Project Based LearningSystems of Equations and Inequalities

Math I

Project Title: Systems of Equations and Inequalities

Project Idea: You are a team of nutrition counselors working for a major hospital. Prepare a written plan for the doctors suggesting nutrition requirements that should be included in the diets for patients with a specific illness. The justification for your recommendations should include both algebraic and graphic components.

Entry Event:Invite the school dietician to come speak to the class about the importance of a nutritious diet. Some examples will be shared that show the negative health effects of a poor diet. Use this opportunity to launch the project idea.

West Virginia College- and Career Readiness Standards:

Objectives Directly Taught or Learned Through Inquiry/Discovery

Evidence of Student Mastery of Content

1. 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.

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.1HS.6 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: Limit to linear and exponential equations and in the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs.

The student meets the teacher set criteria on the Cellular Phone Plans Academic Prompt(includes Rubric)

M.1HS.7 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. (e.g., Represent inequalities describing nutritional and cost constraints on combinations of different foods.) Instructional Note: Limit to linear equations and inequalities.

The student completes Special Diets Project (includes Rubric) at a level of success designated by the teacher.

Performance Objectives:

KnowWhether or not a point lies on a line (given the ordered pair and the equation)Graphing is a method of finding the solution set of systems of linear inequalities.The maximum or minimum value of an algebraic expression occurs at a corner point of the feasible region.Linear programming is the process of finding a feasible region and determining the point that maximizes or minimizes an expression.

DoUse systems of inequalities to model real-world applications.Created by Cheryl Reger, NBCT

Use linear programming techniques to find the maximum or minimum values of a function over the feasible region.Convert verbal statements to algebraic representationsGraph a linear inequalityGraph a system of two or more inequalitiesRecognize information that can be modeled by a linear equation or inequalityDetermine the location of a point on a graph with respect to a given lineDevelop a list of inequalities that represents the constraints of a problemDetermine which region to shade when graphing a system of inequalitiesDetermine the region of feasible solutions for a set of inequalitiesFind the corner points of a feasible region

Driving Question: How can a nutritious diet affect the recovery or well-being of a person with an illness?

Assessment Plan:

Major Group ProductsTeam Group Roles and contractLog Picnic Tables Academic PromptWritten recommendations for Special Diets Project

Major Individual ProjectsCellular Phone Plans Academic PromptDocumentation artifacts (team strengths & weaknesses, email correspondence, interview Q & A, etc.)Linear Programming practice problems - http://www.sonoma.edu/users/w/wilsonst/courses/Math_131/lp/default.html

Assessment and Reflection:

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

Collaboration Written CommunicationSpecial Diets Rubric

X

Critical Thinking & Problem SolvingSpecial Diets Rubric

X Content KnowledgeSpecial Diets Rubric

X

Oral Communication OtherCellular Phone Plans Academic PromptLog Picnic Tables Academic Prompt

X

Other classroom assessments for learning: (Check all that apply)

Quizzes/ tests Practice presentationsSelf-evaluationStudent Self Reflection

X NotesTeam meeting notes

X

Peer evaluationTeam Final Evaluation

X Checklists/observationsCollaboration Checklist

X

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

SurveyProject Debrief

X Focus Group

Discussion Task Management ChartTask Checklist

X

Journal Writing/ Learning Log Other

Map the Project:

Product:

Knowledge and Skills Needed Already Have Learned

Taught Before the Project

Taught During the Project

1. Evaluate expressions X2. Graph lines X

Created by Cheryl Reger, NBCT

3. Solve systems of linear equations X X4. Graph linear inequalities X5. Determine a linear inequality to model a given situation

X

6. Graph systems of linear inequalities X7. Use linear programming techniques X8. Search and navigate the internet X9. Insert graphs from calculator to computer X10. Draw and justify valid conclusions X

Resources:

School-based Individuals:School district dieticianSchool nurse

Technology:Computers with Internet access and printer Computer algebra software Graphing calculators with graphlinkWord processing/spreadsheet/publishing software

Community: Local health care providersHospital dietician

Materials:Lego manipulatives Graph paperRulersColored pencils

Websites:Resource for team building activities: Index to Group Activities, Games, Exercises, & Initiatives – http://wilderdom.com/games/Resource for forming collaborative groups: nsrfharmony.org http://www.nsrfharmony.org/system/files/protocols/north_south_1.pdfFree Algebra II Worksheets - http://www.kutasoftware.com/freeia2.htmlGraph linear inequalitiesSystems of two linear inequalitiesSystems of two equations; word problems

Systems of equations activities: SAS® Curriculum Pathways ® - http://www.sasinschool.com >Introductory Algebra > System of Equations > Classroom Activity 1019Introductory Algebra > System of Equations > Web Lesson 495Introductory Algebra > System of Equations > Project 287

For linear programming step-by-step investigations:HSQR.org: Modules - http://www.hsor.org/modules.cfm Cutting Times/ High Step ShoesIlluminations Web Links: http://illuminations.nctm.org/Lesson.aspx?id=2355 > Dirt Bike DilemmaLinear Programming Model -http://www.teacherlink.org/content/math/interactive/flash/linearprogramming/linearprogramming.htmlAdditional resource: http://www.shelovesmath.com/algebra/advanced-algebra/linear-programming/Linear Programming practice problems - http://www.sonoma.edu/users/w/wilsonst/courses/Math_131/lp/default.html

Diet Websites:Cancer survivors: Care for your body after treatment - MayoClinic.com – http://www.mayoclinic.org/departments-centers/mayo-clinic-cancer-centerEat Smart: Five important goals to improve your diet – CNN.com -

Created by Cheryl Reger, NBCT

 http://www.cnn.com/2007/HEALTH/diet.fitness/02/12/CL.eat.smart/index.html Nutrition in Cancer care (PDQ®) – National Cancer Institute - http://www.cancer.gov/about-cancer/treatment/side-effects/appetite-loss/nutrition-hp-pdqFlavor Your Meals – Must-Have Summer Herbs from the Academy of Nutrition and Dietetics - www.eatright.org/

Manage the Process: See Systems of Equations and Inequalities Storyboard

Week 1Assign students to groups according to their personal style. The activity to determine this style can be found at nsrfharmony.org - http://www.nsrfharmony.org/system/files/protocols/north_south_1.pdf Assign groups by mixing up the styles as much as possible. One of each style per group works well.After assigning groups, give them time to talk to each other about their individual style, strengths, and weaknesses. Students will take notes about members of the group and keep this as an artifact for the project. Plan other activities to help students see the importance and meaning of collaboration. Some team building activities can be found at Index to Group Activities, Games, Exercises, & Initiatives - http://wilderdom.com/games/

Launch the project by inviting a member of the healthcare or nutrition field to speak to the students. Ask the speaker to try to provide the students with some type of demonstration that they will find shocking. Some students may be willing to share personal experiences relating health and nutrition. Share a vision of the project with the students. Make connections between the goals of the project and the lives of the students. Nutrition is an important consideration as we all try to live healthy, productive lives.

Students will discuss the team roles that are outlined on the Team Group Roles sheet. They will use the Internet as a resource to help them develop a contract. The contract will include individual and group expectations and the role each member will assume. Each team member will sign the contract and the team recorder will keep a copy and turn in a copy.Teams will begin to discuss the project scenario. They will complete a first draft of the KWL to guide them as they think through the goals of the project.

Week 2Students will need practice using the Internet to find information that can be modeled using linear equations. The Cellular Phone Plans Academic Prompt will provide them with the opportunity to find information with more focus than the final project. Although students will collaborate on this assignment, each student will produce their own product. After a rough draft is complete, it will be peer reviewed before the final draft is submitted.

