In modern times, Christians all over the world celebrate the birth of Jesus Christ on Christmas, which falls on December 25 (tomorrow). This date may have been chosen to offset pagan celebrations of Saturnalia and Natalis Invicti. Some believe that celebrating the birth of Jesus, the “true light of the world” was set in synchronization with the December solstice because from that point onward, the days began to have more daylight in the northern hemisphere. Read more about the winter solstice on our previous blog post, Celebrating the Winter Solstice.
Christmas is often referred to as Yule, which may have derived from the Norse word jól, referring to the pre-Christian winter solstice festival. Yule is also known as Alban Arthan and was one of the “Lesser Sabbats” of the Wiccan year in a time when ancient believers celebrated the rebirth of the Sun god and days with more light. This took place annually around the time of the December solstice and lasted for 12 days. The Lesser Sabbats fall on the solstices and equinoxes.
The Feast of Juul was a pre-Christian festival observed in Scandinavia at the time of the December solstice. Fires were lit to symbolize the heat, light and life-giving properties of the returning sun. A Yule or Juul log was brought in and burned on the hearth in honor of the Scandinavian god Thor.
A piece of the log was kept as both a token of good luck and as kindling for the following year’s log. In England, Germany, France and other European countries, the Yule log was burned until nothing but ash remained. The ashes were then collected and either strewn on the fields as fertilizer every night until Twelfth Night or kept as a charm and/or as medicine.
This is a yule log or "buche de Noël" my daughter made for her high school French class to eat on their final day of school before the winter break. Here's the recipe.
Christians around the world celebrate the birth of Jesus, God's Son, on Christmas day. Many churches hold special services on Christmas Eve and Christmas Day as well as on the Sundays surrounding those dates. We sing Christmas carols, add more lights to our homes, decorate Christmas trees, hang wreaths and garland, light candles, exchange gifts, and retell the story of St. Nicholas (Santa Claus).
Nicholas was born during the third century in the village of Patara, in Greece (now on the southern coast of Turkey). His wealthy parents raised him to be a devout Christian, but died in an epidemic while Nicholas was still young. Obeying Jesus' words to "sell what you own and give the money to the poor," Nicholas used his whole inheritance to assist the needy, the sick, and the suffering. He dedicated his life to serving God and was made Bishop of Myra while still a young man. Bishop Nicholas became known throughout the land for his generosity to people in need and his love for children. St. Nicholas' feast day on December 6th is the main day for gift giving and celebrating in much of Europe.
In the Netherlands St. Nicholas is celebrated on the 5th, the eve of the day, by sharing candies (thrown in the door), chocolate initial letters, small gifts, and riddles. Dutch children leave carrots and hay in their shoes for the saint's horse, hoping St. Nicholas will exchange them for small gifts. Here in the United States, children sit on Santa Claus' lap and let him know what gifts they would most like to receive when he arrives with his sleigh and reindeer on Christmas. Many stories and legends have sprung up around St. Nicholas. Read more at StNicholasCenter.org.
However you celebrate this special season, we wish you and your family a Merry Christmas and a Joyeux Noel!
December 21 marks the winter solstice this year, the date thought by some to have been forecast as the end of this age by the Mayas who had remarkably accurate lunar and solar calendars.
However, in March 2010 a team of archaeologists from Boston University discovered a portion of a wall in Xultún, once a sprawling Maya city-state in northern Guatemala. Little did they know that the wall held never-before-seen paintings and an ancient Maya calendar. With the discovery of this new calendar, those speculations about today being predicted as the end of this age are largely put to rest. The markings on this wall suggest dates thousands of years in the future. Read about the discovery of the oldest known Maya Calendar on our previous blog post.
The solstices are the results of Earth's north-south axis being tilted 23.4 degrees relative to the ecliptic, the plane of our solar system. This tilt causes different amounts of sunlight to reach different regions of the planet during Earth's year-long orbit around the sun.
The December solstice occurs when the sun reaches its most southerly declination of -23.5 degrees. In other words, it is when the North Pole is tilted 23.5 degrees away from the sun. Depending on the Gregorian calendar, the December solstice occurs annually on a day between December 20 and December 23. On this date, all places above a latitude of 66.5 degrees north (Arctic Polar Circle) are now in darkness, while locations below a latitude of 66.5 degrees south (Antarctic Polar Circle) receive 24 hours of daylight.
Use this Sunrise and Sunset calculator to find the number of daylight hours during the December solstice in cities worldwide. Here's a gorgeous sunset over the Big Island of Hawaii at Kona. What time does the sun set today in your part of the world?
The sun is directly overhead on the Tropic of Capricorn in the southern hemisphere during the December solstice. It also marks the longest day of the year in terms of daylight hours for those living south of the equator. Those living or travelling south from the Antarctic Circle towards the South Pole will see the midnight sun during this time of the year.
For those of us in the northern hemisphere, the December solstice marks the day of the year with the least hours of daylight. People living or traveling north of the Arctic Circle towards the North Pole will not be able to see the sun during this time of the year.
December 20 and December 23 solstices occur less frequently than December 21 or December 22 solstices in the Gregorian calendar. The last December 23 solstice occurred in 1903 and will not occur again until the year 2303. A December 20 solstice has occurred very rarely, with the next one occurring in the year 2080.
In Excel Math, students learn about the Gregorian calendar. Here's a worksheet showing an easy way to remember which months of the year have 31 vs. 30 days (except February, which has 28 or 29 in a leap year). Click here to download this worksheet. Learn more about Excel Math on our website: www.excelmath.com.
Here in San Diego, thewintersolstice marks a popular time of year to hike Cowles Mountain (pronounced "coles"), part of Mission Trails Regional park. From the eastern horizon, at dawn on the days surrounding the solstice, a peak splits the rising sun so it appears for a short time to be divided in two. This part of the mountain was a Kumeyaay Winter Solstice observatory site. You could see hikers start up the mountain with their flashlights before dawn this morning to watch this amazing sight. Read more about hiking up Cowles Mountain on ourFebruary 2, 2012 blog post, Take a Hike.
New to Excel Math? Visit our website to learn more and take a look at sample lessons: www.excelmath.com.
Recent surveys indicate that teachers have lost faith in assessments and doubt that standardized tests in some cases accurately reflect students' achievement. W. James Popham, an assessment expert who serves on the Smarter
Balanced technical-advisory committee, said tests can provide meaningful
information only if teachers and students get more fine-grained
feedback than an overall score in writing or in math “concepts and
procedures.”
“It’s still too broad,” he said. “No one can ferret out what
students need help with. For Smarter Balanced to make a real
contribution, it has to make certain that its other two pieces, the
interim and formative assessments, are instructionally focused, so
educators can do something with the results.”
Excel Math assessments give teachers fine-grained feedback with tables in each Teacher Edition to show which concepts are addressed by each test question and where those concepts are first taught in each Excel Math grade level. So when a student or class misses a certain test problem, the teacher can go back to the lesson where that concept was first introduced and reteach or review it with the students. In addition, the spiraling process and spaced repetition of concepts means that mastery does not occur the first time a concept is introduced. Rather, concepts are reviewed regularly to help students achieve long-term mastery and gain confidence in their math skills.
With Excel Math assessments, teachers not are not only getting data for future instruction (and can see where more review needs to happen and which concepts need reinforcement or clarification), they are also measuring the results of past instruction. With these results, teachers can fine-tune their teaching methods and focus on specific concepts to best meet the needs of the students in each classroom.
Experts now say the transition to the Common Core State Standards could bring better high-school assessments that more accurately depict achievement. Under the common core, states are expected to adopt "longer, more thoughtful exams," said David Coleman, president of the College Board. However, officials say they are still working to determine what length is feasible. Read more at U.S. News & World Report/High School Notes.
Regular assessment is already built into Excel Math curriculum. There is a test after every five lessons. Tests are cumulative. A table in the Teacher Edition shows the test questions and in which lesson each concept was originally introduced. So if a student misses a problem on the test, the teacher can see at a glance which lesson or concept to review with the student.
Each week, new concepts are introduced to the students. At the same time, old concepts are reinforced and practiced as students prepare for assessment a week later. Students are not tested immediately after learning a concept. They have ample time to practice what they have learned—in class and at home, in calculation and in word problems. Excel Math also includes quarterly tests and end-of-year tests.
