I. Curricular Framework

Concept:

Optimization

Essential Question:

What real-life application does the ability to determine maximum/minimum values have?

Bridge:

Problem-solving

Content:

Maximum/Minimum Value Formulas

Outcomes:

II. Standards Aligned

III. Instruction and Assessment

1. Connect: Connecting to the Concept Experientially

Objective: To form an image of doing the most with the least.

Activity: Working in cooperative learning groups, students are given a sheet or two of wrapping paper and several boxes to wrap. They must get the most wrapped with the least amount of paper. Think of the best way to proceed to wrap all packages.

Assessment: Ability to solve the problem.

2. Attend: Attending to the Connection

Objective: Based on the wrapping paper experience, to define what is meant by maximizing and minimizing.

Activity: Teacher leads discussion eliciting feedback on how groups solved the problem. Connect problems to real world: a department store which provides gift wrap service would want to use the least amount of paper to provide the service yet maximize their profit. Another problem for groups to solve: a soup company wants to make soup cans to hold a given amount with the least amount of manufacturing material. How would soup cans be designed to maximize profit? What factors must be considered?

Assessment: Discussion from the students.

Assessment, Phase One, Level of Engagement, Fascination:

3. Image: Creating a Mental Picture

Objective: To broaden students' understanding of maximum/minimum with an emphasis on conditions set for the problem.

Activity: Working in pairs, students are given two Hungry Jack biscuit wrappers or traced copies. Students are to roll wrappers back up to make two different sizes of cylinders. Tape these two cylinders to hold their shape. One is shorter and fatter. The other is longer and thinner. The teacher should have two cylinders of the two different sizes prepared as discussion models to expedite the student activity. Problem: determine if the the two cylinders hold the same amount.

Assessment: Ability of student teams to create the two cylinders and to find a way to solve the problem proposed.

Assessment, Phase Two, Seeing the Big Picture:

4. Inform: Receiving Facts & Knowledge

Objective: To learn to set up and work a maximum/minimum problem.

Activity: Using overhead transparencies, teacher presents problem solving strategies for mathematically determining maximum and minimum under given conditions. Problems are presented for class to set up and solve together using realistic examples from business and industry.

Assessment: Students involvement during instruction and teacher checking for ability to solve sample problems.

Assessment, Phase Three, Success with Acquiring Knowledge:

5. Practice: Developing Skills

Objective: To develop proficiency in setting up and solving a variety of maximum/minimum problems.

Activity: 1. Students work sample problems in text and teacher-prepared worksheet using a prescribed procedure. 2. Student pairs will use Hungry Jack cylinders made in Quadrant Two Right and seal off the bottom with cardboard and tape. Remembering that the two cylinders have the same lateral area, students will mathematically predict the volume. Using rice or dried beans, they will test and verify their predictions. 3. Each team will bring a sample food can from home. The team task is to design an ideal can to hold a given volume but using the least amount of material for total outer surface area.

Assessment: Completion and correctness of assigned work. In- class quiz to check for understanding.

Assessment, Phase Four, Success with Acquiring Skills:

6. Extend: Extending Learning to the Outside World

Objective: To develop a project applying what has been learned.

Activity: 1. Working in teams, students will design a maximum/minimum problem that exists in their own experience. For example, the SGA wants to sell tickets to the faculty talent show to groups to get the most people to attend, and also to make the most money. If they give a discount of 20¢ for each number of tickets sold over 10 tickets with a single ticket price of $5.00, how many must be sold to maximize profit? 2. Students will write to a major can company to find out how they do or do not use the "ideal can" and what factors determine the size of the can that they do use.

Assessment: Student on-task behavior.

7. Refine: Refining the Extension

Objective: To complete project assignments.

Activity: Students make posters for testing their real-life problems with the rest of the class. Findings of correspondence with can companies are put into a brief report.

Assessment: Quality of sample problems and posters.

8. Perform: Creative Manifestation of Material Learned

Objective: To share in final form what has been learned with the rest of the class.

Activity: Students make a presentation to the class of their self-designed problem, including a visual. The rest of class will set up and solve each team problem presented. Responses from can companies are posted in a composite chart and will be shared orally with the rest of the class.

Assessment: evaluation, peer evaluation, self-evaluation done orally with class.

Assessment, Phase Five,Performance, Creative Use of Material Learned: