Points and Angles Online
February 2002

    Table of Contents:

What Would the Graph Look Like if…? A CBL Adventure

Robert Ruzich

Fenton High School, Bensenville, Illinois

    As you travel through the streets of Chicagoland and admire the beauty of the mathematics around you, you are viewing the accomplishments of others.  These architectural accomplishments are the culmination of designs inspired by a vision, a vision brought to life by creatively integrating the disciplines of mathematics and science. In science, and typically in life, these accomplishments start with an idea, followed by an experiment, and then developed into a form that we can admire.  On Friday, February 8th, Robert Ruzich, mathematics instructor at Fenton High School in Bensenville, will take us on an adventure into the world of science and mathematics that starts with a phenomena and then develops a generalization, a mathematical rule, when he speaks at M.M.C. in a talk entitled, “What Would the Graph Look Like if?…A CBL Adventure.”  Bob has been active in T3, Teachers Teaching with Technology, an affiliate of the Educational Division of Texas Instruments.  He has been giving integrated presentations using the graphing calculator, along with various interfaces, to do data modeling activities in the classroom.  Bob will use the CBL (Calculator Based Laboratory), taking on an inductive process, to find mathematical rules.  When teaching mathematics we typically start with a generalized rule and then end a unit with specific data to fit the rule.  Bob will  take us on an adventure that will allow our students to use their ideas to gather data and then discover a mathematical rule. Hope to see you on Friday, February 8th, at Fountain Blue, as Bob Ruzich uses the data gathered from a CBL to visualize mathematics.   

Pat Bowler Johnson

Board Meeting/ Elections

February 11th, your board will be meeting.  If you have anything that you would like to have discussed, please feel free to e-mail Fern Tribbey at:  <ftribbey@d113.lake.k12.il.us>.  You may also call me at school (847)926-9223.  One of the things that the board will be discussing is the ballot.  If you are interested in running for a board position or for president-elect, please e-mail Ron Vavrinek at: <rvav@imsa.edu>.

Pryjma's Pedagogical Pondering Points

If you can't attend Terry Phillips' New Teacher Workshop at New Trier on Saturday February 9 but you still have a question or concern as a new teacher, e-mail George Pryjma at: <gpryjma@aol.com>

Have you seen any vanity license plates that are math related?  For example:  "QT314" or "MR MATH"?  E-mail your sightings to: <tribbeys@excite.com> and I'll include them in a future Points & Angles.



 Five Pentagons and a Constant Sum

    In this year's contest, in the December 2001 Points and Angles, 25 circles were placed on the vertices of 5 pentagons, and 75 circles were placed on the sides between the vertices.  One pentagon had 2 other circles on each side, one pentagon had 3 other circles on each side, and so on up to a pentagon with 5 other circles on each side.  The task was to place 100 different positive integers in the 100 circles so that the sum of the numbers on each of the 25 sides of the pentagons would be the same number.

    If the sum is a large number, such as 1,000,000, then the task is rather easy to do.  But if the sum is a small number, such as 150, then the task is impossible because there are only so many ways to obtain the sum using different numbers.  I believe it can be proved that 224 is the lowest possible sum, and two entries achieved that sum.  They will split the first and second place prizes and each receive $40.  A third entry achieved 226 and will get the third place prize of $20.  The prize winners are:  

1.  Tom Edwalds   224   actuary           Munich-American Reassurance Co.

1.  Gary Genualdi 224   teacher           Schaumburg Christian School

3.  Daniel Choist 226   10th grader Walter Payton College Prep, Chicago

Two other entrants, with sums of 230, receive honorable mention:  James Labuz, a 12th grader at St. Patrick H.S. in Chicago, and Doug O'Roark, a teacher at Walter Payton.  There were only six other entries.  Three of these had sums of 250 and three had sums of 300.  The total of 11 entries indicates that it took a considerable amount of work even to enter this contest.  We congratulate all the entrants.

Tom Edwalds entry uses the integers from 1 to 100.  The vertices of each pentagon are in boldface.

Sides of smallest pentagon:   61, 100, 63, 96, 65, 97, 62, 98, 64, 99.

2nd smallest pentagon:  20, 94, 87, 23, 89, 93, 19, 91, 92, 22, 88, 90, 24, 85, 95.

Middle pentagon:  1, 86, 72, 55, 10, 81, 66, 54, 13, 83, 67, 56, 5, 82, 68, 53, 16, 84, 71, 52.