The Excel Math Score Distribution Chart combined with the test tables mentioned above are included in each Excel Math Teacher Edition. These assessment tools can help the teacher see at a glance where the class as a whole is having trouble grasping concepts and where students need individual help to achieve mastery. Download your free chart here: www.excelmath.com/usingexcel/record_keeping.html. Instructions are included.
Another assessment tool created by Excel Math, are the easy-to-use placement tests, available in English and Spanish.
Use these Placement Tests to determine where a student should start in the Excel Math program. Each Placement Test file contains six tests that evaluate a student's preparedness for Excel Math.
The tests are labeled A - F, which correspond to first through sixth grade. The test questions review the prior year's work. Successful completion of each test shows readiness for that grade level. Instructions for using the tests are included.
Since today's date is 12-12-12 and Christmas is fast approaching, I was reminded of the song, The 12 Days of Christmas. Our family has a brightly-illustrated picture book with the song lyrics that we enjoy reading over again this time of year.
The 12 Days of Christmas is a favorite Christmas carol. There are versions with motions that make it a fun counting song to do with children. We start singing it after Thanksgiving and continue through to Christmas, but the twelve day of Christmas actually refers to the 12 days after Christmas. These are the 12 days between Christmas (the day celebrated as Jesus' birth) and Epiphany (when the wise men arrived at the home of the little child Jesus).
The poem mentions gifts given on each day over a twelve-day period. The letters in parentheses indicate the abbreviation we'll use for each gift:
On the first day of Christmas, my true love gave to me a partridge in a pear tree (p).
On the second day of Christmas, my true love gave to me 2 turtledoves (t) and a partridge in a pear tree.
On the third day of Christmas, my true love gave to me 3 french hens (f), 2 turtledoves and a partridge in a pear tree.
On the fourth day of Christmas, my true love gave to me 4 calling birds (c), 3 french hens, 2 turtledoves and a partridge in a pear tree.
On the fifth day of Christmas, my true love gave to me 5 gold rings (gr, 4 calling birds, 3 french hens, 2 turtledoves and a partridge in a pear tree.
On the sixth day of Christmas, my true love gave to me 6 geese a-laying (gl), 5 gold rings, 4 calling birds, 3 french hens, 2 turtledoves and a partridge in a pear tree.
On the seventh day of Christmas, my true love gave to me 7 swans a swimming, 6 geese a-laying, 5 gold rings, 4 calling birds, 3 french hens, 2 turtledoves and a partridge in a pear tree.
On the eighth day of Christmas, my true love gave to me 8 maids a milking (m), 7 swans a swimming, 5 gold rings, 4 calling birds, 3 french hens, 2 turtledoves and a partridge in a pear tree.
Now that you get the idea, we can skip ahead to the gifts given on the twelfth day:
On the twelfth day of Christmas, my true love gave to me 12 drummers drumming (dd), 11 pipers piping (pp), 10 lords a leaping (ll), 9 ladies dancing (ld), 8 maids a milking (m), 7 swans a swimming (s), 5 gold rings (gr), 4 calling birds (c), 3 french hens (f), 2 turtledoves (t) and a partridge in a pear tree (p).
The first day the true love gives a gift of a partridge in a pear tree. The second day's gifts include another partridge and tree plus two turtledoves. The third day brings yet another partridge and pear tree, two turtledoves, and three french hens. By the end of the 12 days, the poor lover has received 12 partridges in pear trees, 22 turtledoves, 30 french hens, etc. It sounds like a zoo (literally). Can you figure out the total of each item given by the end of the 12-day period?
Each day, a new gift is given, plus all the gifts from the previous day are given once again. We can put together a mathematical equation to represent the total gifts, using the abbreviations shown above for each gift: 12p + 11(2)t + 10(3)f + 9(4)c + 8(5)gr + 7(6)gl + 6(7)s + 5(8)m + 4(9)ll + 3(10)ld + 2(11)pp + (12)dd
Excel Math gives students a strong foundation of elementary math so they can build solid math skills and develop a love for math. Learn more about how it can work for your students at www.excelmath.com.