During this week students will also begin graphing systems of linear inequalities. It will be a prelude to the process known as linear programming. Be sure to make connections to students’ prior knowledge about graphing systems of equations and graphing linear inequalities.

Differentiate instruction by providing resources based on individual need. Topics include graphing lines, graphing systems of equations, solving systems of equations with two unknowns, and graphing linear inequalities.Provide these practice resources:

Free Algebra II Worksheets - http://www.kutasoftware.com/freeia2.htmlGraph linear inequalitiesSystems of two linear inequalitiesSystems of two equations; word problems

Systems of equations activities: SAS® Curriculum Pathways ® - http://www.sasinschool.com >Introductory Algebra > System of Equations > Classroom Activity 1019Introductory Algebra > System of Equations > Web Lesson 495Introductory Algebra > System of Equations > Project 287

Week 3If possible, provide students with a history of how linear programming evolved. Students work in collaborative groups and use manipulatives to explore the linear programming process by using the Log Picnic Table Academic Prompt. This product should be completed as a group. This task will be particularly helpful to the kinesthetic learner who needs something concrete to help make connections.

Created by Cheryl Reger, NBCT

Use online investigations to introduce linear programming and the related vocabulary. Start with the problem titled “High Step Shoes.”HSQR.org: Modules - http://www.hsor.org/modules.cfm Cutting Times/ High Step ShoesIlluminations Web Links: http://illuminations.nctm.org/Lesson.aspx?id=2355 > Dirt Bike DilemmaAdditional Resource: http://www.shelovesmath.com/algebra/advanced-algebra/linear-programming/

To make connections, outline the linear programming steps as you model the algebraic process known as linear programming for the Log Picnic Table Academic Prompt revisited.

Provide opportunities for students in their teams to investigate and solve real-world applications of linear programming. See (Linear Programming practice problems - http://www.sonoma.edu/users/w/wilsonst/courses/Math_131/lp/default.html

Discussion questions might include the following: Why do we look at feasible regions? Are all the points in the feasible regions solutions or only those with integer coordinates? What types of real-world problems use linear programming as a method to find the solution?Plan to have a test on the linear programming process during the coming week.

Progress check (One minute prompt): What is the progress of your group? Are there problems you have encountered?

Week 4Students will spend time on research for the special diet project. They will work with their group to choose an illness and search for relevant constraints related to the driving question. Some teams will require more help with resource information. Differentiate for these groups by supplying them with suggested resources to explore. The final project allows students to showcase their comprehensive understanding of linear programming by connecting systems of equations and inequalities to the real-world.

Week 5Students will prepare their recommendations.

Essential questions for reflection: How can we use systems of inequalities in real-world situations? What makes a solution feasible? When does it make sense to have solutions other than exactly one point?

Project Evaluation:

After final projects are submitted, students will complete the Student Self Reflection, Team Final Evaluation and Project Debrief.

Key Words: Linear programmingSystems of equationsSystems of inequalitiesGraphing systemsConstraintsFeasible regionObjective function

Created by Cheryl Reger, NBCT

Systems of Equations and Inequalities Storyboard

Week 1

Form groups

Team building activities ==>

Launch the project

Goals ==>KWL

Team roles

Team contract

Week 2

Research for Cell Phone Prompt

==>

Rough draft for Cell Phone Prompt

==>Peer review of rough draft

Practice graphing systems of linear inequalities

Final draft of Cell Phone Prompt

Week 3

Explore linear programming with manipulatives (Log Picnic Table Prompt)

==>Linear programming investigation: High Step Sport Shoes

Develop vocabulary and process

==>Explore real-world applications

More practice with the linear programming process

Progress check with each team

Week 4

Research for Special Diet project

KWL revised ==>

Test on linear programming process and skills

==>Continue research and choose the illness for the group project

Continue research to develop nutrition constraints

Week 5

Prepare presentation of project

==>

Peer review of projects

Prepare final written ==>plan

Final projects are due

Peer evaluation

Project reflection

Created by Cheryl Reger, NBCT

Mathematical Reasoning

and Problem Solving Graphs

Objective Function

Mathematical Concepts

Technology Neatness and Organization

20% 20% 20% 20% 10% 10%

4

I use complex and refined mathematical reasoning to research and find at least three nutritional constraints for a specific illness.