2nd largest pentagon:  2, 80, 75, 30, 29, 8, 76, 70, 32, 26, 12, 79, 74, 37, 18, 4, 77, 69, 31, 28, 15, 78, 73, 35, 21.

Largest pentagon:  3, 60, 48, 39, 34, 33, 7, 58, 50, 45, 44, 9, 11, 59, 46, 43, 42, 17, 6, 51, 49, 41, 36, 27, 14, 57, 47, 40, 38, 25.

Gary Genualdi's entry uses the integers from 1 to 101 but not 91.

Sides of smallest pentagon:   61, 101, 62, 99, 63, 97, 64, 95, 65, 98.

2nd smallest pentagon:  20, 100, 81, 23, 90, 89, 22, 87, 94, 21, 96, 88, 19, 93, 92.

Middle pentagon:  11, 71, 57, 73, 12, 74, 75, 50, 13, 40, 78, 79, 14, 80, 33, 82, 15, 8963, 30, 85

2nd largest pentagon:  6, 51, 52, 53, 55, 7, 68, 56, 27, 58, 8, 59, 35, 76, 37, 9, 70, 24, 25, 86, 10, 67, 54, 69, 18.

Largest pentagon:  1, 26, 49, 16, 46, 84, 2, 31, 28, 66, 34, 60, 3, 36, 17, 48, 39, 77, 4, 41, 42, 43 44, 45, 5, 29, 47, 38, 72, 32.

We hope that all of you who worked on this contest or used it in your classes found it to be a worthwhile activity.  Please address any comments to Zalman Usiskin, University of Chicago, 5835 S. Kimbark, Chicago, IL  60637.


MMC Scholarship

The Metropolitan Mathematics Club of Chicago is offering $1,000 in scholarships for high school students who plan a career in the teaching of mathematics.  Contact Conrad Wayne at Rich South High School,   708-679-3150

The Metropolitan Mathematics Club of Chicago is offering $1,000 in scholarships for high school students who plan a career in the teaching of mathematics.  The selected students, their parents, and their sponsoring teachers will also be invited to the May meeting of the MMC at which time the recipients will be honored. 

    The guidelines used for selection shall be:
I.    A. Demonstration of overall academic scholarship with inclusion of at least eight semesters of college preparatory mathematics.  (A minimum cumulative grade point average of 3.0, with A = 4.)

       B. A statement of the intention to pursue a career in mathematics teaching.
       C. Indication of participation in extra curricular activities, especially those which may have a positive influence on a teaching career.

II. Applicants must have a letter of recommendation from a member of the Metropolitan Mathematics Club who is familiar with the applicant’s academic performance and his or her potential as a mathematics teacher.

III. Applicants must submit an essay of at most 400 words explaining why they would like to be a mathematics teacher.

 The scholarship award or awards will be determined by a selection committee of MMC members appointed by the Executive Board.  To be eligible, an applicant must submit the application, have an official transcript sent, and request a letter of recommendation from a member of the MMC such that all of the materials are received by the date on the application.

 The committee will establish its own guidelines for evaluating applications, and will make a recommendation to the Executive Board as to the awarding of the scholarship.  No member of the selection committee may nominate nor recommend a candidate.


Application Deadline:  March 18, 2002
Name:__________________________________________  Date:_________________




School Address:________________________________________________________


Home Phone:(____)___________________  School Phone:(____)________________

Sponsoring Teacher (Must be MMC member):_________________________________

Please complete the following:

Overall Grade Point Average:_________  (A = 4, B = 3, C = 2, D = 1, F = 0)

 Mathematics Courses       Grade          Mathematics Courses       Grade

___________________      _____         __________________      _____

___________________      _____         __________________      _____

___________________      _____         __________________      _____

___________________      _____         __________________      _____

Extracurricular Activities:____________________________________________



In addition applicants must also send:

1.  A letter of recommendation from the sponsoring teacher, who is a member
     of the Metropolitan Mathematics Club of Chicago: **

2.  A current transcript for seven semesters of high school.**

3.  An essay not to exceed 400 words on:  “Why I would like to teach

Please send all information to: Conrad Wayne
      Mathematics Department
      Rich South High School
      5000 Sauk Trail
      Richton Park, IL  60471
      phone: 708-679-3150; fax: 708-679-3168
**(Letters of recommendation and transcripts may be sent by separate mail.)
(Photocopy as needed)

The Chicago Area All-Star Math Team will be holding tryouts for the 2002 team on Thursday, February 28, from 4 to 9:30 p.m. in room S112 at Evanston Township High School. The All-Star Team will practice in the Spring and will travel to Iowa to compete in the ARML (American Regions Math League) competition on June 2, 2002. Any high school student is eligible to tryout for the team. About 35 students will be selected the night of the tryout and about 30 more will be selected a few days later. Please contact Mary Lappan (lappanm@newtrier.k12.il.us, 847.784.6608) with any questions.