Now back to our twelve days of Christmas math. The first number indicates the days the gift was given (12 days for the partridge, 11 for the turtledoves, etc.) and the number in parentheses indicates the quantity of that gift given each day (1 partridge, 2 turtledoves, 3 french hens, etc.) Over the 12 days of Christmas, the true love would give:
12 partridges and pear trees 22 turtledoves 30 french hens 36 calling birds 40 gold rings 42 geese laying eggs 42 swans 40 maids and cows 36 lords 30 ladies dancing 22 pipers with pipes 12 drummers with drums
Notice the pattern. There are 12 each of partridges and drummers, 22 each of turtledoves and pipers. So the easiest way to count the gifts is to add the first half of the list and then multiply it by two:
12 + 22 + 30 + 36 + 40 + 42 = 182
182 x 2 = 364
There are a total of 364 gifts given in this song (not counting the additional items such as pear trees, drums, pipes, cows, etc.)
If you'd like to hear a new slant on this old favorite, you might enjoy "The 12 Days After Christmas" written by Frederick Silver in 1968. It begins, "The twelve days after Christmas, my true love and I had a fight. And so I chopped the pear tree down and burned it just for spite!" Listen to the Desert Chorale perform the song:
Whichever version you prefer, take some time to enjoy this once-in-a-lifetime day of 12-12-12. Sit back, relax, and read one of the following books about twelve:
The Twelve Days of Christmas The Gershwins and Me: A Personal History in Twelve Songs Twelve Angry Men Polar Bears Past Bedtime (Magic Tree House #12) Twelfth Night The Labours of Hercules: Twelve Hercule Poirot Mysteries The Twelve Tribes of Hattie
Hanukkah or the "Festival of Lights," also known as Chanukah, began on Saturday, December 8 and continues through December 16 this year. The eight-day festival commemorates the rededication of the Second Temple in Jerusalem during the second century B.C.
According to legend, Jews had risen up against their Greek-Syrian oppressors in the Maccabean Revolt. Afterward, Judah led his followers to cleanse the Second Temple, rebuild its altar and light its menorah (pictured above, under the Hebrew letters). The gold candelabrum had seven branches representing knowledge and creation. Its candles were intended to be kept burning every night. Today Hanukkah menorahs have 8
branches plus one servant light. The servant candle lights the
other 8 candles. The eight days and nine candles of Hanukkah give us an opportunity to use counting along with addition and subtraction in a fun multi-cultural lesson.
For over 35 years, Excel Math has provided teachers (Kindergarten through Grade 6) with proven lessons that help students excel at mathematics. Learn more and watch Excel Math in action at www.excelmath.com. Read some Excel Math success stories at excelmath.com/about/successes.html. Now back to Hanukkah.
Dreidel (wooden toy)
Hanukkah means "dedication" in Hebrew. It begins on the 25th of Kislev on the Hebrew calendar and usually occurs in November or December. According to the Talmud, one of Judaism’s most revered texts, Judah Maccabee and the others who took part in the rededication of the Second Temple witnessed a miracle. Even though there was only enough olive oil to keep the menorah’s candles burning for a single day, the flames continued for eight nights, giving the people time to find a fresh supply. This amazing event inspired them to proclaim a yearly eight-day festival.
Each day during Hanukkah, children receive presents, and a candle on the Menorah is lit. Traditional Hanukkah foods are fried in oil. Potato pancakes (known as latkes) and jam-filled donuts (sufganiyot) are popular in many Jewish households. Other Hanukkah customs include playing with four-sided spinning tops called dreidels, exchanging gifts and singing Hanukkah songs.
If your students celebrate Christmas or Kwanzaa, download the blank candles worksheet and have them draw a wreath or a Kinara (Kwanzaa candle holder) beneath the top candle. Students who don't celebrate holidays can simply draw a fireplace mantel or table (or candle holders of their choice) beneath the candles. Have your students print their names in the box at the top and cut out the candles on the bottom of the page. For Kwanzaa, let students color the candles red, black and green before cutting them out. You will only need 7 candles for Kwanzaa. Point out that there is already one candle on the Menorah (or Kinara or wreath):
Then ask, "If you add two more candles, how many will you have on the Menorah?" (3) Let your students add two more candles to the Menorah. Continue until all the candles are glued to the Menorah. Then show your students how to slide the paper strip through the slits to cover the candle flames. Let them slide the strip the other way to uncover some of the flames. Ask them how many candles are on the Menorah. (9) Ask them how many candles will be lit if they cover 5 flames. (4) Have them cover 5 flames to check. Continue in this way using addition and subtraction.
