GoingGreenProject.pdf

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UVA-SCPS Office of Mathematics Outreach with support from VADOE Mathematics and Science Partnership Grant Program NCLB Title II Part B

Revised 11/27/12 1

Mathematics Senior Level Capstone Course

Unit Overview

Title of Unit: Going Green … to the Max! Unit Designers:

Amy Bianco - Fauquier Blaire Conner - Fauquier Dennis Myers - Warren Stephanie O’Brokta - Fauquier Kim Raines - Fauquier

Lisa Rogers - Warren Bev Wynn - Fauquier Context: Package Design: Maximizing volume of a package while keeping the amount

of material eco-friendly. Number of Class

Hours:

16-18 hours Unit

Design: Task Based X Project Based Other Subject Areas/Disciplines Addressed: Environmental Science Engineering/Design Decision Making

Driving Question: How can you get the most volume out of a package while minimizing surface area? Mathematics Content

Addressed:

Perimeter, Area, Surface Area, Lateral Area, Volume, Measurement,

Functions (Linear, Quadratic, & Cubic), Observing Patterns, Collecting Data

MPE Addressed: Problem Solving, Decision Making, and Integration, Understanding and Applying Functions Assumption of Prior

Knowledge:

Basic understanding of perimeter, area, surface area and volume, two-dimensional and three-dimensional representations

College and Career Readiness/21st Century Skills to be taught(T) during this unit or expectation (E) for student use during this unit and assessed (A): BIE Page 35-37

Collaboration – Students will work in groups

E Research – Green effort and “Greenopia” ratings

E & A

Communication (Oral and/or Written) – final presentation (poster, video, etc.)

A Technology – Excel spreadsheets, power point, video

E & A

Critical Thinking/Decision Making – Deciding on package design and ranking other packages from most to least eco-friendly, efficient, etc.

T, E, & A Other: (Describe)

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Revised 11/27/12 2

Performances: X Class

School Individual – Student ranking and justification Expert

Community

X Other: School Green Club

Launch: Event or experience used to engage the students interest and inquiry:

Teacher shows the following video clip about a new Coca-Cola package:

Video provides an introduction to possible factors that influence package design. (0:01:05) http://www.youtube.com/watch?v=RPh6QD2MFqk

Evaluation: Formative Assessments (During the Unit)

Interview Practice Presentations X

Mathematicians Journal X Notes X

Preliminary

Plans/Outlines/Prototypes

X Checklists

Rough Drafts X Concept maps

Field Tests Other:

Summative Assessment (End of Project)

Written Products, with a rubric X Peer Evaluation, with a rubric

X

Oral Presentation with a rubric X Self Evaluation, with a rubric

Other Product(s) or

Performance(s), with a rubric

Other:

Resources Needed: On-site people, facilities: Facilitator/Teacher

Equipment/Technology: Computers for research, excel, and PowerPoint access Graphing Calculators

Video Cameras (optional) Materials: See unit outline below Community Resources: None

Reflection Methods: Individual, Group, and/or Whole Class

Mathematicians Journal x Small/Focus Groups Whole Class Discussions x Fishbowl Discussions

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Revised 11/27/12 3

Material Adapted From: (Provide credit for any materials or activities adapted from other sources.)

Explore Activity 1: https://sites.google.com/site/mathematicscapstonecourseunits/home/powerpoints-other-resources ; Rectangles Excel file Explore Activity 2: http://illuminations.nctm.org/LessonDetail.aspx?ID=L726;

http://education.ti.com/calculators/downloads/US/Activities/Detail?id=4918

Explore Activity 4: http://illuminations.nctm.org; http://illuminations.nctm.org/LessonDetail.aspx?id=L797 Teacher Rubric for Presentation: Mathematics Teacher, October 2011, Vol. 105, No.3

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UVA-SCPS Office of Mathematics Outreach with support from VADOE Mathematics and Science

Partnership Grant Program NCLB Title II Part B Revised 11/27/12 4 Virginia’s Senior Level Capstone Course

Instructional Plan Unit Title: Going Green … to the Max!

Driving Question: How can you get the most volume out of a package while minimizing surface area? Task/Project/Problem: ENGAGE How will student’s interested be piqued so they want to engage in the inquiry in this unit? Time: 10-15 minutes

Teacher shows the following video clip about a new Coca-Cola package: http://www.youtube.com/watch?v=RPh6QD2MFqk

Video provides an introduction to possible factors that influence package design. (0:01:05)

Whole class discussion of the video clip – What are some factors that influence package design?

Mathematician Journal Prompts Choose a

product you have seen in the grocery store. What is the package’s shape? Why do you think it is shaped this way?

