## Cutting Corners — Parts 1 & 2

By Delaina Martin, October 15, 2009

• High School

### Category

• City of Neighborhoods

• Mathematics

### Lesson Time

270 minutes for classroom activities and 120-180 minutes for homework

### National Standards

Mathematics

Standard 1. Level IV. Uses a variety of strategies in the problem-solving process

1. Uses a variety of strategies (e.g., identify a pattern, use equivalent representations) to understand new mathematical content and to develop more efficient solution methods or problem extensions

5. Uses formal mathematical language and notation to represent ideas, to demonstrate relationships within and among representation systems, and to formulate generalizations

7. Understands connections between equivalent representations and corresponding pro cedures of the same problem situation or mathematical concept (e.g., a zero of a function corresponds to an x-intercept of the graph of the function, the correspondence of binary multiplication to a series electrical circuit and the logical operation "and")

8. Understands the components of mathematical modeling (i.e., problem formulation, mathematical model, solution within the model, interpretation of solution within the model, validation in original real-world problem situation)

Standard 2. Level IV. Understands and applies basic and advanced properties of the concepts of numbers

1. Understands the properties (e.g., relative magnitude, density, absolute value) of the real number system, its subsystems (e.g., irrational numbers, natural numbers, integers, rational numbers), and complex numbers (e.g., imaginary numbers, conjugate numbers)

2. Understands the properties and basic theorems of roots, exponents (e.g., [bm][bn] = bm+n), and logarithms

Standard 3. Level IV. Uses basic and advanced procedures while performing the processes of computation

1. Adds, subtracts, multiplies, divides, and simplifies rational expressions

2. Adds, subtracts, multiplies, divides, and simplifies radical expressions containing positive rational numbers

3. Understands various sources of discrepancy between an estimate and a calculated answer

4. Uses a variety of operations (e.g., finding a reciprocal, raising to a power, taking a root, taking a logarithm) on expressions containing real numbers

Standard 4. Level IV. Understands and applies basic and advanced properties of the concepts of measurement

3. Selects and uses an appropriate direct or indirect method of measurement in a given situation (e.g., uses properties of similar triangles to measure indirectly the height of an inaccessible object)

4. Solves real-world problems involving three-dimensional measures (e.g., volume, surface area)

Standard 8. Levels IV and V. Understands and applies basic and advanced properties of functions and algebra

1. Understands appropriate terminology and notation used to define functions and their properties (e.g., domain, range, function composition, inverses)

2. Uses expressions, equations, inequalities, and matrices to represent situations that involve variable quantities and translates among these representations

4. Understands properties of graphs and the relationship between a graph and its corresponding expression (e.g., maximum and minimum points)

5. Understands basic concepts (e.g., roots), applications (e.g., determining cost, revenue, and profit situations), and solution methods (e.g., factoring, approximation using sign changes) of polynomial equations

8. Understands the general properties and characteristics of many types of functions (e.g., direct and inverse variation, general polynomial, radical, step, exponential, logarithmic, sinusoidal)

4. Adds, subtracts, multiplies, and divides polynomials

5. Factors polynomials using a variety of methods (e.g., difference of squares, perfect square trinomials)

6. Knows how to compose functions

Standard 9. Level IV. Understands the general nature and use of mathematics

1. Understands that mathematics is the study of any pattern or relationship, but natural science is the study of those patterns that are relevant to the observable world

2. Understands that mathematics began long ago to help solve practical problems; however, it soon focused on abstractions drawn from the world and then on abstract relationships among those abstractions

4. Understands that theories in mathematics are greatly influenced by practical issues; real-world problems sometimes result in new mathematical theories and pure mathematical theories sometimes have highly practical applications

10. Understands that mathematicians commonly operate by choosing an interesting set of rules and then playing according to those rules; the only limit to those rules is that they should not contradict each other

Common Core Standards

Integration of Knowledge and Ideas:

Integrate and evaluate content presented in diverse media and formats, including visually and quantitatively, as well as in words.

Delineate and evaluate the argument and specific claims in a text, including the validity of the reasoning as well as the relevance and sufficiency of the evidence.

Range of Reading and Level of Text Complexity:

Read and comprehend complex literary and informational texts independently and proficiently.

Anchor Standards for Writing:

Research to Build and Present Knowledge:

Conduct short as well as more sustained research projects based on focused questions, demonstrating understanding of the subject under investigation.

