I follow in testing the validity of at least three theorems or postulates of Euclidean geometry in the non-Euclidean geometry.

___ /30

5 I accurately justify my conclusions regarding the validity of the Euclidean theorems or postulates in the non-Euclidean geometry.

___ /30

6 I provide accurate diagrams and models to support my conclusions.

___ /10

7 My end product uses correct grammar and spelling. ___ / 5

Total ___ /100

Clinometer ActivityConstruct and use a clinometer to measure the angle of elevation of several inaccessible objects outside of the school; create a report that demonstrates and explains the use of trig ratios to determine the heights of these objects.

Rubric

1 We provide accurately constructed clinometers. ___ /15

2 Our report includes a clear and logical explanation of the process we use to determine the angle of elevation.

___ /30

3 We accurately demonstrate the use of trig ratios to determine the heights of inaccessible objects.

___ /30

4 Our diagrams are accurate and support our calculations and explanations.

___ /15

5 Our report is neat and uses correct grammar and spelling.

___ /10

Total ___/100

Sample Truncated Globes The manufacturing division needs to plan the construction phase of the truncated globe souvenirs. It has requested that your engineering division construct sample models of two different truncated globes. The models must adhere to company standards mandating that they be as economical as possible, i.e. use a minimal amount of material. The base of the scale models must also fit into the cup holder of the airline’s pull-down tray. Create a report to explain the method your engineering team used to determine the dimensions of the truncated globes. Include diagrams and a mathematical justification for each of the models.

Rubric

1 We provide two accurate models of truncated globes that adhere to all requirements.

___ /20

2 Our report includes an accurate explanation of the process I use to determine the dimensions of the truncated globes.

___ /25

3 Our report includes mathematical justification that accurately justifies the dimensions of the truncated globes.

___ /25

4 Our diagrams accurately support our calculations and explanations.

___ /15

5 Our report follows a clear and logical sequence that is complete and easy to understand.

___ /10

6 Our report is neat and uses correct grammar and spelling.

___ / 5

Total ___/100

Pythagorean Theorem ApplicationYou are an engineer for a company that manufactures large screen televisions. Company research has determined that the optimum dimensions for television screens approximate the Golden Ratio. The Sales Department needs to be able to provide customers with the dimensions of televisions based on its size. Provide the Sales Department with a formula to determine screen dimensions given its size (a television’s size is its diagonal measure). Include diagrams, examples and mathematical justifications in your report.

Rubric

1 I provide accurate algebraic expressions to represent the television’s dimensions. ___ /10

2My report includes an explanation that includes a mathematical justification of the process I use to determine the television’s dimensions.

___ /20

3 I accurately determine a formula that relates a screen’s dimensions to its size. ___ /10

4My report includes an explanation that includes a mathematical justification of the process I use to determine a formula that relates screen dimension to size.

___ /20

5 My diagrams support my calculations and justifications. ___ /106 I provide accurate examples that demonstrate the use of my

formula. ___ /10

7 I present my findings in a clear and logical sequence that is complete and easy to follow. ___ /15

8 My report uses correct grammar and spelling. ___ / 5Total ___/100

Pythagorean Theorem DemonstrationResearch and present a proof of the Pythagorean Theorem. Present a clear explanation of the proof that is supported with diagrams and/or models.

Rubric4 3 2 1

Organization

10%

We present our comparisons in a clear and logical sequence that is complete and easy to follow.

We present our comparisons in a logical sequence that is complete and relatively easy to follow.

We present our comparisons in a manner that is complete, but its disorganization often makes it difficult to follow.

Our presentation is incomplete and/or disorganized.

Subject Knowledge

60%

We compare the characteristics of points, lines and planes in the non-Euclidean and Euclidean geometries with explanations that are clear, thorough, and mathematically correct; we test the validity of three Euclidean theorems in the non-Euclidean geometry and demonstrate our understanding of the mathematical concepts with explanations that are clear, thorough, and mathematically correct.

