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## Perfumania

By Tonya Adison, November 5, 2006

• High School

### Category

• Product Design

• Mathematics

### Lesson Time

Two sixty-minute class periods

### Introduction

In this lesson, students will be finding the volume of various 3-dimensional figures. Volume is an important consideration when planning for a variety of projects and decision making processes. Students will be able to touch and consider actual perfume/cologne bottle designs and use their observations to find the volume of specific designs. Ultimately, they will design their own bottle for Jennifer Lopez’s new fragrance.

### National Standards

Math Standards
Numbers and Operations
• Understand meanings of operations and how they relate to one another
Algebra
• Use mathematical models to represent and understand quantitative relationships
Geometry
• Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships
Measurement
• Understand measurable attributes of objects and the units, systems, and processes of measurement
• Apply appropriate techniques, tools, and formulas to determine measurements
Problem Solving
• Build new mathematical knowledge through problem solving;
• Solve problems that arise in mathematics and in other contexts
• Apply and adapt a variety of appropriate strategies to solve problems
Reasoning and Proof
• Recognize reasoning and proof as fundamental aspects of mathematics
• Make and investigate mathematical conjectures
Communication
• Communicate their mathematical thinking coherently and clearly to peers, teachers, and others
Connections
• Recognize and use connections among mathematical ideas;
• Understand how mathematical ideas interconnect and build on one another to produce a coherent whole
• Recognize and apply mathematics in contexts outside of mathematics
Representation Use representations to model and interpret physical, social, and mathematical phenomena

Common Core Standards

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.

Evaluate a speaker's point of view, reasoning, and use of evidence and rhetoric.

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.

Vocabulary Acquisition and Use:

Acquire and use accurately a range of general academic and domain-specific words and phrases sufficient for reading, writing, speaking, and listening at the college and career readiness level; demonstrate independence in gathering vocabulary knowledge when encountering an unknown term important to comprehension or expression.

### Objectives

Students will be able to:
• identify various geometric shapes
• apply the given formulas to determine the volume of these shapes and other shapes in relevant situations
• design their own container to conform to specifications provided
• use their knowledge of volume formulas and shapes to compute volumes of other shapes using proportions

### Materials

• rulers (one for each student)
• several perfume/cologne bottles (at least one per group)
• calculators
• construction paper
• graph paper
• student handout
• formula handout

### Vocabulary

Volume–the measure of the amount of space a solid figure occupies Cube–A solid figure with six square faces Rectangular Prism–A solid figure with six rectangular faces Pyramid–A three-dimensional figure that has a polygon for its base and whose faces are triangles having a common vertex Sphere–A three-dimensional figure with all points in space a fixed distance from a given point, called the center Cylinder–A three-dimensional figure having two parallel bases that are congruent circles Cone–A three-dimensional figure with one vertex and a circular base

### Procedures

Set-up: The teacher should predetermine groups of 3-4 students and have a bottle of perfume/cologne for each student group. (Given the safety considerations, if the teacher does not feel comfortable with students handling the usually glass bottles, teacher can use Styrofoam shapes and label them with various perfume/cologne fragrances.) Teacher Motivation: Students are commissioned by Jennifer Lopez to design a bottle for her new fragrance (“Still”). They will use the knowledge gained in this lesson to complete this task.
Teacher Presentation: Teacher starts with a question about the students’ favorite fragrances and probes the students to elicit more descriptions of the bottles in which these fragrances are packaged. The teacher then asks the students what these bottles have in common.
If the group does not answer, the teacher should introduce the concept of three-dimensional figures. From here, introduce the various three dimensional shapes that will be covered in the lesson: cube, rectangular prism, pyramid, sphere, cylinder, and cone. The teacher will define each shape and provide the formula to find the volume of each figure.
The teacher will then display several perfume/cologne bottles with various volume calculations. Students are asked to match the bottles with the volume. Class “consensus” is posted and left on the board. After this 15-20 minute presentation, the teacher arranges the students in their groups and distributes the various perfume/cologne bottles along with rulers.
Students in Groups:
1. Students sketch their bottle in the space provided on their handout and identify the shape of their bottle.
2. Students identify the formula they will use to find the volume of their figure.
3. Students take the necessary measurements to compute their volume.
4. Students work together to find the volume of the given bottle by substituting values into the correct formula.
5. Each group shares their findings.
6. The class reorganizes the volumes on the board, matching the appropriate volume to the appropriate bottle, to reflect their findings if necessary. Students are provided with the specifications for the perfume bottle they will design.
7. Students sketch designs that fulfill the requirements on graph paper.
8. Students create a prototype of the bottle using construction paper.
9. Students should also design a graphic or logo that will be featured on the front of the bottle. Remind students that the logo should represent the fragrance and spark a buyers interest. 10. Student groups present their ideas to the class.
Wrap-Up: Teacher/student discussion on what, if anything, surprised the students. Which designs do you like and why?

### Assessment

Ask each student to select an item in the classroom or in their book-bag and determine its volume. This lesson can be easily differentiated by the predetermined group selection. You can then give a less complicated formula to the groups who may struggle more. Additionally, the creative process may allow students who struggle mathematically to engage in the activity in areas in which they excel.

### Enrichment Extension Activities

Activity 1: A new shape? Consider the wedge shaped bottle. How can we determine the volume of this shape using what we already know? (One-quarter of a cylinder!)
Activity 2: Conversion using Proportions 1. Students are asked to look at the underside of their bottles and see if their calculations were correct. (How close are you?) 2. Students will notice that their answers are not consistent with the numbers stated on the bottle. TEACHING INTERJECTION: Explain to students that their volume calculations need to be converted into either ounces or milliliters. This can be done using proportions (previously covered material). Provide students with conversion information. 3. Students convert their findings to milliliters and ounces. 4. If wrong, students discuss the reasons their answers still do not equal the measurements on the underside of their bottles.