Points and Angles Online
November 2001

    Table of Contents:

 An Introduction to TI Interactive!™

Stephanie Casey                   Jim Casey
Deerfield High School    New Trier High School

Many of us embody within creative ideas of how to develop and implement interdisciplinary activities, but sometimes we are unable to put our plans into action for various reasons.  On Friday, November 16th, Stephanie and Jim Casey will introduce the Texas Instrument’s software, TI Interactive!™  This software is used by educators to assist them with developing interdisciplinary activities.  TI Interactive!™ was developed on the keystroke commands of the TI- 83 and 89 calculators.  Thus, if you are “comfortable”using your TI calculator, then you have enough knowledge to use TI Interactive!™  You only have to be somewhat creative to use this software to enhance student learning.  Stephanie, a mathematics instructor at Deerfield High School, and Jim, a biology and science instructor at New Trier High School, will use the TI Interactive!™ software to illustrate how one can investigate and visualize interdisciplinary concepts.  In other words, these two young, energetic individuals will use mathematical techniques, scientific data, and writing skills which will allow students to discover ideas, using a computer.  TI Interactive!™ is billed as an integrated learning program for both the math and science disciplines.  With the different educational backgrounds that both of the Casey’s have, they are able to integrate the subjects of math and science in a demonstration of how a similar integration can take place in our classrooms.  Please join us on Friday, November 16th, for a night of visualizing mathematics, using interactive software that will allow each of us to develop and implement interdepartmental activities for students to discover another dimension of mathematics.
       --- Pat Bowler Johnson

What follows is my first “Pryjma’s Pedagogical Pondering Points” for new teachers.  I have been asked to write a monthly column responding to questions posed by those new to the teaching of mathematics. Because my AOL account has been overwhelmed and paralyzed by the sheer volume of e-mail queries sent to me by eager young teachers, I must write this first column without the benefit of having any actual questions (really, just a minor inconvenience for one who can easily lecture for 40 minutes on the topic of parentheses and their impact upon human thought). I’ll just make up a question or two.

Question: How do I respond to a student who complains that “I was just 3 points away from an A. Can’t you give it to me?”
Answer: Here I rely upon the advice given to the MMC by Wade Ellis about a decade ago. Wade counseled that instead of giving 100 point test and 50 point quizzes, make the tests 200,000 point and the quizzes 50,000 or 100,000 points. Then when a student comes up and says “I’m only 2,387 points away from an A, can’t you give it to me?”, your compassionate respond can be “well if it were only a few hundred points, I wouldn’t see a problem --- but with 2,387 points my hands are tied!”.

Question: How do I motivate students to ask questions in class?
Answer: I have found that if I ask “Any questions?” or “Anything else?” I am far less likely to get questions than if I ask “What questions do you have?” or “What else can I help you with?”  It appears that the latter phrases seem to obligate students to look more carefully at what they need, while the former act as indicators that question time is over!

Points to Ponder:
A recent survey revealed that 7 out of 5 Americans have trouble understanding fractions.
Did you know that there are three kinds of math teachers? Those that can count and those that can’t!

Seriously though, please address any questions to me at: <gpryjma@aol.com>

       -George Pryjma

The winners for the drawing at October’s MMC meeting for a copy of Mathpedia TM are:
Katie Lineham -Evanston Township High School
Judith Meckley -Joliet Junior College
Steven Tribbey

Conrad Wayne has announced the two winners of last year’s MMC Scholarship.  They are:

Hannah Barker, Evanston Township High school, who is enrolled at University of Chicago. Her high school math curriculum included B/C Calculus and Multivariable Calculus and Linear Algebra. She was also very active in service projects in and out of school and has a number of accomplishments in music (choir/guitar).

Rose Chen, Highland Park High School, who is enrolled at University of Illinois at Chicago. Her high school math curriculum included B/C Calculus. Rose was able to successfully overcome the challenge of moving to Highland Park from China in her sophomore year and having to learn a new language.

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.  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.

The MMC Conference of Workshops to be held January 26, 2002 at Homewood Flossmoor High School has been CANCELLED due to a very unusual lack of response from potential presenters. We are considering running this conference every two years, rather than annually. If you are a speaker who did respond, save all those great ideas until next year. A BIG thank you goes out to all those who participated in organizing this conference.    -Suzanne and Toni


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)

Points from the Interior

       All of us receive gifts from various people at various times through out our career.  Last year I had received a “Teacher” calendar where for each day, there was a quote.  Some are quite funny and some are quite serious and thought provoking.  Here are some quotes from this calendar that could lead to a great discussion about what teachers are all about.  Enjoy.

“There are no secrets to success.  It is the result of preparation, hard work, and learning from failure.” (Colin Powell)  A very good friend of mine once told me that from failing, he became the type of mathematics educational leader that he is today.  I think what he was trying to say to me is that it is OK to fail.  I admire this person and he has been one of my mentors throughout my career.  It does develop character, though!

“A great teacher never strives to explain his vision.  He simply invites you to stand beside him and see for yourself.” (R. Inman) Have you ever been there?  Or are you there now?

Julius Lester wrote, “A good teacher is one who helps you become who you feel yourself to be.  A good teacher is also one who says something that you won’t understand until ten years later.”  I do not know about you, but I have been there!

