Author: Jeff Noble
E-Mail Address: hordes_illini90@hotmail.com
School: Crete-Monee High School : Crete, Illinois
Home Page URL:  http://www.angelfire.com/il/mystudents/COMBOS.html

                                links are on this page for the introductory handout, resource evaluation, and rubric

THEME : ratios, measurements, geometry, golden ratio, internet research, use of software to report results

BROAD CONCEPT: Teamwork, project building, presentation skills, transfer of school to real life

GRADES: 9-12 algebra

INTEGRATED SUBJECTS: art, science, english, geometry

UNIT GOALS AND PURPOSE:

The purpose of this project is for the students to work as a team to use internet resources in order to discover what the golden ratio is well enough to explain it to others.  They will have to sort through and evaluate various pieces of information as they figure out what the golden ratio really is and how it works.  They will learn how the golden ratio is both defined by algebra as well as geometry.  In addition, the students will look at the world around them in order to see if the ratio is at work in man made items and nature itself and conjecture why that may or may not be. In the classroom, the students will pick some items to measure, and using a spreadsheet, report them as golden or not.  Finally the teams will make a scale drawing of a swimming pool deck using the golden ratio in as many ways as possible and using a given scale.  All the dimensions of the figures created will be reported on the spreadsheet as well.  Each person will have a defined role and successful completion of the project depends on all members of the team working together.  Each team will use the assignment introduction as a checklist that the teacher will initial as the tasks are completed.  A rubric is included which details all that is expected including the research, the paper, the spreadsheet, the project, and the presentation.  Fellow classmates as well as the teacher will use the rubric to determine the grade.
 
 

TIME FRAME:

10 school days, of which 5 are necessary to have a computer lab with internet access
 
 

OBJECTIVES:

In this lesson, students will:
 

I. Use the internet resources given below in order to accomplish:

*Defining what the golden ratio is mathematically and practically

*Describing what geometric shapes golden ratios are found in and how they might be applied

*Evaluating the internet resources as to their usefulness

*Finding objects from the world that are in proportion to the golden ratio and giving reasons why

   they might have been created that way

II. Use teamwork to accomplish:

*Finding objects around you that are in proportion to the golden ratio and giving reasons why they

   might have been created that way

*Writing a group paper that puts together the math of the golden ratio and the geometry as well.

*Creating and drawing a swimming pool area using the golden ratio in as many ways as possible

*Create a spreadsheet clearly showing the dimensions of the objects you measured in the classroom.

    You will also include all the objects from your swimming pool diagram, and use functions to show

    their ratios as being golden or not

*Present your creation to the class including brief explanation of how the objects fall under the golden
    ratio

*Write a paper presenting all the work you did individually for the group to ensure success for the

    whole group
 
 

MATERIALS:

• General materials and equipment: rulers, posterboard, markers, handouts of assignment introduction, Rubric, and evaluation of resources  

• Technology tools: computer lab with internet access, computer with spreadsheet program and word processing

• Software Used in this Unit

Geometry:

• Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical

arguments about geometric relationships

• Understand measurable attributes of objects and the units, systems, and processes of measurement

• organize and consolidate their mathematical thinking through communication;
• communicate their mathematical thinking coherently and clearly to peers, teachers, and others;
• analyze and evaluate the mathematical thinking and strategies of others
• use the language of mathematics to express mathematical ideas precisely.

• 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.

• create and use representations to organize, record, and communicate mathematical ideas;
• select, apply, and translate among mathematical representations to solve problems
• use representations to model and interpret physical, social, and mathematical phenomena. 

 

Lesson 1: (Introduction)

        Have the students get into groups of three.  If the class isn't divisible, then groups of two work can work but keep them to a minimum.  Pass out the introductory handout and the resource evaluation handout.  Impress upon them the importance of these two documents to the success of the group.  The introductory handout has the Webquest's web address and will be checklist that they can use, while the resource evaluation will help them insure that they have read through all the relevant websites and understand them.  Take the class to the computer lab and the goal of the day is just to let the students get acquainted with the main webpage,  how to use it, and what the golden ratio is and its importance.  The students will also want to start checking the links out for what is on them.  They do not want to start filling anything out other than the occasional note to themselves on what looks like "cool" websites on their first impression.
 

