Collision Course Ahead by Jason Kozel

Transformation 2013 PBL 5E Planning Form Guide PBL Title: Collision Course Ahead Teacher(s): Bonnie McClung School: Transformation 2013 T-STEM Center Subject: Algebra II Abstract: During this PBL, students will apply their knowledge of systems of equations and inequalities as they design their own company.

MEETING THE NEEDS OF STEM EDUCATION THROUGH PROBLEM BASED LEARNING ÂŠ 2008 Transformation 2013

Last Updated: 9/25/08

Begin with the End in Mind The theme or “big ideas” for this PBL: Students will work with systems of equations and inequalities, learn how the concepts are used in real life, and use the concepts to set up a business situation. TEKS/SEs that students will learn in the PBL: (2A.3) Foundations for functions. The student formulates systems of equations and inequalities from problem situations, uses a variety of methods to solve them, and analyzes the solutions in terms of the situations. The student is expected to: (A) analyze situations and formulate systems of equations in two or more unknowns or inequalities in two unknowns to solve problems; (B) use algebraic methods, graphs, tables, or matrices, to solve systems of equations or inequalities; and (C) interpret and determine the reasonableness of solutions to systems of equations or inequalities for given contexts. Key performance indicators students will develop in this PBL: Develop vocabulary (systems of equations, systems of inequalities, function, solutions, variables, unknowns), analyze graphs, determine feasibility regions within inequalities, create a business situation, solve systems of equations by using graphs and tables, classify systems of equations as consistent, inconsistent or dependent, solve systems of equations algebraically, solve systems of inequalities, solve linear programming problems, and solve systems of equations in three variables algebraically. 21st century skills that students will practice in this PBL: www.21stcenturyskills.org Creativity, teamwork, leadership, interdependence, flexibility, adaptability STEM career connections and real world applications of content learned in this PBL:

Last Updated: 9/25/08

Careers: Business, technology Application: design logo, company name, write a business plan, create a profit statement

The Problem You and three of your friends have decided to take part in a business venture. Your team needs to apply for a small business loan for the business’ start-up. You all want the major portion of the business to be based upon two products. As you are putting together your business plan and presentation, keep in mind the following:  Your design will need to include the maximum and minimum number of items you expect to sell from each product.  You must include the total number of products you expect to sell.  You must write an objective function representing the profit statement from the price of each product.  As marketing research shows, products sell best when their logos and names are appealing to the eye, so you will need to develop a name and logo for your company.  You will need to assemble a formal presentation of the business plan to present to the bank. Good luck with your endeavor!

Last Updated: 9/25/08

Map the PBL Already Learned

Taught before the project

Taught during the project

1. Develop vocabulary (systems of equations, systems of

inequalities, function, solutions, variables, unknowns) 2. Analyze graphs

Performance Indicators

3. Determine feasibility regions within inequalities

4. Create a business situation

5. Solve systems of equations by using graphs and tables

6. Classify systems of equations as consistent, inconsistent or

dependent 7. Solve systems of equations algebraically

8. Solve systems of inequalities

9. Solve linear programming problems

10. Solve systems of equations in three variables algebraically

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Team-Building Activity It is important that teachers provide team-building activities for students to help build the 21st Century Skills that are necessary for success in the workforce. Team-building helps establish and develop a greater sense of cooperation and trust among team members, helps students adapt to new group requirements so that they can get along well in a new group, serves to bring out the strengths of the individuals, helps identify roles when working together, and leads to effective collaboration and communication among team members so that they function as an efficient, productive group. Our students are often not taught how to work in groups, yet we assume that they automatically know how. Use team-building activities with your students so that you can see the benefits which include improvement in planning skills, problem solving skills, decision making skills, time management skills, personal confidence, and motivation and morale. Cup Stack Team-Building Activity

Objectives:  Participants work together in teams to accomplish a timed task.  Participants practice effective communication skills.  Participants reflect on one’s participation in a teamwork setting. Group Size: 3 to 4 participants (ideal is 4 participants) Materials: You will need a watch or clock with a second hand or a timer/stopwatch to time 1 minute 15 seconds. Each team will need 15 foam cups and a rubber band with 4 strings attached like rays of sun.

