## Wednesday, November 28, 2012

### Mapping Out the Compass Rose

The compass rose has appeared on charts and maps since the 1300's when the portolan charts first made their appearance. The term "rose" comes from the figure's compass points resembling the petals of a rose. Can you see the resemblance?

Originally, the compass rose was used to indicate the directions of the winds (and it was then known as a wind rose), but the 32 points of the compass rose come from the directions of the eight major winds, the eight half-winds and the sixteen quarter-winds. Each point is indicated by degrees, with 0º for North, 90º for East, 180º for South and 270º for West.

The 32 points are therefore simple bisections of the directions of the four winds (but the Chinese divided the compass into 12 major directions based on the signs of the Zodiac). North is usually at the top, and each direction is abbreviated using its first letter (N for North, E for East, etc.)

The compass rose above is divided into subsections so NE is northeast, NNE is north-northeast, NbE is north by northeast, etc. One of the first things western apprentice seamen had to know were the names of the points.

Here are some simpler but very colorful versions of the compass rose:

You may also like these articles:

## Monday, November 26, 2012

### Fibonacci's Spiral and Excel Math

Today we'll take a look at the Fibonacci spiral and rectangle. Fibonacci, or more correctly Leonardo da Pisa (also called Leonardo Bigollo), was born in Pisa, Italy in 1175. He travelled widely in Barbary (Algeria).

In 1200 he returned to Pisa and used the knowledge he had gained on his travels to write Liber abaci.

Fibonacci: 810 Years of Mathematical Magic and
Celebrating Fibonacci Day

The famous Fibonacci sequence is as follows:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, ...

where the first two numbers of the series are 1 and 1 and each number afterward is defined as the sum of the two previous terms, Fn = Fn - 2 + Fn - 1. (Though in Fibonacci's sequence the first number was 1 and the second number was two, the first one was assumed.)
We can make a picture showing the Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, 21... if we start with two small squares of size 1 next to each other. On top of both of these draw a square of size 2 (= 1 + 1).

We can now draw a new square—touching both a unit square and the latest square of side 2—so having sides 3 units long; and then another touching both the 2-square and the 3-square (which has sides of 5 units). We can continue adding squares around the picture, each new square having a side which is as long as the sum of the latest two square's sides. This set of rectangles whose sides are two successive Fibonacci numbers in length and which are made up of squares with sides which are Fibonacci numbers, we will call the Fibonacci Rectangles.

Here is a spiral drawn in the squares, a quarter of a circle in each square. The spiral is is made up of fragments which are parts of circles and does not go on getting smaller and smaller) but it is a good approximation to a kind of spiral that does appear often in nature. Such spirals are seen in the shape of shells of snails and sea shells and in the arrangement of seeds on flowering plants. The spiral-in-the-squares makes a line from the centre of the spiral increase by a factor of the golden number in each square. So points on the spiral are 1.618 times as far from the centre after a quarter-turn. In a whole turn the points on a radius out from the centre are 1.6184 = 6.854 times further out than when the curve last crossed the same radial line.

This golden ratio 1·618034 is also called the golden section or the golden mean or just the golden number. It is often represented by a Greek letter Phi. The closely related value which we write as phi with a small "p" is just the decimal part of Phi, namely 0·618034.

This is an image of a Nautilus sea shell. You can see the spiral curve of the shell. The internal chambers that the animal using it adds on as it grows. The chambers provide buoyancy in the water. Now draw an imaginary line from the center of the shell out in any direction and find two places where the shell crosses it so that the shell spiral has gone round just once between them. The outer crossing point will be about 1.6 times as far from the centre as the next inner point on the line where the shell crosses it. This shows that the shell has grown by a factor of the golden ratio in one turn.

This mathematical spiral can also be seen in music. According to a recent article in the Wall Street Journal, Debussy worked in spirals. "Musicologists have detected evidence of spiral-like mathematical structures in this music, related to the Fibonacci sequence and the Golden Mean."

Read more at Ron Knott's web site: Fibonacci Numbers and the Golden Section.

Excel Math uses a spiraling strategy similar to that found in the nautilus shell. This unique spiraling strategy introduces new math concepts to students while reviewing previously-taught concepts. It gives students the opportunity to master the old through spaced repetition, while being challenged with the new. Once a concept is introduced, it literally stays in front of the students for the rest of the school year. The spiraling strategy of repeating concepts at regular intervals throughout the curriculum is an integral part of Excel Math.

This spiraling strategy is a sophisticated process of introducing new concepts, reinforcing the concepts regularly, and then assessing the concepts. It leads to mastery and long-term competency for each student. In other words, the spiraling strategy helps move new concepts into the child's long-term memory and keep them there. Read more on our previous blog post: Spiraling Into Control.

Excel Math curriculum continually brings in new topics while refreshing math concepts the students have learned before. Students aren't tested on a subject until they've had multiple chances to succeed in Guided Practice and Homework. Here's a visual road map explaining this spiraling strategy:

Constant review and spaced repetition of the math concepts ensure students remember those concepts long after they are first introduced. Lessons build upon previous learning and often blend math and literacy, producing well-rounded and confident students. To learn more, visit the Excel Math website.