You can also use this worksheet to help your students recognize left and right. Ask each student to put a finger on a candle to the right of the center candle. Do the same with a candle to the left of the center candle (the tall one is the center). Then have them use the paper strip to cover two flames on the right of the center candle. Let your students make suggestions for additional math problems or story problems about the candles.
How do you help your students relate math to customs and traditions? How do you help them see math in everyday life? Leave a comment below with your teaching suggestions.
Louisiana state education officials say new textbooks do not match up with the Common Core State Standards. (Read the full blog post here.) As a result, they are thinking about rejecting proposed textbooks from publishers submitted for the most recent adoption cycle. Among the textbooks specifically cited for insufficient common core alignment were math books geared for students in K-2. Here's where Excel Math can help fill the gap.
One of the great benefits of Excel Math is that it is continually enhanced. We print Excel Math throughout the year so teachers always have the latest lesson sheets, unlike textbooks, which are static until you buy a new book. Over the years, Excel Math has been updated with new teaching tools, but the proven math teaching techniques, in-depth spiraling, the CheckAnswer feedback loop, and other time-tested benefits of this program have not changed.
Excel Math Gives Your Students More
Excel Math uses proven instruction strategies in the Daily Lessons that include:
Daily Instruction of specific concepts
Basic Fact Practice (including online timed practice)
Guided Practice
Critical thinking activities
Interactive manipulatives
Challenging story problems
Stretches (brainteasers)
Projectable lessons (interactive way to engage the class using an electronic format)
Preprinted lesson sheets (students don't need to waste time copying problems)
Formal assessment is incorporated into the Excel Math program and is organized into weekly, quarterly, and year-end tests. Our unique spiraling strategy ensures that students are not tested on concepts that have just been taught. Instead, concepts are introduced and then reviewed and practiced before they are included on tests. In addition, concepts spiral back into the lessons on a regular basis so students are able to retain them for the long term.
Built into this structure are powerful self-checking feedback loops that keep students on task and engaged. Students can work independently, catching mistakes as they occur and correcting them before moving on. They are given the opportunity to build higher-level thinking skills through hands-on activities, Create-A-Problem stories, stretches, and more. These varied activities extend opportunities for differentiated instruction and let teachers merge math with literacy. Mastery is achieved over time by exposing students to concepts and then using spaced repetition to keep those concepts revolving back to the forefront, helping students gain complete understanding. Excel Math Lessons Correlate with Common Core
We were pleased, but not surprised at how well our Excel Math materials correlate to the Common Core and new state standards. Take a look for yourself. View and download the correlations by grade level. Simply click on the grade level button to see the correlation between that set of educational standards and Excel Math. With Excel Math, students learn higher-order thinking skills beyond what is required of the Common Core. Our correlations allow you to focus on the standards or go even further, with additional concepts we provide.
Within each grade level, we correlate the Excel Math Lessons, Stretches, Activities and Exercises to each Common Core standard. At the bottom of each Common Core (CCS) correlation, we list additional concepts covered by Excel Math. You can use these additional concepts to provide accelerated learning for your students who are ready for more.
This year we added some Common Core activities to our Excel Math lessons to expand on those concepts already included in the Excel Math Teacher Edition, Student Sheets and Projectable CDs. Click here for our Tools for Teachers and these new lesson pages (available for viewing and download as PDF files).
Does Your Curriculum Need a Boost?
If you're not already using Excel Math, why not? Excel Math is proven as a core curriculum and also helps increase student test scores in a supplement position. Even though most textbooks don't meet or correlate to the new Common Core Standards, Excel Math correlates and even goes beyond CCS requirements. Projectable lessons let your flip your class or use an electronic whiteboard (or just a screen) to engage the entire class in learning.
Why sink large amounts of money into textbooks or programs that don't achieve the necessary results but can become a huge financial drain? Excel Math is extremely affordable and it really works. Students gain confidence as they achieve mastery and really begin to understand and enjoy mathematics. Excel Math Professional Development is also available for a nominal fee. (Read more about hosting a PD session at your school.)
Excel Math Really Works
Excel Math works equally well for remediation and resource students as for GATE and more advanced students. These unique lessons for K-6 can help teachers transition to the Common Core and new state standards, building student confidence in the near term and advancing students with proven test results and reasonably priced materials that really work in the math classroom. Excel Math solves some of the problems faced by educators and provides an excellent program supporting the needs of students and teachers alike.