Lesson2&3: (Research)

    Using the resource evaluation handout as a guide, the students have two days to do the "meat" of their research.  They must visit every site and rate them based on their quality of information, appearance, and usefulness.  It will take all their time to do this.  All the webpages have vast information on related links and the topic of the golden ratio is new to the students.  Avoid answering student's questions directly about the content itself, instead lead them to answer their own questions using the resources available.  Be very helpful though on troubleshooting problems that come up not related directly to the learning itself as frustration can keep groups from getting their work done.
 

Lesson4: (Accomplish tasks)

    Pass out the rubric to all groups.  They have hopefully already looked at it as it is linked on the main webpage.  This is the day for the mathematician and geometric technician to really lead the way.  They should know by now what they know and don't yet know.  The leader should have a good idea of what information is on the sites based on the resource evaluation handout.  This is the last day for the group to really polish up their information and the second half of the day each group should start composing their paper and doing the necessary measurements around the room.
 

Lesson5: (Complete paper)

    This is the last day the groups should have to be in the computer lab.  This day is for typing the final version of the group paper and creating the spreadsheet that will report their measurements.  The teacher may want to collect the group's papers in order to give a final grade, or give a partial grade that can be improved upon in the student's own time the following week.  By using the computer lab, no student has the excuse of not having home access to a computer for typing the paper.
 

Lesson6&7: (Swimming pool project)

    The project now starts the alternative assessment section.  The students now have the assignment of creating a swimming pool deck using the golden ratio and golden geometry in as many ways as possible.  This assignment checks whether the students really understood what they learned the previous week.  It also allows more creative students to put their knowledge and ideas into practice.  This open ended assignment still has mathematical guidelines but will lead to higher level thinking if the students choose to.  Have poster boards and markers available for each group.  Impress upon the groups though to first sketch what they want to draw on scratch paper.  They will be graded on their actual drawings for accuracy in measurements.  The mathematician really needs to use their knowledge of ratios at this point to ensure proper dimensions.  The geometric technician will be in charge of the shapes and their accuracy as to scale.
 

Lesson8: (Polish and prepare to present)

    This is the final day for groups to finish their drawings.  They are also to create a spreadsheet listing all the real dimensions of their shapes, as well as their scaled dimensions.  For instance: 5x3 inch rectangle is kiddy pool on drawing and using scale of 1in=5ft that would create a real sized kiddy pool of 25x15 feet.  25/15 is 1.66 which is approximately the golden ratio.  All this information should be in the spreadsheet.  The use of the classroom computer is necessary for one group at a time to create their spreadsheet.  If there is not enough time for all groups, it may be necessary to use the computer lab this day as well.
 

Lesson9&10: (Present and summarize)

    The groups present their projects these final two days.  Hand out as many copies of the rubrics as there are groups to each student.  They will circle whichever section they feel the presenting group achieved.  The teacher can choose to average all rubrics scored, or use just their own.  The groups can use their final paper to help them give the information noted on the rubric.  Encourage them to "sell" their drawings and have fun presenting their ideas.  Warn each group to read the rubric carefully before coming to class so that they know all the information you are looking for.  Collect their final drawings, spreadsheets, and papers.  Have each student write a short paper listing all the work they did to help the group successfully accomplish the project.  This gives the teacher information that may explain poor group performance or outstanding performers.  Use leftover time for feedback on the project.  The students will want to talk about their experiences and what they liked in each other's presentations.  This activity gives closure to the project and is a great learning opportunity.
 