## Wednesday, November 21, 2012

### Celebrating Fibonnacci Day

Fibonacci, or more correctly Leonardo da Pisa (also called Leonardo Bigollo), was born in Pisa, Italy in 1175. He travelled widely in Barbary (Algeria) and was later sent on business trips to Egypt, Syria, Greece, Sicily and Provence.

In 1200 he returned to Pisa and used the knowledge he had gained on his travels to write Liber abaci in which he introduced the Latin-speaking world to the decimal number system.

The Fibonacci Series is a sequence of numbers first created by Leonardo Fibonacci in 1202, introduced in Liber abaci and later named the Fibonacci sequence in his honor. Read more about Fibonacci, Roman numerals and the decimal system on our previous blog post: Fibonacci—810 Years of Mathematical Magic.

 Leonardo da Pisa (Fibonacci)
Mathematicians celebrate Fibonacci Day on November 23 or 11/23, taking the date from the first four numbers in the Fibonacci sequence shown below. The Fibonacci series begins with 0 or 1. After that, use the simple rule: Add the last two numbers to get the next number in the sequence. See the series in action at thinkquest.org.

This is readily translatable into the following set of equations [3, p4]
• 1 = 1²
• 1 + 3 = 2²
• 1 + 3 + 5 = 3²
• 1 + 3 + 5 + 7 = 4²
• and suggests the general formula:
• 1 + 3 + ... + (2n-1) = n²
From this statement the famous Fibonacci numbers can be derived.
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, ...

where the first two numbers of the series are 1 and 1 and each number afterward is defined as the sum of the two previous terms, Fn = Fn - 2 + Fn - 1. (Though in Fibonacci's sequence the first number was 1 and the second number was two, the first one was assumed.) Read more at plus.maths.org.

If it seems too time-consuming to have to calculate the Fibonacci numbers yourself, use this Fibonacci Calculator. It calculates thousands of Fibonacci numbers exactly and millions upon millions to the first few digits.

 Click this image to view correlations
With Excel Math lessons, students learn how to calculate numbers, do mental math, check their own work, and become confident and successful at mathematics. Excel Math is a proven method that gives students a solid foundation of elementary math. Our math lessons help students actively participate and interact with the teacher, with Projectable lessons, with each other, and with manipulatives. We help students develop higher-order thinking skills as they use math curriculum designed to keep their minds engaged in the learning process.

Excel Math Student Lesson Sheets are much more than just math worksheets. Using strategically placed spaced repetition, Excel Math gives you a proven approach to teach math concepts for long-term retention, with powerful features and advantages, including our proprietary Spiraling Strategy. Much like the Fibonacci spiral (which we'll visit in our blog post next week), concepts are reintroduced on a regular basis (with our unique method of spaced repetition) so they become a part of the student's longterm memory. Read the glowing reports from teachers, parents and principals around the country.

This year, Fibonacci Day falls on the day after Thanksgiving. If you're celebrating, leave a comment in the box below to tell us what you'll be doing. From everyone at Excel Math, Have a wonderful Thanksgiving and Happy Fibonacci Day!

## Monday, November 19, 2012

### Aged to Perfection: Numbers

Really? Perfect numbers? How can one number be more or less perfect than another?

Perfect numbers were thought to have important numerological properties by the ancients, and were extensively studied by the Greeks, including Euclid. The ancient Greeks held a great reverence for the mysticism of numbers.

St. Augustine argued that "Six is a number perfect in itself, and not because God created all things in six days; rather the inverse is true; God created all things in six days because the number is perfect.  And it would remain perfect even if the work of the six days did not exist."  (The City of God, Book 11, Chapter 30).

A perfect number is a number where the sum of the number's proper divisors adds up to the number itself. For example:

The first and smallest perfect number is 6. The proper divisors of 6 are 1, 2 and 3. 1 + 2 + 3 = 6 so 6 is a perfect number. (Six is also a factor of 6, but 6 is considered an improper divisor.)

The next perfect number is 28, as its proper divisors are 1, 2, 4, 7, and 14. 1 + 2 + 4 + 7 + 14 = 28 so 28 is also a perfect number. Again, the sum of those integers is 28.
496 =
1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248

8128 is the next perfect number.

Euclid’s theorem: IF (2^k-1) is a prime number THEN 2^{k-1}\times (2^k-1) is a perfect number.

In Euclid's day, these four perfects were all that were known. Incredibly, he saw that for each of these perfects, the formula 2n - 1(2n - 1) produces perfects for the prime n values of 2, 3, 5, 7. Example:

For n=2, 21(22 - 1) = 2(3) = 6.

Notice that 2n-1 is also prime. Euclid brilliantly proved that 2n - 1 (2n - 1) yields an even perfect number when 2n - 1 is a prime number. Note also that not all primes work for n to yield a perfect number.