We compare the characteristics of points, lines and planes in the non-Euclidean and Euclidean geometries with explanations that are mathematically correct; we test the validity of three Euclidean theorems in the non-Euclidean geometry and demonstrate our understanding of the mathematical concepts with explanations that are mathematically correct.

We compare the characteristics of points, lines and planes in the non-Euclidean and Euclidean geometries with explanations that are mathematically correct; we test the validity of three Euclidean theorems in the non-Euclidean geometry and demonstrate our understanding of the mathematical concepts with explanations that are primarily mathematically correct.

We are unable to adequately answer questions related to the mathematical concepts of the project.

Public Speaking

10%

We speak so that our presentation can clearly be heard; we use proper grammar and correct pronunciation; we appropriately use mathematical vocabulary to demonstrate an understanding of the terms.

We speak so that our presentation can generally be heard; we generally use proper grammar and correct pronunciation; we use mathematical terms appropriately.

We speak softly so that our presentation is difficult to hear; at times, our use of grammar and pronunciation detracts from the presentation; we use mathematical terms appropriately, but infrequently.

We speak softly so that our presentation is difficult to hear; our use of grammar and pronunciation detracts from the presentation; we seldom use mathematical terms, or use them inappropriately.

Group Participatio

n10%

Each member of our group participated with relatively equivalent roles.

Each member of our group participated, but our roles were not equivalent.

Each member of our group participated, but not all spoke.

Not all members of our group participated in the presentation.

Diagrams and/or Models

10%

Our graphics and/or display materials are designed to explain and support our presentation.

Our graphics and/or display materials are related to our presentation.

Our graphics and/or display materials often distract from our presentation.

Our graphics and/or display materials are unrelated to our presentation or none are provided.

Research Documentat

ion

We present a thorough list of resources and a description of the discoverer of the proof.

We present some resources and the name of the discoverer of the proof.

We present some resources or the name of the discovered of the proof.

We do not provide resources or the name of the discovered of the proof.

Collaboration Rubric

Criteria 4 3 2 1

Quality of Work /Pride(30%)

I provide work of the highest quality; my work reflects my best efforts. I can be relied on to complete my share of the group work by deadlines.

I provide high quality work; my work reflects a strong effort. I completed my share of the group work, but had difficulty meeting a deadline.

I provide work that occasionally needs to be checked/redone by other group members to ensure quality; my work reflects some effort. I completed my share of the group work, but had difficulty meeting deadlines.

I provide work that usually needs to be checked/redone by others to ensure quality; my work reflects very little effort. I relied on others to complete my share of the group work.

Time-managemen

t(30%)

I routinely use time well throughout the project to ensure things get done on time. My group does not have to adjust deadlines or work responsibilities because of this my procrastination.

I usually use time well throughout the project, but may have procrastinated on one thing. My group does not have to adjust deadlines or work responsibilities because of this my procrastination.

I tend to procrastinate, but always gets things done by the deadlines. My group does not have to adjust deadlines or work responsibilities because of this my procrastination.

I rarely get things done by the deadlines AND my group has to adjust deadlines or work responsibilities because of my inadequate time management.

Contributions

(10%)

I routinely provide useful ideas when participating in the group discussion. I am a definite leader who contributes a lot of effort.

I usually provide useful ideas when participating in the group discussion. I am a strong group member who tries hard.

I sometimes provide useful ideas when participating in the group discussion. I am a satisfactory group member who does what is required.

I rarely provide useful ideas when participating in the group discussion. I sometimes refuse to participate.

Focus on the Task

(10%)

I consistently stay focused on the task and what needs to be done. I am very self-directed.

I focus on the task and what needs to be done most of the time. Other group members can count on me.

I focus on the task and what needs to be done some of the time. Other group members must sometimes nag, prod, and remind to keep me on-task.

I rarely focus on the task and what needs to be done. I let others do the work.

Monitors Group

Effectiveness

(10%)

I routinely monitor the effectiveness of the group, and make suggestions to make it more effective.

I routinely monitor the effectiveness of the group and work to make the group more effective.