What are some of the factors that influence package design? EXPLORE Teacher provides

guidance for the explorations to prepare students with the

knowledge and skills to engage in the task. Students will self-assess on the prior knowledge and skills assumed for the unit Time: Activity 1: 45 minutes

The teacher selects from the following activities based on the needs of the students (any or all).

Activity 1: Perimeter and Area Optimization Excel Demo Materials:

 Student HO #2

 Student access to Excel spreadsheet Description:

Microsoft Excel is used to examine various properties of area and perimeter by using a dynamic rectangle. Students explore the relationship between area and perimeter by experimenting with rectangles having sides of various lengths. The rectangles can be manipulated by the students using scroll bars.

Class Discussion: What is the relationship between Area and Perimeter? Note: Activity is from:

https://sites.google.com/site/mathematicscapstonecourseunits/home/powerpoints-other-resources ; download the Rectangles Excel file (macro-enabled) to complete the activity.

Mathematician Journal Prompts How would you describe

perimeter, area, and volume to an elementary student? What have you observed about the relationship of the

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Partnership Grant Program NCLB Title II Part B Revised 11/27/12 5 Activity 2:

60-90 minutes Activity 3: 45 minutes Activity 4: 60 – 90 minutes

Activity 2: Investigate the relationship between Surface Area and Volume of various packages

Materials:

 Empty cardboard packages of various shapes (ex: cereal box, oatmeal container, etc.) that can be cut into nets

 Scissors

 Two packages with equal surface area and different volume (or equal volume and different surface area)

Description: Students work in assigned pairs. Each pair chooses a package from the class collection and calculates its surface area and volume. Students cut apart package to visualize the net and to find the surface area.

Class Discussion: Before doing surface area and volume measurement and calculation, students create hypothesis related to the following question: Do you think packages with the same surface area also have the same volume? Why or why not? Students also discuss accuracy of their hypothesis after completion of the activity.

Note: Collect measurements from the class and find two different packages with approximately the same surface area (if no such packages exist, teacher should have an example on hand).

Activity 3: Volume of an Open Box Materials:

 Volume of a Box HO # 3A and HO #3B (answer key)

 Graph paper HO # 3C (calculations are based on a 28 units by 40 units grid)  Scissors

 Graphing Calculator

Description: Students cut around the perimeter of the (28 x 40) graph paper and discard the blank margins. Congruent squares are cut from each corner of the graph paper. The remaining graph paper is folded to form an open box. See HO # 3A for further

instruction.

If the teacher would like to expand this activity, each person in the group could create 2 or more different open boxes. Then compile the groups’ information on the data collection sheet.

Class Discussion: What are the possible measurements for the height? For what height (approximately) does the greatest volume occur?

Note: Volume of Box Activity is from http://illuminations.nctm.org For more information go to http://illuminations.nctm.org/LessonDetail.aspx?ID=L726 Click on the Lesson 2 tab. Click on “Materials”. Click on “Activity Sheets”. The Volume of a Box Activity is the fourth activity.

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Partnership Grant Program NCLB Title II Part B Revised 11/27/12 6 navigator, the website below has resources. You may want to look at both sites and

combine the resources. See (HO # 3D) and:

http://education.ti.com/calculators/downloads/US/Activities/Detail?id=4918

Activity 4: Popcorn, Anyone? Materials:

 8 “ x 11.5” White paper & colored paper  Popcorn

 Paper plates  Cups  Tape  Rulers

 Popcorn Prisms Anyone? (HO # 4A and #4C (answer key))  Popcorn Cylinders Anyone? (HO # 4B and #4D (answer key))

Description:

Students create popcorn containers shaped like two rectangular prisms and two

cylinders, each having the same lateral area. Students compare the volume of each and investigate which measurements, when changed, affect the volume the most.

Note: Activity is from http://illuminations.nctm.org For more information go to: http://illuminations.nctm.org/LessonDetail.aspx?id=L797 Click on “Materials” to find the activities and answer keys.

EXPLAIN Teacher introduces the main task of the unit and

prepares students to in small group independent work. Time: 15 - 20 min

Teacher divides students into groups of 3-4 and explains the task to students:

Description: According to www.greenopia.com, Powerade and Vitamin Water (both of which are owned by Coca-Cola) are two of the worst environmentally friendly

beverages, each receiving a green rating of zero. Greenopia sites many reasons for the poor rating, including, heavy packaging. Coca-Cola has hired you to remedy this

situation as they have been putting forth their best effort to reduce their carbon footprint. Your goal is to make the packaging as environmentally friendly as possible without reducing the amount of beverage contained within the packaging. You may change anything you would like about the packaging (material, shape, et cetera) but you must keep in mind that the package must also be ideal for shipping and display (on shelves in grocery stores, in vending machines, etc.) You and your group must turn in a 2-D drawing of your design as well as a 3-D model of your new container. Calculate the surface area and volume of your new container and compare it to that of the current Powerade and Vitamin Water containers. Present your findings to the class proving that your design is more environmentally friendly than the current design. Be sure to explain what materials you used in your design. You may present your findings to promote your product to the class via poster, PowerPoint, video or other media. HO #5

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Partnership Grant Program NCLB Title II Part B Revised 11/27/12 7 ELABORATE The student groups are working independently with teacher consultations. Time: 10.5 hours

Resources/Materials:

 Powerade and Vitamin Water bottles

 Access to computers for research, access to Microsoft Office (specifically Word and Power Point) so students can prepare presentations

 Video camera for groups who want to make a video presentation  Poster board or poster paper

 Rulers

 Materials needed for 3D model

Teacher monitors each group’s progress during this time and sets a due date for each component. (The initial research will be done during the first segment; however additional research will be required throughout the project.)

 Research and 2D drawing: Approximately 180 minutes

 3D drawing, volume and surface area calculations: Approximately 180 minutes  Preparing final presentation: Approximately 270 minutes

Mathematician Journal Prompts What is the shape of your new bottle design? How did your group decide on a shape? What research did you find that helped you make your decision? EVALUATE Working groups submit products or make presentations Time: 15-20 minutes per group

Students promote their product to the class. They present the 2-D drawing, 3-D model, surface area and volume calculations. The audience completes a student rubric for each package.

 Teacher rubric (HO # 1)  Peer Rubric (HO #1a)

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Partnership Grant Program NCLB Title II Part B Revised 11/27/12 8 Map the Unit

What do students need to know and be able to do to complete the task/project/problem

successfully? How and when will they assess their own necessary knowledge and skills? How will they remediate their own gaps or weaknesses in knowledge and skills? Look at each major task for the unit and analyze the tasks necessary to produce a high-quality product.

Task:

KNOWLEDGE AND SKILLS NEEDED Assumed already learned Students will self-assess Will be taught during the unit 1. Calculating perimeter and area of 2D

figures √

2. Measuring and calculating surface area and

volume of 3D solids √ √

3. Making judgments relating surface area and

volume √ √ √

4. Presentation creation and skills _√ _√

5. Internet research _√ _√

6. Recognizing patterns and relationships _√ _√

7.

8.

9.

10.

11.

What project tools will student’s use?  Know/need to know lists  Daily goal sheet

√ Mathematician’s Journals  Briefs/Memos

 Task lists

 Planning Calendar

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Partnership Grant Program NCLB Title II Part B Revised 11/27/12 9 HO # 1

RUBRIC – NEW BOTTLE DESIGN

Exemplary 4 Competent 3 Limited 2 Needs Improvement 1 SURFACE AREA Surface Area is

calculated correctly using correct units; all work is shown

Surface area calculation includes minor arithmetic error or incorrect units, or all work is not shown. Surface area calculation includes significant errors, units are

incorrect, or all work is not shown.

Incorrect formula is used to

calculate surface area.

VOLUME Volume is calculated correctly using correct units; all work is shown

Volume calculation includes minor arithmetic error or incorrect units, or all work is not shown. Volume calculation includes significant errors, units are

incorrect, or all work is not shown.

Incorrect formula is used to

calculate Volume.

2-D DRAWING 2-D drawing of package design is accurate and neat.

2-D drawing of package design is accurate but not neat.

2-D drawing of package design is inaccurate but neat.

2-D drawing is inaccurate and not neat.

3-D VISUAL REPRESENTATION

3-D

representation of package design is accurate and neat.

3-D

representation of package design is accurate but not neat.

3-D

representation of package design is inaccurate but neat.

3-D representation is inaccurate and not neat. OVERALL PRESENTATION Overall presentation is neat and organized. Overall presentation is somewhat neat and organized. Overall

presentation is not neat or is

disorganized.

Overall presentation is sloppy and disorganized. COMMUNICATION Explains the

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Partnership Grant Program NCLB Title II Part B Revised 11/27/12 10 HO #1a

Group Members’ Names :____________________________________ Ranking of Product: Evaluator : _________________________ Date: _________________ #______ out of _____ Directions: As each group presents their project, complete the following rubric.