In the 1700's, Euler expanded on Euclid's formula and proved that it will yield all of the even perfect numbers. Read more at http://trottermath.net/perfect-numbers-et-al/

The first few perfect numbers  are summarized in the following table together with their corresponding indices :
 1 2 6 2 3 28 3 5 496 4 7 8128 5 13 33550336 6 17 8589869056 7 19 137438691328 8 31 2305843008139952128

Just as we have a scarcity of perfect people, there are not many perfect numbers, which makes them special and worthy of our attention. The third perfect number does not appear until nearly 500; the fourth one is over 8,000; and the fifth one is over 33 million! Read more at http://mathworld.wolfram.com/PerfectNumber.html

In Excel Math, we recognize that perfect numbers are scarce and perfect students perhaps even more so. But all students can develop a love of mathematics and can be given the tools to succeed at math.

Visit our web store to order Student Lesson Sheets (155 lesson sheets plus tests) and a Teacher Edition set of 155 Lessons with brainteasers, teaching suggestions, the answer key, activities, and reproducible manipulatives for each grade level. Excel Math is available for Kindergarten through Grade 6.

Since an entire year of Excel Math curriculum is as low as \$11.00 per student, many schools use it as their core curriculum. Other schools find it's a powerful supplement to their adopted curriculum. In both situations, students gain confidence in mathematics as test scores soar.

New to Excel Math? Visit our website at www.excelmath.com for elementary math lessons that really work!

## Tuesday, November 13, 2012

### Targeted Professional Development, Part II

At Excel Math, we focus on both the profession of teaching and "the professional" (the teacher) when creating professional development seminars. These information-packed sessions are intended for administrators and teachers who want to learn more about using Excel Math and discover best practices for implementing elementary math lessons in the classroom.
Read Part I of this series.

Reports such as the 2009 “Professional Learning in the Learning Profession,” by the National Staff Development Council and the School Redesign Network at Stanford University affirm a direct link between highly effective, sustained professional development and differentiated approaches to teacher training, collegial collaboration and risk taking. Read more at http://www.districtadministration.com/article/sustainable-professional-development

Choose between two Excel Math Professional Development seminars: one for users new to Excel Math (Initial P.D.) and the other (Next Step P.D.) for schools that have used Excel Math for a period of time.

For the Initial P.D.: The Excel Math Professional Development gives teachers instructional strategies (best practices) for effective direct instruction and tips to take full advantage of the three components of Excel Math. Plus, you will learn how to utilize the regular assessment in Excel Math for maximum instruction. Because Excel Math emphasizes Critical Thinking instead of fill-in-the-blank answers, the Excel Math lessons are an outstanding bridge to the new requirements of the Common Core Standards. Additionally, this seminar includes how to effectively blend Excel Math with an adopted core curriculum for maximum instruction in cases where Excel Math is used as a supplement.

For the Next Step P.D.: Excel Math—Your Bridge to the Common Core explores in more depth the specifics of Excel Math as an outstanding bridge to the new requirements of the Common Core Standards (CCS) and reviews Instructional Strategies outlined in the Initial Professional Development. View CCS correlations for Kindergarten through Grade 6 here.

Texas P.D.: We also offer a separate Texas Professional Development seminar. Because Excel Math emphasizes rigor as well as in-depth learning (with our unique spiraling process) and regular assessment, Excel Math lessons are an outstanding bridge to the new requirements of the STAAR Readiness Standards. For Texas Professional Development and Excel Math correlations to TEKS click here.

Contact Bob Parrish via email (bob@excelmath.com)  or by phone (1-866-866-7026) for more information and to schedule your seminar(s). Watch Bob in action here.

Take a look at the glowing report one principal sent us after an Excel Math P.D. seminar:
“Robbinsville Elementary School has used Excel math in previous years with great results (4th grade 96% of students passing state tests) at some grade levels, but this year we adopted Excel Math school wide (K-6). To ensure effective math instruction and consistency across grade levels we requested a one day training that involved a school wide presentation and breakout sessions for each grade level for follow-up questions or concerns.

We are extremely fortunate to have had Mr. Bob Parrish to provide this training. Mr. Parrish was very informative, approachable, and well prepared using PowerPoint presentation of Excel Math. Mr. Parrish is an extremely competent and engaging presenter and extremely knowledgeable of the needs of math instructors and their students. Thank you, Mr. Parrish and AnsMar Publishing for enlightening and energizing the staff at Robbinsville Elementary School. ”
— William Laughter, Principal
Robbinsville Elementary School

Available for No Charge — Watch the Excel Math Overview DVD to prepare for your P.D. session. Our DVD contains an overview of Excel Math plus training and instructional strategies.

Part one of the DVD introduces you to Excel Math. Watch the intro video here [7:00 minutes]:

Part two guides you through the implementation of Excel Math in the classroom.

Part three includes instructional strategies and best practices. This segment helps you get the most out of the Excel Math lessons.

The contents of the DVD are available on the Excel Math website. The videos describe how to teach Excel Math with actual classroom examples. You may want to view them at the beginning of the year for an overview of the program, and review again in 4-6 weeks for a refresher course.

The Excel Math DVD is available as a part of our free Sample Packet and is also available separately at no charge. Email us for a DVD or Sample Packet, and we'll send them to you immediately.

If you've attended P.D. seminars in the past, let us know what you thought. Leave a comment in the box below. We love hearing your feedback and suggestions!