string

Setup:  Cut string into 2-foot lengths. Tie four strings to the rubber band evenly spaced around the circle. It should look like a sun with four rays coming out. Rubber band  Divide the cups into stacks of 15. Procedures:  Explain to the class that they will participate in a team-building activity that focuses on accomplishing a task and communication.  Distribute a set of materials to each team. Explain that the task is to build a pyramid using the cups with a 1 minute 15 second time limit. The pyramid will begin with 5 cups in a row at the base, 4 cups on the next row, 3 cups in the middle row, then 2 and finally 1 cup at the top. Group members may not touch the cups with their hands or any part of their body, even if the cups fall. Each person may only hold the end of one string attached to the rubber band (unless group size is 3 and then one participant may hold 2 strings). Group members must work together to stretch and relax the rubber band to grab each cup and place the cup in the right place.  When groups are ready, give them 30 seconds to practice and plan.  At the end of 30 seconds, have them disassemble their practice pyramid. When they are ready, start timing 1 minute 15 seconds. When time is up, stop the activity and check each team’s progress.  Debrief the activity with these questions: o Was anyone frustrated at all during the activity? If so, how was it handled?

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Why is teamwork so important for this activity? Did any team come up with a strategy for working together as a team? If so, what was the strategy? o Are you ever in a situation where you must use teamwork? Is it always easy for you? Why or why not? o What are some skills needed to be good at teamwork? o How did you contribute to your team? Did you give suggestions? Lead or follow? Encourage or cheer? o How would you do the activity differently if you were asked to do it again? ď ˛ Reset and repeat the activity. Give teams 30 seconds to strategize before starting the time. After the task, debrief with these questions: o Did your teamwork improve this time? How and why did it improve? o Why is good communication important to accomplishing this task? o How would you use this in your classroom, on your campus, or with other teams? o o

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5E Lesson Plan PBL Title: Collision Course Ahead TEKS/TAKS objectives: 2A.3A,B,C Engage Activity Introduce the unit by showing the “Algebra in Simplest Terms – 20 – Systems of Equations” video: http://video.google.com/videosearch?source=ig&hl=en&rlz=&q=systems+of+equations &um=1&ie=UTF (to access this video, see the links on the left of this screen). This video is about 29 minutes long, so preview it and use your discretion with regards to showing it in its entirety or just the first few minutes of it. If you choose to show the entire video, have students take notes in their journals and debrief them upon completion of the video. Solidify the concepts of consistent, inconsistent, and dependent systems. Discuss additional examples of real-world applications of systems of equations. Record these examples on large chart paper for future reference during the unit. After debriefing, have students work in pairs and use the following website to reinforce concepts and offer students an opportunity to work through some problems together: http://www.purplemath.com/modules/systlin2.htm Introduce the students to the project: You and three of your friends have decided to take part in a business venture. Your team needs to apply for a small business loan for the business’ start-up. You all want the major portion of the business to be based upon two products. As you are putting together your business plan and presentation, keep in mind the following:  Your design will need to include the maximum and minimum number of items you expect to sell from each product.  You must include the total number of products you expect to sell.  You must write an objective function representing the profit statement from the price of each product.  As marketing research shows, products sell best when their logos and names are appealing to the eye, so you will need to develop a name and logo for your company.  You will need to assemble a formal presentation of the business plan to present to the bank. Good luck with your endeavor! Provide students with an opportunity to start brainstorming regarding their business. Have them record ideas from their brainstorming session in their journals. Present students with a copy of the rubric so that they will know the expectations of the project. When students are finished brainstorming, provide them with additional practice to ensure a firm foundation of the math concepts.

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Engage Activity Products and Artifacts Math Journal: Vocabulary list Examples of dependent, independent and inconsistent systems Examples of real life applications Brainstorm ideas Solutions to practice problems Engage Activity Materials/Equipment Math Journal, computer access, pencil/paper for note taking, large chart paper, practice problems Engage Activity Resources http://video.google.com/videosearch?source=ig&hl=en&rlz=&q=systems+of+equations &um=1&ie=UTF http://www.purplemath.com/modules/systlin2.htm