Our organization is sponsoring a new teachers’ workshop on Saturday, February 9th at New Trier High School.  Terry Phillips is doing a great job in making this event for our new (within the first two years) teachers an exciting and informative one.  Can you remember your first year of teaching?  Even though I am in my twenty-seventh year, I still can remember my first year as if it was yesterday!  I remember being so excited before the school year began and thinking about how I was going to set up my classroom.  I was to share my room with another teacher for one of the periods of the day.  One day before the year began I was stopped by school personnel while walking in with a box of supplies, mistaken for a student.  I was told to come back the first day when students are to report.  I proudly stated that I am one of the new teachers and then I was able to proceed to my classroom.  I remember the first day of classes that year and how excited I was to finally begin my lifelong dream of being a mathematics teacher, I could hardly sleep.  I realized that using humor got me further with my students.  Then I can remember one day in November of that year, walking out of the building at the end of a very long day questioning whether I chose the correct profession.  The principal had asked me how things were going.  After I shed a few tears, he told me that the first year is the hardest and that the following years will be better.  He also encouraged me to continue with my dream because I was enabling my students to learn.  Looking back on that day in November, I realized that he was quite right in what he told me.  I am glad that I continued with my dream.  I still have a lot to learn and am not afraid to ask for assistance and am not afraid to share my thoughts with the new teachers in my department. I am asking you to share your memories with a new teacher that you know.  They might not feel so alone in what and how they are feeling about the job that they are doing in such a rewarding profession such as ours.  Be a mentor.

February 11th, your board will be meeting.  If you have anything that you would like to have discussed, please feel free to e-mail me at:  ftribbey@d113.lake.k12.il.us.  You may also call me at school (847)926-9223.  One of the things that the board will be discussing is the ballot.  If you are interested in running for a board position or for president-elect, please e-mail Ron Vavrinek at: rvav@imsa.edu.

I am looking forward to seeing everyone at the Fountain Blue on February 8th to hear what Bob Ruzich of Fenton High School has to say.

"The Mathematics of Frank Lloyd Wright's Architecture"

Mary Wiltjer

Addison Trail High School

At the largest dinner-meeting this year, and our first at the Fountain Blue, Mary Wiltjer treated a very enthusiast crowd to a delightful virtual tour of many of Frank Lloyd Wright's houses.  Each house was designed to relate to the person who was to live in it.

Text Box:  
Robie House
The Robie House is Wright's most famous "prairie house" and an excellent example of Wright's use of the straight line to make the house part of its natural setting.  The courtyard is enclosed with a brick wall and is covered by overhanging eaves, which parallel the wall.  The eaves extend around to the front of the house, keeping the same horizontal line, where it is accentuated by a series of casement windows. 


Text Box:  
Home - Playroom
In designing his own home, Frank Lloyd Wright used the circle to create open space.  Always emphasizing the human scale, he scaled the playroom to a child's size.  In some homes, he went beyond the architecture of the home and designed furniture, china, and even dresses.  He was insistent that everything should fit together.

 Frank Lloyd Wright's architecture reflects his concern that a house provides shelter with warmth.  He would put something "light" where the building is heaviest, placing windows immediately under the roof or around corners.  He used pigment in the concrete used to separate the bricks vertically so that the lines would flow only in a horizontal direction. 

 In designing the Guggenheim Museum, Wright used circles and spirals and an elegant skylight to connect to nature.  Visitors take an elevator to the top and spiral back down, with each spiral smaller than the one above.  In other homes, he played with hexagons, pentagons, octagons and even half circles.

 Within his homes, Frank Lloyd Wright was concerned about the flow of space.  He wanted to define space without confining it.  Unlike other houses of his time, he took small spaces and made them larger, at times creating a room within a room.  Seeing out, but not seeing in, and being in touch with nature were important.  There is an emphasis on beautiful, geometric shapes, not orientation. 

 Mary's talk, an excellent start for 2002 and for our stay at the Fountain Blue, left us all fascinated with the architecture of Frank Lloyd Wright and his incorporation of mathematics into each house.  Those who are going (or should we say, "have gone") on the tours are looking forward to an additional treat.

                                                                                                                         -Ron Vavrinek

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