My graph of the feasible region is clear and all important parts are accurately labeled.

I use complex and refined mathematical reasoning to identify an objective function and accurately minimize or maximize the expression

My justification shows complete understanding of the algebraic concepts used to solve linear programming problem(s) and is presented clearly and persuasively.

I demonstrate exceptional skill in the use of multiple technology tools and electronic resources to create my final product

My work is presented in a neat, clear, organized fashion that is easy to read.

3

I use effective mathematical reasoning to research and find at least two nutritional constraints for a specific illness.

My graph of the feasible region is clear and most labels are accurate.

I use effective mathematical reasoning to identify an objective function and accurately minimize or maximize the expression

My justification shows substantial understanding of the algebraic concepts used to solve linear programming problem(s).

I effectively used multiple technology tools and electronic resources to create my final product

My work is presented in a neat and organized fashion that is usually easy to read.

2

I show some evidence of mathematical reasoning to research and find nutritional constraints for a specific illness.

My graph of the feasible region is somewhat inaccurate.

I show some evidence of mathematical reasoning to identify an objective function to minimize or maximize.

My justification shows some understanding of the algebraic concepts needed to solve linear programming problem(s).

I show some evidence of using technology tools and electronic resources to create my final product.

My work is presented in an organized fashion but may be hard to read at times.

1

I show little evidence of mathematical reasoning used to find constraints for a specific illness.

My graphs are inaccurate, difficult to understand, or are not used.

I show little evidence of mathematical reasoning or no objective function is identified.

My justification shows very limited understanding of the underlying concepts needed to solve linear programming problem(s) or is not written.

I show little evidence of using technology tools and electronic resources to create my final product.

My work appears sloppy and unorganized. It is hard to know what information goes together.

Special Diets:

You are a team of nutrition counselors working for a major hospital. Prepare a written plan for the doctors suggesting nutrition requirements that should be included in the diets for patients with a specific illness. The justification for your recommendations should include both algebraic and graphic components.

Analytic-Trait Rubric for Special Diets

Created by Cheryl Reger, NBCT

Created by Cheryl Reger, NBCT

Team Group Roles

Members of the group will assume the following roles. The responsibilities of each role include, but are not limited to the following:

Team leader: The team leader keeps all members of the group on task. They make sure that each member participates fully and shows respect for one another.

Recorder: The recorder maintains the group work folder, writes out solutions to problems, and facilitates the preparation of materials for presentations.

Reporter: The reporter reports progress to the group, class, and instructor. They facilitate any oral presentations required for the project.

Process director: The process director maintains the ground rules, monitors time, cleans up materials, and makes sure each group member is mastering concepts.

Floater (in groups of five): The floater assists the leader or other group members as needed. They will substitute for any absent member and assume their duties.

Participant: All members of the group are expected to participate fully. As a participant, they will take part in all activities of the group and keep a journal and a task management chart.

Team Expectations

Each member of the group is expected to pull their weight as part of the group. To prevent any member of the group from receiving a lower grade as a result of group learning, you will compose a group contract that every member will sign. Use the Internet to research contracts that could help you with some ideas. Be sure to include behaviors you expect from each individual as well as the group as a whole. Each member must sign the contract and designate which of the above roles they will be assuming.

Created by Cheryl Reger, NBCT

Collaboration Checklist

Not at all 1 Sometimes 2

Often3

Displays a positive attitude

Participates in discussion

Stays on task/uses time wisely

Organizes results/completes tasks

Engages in cooperative behavior

Comments _____________________________________________________________

_____________________________________________________________

_____________________________________________________________

_____________________________________________________________

Created by Cheryl Reger, NBCT

What do we know? What do we need to know? What should wedo to learn more?