1. Circle the level each group reaches in each category.

2. Under the level you circled, give examples of why you chose that category. Peer Evaluation Rubric

Needs Improvement Acceptable Excellent

Topic Not on the topic assigned

Partly on topic Exactly on Topic

Organization None of the information organized

Information is partly organized

Information is well organized and easily

understood Facts Calculations are done

incorrectly

Calculations include errors and/or incorrect

units

Calculations are done correctly including

correct units Visual Presentation Extremely messy and

unclear

Only a few problems Very neat and clear

Explanation No explanations given Only a few explanations given

Everything well explained Word Choice Words are dull and

interesting

Nice everyday words Exciting and interesting words Mechanics Lots of mathematical

errors

A few mathematical errors

No mathematical error

Overall Presentation

Presentation was not convincing

Presentation was somewhat convincing

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Partnership Grant Program NCLB Title II Part B Revised 11/27/12 11 HO # 2 Perimeter & Area Optimization

Name: ____________________________________________ Date: _________________ https://sites.google.com/site/mathematicscapstonecourseunits/home/powerpoints-other-resources Download Rectangles Excel file – be sure macros are enabled.

Section I.

For this section, we will be working with the spreadsheet labeled “Introduction.” 1. Move the length scroll bar to the left. What happens?

2. Move the length scroll bar to the right? What happens?

3. Move the width scroll bar up and down. How does the rectangle change?

4. What do you think would happen if on the spreadsheet, you made a rectangle with width 10 and length 0? Draw what you think will happen and explain what you drew.

5. For this “rectangle,” what do you think the perimeter would be? Why?

6. Try it on Excel. What actually happened? Draw it. Is this different in any way from your prediction? If yes, how is it different and why do you think this is so?

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Partnership Grant Program NCLB Title II Part B Revised 11/27/12 12 HO#2 (continued)

Section II.

For this section, we will be looking at the spreadsheet labeled “Fixed Perimeter.”

1. What does it mean to have the perimeter fixed on this rectangle?

2. Slide the length scroll bar to left and right. a. What happens to the width value? Why?

b. What happens to the rectangle?

c. The area?

d. The perimeter?

3. Does the area always increase as you increase the length? Why or why not?

4. Is there a largest area that you can obtain with this rectangle? If so, how do you get it? If not, why isn’t there a largest area?

Section III.

For this section, we will look at the spreadsheet titled “Fixed Area.” 1. What does it mean to have the area fixed on this rectangle?

2. Slide the length scroll bar to left and right. a. What happens to the width value? Why?

b. What happens to the rectangle?

c. Area?

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Partnership Grant Program NCLB Title II Part B Revised 11/27/12 13 HO#2 (continued)

3. Try moving the length scroll bar all the way to the right. What happens to the width?

4. Try moving the length scroll bar all the way to the left. What happens to the width?

5. Why do you think we can’t get a side length of zero as in the previous two spreadsheets? What do you think would happen to the area if you had a side length of zero?

6. Is there a smallest perimeter you can find for this rectangle? If so, how do you get it? If not, why isn’t there a smallest perimeter?

Section IV.

1. In sections 2 and 3, we were only able to manipulate length. Do you think it would be different if we were only able to manipulate width? Why or why not?

2. What is the relationship between the length and the width of the rectangle with maximized area?

3. What is the relationship between the length and the width of the rectangle with minimized perimeter?

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Partnership Grant Program NCLB Title II Part B Revised 11/27/12 14 HO # 3A

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Partnership Grant Program NCLB Title II Part B Revised 11/27/12 15 HO # 3B

Answer KEY

1 38 26 988

2 36 24 1728

3 34 22 2244

4 32 20 2560

5 30 18 2700

6 28 16 2688

7 26 14 2548

8 24 12 2304

13 14 2 364

Cubic Function

( )

(28 2 )(40 2 )

f x



x



x



x

Or

( )

4 (14

)(20

)

f x



x



x



x

Or

3 2

( )

4

136 1120

f x



x



x



x

Maximum volume occurs when x



5.4 units (the height) Maximum volume is approximately 2,712 cubic units

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Partnership Grant Program NCLB Title II Part B Revised 11/27/12 16

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Partnership Grant Program NCLB Title II Part B Revised 11/27/12 34 HO#5

Name: _______________________________________

Group Members Names: _______________________________________________________________

According to www.greenopia.com, Powerade and Vitamin Water (both of which are owned by Coca-Cola) are two of the worst environmentally friendly beverages, each receiving a green rating of zero. Greenopia sites many reasons for the poor rating, including, heavy packaging. Coca-Cola has hired you to remedy this situation as they have been putting forth their best effort to reduce their carbon footprint. Your goal is to make the packaging as environmentally friendly as possible without reducing the amount of beverage contained within the packaging. You may change anything you would like about the packaging (material, shape, et cetera) but you must keep in mind that the package must also be ideal for shipping and display (on shelves in grocery stores, in vending machines, etc.) You and your group must turn in the following:

 2-D drawing of your design  3-D model on your new container

 Surface area and volume calculations of your new container along with a comparison to the current surface area and volume of the Powerade and Vitamin Water containers showing that you decreased the surface area without compromising the volume.