I occasionally monitor the effectiveness of the group and work to make the group more effective.

I rarely monitor the effectiveness of the group and do not work to make it more effective.

Working with Others

(10%)

I almost always listen to, share with, and support the efforts of others. I try to keep people working well together.

I usually listen to, share with, and support the efforts of others. I do not cause "waves" in the group.

I often listen to, share with, and support the efforts of others, but sometime am not a good team member.

I rarely listen to, share with, and support the efforts of others. I am often not a good team player.

Developed using Rubistar (rubistar.4teachers.org)

Name: ____________________________

1) How well do you think you met the requirements for the project?

Component

Your Estimat

e (of

100%)

Actual Comments

Geometries Comparison

Pythagorean Theorem DemonstrationPythagorean Theorem Application

Clinometer Activity

Sample Truncated Globes

Project Scenario Presentation

2) How well did you work with your team?

Component

Your Estimat

e (of

100%)

Actual Comments

Work Distribution

Participation in Problem-Solving

Methods for Insuring Accuracy

3) As you reflect on this long-term group project, in what ways was your team most successful? In what areas does your team still need improvement? Explain your answers.

WORK DISTRIBUTION GRAPHAs you think about the entire project, complete a circle graph to show the approximate percentage you think each member contributed.

Team Final Evaluation of ProjectName ______________________________________ Role _________________ Score _____Name ______________________________________ Role _________________ Score _____ Name ______________________________________ Role _________________ Score _____As you read each job description, rate yourself and your colleagues using the following levels: A: Accomplished C: Competent E: Emerging N: Novice

Unusually good Consistent Inconsistent, but evident Not evident

Project ManagerResponsibilities: Maintain the team schedule Take the lead on written work, including the community

survey, proposals and group presentations Track team progress Ensure completion of team products Recognize and use the strengths of other members of the

team Check up on progress Deal with schedule changes or setbacks in the research,

design and creation process Assist the design engineer and computer engineer to ensure

successful completion of project tasks

Rating: _______

Comments:

Design Engineer

Responsibilities: Conceptualize and think though alternate recreation area

design ideas and solve related problems Lead in the development of design of all models, including

sample truncated globes Analyze the design to ensure accuracy Base decisions on sound mathematical reasoning Analyze ideas with experimentation and testing Think “outside the box” for a better, cheaper, more efficient,

or elegant way to accomplish a task or solve a problem Assist the project manager with scheduling, writing, and

organization Help the computer engineer develop a sound research plan

and produce reports

Rating: _______

Comments:

Computer Engineer

Responsibilities: Lead in the creation of the community survey and design a

method for organizing, interpreting, and presenting the results Lead and organize research on non-Euclidean geometries and

the Pythagorean Theorem Lead and organize research on topics related to the design Organize the results of the research into a report Create an effective presentation of the sample truncated

globes to highlight the cost efficiency of the design Assist the project manager with scheduling, writing, and

organization Help the design engineer develop designs that are in accord

with the space and cost requirements Help the design engineer ensure the accuracy of the design

Rating: _______

Comments:

Final Project EvaluationName: ___________________________________

Self Strongly Disagree

Disagree

Neutral Agree Strongly Agree

1. Overall I did an outstanding job on the project.Comments:

2. I learned important mathematical concepts.Comments:

3. I enjoyed the process and the project.Comments:

Team

Strongly Disagree

Disagree

Neutral Agree Strongly Agree

1. I worked effectively with my team.Comments:

2. I am satisfied with my team’s performance.Comments:

3. I learned useful lessons about team work through this project.Comments:

Eliminate

Keep With

Changes

Keep Comments

Know - Need to Know/Wish I Could – Need to Do Chart

Vocabulary Knowledge Rating Scale

Daily Journal

Daily Team Assessment

Geometries Comparison

Internet Research

Pythagorean Theorem Demonstration

Pythagorean Theorem Application

Clinometer Activity

Sample Truncated Globes

Project Scenario Presentation