KWL

Created by Cheryl Reger, NBCT

Task Checklist

Individual o Team summary of individual styles, strengths, and weaknesses

o Cellular Phone Plan Prompt

o Cutting Times

o High Step Sports Shoes

o Linear programming practice problems

o Peer evaluation

o Student reflection report

o Other artifacts

Groupo Team contract

o KWL

o Information sources

o Documentation of communication (email, interview, etc.)

o Team notes

o Log Picnic Table Prompt

o Peer review confirmation (who did what)

o Final nutrition project

Created by Cheryl Reger, NBCT

Cellular Phone Plans

You are a sales person for a major cell phone service company. Prepare a summary report that compares two of your company‘s calling plans to help the doctors at Valley General Hospital decide which plan is best for them. Your report should include algebraic and graphical representations of each plan, conditions where the two plans are the same cost, and advantages/disadvantages of each plan.

Performance List for Cellular Phone Plans

1 I choose a phone company and describe two calling plans using words. ___ / 52 I accurately represent each plan algebraically. ___ /103 I accurately graph each of my algebraic representations. ___ /104 I accurately find the conditions where the two plans are the same cost from my

algebraic or graphical representations. ___ /105 I accurately find at least one advantage of each plan. ___ /106 I accurately justify from the results of my representations at least one advantage

of each plan. ___ /207 I accurately find at least one disadvantage of each plan. ___ /108 I accurately justify from the results of my representations at least one

disadvantage of each plan. ___ /209 My end product uses correct grammar and spelling. ___ / 5

Total ___/100

Created by Cheryl Reger, NBCT

Performance List for Cellular Phone Plans

1 I choose a phone company and describe two calling plans using words. ___ / 52 I accurately represent each plan algebraically. ___ /103 I accurately graph each of my algebraic representations. ___ /104 I accurately find the conditions where the two plans are the same cost from my

algebraic or graphical representations. ___ /105 I accurately find at least one advantage of each plan. ___ /106 I accurately justify from the results of my representations at least one advantage

of each plan. ___ /207 I accurately find at least one disadvantage of each plan. ___ /108 I accurately justify from the results of my representations at least one

disadvantage of each plan. ___ /209 My end product uses correct grammar and spelling. ___ / 5

Total ___/100

Created by Cheryl Reger, NBCT

Log Picnic Tables:

The Valley General Hospital is building an area for patients and their families to eat in a cheerful outside environment. They have put out an RFP (request for proposals) to companies that build picnic furniture. You are a shop foreman for a company that builds log picnic tables and benches. The present inventory includes twelve long and fifteen short logs. A table is made from two long and three short logs. A bench is made from one long and one short log. The profit on a table is $40.00, while the profit on a bench is $12.00. Write a letter to the CEO of the company to recommend the number of tables and the number of benches the company should produce using the current inventory. Justify your recommendations.

Performance List for Log Picnic Tables

1 My letter includes an explanation of how I used an organized method to be sure I considered all possible solutions. ___ /20

2 My letter includes an explanation of how I tested all possible solutions from my list of possible solutions. ___ /20

3 My letter includes the best option. ___ /204 My letter includes appropriate justification of my choice. ___ /305 My end product follows correct form for a letter and uses correct grammar and

spelling. ___ /10Total ___/100

Created by Cheryl Reger, NBCT

Student Self Reflection

Not at all Sometimes Often

I listened to the ideas of others

I offered suggestions

I worked on the project at home

I stayed on task

I met deadlines

I encouraged and complimented others

I did my best work

I enjoyed the project

I cooperated with my group

Some ways we worked well were

_____________________________________________________________

_____________________________________________________________

We could have worked better by

_____________________________________________________________

_____________________________________________________________

I am proud of the project because

_____________________________________________________________

_____________________________________________________________

This project could have been better if I had

_____________________________________________________________

_____________________________________________________________

Created by Cheryl Reger, NBCT

Team Final EvaluationSpecial Diets Project

Name ________________________________ Role ___________________ Score ______

Name ________________________________ Role ___________________ Score ______

Name ________________________________ Role ___________________ Score ______

Name ________________________________ Role ___________________ Score ______

Name ________________________________ Role ___________________ Score ______

How well do you think you met the requirements of the project?

Component

Level

CommentsYour

Estimate ActualCombined research with mathematical reasoning to find nutritional constraints for a specific illness.

20

Created an accurate graph of the constraints to find the region of feasible solutions

20

Identified a reasonable objective function and accurately used linear programming to minimize or maximize the function

20

Justified the recommendations using both algebra and graphing components

20

Demonstrated skill in the use of technology

10

Created a neat, clear, organized product

10

Total 100

Created by Cheryl Reger, NBCT

Rate yourself and your colleagues using the following levels: A: Accomplished (unusually good) C: Competent (consistent) E: Emerging (inconsistent, but evident) N: Novice (not evident)

How well did you work with your team?

Component

Level

CommentsYour

Estimate ActualWork distribution and task completion 10

Participation in the problem-solving process and discussion

20

Cooperative behavior 20

Total 50

WORK DISTRIBUTION GRAPH: As you think about the project, make a circle graph that shows the percentage you think each of your team members contributed.

Created by Cheryl Reger, NBCT

As you reflect on this project, rate each person in your team?

Team LeaderPerson responsible:

Component

Level

CommentsYour

Estimate ActualKeeps all members of the group on task

10

Make sure that each member participates fully

10

Makes sure that each member shows respect for one another

10

Participates fully 10

Takes part in all activities of the group 10

Keeps a journal and a task management chart

10

Pulls their weight as part of the group 10

Meets the requirements included in the group contract

30

Total 100

Created by Cheryl Reger, NBCT

RecorderPerson responsible:

Component

Level

CommentsYour

Estimate ActualMaintains the group work folder 10

Writes out solutions to problems 10

Facilitates the preparation of materials for presentations

10

Participates fully 10

Takes part in all activities of the group 10

Keeps a journal and a task management chart

10

Pulls their weight as part of the group 10

Meets the requirements included in the group contract

30

Total 100

Created by Cheryl Reger, NBCT

ReporterPerson responsible:

Component

Level

CommentsYour

Estimate ActualReports progress to the group 10

Reports progress to the class and/or instructor

10

Facilitates any oral presentations required for the project

10

Participates fully 10

Takes part in all activities of the group 10

Keeps a journal and a task management chart

10

Pulls their weight as part of the group 10

Meets the requirements included in the group contract

30

Total 100

Created by Cheryl Reger, NBCT

Process directorPerson responsible:

Component

Level

CommentsYour

Estimate ActualMaintains the ground rules and monitors time

10

Cleans up materials 10

Makes sure each group member is mastering concepts

10

Participates fully 10

Takes part in all activities of the group 10

Keeps a journal and a task management chart

10

Pulls their weight as part of the group 10

Meets the requirements included in the group contract

30

Total 100

Created by Cheryl Reger, NBCT

FloaterPerson responsible:

Component

Level

CommentsYour

Estimate ActualAssists the leader or other group members as needed

20

Substitutes for any absent member and assumes their duties

10

Participates fully 10

Takes part in all activities of the group 10

Keeps a journal and a task management chart

10

Pulls their weight as part of the group 10

Meets the requirements included in the group contract

30

Total 100

Created by Cheryl Reger, NBCT

Project Debrief: Systems of Equations and Inequalities

Name _____________________________________

Reflection is an important part of any project. Please answer the following questions in a thoughtful manner. Be specific with your comments.

What are some things you learned during the project?

Was there a part of the project you liked or did not like? Explain.

What could be added or dropped to make the project better?

Was enough time allocated for each task in the project? Give details.

Are there any assignments you still don’t understand?

Other comment

Created by Cheryl Reger, NBCT