Explore Activity Many times a system of equations is just not enough to describe a situation. You will need a system of linear inequalities to describe an entire region of possible solutions (called the feasibility region). Explore a given problem with the students. Before the gasoline crunch hit this year, a boat dealer was having a combined annual sales from boats and jet skis of approximately $780,000. Jet skis have become increasingly more popular with the younger generation and they account for about 75% of the dealer’s total sales. We are assuming that he sells both products so that becomes part of our problem situation. You have been asked to sketch a region showing all the possible combinations of sales for the company because the owner has to decide on how many of each to order for his inventory next year and still make a profit. Have students work in groups of 3-4 and give them 20 minutes to work on the problem before discussing it as a class. At the end of the 20 minutes, have each group present their sketch on a dry erase board or large grid paper to the class and see how many agree or disagree with each other’s solutions. Give them more time if necessary to work on the problem. x + y < 780,000 y ≥ .75( x + y) x > 0 and y > 0

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Window Settings: XMIN 0 *Note: Students can input data in thousands, thus entering 780 instead XMAX 300 of 780000 but the point of intersection is also in thousands. Xscl 10 YMIN 0 YMAX 800 Yscl 10 Provide a debriefing period and ask the following questions:  What window did you use?  What did you have to do to the second inequality before entering it into your calculator?  Did you use the shading function on your calculators or did you do it by hand?  Approximately how much money was obtained from boat sales? From jet ski sales?  If a boat sold for an average of $24,000 each and jet skis sold for an average of $18000 each, approximately how many boats and jet skis did they sell?  What are some other factors the owner has to consider before he orders inventory for next year?  What gave you the most trouble with this problem? Explore Activity Products and Artifacts Math Journal:  vocabulary (feasibility region)  answers to questions posed in the explore activity  graph of the inequality region Teacher observation notes Explore Activity Materials/Equipment Graph paper, graphing calculator, dry erase boards, pens, pencil scratch paper, math journal, large chart paper, markers Explore Activity Resources None Explain Activity Debrief as necessary about the explore activity and allow students to ask questions. Explain that the type of problem they just explored was called a linear programming problem which is a method of maximizing or minimizing a problem and the inequalities © 2008 Transformation 2013

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formed are called constraints. More information is necessary in order to actually maximize/minimize. Use the following website to explain how a system of linear inequalities forms a feasibility region using an example or a problem of your choice. http://people.hofstra.edu/Stefan_Waner/realworld/LPGrapher/lpg.html Have the students work in pairs to solve some linear programming problems using the attached worksheet. (Problems chosen from Holt Algebra 2, 2007) Observe the activity and see if they have any questions. Ask them to think of areas of our world in which linear programming applies. Record their responses on large chart paper so they can refer to it during the design challenge. Explain to them that they will be using the skills they are learning to design their own business situation. Explain Activity Products and Artifacts Math Journal:  What is linear programming?  Name three areas of our world in which linear programming applies.  What is a constraint? Completed Practice Problems Explain Activity Materials/Equipment Computer, math journal, worksheet problems, graphing paper, graphing calculators Explain Activity Resources http://people.hofstra.edu/Stefan_Waner/realworld/LPGrapher/lpg.html

Elaborate Activity Reintroduce students to the project: You and three of your friends have decided to take part in a business venture. Your team needs to apply for a small business loan for the business’ start-up. You all want the major portion of the business to be based upon two products. As you are putting together your business plan and presentation, keep in mind the following:  Your design will need to include the maximum and minimum number of items you expect to sell from each product.  You must include the total number of products you expect to sell.  You must write an objective function representing the profit statement from the price of each product.

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As marketing research shows, products sell best when their logos and names are appealing to the eye, so you will need to develop a name and logo for your company.  You will need to assemble a formal presentation of the business plan to present to the bank. Good luck with your endeavor! Students should research what formal business plans that are submitted to banking organizations look like and encompass. Ideally, the instructor should provide students with a list of items that must be included in their plans. Remind them to apply their knowledge of systems of equations and inequalities to their project. Students will need to create a formal presentation of their finished product to share with the rest of the class and possibly an outside panel made up of local business men and women and/or bankers. Elaborate Activity Products and Artifacts Business Plan, presentation Elaborate Activity Materials/Equipment Computers, word processing software, presentation software, graphic design software (optional), Internet access, sample business plans Elaborate Activity Resources None Evaluate Activity Groups will formally present their company’s business plan. They will need to provide the panel members (if from the community) with copies of their business plan and handouts for their presentations. Remember, the presentation must be professionally done, as outside guests will be evaluating student performance and professionalism. Evaluate Activity Products and Artifacts Presentation of business plans Evaluate Activity Materials/Equipment Computer, LCD projector, handouts, copies of business plans Evaluate Activity Resources None © 2008 Transformation 2013

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Systems of Equations/Inequalities Worksheet Solve each of the following by (1) graphing, (2) table and (3) algebraically. If the system is inconsistent or consistent and dependent, so state. 1.

x – y = -4 3x – 6x = -12

y=x–1 x + 4y = 6

y – x = -3 2x + 3y = 6

3x – 9y = - 21 x – 3y = 6

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5x – y = 33 7x + y = 51

Graph each linear inequality: Use a dotted line when needed. 6.

