**High School Mathematics-Physics SMILE Meeting
1997-2006 Academic Years
Mathematics: Miscellaneous**

**11 September 2001: Monica Seelman (Williams and St James Schools)
Venn Diagrams**

Discussed an example of a Venn Diagram. In Particular, she
considered the
following sets:

**A**: set of all integers from 1 to 1000**B**: set of all multiples of 11**C**: set of all numbers below 500 with identical digits; eg, 11, 66, 111, 444

She pointed out that all two-digit numbers in set ** C** are
also in set **B**, but
that the three-digit numbers in ** C** are not in **B**,
because they are not divisible by
**11**. But the four digit numbers with identical digits **
are**
divisible by ** 11** [**5555 = 101 ´
55**].
As an extension, all numbers with an even number of identical digits
are divisible by **11**, whereas those with an odd number of digits
are not.
Interesting results in "eleven-ology"!

**11 September 2001: Earnest Garrison (Robeson HS, Physics)**

handed out a write-up of a ** Paper Clip Lab**, in which fatigue
and fracture of
solids was studied using paper clips. The idea is to determine
the
distribution in the number of times one must bend a paper clip back and
forth in
a controlled fashion to get it to break at the "little loop", the
"big loop", and on the "straight section".

Earnest also showed us an exercise in estimating the area of an irregularly shaped lake on a map, by overlaying a square lattice of dimension 1 cm. The idea is to estimate the area as follows:

**J:**Number of squares completely covered by the lake**K:**Number of squares more than 50 % covered by the lake.**L:**Number of squares 50 % covered by the lake.**M:**Number of squares less than 50% covered by the lake.**N:**Number of squares not covered at all by the lake.

The area of the lake is estimated to be **J + K + L/2**.
This
estimate is fairly accurate, in practice! And, students ware surprised
at how
closely their results agree with one another.

**11 September 2001: Fred Schaal (Lane Tech HS, Mathematics)**

suggested that "conditional logic" should be considered as an
alternative to "deductive logic", and "inductive
logic". In conditional logic we have the statements ** A ®
B** and ** B ® C**, from which we
conclude that ** A ® C**.
As an example, he considered the syllogism

**All animals named Flicka are
horses.
All horses have four legs:
\ All animals named Flicka have four legs.**

Note
that ** A ®
B** and **C ® B** does not permit
the conclusion **A ® C**,
so that the following syllogism is** incorrect.**

**All horses have four legs.
All animals named Flicka have four legs:
\ All horses are named Flicka.**

**20 November 2001: Estellvenia Sanders (Chicago Vocational HS)
Digital Numerics**

**Estellvenia** uses these activities with her high school students.

She gave us a sheet with the numbers 1-20 located randomly on it, and we were told to touch as many of the numbers as possible over a given time period (10-20 seconds), timed by a partner using a stopwatch. We were to touch the numbers in increasing order (1 ... 2 ... 3 ... ) with an index finger. We recorded the total number touched by each of us over three trials. Then, we analyzed and compared the data. By this exercise, some of the students will be able to remember and identify the numbers more quickly.

We then saw how sign language digits (numerically) can be combined with standard

American Sign Language[http://www.lessontutor.com/ASLgenhome.html] symbols to speed up sign language, in that some letters have both "letter" and "number" signs in the 1867 version. In the modern version of sign language, all letters haveletter symbolshttp://www.masterstech-home.com/The_Library/ASL_Dictionary_Project/ASL_Tables/Alphabet.html, and numbers have separatenumber symbols, http://www.masterstech-home.com/The_Library/ASL_Dictionary_Project/ASL_Tables/Numbers.html, so that no mixing of numbers and letters occurs. Very interesting,Estellvenia.

**04 December 2001: Bill Colson (Morgan Park HS, Mathematics) New Toys
Bill **used his $100 equipment allotment from CPS to obtain
blackboard drawing
materials from the K-12 Mathematics and Science Catalog for Fall 2001
of the EAI
http://www.eaieducation.com/.

Eric Armin Inc (EAI) Education

567 Commerce Street

PO Box 644

Franklin Lakes NJ 07417-0644

1 - 800 - 770-8010

In particular, he purchased these items:

- the Overhead Safe-T Compass [part number 530081 - semi-clear] $1.40 each
- Clever Catch Balls [Algebra and Geometry: http://www.24hours7days.com/Games/All_Clever.html] with questions printed on it. When a student catches the ball with both hands, s/he should answer the question closest to the thumb on the left hand.

He showed us how well the compass worked on the blackboard; then we
tossed
the ball around for a while and answered the questions. Useful stuff.
Thanks, **Bill!**

**05 March 2002: Roy Coleman (Morgan Park HS Physics)** -- **Probabilities**

- He had us pick a number from the list
**1 - 2 - 3 - 4**:

He claimed that, in a free choice, about**75%**of people pick**#3**, "reasoning" that**#4**is "last",**#1**is "first", and**#2**is "too close to first".**Believe it or not!**

- Suppose that, in a given combat arena, the following
probabilities are correct:
- 60% Probability of a shell hitting its target
- 60% Probability of a shell exploding on impact
- 50% Probability of being killed when hit by an exploding shell

- 36% probability of being hit by an exploding shell
- 30% probability of being killed by an exploding shell aimed at you
- 18% probability of dying
- 6 shots required per fatality are required, in general

**Roy**, we hope you get help soon!

**11 March 2003: Bill Colson [Morgan Park HS,
Mathematics]
T. G. I. P. --- Thank God It's Pi Day!
Bill** called our attention to the following websites from a recent

**The Joy of Pi**: http://joyofpi.com/pilinks.html**The Exploratorium Ridiculously Enhanced Pi Page**: http://www.exploratorium.edu/pi/.**Math Forum Pi Day Activities**: http://mathforum.com/t2t/faq/faq.pi.html

**Thanks for the timely reminder, Bill!**

**22 April 2003: Leticia Rodriguez
[Peck Elementary School]
Fraction Game
Leticia ** showed us how to play

- Row 1: One circle, marked 1/1
- Row 2: Four circles, each split into two
**regions**of equal halves, with each**region**marked 1/2 - Row 3: Four circles, each split into three
**regions**of equal thirds, with each**region**marked 1/3. - Row 4: Four circles, each split into four
**regions**of equal fourths, with each**region**marked 1/4. - Row 5: Four circles, each split into five
**region**s of equal fifths, with each**region**marked 1/5. - Row 6: Four circles, each split into six
**regions**of equal sixths, with each**region**marked 1/6.

**Good lessons and a good game! Thanks, Leticia!**

**21 October 2003: Bill Colson [Morgan Park HS,
mathematics]
Molecular Expressions/ Florida State U Website
Bill** passed around information from their website, http://micro.magnet.fsu.edu/,
on the following topics:

**Secret Worlds: The Universe Within:**http://micro.magnet.fsu.edu/primer/java/scienceopticsu/powersof10/index.html**Online Optics Activities for Students**: http://micro.magnet.fsu.edu/optics/activities/students/index.html**Online Activity Guidebook for Teachers**: http://micro.magnet.fsu.edu/optics/activities/teachers/index.html**Photo Gallery**: http://micro.magnet.fsu.edu/micro/gallery.html

**Bill** also passed around some geometrical questions concerning
**Tumbling,
Spinning, and Plummeting**, which appeared in the **October 2003**
issue of
**Discover Magazine** in the feature article** "bogglers" by scott
kim"**: http://discovermagazine.com/.

**14 September 2004: Bill Colson [Morgan Park HS,
Mathematics]
Philately**

**Bill** first showed us the new postage stamps issued in honor of **R.
Buckminster Fuller, the Man and Mind behind the Geodesic Dome**:
http://www.usps.com/communications/news/stamps/2004/sr04_043.htm.
The top of his head is shown as a geodesic dome on the stamp.

**14 September 2004: Fred Farnell [Lane Tech HS,
Physics]
SPECIAL BELL SCHEDULE**

**Fred** passed around a copy of the bell schedule at **Lane
Tech**, which he had written on the board as shown here:

Division |
8^{30} to 9^{00} |

1^{st} |
9^{04} to 9^{39} |

2^{nd} |
9^{43} to 10^{18} |

3^{rd} |
10^{22} to10^{57} |

4^{th} |
11^{01} to 11^{36} |

5^{th} |
11^{40} to 12^{15} |

6^{th} |
12^{19} to 12^{54} |

7^{th} |
12^{58} to 1^{33} |

8^{th} |
1^{37} to 2^{12} |

9^{th} |
2^{16} to 2^{41} |

**Fred Schaal**, his **ever-alert **colleague, saw the
notice on the blackboard, and immediately identified a connection with
mathematics. We could view these numbers as exponentials, such as **9 ^{04 }
= 9 raised to the power 4 = 6561**.

- Which would represent the largest number? The next largest number? ...
- Which would represent the smallest number? The next smallest number? ...
- Which line contains numbers that have the least differences?
- Can you develop other meaningful questions that could be asked about these numbers, if the times represent numbers written in exponential notation?

**09 November 2004: Leticia Rodriguez [Peck Elementary
School]
Interactive Fractions **

**Leticia **passed around sheets showing wheels divided into **halves,
thirds, fourths, sixths, ninths, and twelfths**, as well as a **spinner**
for generating the numbers **1, 2, 3, 4, 6, 9, and 12**. We
were to spin twice (**say 3, 12)**, form a fraction by putting the
smaller number over the larger (**3/12**), and color that fraction
in one of the wheels. The person who first colors all the wheels
is the winner. **What a neat way for students to learn
fractions!**

**Very interesting game, Leticia!**

**Bill ** also reported on **Nextfest 2005** [http://www.technovelgy.com/ct/Science-Fiction-News.asp?NewsNum=409],
sponsored by **Wired Magazine** -- a neat expo held at ** Navy
Pier, June 24-26,
2005**.
It was a festival of new innovations of various types, including visits
by the **Cloned Cats**! ** Bill
**commented that teachers are able to see** IMAX** films at **
Navy Pier ** at
no charge a few weeks
after their opening. **Great, Bill!**

**04 October 2005:
Paul Fracaro (Joliet Central HS,
math/physics)
Paper Plate Fractions + Whiteboard Demonstration
Paul
** used paper plates to explore fractions. The thin (inexpensive)
paper plates can be folded into

**
Paul** then showed us some white boards the size of typical letter
paper
printed with a
rectangular grid, as well as ** X and Y axes**. With markers, it is
a convenient way for the students to graph. They are available from **
ETA Cuisinaire**: http://www.etacuisenaire.com/.
**Neato! Thanks, Paul.**

**18 October 2005: Walter McDonald (VA and CPS substitute teacher)
Hidden Magic Coin
Walter
** handed
out a sheet which contained directions for the hidden coin trick (from

Five coins were shaken and then scattered onto the
table. **Walter** looked at them, and asked a volunteer to
turn over
any two coins, and then cover up any one coin from view. **Walter t**hen
looked at the four coins in view, and told us that the hidden coin was "**heads**".
We looked. **He was right -- and we applauded Walter! Walter **repeated
this feat a second time, and he made us all curious to know how he did
this. **Marilynn Stone** figured it out. You need at
first
to count the ** original number of heads** and remember it.
Then count
the number of heads in the final configuration with one coin
covered. When you then flip
two coins, only the following three cases can occur:

Initial |
Final |
Change in #Heads |

HH |
TT |
- 2 |

HT |
TH |
0 |

TT |
HH |
+ 2 |

**07 February 2006:
Fred Schaal (Lane
Tech)
Prose and Poetry Day
Fred** talked about a class period between the first and second
semesters,
which he uses as a prose and poetry day. He read us a poem with a
whimsical tribute to the number three --

**04 April 2006:
Fred Schaal (Lane
Tech)
Looking for 6174
**
Years ago, when
calculators were new,