Anchor standards for Speaking and Listening:

Comprehension and Collaboration:

Prepare for and participate effectively in a range of conversations and collaborations with diverse partners, building on others' ideas and expressing their own clearly and persuasively.

Integrate and evaluate information presented in diverse media and formats, including visually, quantitatively, and orally.

Presentation of Knowledge and Ideas:

Present information, findings, and supporting evidence such that listeners can follow the line of reasoning and the organization, development, and style are appropriate to task, purpose, and audience.

Adapt speech to a variety of contexts and communicative tasks, demonstrating command of formal English when indicated or appropriate.

Anchor standards for Language:

Conventions of Standard English:

Demonstrate command of the conventions of standard English grammar and usage when writing or speaking.

### Objectives

After this lesson, students will be able to:

• interpret, extrapolate, and then transform what they know about optimization
• better understand the optimization concept critical to AP Calculus
• design their own container using optimizations
• understand the design process, especially the “understanding the client step”
• apply the design process to any problem solving situation that they encounter
• have mathematical knowledge of concepts and techniques at a working level, not just a superficial familiarity
• understand why products we use everyday are packaged as they are

### Resources

graphing calculator

### Materials

• one cylindrical and one box food item per student
• cardboard or poster board to make prototype
• poster board
• markers
• various craft items
• rubric for each group

N/A

### Procedures

Before the design project is introduced, students will have had two days of optimization lessons.  Students will work in groups of three.

PART 1

The point of Part 1 is for students to get some hands-on experience with finding simple optimizations.  Up until this point students will have only completed problems from worksheets or from the book.  In Part 1 of the design lesson, each group must bring in one item in a cylindrical container and one item in a rectangular prism container (a box).

1. Each group will find the measurements necessary to calculate the volume of each container.

2. They actually find the volume.

3. They then use the optimization to find the dim ensions of each container (radius and height for cylinder or length, width, and height for box) that will hold the same volume but use the least amount of material.  During this time, students will determine whether the most cost effective container would be a maximization or a minimization.  [Answer: This optimization will be a minimum because they want to find the most to find the most cost-effective container: that is, the smallest total surface area.]

Each group will record all information and calculations for each container in an organized manner to be included in their final report. (Note: If desired, each group member could be assigned a job to make the project go smoother and to keep every student accountable for some part of the project.)

PART 2

In Part 2, each group will be contracted to design a container to hold a specified volume of food. To help prevent copying, each group will get a different volume with which to work.  They will be able to choose from a cylinder or box for their container.  Some students may be wary of wet food in a box container, so you can be more specific and state that the food is dry. Whether a cylinder or box, the stated volume must be the same in order to make an accurate comparison for the optimization.

1. The first thing the students are to do is optimize each shape to be the most cost-effective as in Part 1 (cost of materials is the same no matter the shape).

2. From the two optimized dimensions, students will calculate the area of each to find out which one is lower.

3. From this, students would be ready to recommend a shape.  However, the teacher should then introduce other matters for them to consider.  Some of these concerns are as follows:

• stackability on shelf
• stackability to ship
• space to ship
• space to display
• cost of material
• getting all pertinent information on packaging

As stated in the design process, the students must clearly define the challenge and then investigate the problem.  They will probably want to interview the manager at the local grocery store to find out information on how products are displayed and maybe sales of certain products.

4. Students then reframe the problem to find that it is now different from the simple optimization that they have already done.

5. The next step is to generate possible solutions and develop ideas for the design of the container.  Maybe the students should go with the original optimization results.  Or perhaps they should make their container larger than the optimization says so that it will outperform the competitor’s packaging. More scenarios will arise, to which they must justify a response.  Throughout the process, the students should  bounce their ideas off of the teacher.

Once they settle on a container (shape and size), the students will finalize their project.  They will be required to keep all information and calculations in an organized manner to be included in their final report along with calculations from Part 1.  They also have to produce a net of their solution.  They will create a prototype of their solution and defend their decisions in a presentation to the teacher and the rest of the class.

### Assessment

Students will be assessed on their presentation: the decisions they made, and how well they can defend their decisions.

### Enrichment Extension Activities

This project could be taken to another level by changing the shape of the container from a simple cylinder or box to something that is harder to mathematically define.