2y – 4x ≥ 12 - y + x ≤ -1

x+y≥3 y–5>0

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Linear Programming Problem Set 1. A cafe sells cold sandwiches and hot entrees. The range of items sold is shown in the table. The cafe has never sold more than a total of 125 sandwiches and entrees in one day. If the cafe makes a profit of $0.75 on each sandwich and $1 on each hot entree, how many of each item would maximize the cafe profit? Menu Item Cold Sandwiches

Minimum Sold 60

Maximum Sold 80

Hot Entrees

Write the constraints. Graph the feasibility region and identify vertices (points of intersection). Write the objective function. Evaluate the objective function and determine the number of each item to be sold that would maximum profit. How much profit would be made?

Graph each feasibility region a) x≥0 y≥0 y ≤ 3x + 1 y ≤ - (3/4) x + 6

x<3 y≥ 0 y < 2x + 1 y ≤ -x + 4

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Plan the Assessment Engage Artifact(s)/Product(s): Math Journal: Vocabulary list, Examples of dependent, independent and inconsistent systems, Examples of real life applications, Brainstorm ideas, Solutions to practice problems Explore Artifact(s)/Product(s): Math Journal: vocabulary (feasibility region), answers to questions posed in the explore activity, graph of the inequality region, teacher observation notes Explain Artifact(s)/Product(s): Math Journal: What is linear programming? Name three areas of our world in which linear programming applies. What is a constraint? Completed Practice Problems Elaborate Artifact(s)/Product(s): Business Plan, presentation

Evaluate Artifact(s)/Product(s): Presentation of business plans

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Rubrics Team Rubric Expert (4) Completeness

Competent (3)

Beginner (2)

Novice (1)

Product meets all criteria and constraints Clearly stated profit statement and realistic

Product meets most of criteria and constraints Profit statement present but not realistic

Product meets some of criteria and constraints Profit statement stated but lacks clarity

Product is incomplete

Quality

Design is clever and thought provoking Problems are correctly worked

Design is clever and neat Problems are mostly correct with minor errors

Design is mostly neat with a few flaws Problems are mostly but several errors occur

Design is unattractive with little thought Several problems incorrect

Communication

Statement of design clearly related and variables clearly stated Problems are written clearly and neatly

Statement of design clear but variables not clearly stated Problems are well written and mostly neat

Statement of design fairly clear but variables not clearly stated Problems are somewhat well written and somewhat neat

Statement of design unclear or not feasible

Product is extremely creative and inviting

Product is very creative and mostly inviting

Product is somewhat creative

No attempt to be creative or inviting

Creativity

No profit statement

Problems difficult to read

Individual Rubric Leadership

Played an essential role in organizing information

Took a lead more than once Took a lead at least once Never took lead and helped others Accepted some responsibility

Cooperation

Extremely good listener and organizer

Listened but talked some Rarely argued

Talked a bit too much and argued quite a bit

On Task

Used time very effectively and very productive

Productive most of the time

Productive some of the time Often distracting to the team Needed some reminders

Creativity

Contributed a lot of the ideas

Contributed many of the ideas

Contributed a few of the ideas

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Never talked unless off task and argued with team

Did not contribute any ideas

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Story Board  Week 1 Activities

Week 2 Activities

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Day 1 Engage (60 minutes) Explore (30 minutes) Day 6 Elaborate (90 minutes)

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Day 2 Explore (cont. 30 minutes) Explain (60 minutes) Day 7 Evaluate (90 minutes)

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Day 3 Explain (90 minutes)

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Day 8 Evaluate (90 minutes)

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Day 4 Elaborate (90 minutes)

Day 9

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Day 5 Elaborate (90 minutes)

Day 10

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