Malcolm X said, “Without education, you’re not going anywhere in this world.”  My parents said this quite a bit during my adolescent years.

“I am enough of an artist to draw freely upon my imagination.  Imagination is more important than knowledge.  Knowledge is limited.  Imagination encircles the world.” (Albert Einstein) Quite powerful!

We know that using our mind is the most important asset that we have.  “I cannot teach anybody anything, I can only make them think.” (Socrates) Here is another quote that I still can hear my mother saying to me, “It is not enough to have a good mind; the main thing is to use it well.” (Rene Descartes)

“Nine-tenths of education is encouragement.” (Anatole France)

“You never know when someone might catch a dream from you.  You never know when a little word or something that you might do, may open up a window of a mind that seeks the light.  The way you teach may not matter at all, but you never know, it might.” (You Never Know from CanTeach website)

See you on November 16th at Berghoff’s!                             -Fern Tribbey

Taking Visualizing Mathematics One Step Further:
Dynamic Visualization
Wade Ellis - October 26, 2001

A decade (not to mention century and millennium) ago, Wade Ellis introduced many MMC members to the power of visualization using the TI-81 graphing calculator.  His insights, pedagogy, and examples were exciting and thought-provoking.  On October 26th, the MMC welcomed Wade Ellis to The Berghoff Restaurant for an updated look at “Dynamic Visualization in the High School Curriculum.”  Initiating our Berghoff meetings, Wade again brought us a sampling of the latest technology for helping students understand mathematics through dynamic visualization.

Wade began by using the TI-83 to help students understand the solving of quadratic inequalities.  Hoping that students could factor  x^2  - 3x – 10 >= 0 into
(X + 2)(x – 5) >= 0,  Wade entered the following into the  y=  option of the TI-83:
Y1 = (x+2),  Y2 = (x – 5),  and  Y3 = Y1*Y2 with a window of X[-3.4,6,1] and
Y[-10,10,1] with Y1 and Y2 selected and Y3 deselected.  After the TI graphed the resulting two parallel lines, Wade used the TRACE feature to find points that did and did not satisfy the quadratic inequality by observing the signs of the respective y-coordinates.  Wade emphasized the importance of having students create a table of ordered pairs and the products of their Y coordinates   --- students must record what they observe in order to be able to analyze and learn from what they see.  Then Wade graphed Y3 (the product) “extra thick” to compare where this graph is above and below the x-axis compared to previous conclusions regarding solutions to the quadratic inequality.  Wade noted that the bottom of the Y3 parabola was not in our window.  He said that this window choice was deliberate and that students need to know that windows may produce incomplete graphs.  Wade then “kicked it up a notch” by changing Y1 to  -(x + 2).  Now the graphs of Y1 and Y2 intersected and the x-coordinate of the intersection point corresponded to the abscissa of the vertex of the Y3 parabola.  Noting that he didn’t have enough room in the margin to prove (or disprove) the generalization of this result, Wade challenged us to do so.

Wade Ellis then brought us to a higher level of technology and greater power for visualization.  Attaching a TI-92 to his magic projection device, he selected PARAMETRIC mode and entered the following:
XT1 = t,  YT1 = (t+3)(t+1)(t-2)  with window t[-4,3,.1], x[-4,3,1], y[-10,10,1]
The resulting graph was analyzed with regard to how close the plotted points were.  Statements like “increasing at a decreasing rate” and “decreasing at an increasing” rate were deduced from the graph.  Algebra students can easily visualize these concepts, once reserved to Calculus!  Wade then changed the XT1 setting to  XT1 = 2  in order to focus on the behavior of  Y as  t went through its values (the smaller the t, the slower the graphing and the denser the points).  Animating Y helped us visualize in another way the behavior of the function.  Finally, Wade reset  XT1  to  XT1 = t  and animated the graph.  Wow!

“Kicking it up still another notch”, Wade Ellis brought out his new Macintosh iBook and demonstrated its built-in graphing program.  This program allowed us to see visualization (no pun intended) at a still higher level.  Because this program (available from Pacific Tech if not built into your Mac) allows for the animation of an entire graph (not just a point), Wade used it to show how the  “h”  in y = x – h and y = abs( x – h ) affect these graphs, followed by  y = 2x – h versus  y = abs(2x – h)   and   y = 2(x-h) versus
y = abs(2(x-h)).  The graph of  y = x  was “frozen”  on the screen for perspective.  Wade Ellis also graphed, for our edification, y = (x – h)^2  versus  y – h – x^2  (yes, the program was perfectly happy to graph the function written in this form).

Wade concluded our journey into visualization by graphing (using the iBook program) a graph of   cos(x)*cos(y) > 0.  The graph was (surprise) a checkerboard!  Wade asked us WHY? And stated that we must have our students go beyond just graphing and into analyzing and reflecting upon what they see.

On a personal note, I was the MMC program chair who first invited Wade Ellis to speak to an MMC dinner meeting.  His talk then inspired me to look further into the complexities, problems, and rewards in using graphing technology to help students understand and internalize certain concepts in mathematics.  I have made a point of seeing any presentation by Wade Ellis at any conference that was possible for me.  I have always come away with a smile on my face and a great many ideas on how to improve my teaching.  Thank you, Wade.

        -George Pryjma

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