SHARE YOUR IDEAS:

The teams will present their research evaluations on the handout included, their research findings in a paper, and their measurements in a spreadsheet.  Each team will make a scale drawing of a swimming pool deck using the golden ratio in as many ways as possible and using a given scale.  They will present their drawing according to the guidelines given in the rubric
 
 

STUDENT AND TEACHER ROLES:

Teacher:

The teacher's role is to just be a facilitator.  The activities are really self explanatory.  That is the purpose of putting the students into teams with exact roles.  Let them be in charge of their learning, just help them if they get stuck with details.  Do not feed them the information, you should be familiar with all the webpages, but let them do the exploring.
 

Students:

The first week of the webquest is to do the internet research and get all the information needed. The following week's first three days, the teams will work together to draw and design the swimming pool deck. The last two days are for class presentations.

To accomplish the task, each student will need to each take on a role in the group. The 3 roles are: team leader, mathematician, and geometrical technician. Each role has different responsibilities so make sure to read them before deciding. Even though each role has different jobs, each member of the group will have to know what their partners are doing. Communicating is vital and checking each others work will be invaluable to you.

Group:

1. Read through the duties of each role
2. Pick your roles
3. Leader takes main seat in front of computer and other 2 prepare to take notes on the resources section of the introductory handout
4. When mathematician and geometric technician have finished steps 3 from their lists, discuss and record in the report why the objects they found might have been made in the golden ratio
5. Put together the work of the mathematician and the geometric technician as a finished paper
6. Put together the dimensions of the objects measured into a spreadsheet showing their ratios.
7. (Second Week) Design a swimming pool in the golden ratio. You may also design accessories for the pool that are also in the ratio. The more you have the better. The more creative you are the better.
8. Put the dimensions of the objects in the swimming pool deck into the spreadsheet started before
9. Present your design to the class and illustrate all uses of the golden ratio. The more common sense it makes the better.

Mathematician:

1. In your report, define the Golden Ratio in exact terms and its rounded approximate. Be thorough in your explanation
2. In your report, define how the Golden Ratio is related to Fibonacci's Sequence and show an example
3. Measure the computer screen and at least 2 other items in the room. In your report, show their ratios as compared to the golden ratio
4. (Next Week) Contribute to the swimming pool design numbers that will make it in the golden ratio
5. Demonstrate to the class a golden example from the numbers you chose in the design of the pool

Geometric Technician:

1. In your report, complete a picture of the Golden Rectangle showing what is true about the sides and how it is created
2. In your report, draw at least 2 other shapes that fall under the golden ratio and explain briefly why they do
3. Find a worldly application of the golden ratio as it applies to art, nature, or architecture. In your report, sketch the object, get its dimensions if possible, and explain it's golden tendencies
4. (Next Week) Contribute to the swimming pool project different shapes you learned are in the golden ratio
5. Illustrate the shapes you chose for the class in the presentation

Team Leader:

1. Navigate the team through the websites efficiently and fill out the evaluation handout
2. Based on your evaluations, lead the group through the websites so that they can get their information
3. When both your partners say they are done, check their work and the group checklist to make sure it is all finished
4. Type the group paper presented to you by your mathematician and geometric technician
5. Type the spreadsheet using the numbers given to you
6. (Next week) Lead the group through the design and drawing of the swimming pool part of the project
7. Present the project to the class using the mathematician to explain the ratios, and the geometric technician to explain the shapes

ADDITIONAL REFERENCE PRINT RESOURCES STUDENTS MAY USE :

Algebra or Geometry textbook

Pappas, T. "The Golden Rectangle." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 102-106, 1989.
 
 

THE FOLLOWING BOOKS CONTAIN INFORMATION ABOUT THIS TOPIC:

Cook, T. A. The Curves of Life, Being an Account of Spiral Formations and Their Application to Growth in Nature, To Science and to Art. New York: Dover, 1979.
 

FOR ADDITIONAL ONLINE INFORMATION AND MEDIA PERTAINING TO THIS UNIT, TEACHERS MAY WANT TO USE THE FOLLOWING INTERNET ARTICLES: