Finding Joy in the The Pythagorean Theorem

Read this article  on Daily Kos from Mar. 18 2012
Read this article on the Idaho Press Tribune website from Mar. 22, 2012
Read this article on Yahoo! from March 26, 2012

Pythagorean Theorem may be the best-known equation in mathematics. Its origins reach back to the beginnings of civilization, and today every student continues to study it. What most nonmathematicians don’t understand or appreciate is why this simply stated theorem has fascinated countless generations.

I first met Alfred S. Posamentier in 2001, when he was the Dean of Education at City College in New York. The energy level of his intellectual engagement with my work is part of an experience I will never forget.

In his entertaining and informative book, The Pythagorean Theorem: The Story of Its Power and Beauty , Posamentier, a veteran math educator, makes the importance of the Pythagorean theorem delightfully clear.

The Pythagorean theorem has attracted interest outside mathematics as a symbol of mathematical abstruseness, mystique, or intellectual power. Scores of references to it can be found in literature, plays, musicals, songs, stamps and cartoons. It can be introduced to students during the middle school years and becomes increasingly important during high school. It is not enough to merely state the algebraic formula for the Pythagorean Theorem.

According to University of Georgia graduate student Stephanie Morris:

Students need to see the geometric connections as well. The teaching and learning of the Pythagorean Theorem can be enriched and enhanced through the use of dot paper, geoboards, paper folding, and computer technology, as well as many other instructional materials.

The Pythagorean theorem or Pythagoras’ theorem is a relation in Euclidean geometry among the three sides of a right triangle (right-angled triangle). In terms of areas, it states:

In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle).
The theorem can be written as an equation relating the lengths of the sides a, b and c, often called the Pythagorean equation:  a^2 + b^2 = c^2\!\,  where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides.

The theorem is named after the Greek mathematician Pythagoras, who by tradition (only) is credited with its discovery and proof. The theorem has numerous proofs, possibly the most of any mathematical theorem. These are very diverse, including both geometric proofs and algebraic proofs, with some dating back
 thousands of years. The theorem can be generalized in various ways, including higher-dimensional spaces, to spaces that are not Euclidean, to objects that are not right triangles, and indeed, to objects that are not triangles at all, but n-dimensional solids.

Posamentier begins his book with a brief history of Pythagoras and the early use of his theorem by the ancient Egyptians, Babylonians, Indians, and Chinese, who used it intuitively long before Pythagoras’s name was attached to it.

He then shows the many ingenious ways in which the theorem has been proved visually using highly imaginative diagrams. Some of these go back to ancient mathematicians; others are comparatively recent proofs, including one by the twentieth president of the United States, James A. Garfield.

After demonstrating some curious applications of the theorem, Posamentier then explores the Pythagorean triples, pointing out the many hidden surprises of the three numbers that can represent the sides of the right triangle (e.g, 3, 4, 5 and 5, 12, 13). And many will truly amaze the reader.

Next, Posamentier turns to the “Pythagorean means” (the arithmetic, geometric, and harmonic means). By comparing their magnitudes in a variety of ways, he gives the reader a true appreciation for these mathematical concepts.

The final two chapters view the Pythagorean theorem from an artistic point of view–namely, how Pythagoras’s work manifests itself in music and how the Pythagorean theorem can influence fractals.
This lucid presentation and gift for conveying the significance of this key equation to those with little math background will inform, entertain, and inspire the reader, once again demonstrating the power and beauty of mathematics!

The Juice Box Bully: Empowering Kids to Stand Up For Others … plus other books and resources

“Every child deserves the opportunity to have a successful early learning experience, which is the foundation for becoming a lifelong learner,” writes Bob Sornson, PhD, author of The Juice Box Bully: Empowering Kids to Stand Up For Others. “My work with the Early Learning Foundation is all about helping parents and schools be more effective building the skills and behaviors that help kids be successful.”

Juice Box Bully is part of my effort to teach kids to stand up and speak up for themselves and others, build healthy classroom and school cultures, and make the world a better place! Only by building safe and connected cultures will we get rid of the bullying behaviors that harm children. Punishing bullies is not enough. We must teach our children a better way.

Have you ever seen a bully in action and done nothing about it? In Juice Box Bully, the kids at Pete’s new school get involved, instead of being bystanders. When Pete begins to behave badly, his classmates teach him about “The Promise”. Will Pete decide to shed his bullying habits and make “The Promise”?
Bullying behaviour comes in many shapes and sizes, and being bullied in childhood can have lifelong effects, according to author Dr Sabina Dosani in Bullying (52 Brilliant Ideas) to Keep Your Child safe and Happy. Recent UK research indicates that 1 in 4 primary school children and 1 in 10 secondary school children are bullied at least once a term. Bullying makes children lonely, unhappy and frightened. Tackling bullying and its side-effects can be a very delicate business, and each case is unique. Often parents don’t know anything is wrong until events overtake them, or they can’t think of the best way to help their child survive the experience. Being armed with the right information about bullying, recognising the symptoms to look out for, and knowing effective ways of breaking the cycle of abuse can be crucial factors in ensuring your child deals with their tormentors in a calm, positive and proactive way. In Bullying Dr Sabina Dosani has put together 52 brilliant ideas to enable parents to help their children survive being bullied and become stronger and more confident people as a result. Including advice on identifying different types of bully, clever tips for not reacting to taunts, self-defence ideas for increased confidence and methods to take the wind out of a bully’s sails, Bullying will help you to help your child find their own empowering way to take control of the situation and rid themselves of the fear that being bullied can bring. In Bullying Dr. Sabina Dosani has put together 52 brilliant ideas to enable parents to help their children survive being bullied and become stronger and more confident people as a result.

In Tease Monster: A Book About Teasing Vs. Bullying (Building Relationships), when One-of-a-Kind”is laughed at by Purple for being weird and Green playfully calls One a klutz after tripping on the stairs, is the Tease Monster to blame? With words of wisdom from Mom about the Tease Monster, One discovers that teasing is part of life. And not all teasing is the same. One learns that laughing at someone (mean teasing)has a hurtful bite, but laughing with someone is alright when it’s not done out of spite. This the newest book in the Building Relationships series by Julia Cook.
Author Julia Cook has a phenomenal gift of writing books for children! Her books teach important life skills and are written through a child’s view of the world,” wrote Erin E. Bowden, M.S., Elementary School Counselor.
As an Elementary School Counselor, I have incorporated many of Julia’s books into my curriculum and must say Tease Monster is a powerful relationship-building tool. Children will quickly learn the power of words, how people hear things in different ways, and the importance of laughing with someone not at them. Tease Monster should certainly find its way onto your bookshelf if you’re a parent, school counselor, teacher, social worker, psychologist, or pediatrician.
*****
Social issues: About one fourth of all students from elementary age through high school are the victims of bullying and harassment while on school property because of their race, ethnicity, gender, disability, religion or sexual orientation. Unfortunately the primary reason for bullying is due to something that may set themselves apart from the norm.
According to recent statistics, ‘gay and lesbian teens are two to three times as more likely to commit teen suicide than other youths.’ The Bullying Statistics website continues:
About 30 percent of all completed suicides have been related to sexual identity crisis. Students who also fall into the gay, bisexual, lesbian or transgendered identity groups report being five times as more likely to miss school because they feel unsafe after being bullied due to their sexual orientation. About 28 percent out of those groups feel forced to drop out of school altogether. Although more and more schools are working to crack down on problems with bullying, teens are still continuing to bully each other due to sexual orientation and other factors. 

In a … survey about gay bullying statistics, teens reported that the number two reason they are bullied is because of their actual or perceived sexual orientation or gender expression. The number one reason reported was because of appearance. Teens are at a pivotal point in their young adult lives when they are trying to find out who they are and who they are about to become as adults. This is why being teased, bullied and harassed is something that could negatively affect a person’s self-esteem and view of themselves for the rest of their life.
In fact, about 9 out of 10 LGBT teens have reported being bullied at school within the past year because of their sexual orientation, according to the most recent gay bullying statistics. Out of those numbers, almost half have reported being physically harassed followed by another quarter who reported actually being physically assaulted.  Unfortunately most teens who experience bullying of any kind are reluctant to share their experience or report the incident to a teacher or trusted adult. Even more unfortunate are the gay statistics that report a lack of response among those teachers and school administration. According to a recent statistic, out of the students that did report a harassment or bullying situation because of their sexuality, about one third of the school staff didn’t do anything to resolve the issue.

Here are some other numbers:

• About 42 percent of kids have been bullied while online with one in four being verbally attacked more than once.
• About 35 percent of kids have been threatened online.
• About 58 percent of kids and teens have reported that something mean has been said about them or to them online.
• Other bullying statistics show that about 77 percent of students have admitted to being the victim of one type of bullying or another.
• The American Justice Department bullying statistics show that one out of ever 4 kids will be bullied sometime throughout their adolescence.
• 46 percent of males followed by 26 percent of females have admitted to being victims in physical fights as reported in one report of bullying statistics by the Bureau of Justice School.

Other bullying facts:

• As these bullying statistics indicate, bullying is just getting worse in American schools. Many studies have shown that increasing domestic violence at home are leading to an increase in bullying online and at school.
• Researchers note that one way to help begin to lower these bullying statistics is to tell an adult when it is happening.
• According to the i-Safe American survey of students bullying statistics, about 58 percent of kids admit to never telling an adult when they’ve been the victim of a bullying attack.
• Another way to stay safe from bullies is to inform the school if the attacks are taking place on school property or have something to do with the school.
• Ignore messages sent by cyber bullies.

See also: Bullying 101: The Facts About Bullying, by E.M. Lemus. Are you tired of sitting idly by while bullying becomes more and more frequent in the lives of our young people? The statistics around bullying are staggering. This book will provide information that you need to stop being a bystander and become a key player in helping to end the bully cycle. Learn about the different types of bullying, how to identify it and the laws that currently surround this epidemic by becoming a part of the revolution that says enough is enough.
As an added bonus, the last chapter will give you specific information regarding your state’s laws and what actions they are taking, if any, to combat this issue.