Excel Math is a rigorous curriculum that results in skill mastery, long-term retention of concepts and advanced understanding. Students progress significantly in math skills and aptitude, preparing them for higher level math courses in middle school, high school and college and helping them succeed in the workforce.
The dictionary defines amicable as "having a spirit of friendliness." It comes from the Latin word amicus, meaning "friend."
Mathematicians say that two positive integers are amicable or friendly if each of them is equal to the sum of all the proper divisors of the other, including 1. Proper or aliquot divisors are all the positive divisors of a number, excluding the number itself. The numbers 220 and 284 are called amicable.
Take a look at the proper divisors of these two numbers:
220 284
Divisors Divisors
1 (220) 1 (284)
2 110 2 142
4 55 4 71
5 40
10 22
11 20
When we add together the proper divisors of 220 (1, 2, 4, 5, 10, … , 110) we get the other number (284). Likewise, the sum of the divisors of 284 (1, 2, 4, 71, 142) is equal to the first number (220). Each amicable number has the power to generate another, thus symbolizing mutual harmony, perfect friendship and love.
The amicable numbers were thought by mystics to possess magical powers. Astrologers used these numbers for preparing talismans and horoscopes. They believed that amicable numbers had the power to create special ties between individuals. They also saw these number pairs as symbols of friendship.
Antreas P. Hatzipolakis quotesMathematical Magic Show by Martin Gardener (Viking. London 1984):
The Pythagorean brotherhood regarded 220 and 284 as symbols of
friendship. Biblical commentators spotted 220 in Genesis 32:14 as
the number of goats given Esau by Jacob. A wise choice, the
commentators said, because 220, being one of the amicable pair,
expressed Jacob's great love for Esau. During the Middle Ages this
pair of numbers played a role in horoscope casting, and talismans
inscribed with 220 and 284 were believed to promote love. (p. 167)
For over a thousand years, only this first pair (220 and 284) was known. It was attributed to the philosopher Iamblichus (c.250-330), who credited Pythagoras. Eventually, the mystical quality of amicable numbers caused them to be studied more carefully by number theorists.
The next amicable pair (17,296 and 18,416) was probably discovered in the 9th century by Arab mathematician Thabit ibn Qurra. In 1636, Pierre de Fermat rediscovered this pair. Later, René Descartes discovered a third pair: 9,363,584 and 9,437,056.
Then, in 1747, Leonhard Euler published a short paper [E100] mentioning the technique that Descartes and Fermat had used, and listing 30 amicable pairs, including the three already known, and including one “pair” that was not actually amicable.
A total of 59 pairs were found by Euler, among them the pair (6,232 ; 6,368) and the pair (10,744; 10,856). B.N.I. Paganini, a sixteen-year-old Italian youth, startled mathematicians in 1866 when he found a smaller, overlooked pair of amicable numbers: 1184 and 1210. In 1946 E. Escott wrote a long paper dedicated to the amicable numbers, offering an inventory of 390 amicable pairs. By 1999 there were over 450,000 pairs of amicable numbers and by 2007 there were over 11 million pairs. Here are just a few:
Pythagoras when asked, "What is a friend?" replied that a friend is one "who is the other I" such as 220 and 284. The numbers 220 and 284 form the smallest pair of amicable numbers (also known as friendly numbers) known to Pythagoras.
Amicable numbers serve no practical purpose, but professionals and amateurs alike have enjoyed uncovering them and exploring their properties. This chart lists all amicable pairs under 1014.
Here at Excel Math, we have an atmosphere of friendliness and a very amicable workplace. When you call us (1-866-866-7026) between the hours of 8:00 a.m. and 4:30 p.m. West Coast time, a friendly person will answer the phone, never a machine.
New to Excel Math? Learn more on our website www.excelmath.com. Watch this video to see how Excel Math works:
The Feast of Juul was a pre-Christian festival observed in Scandinavia at the time of the December solstice. Fires were lit to symbolize the heat, light and life-giving properties of the returning sun. A Yule or Juul log was brought in and burned on the hearth in honor of the Scandinavian god Thor.
This is a yule log or "buche de Noël" my daughter made for her high school French class to eat on their final day of school before the winter break. Here's the recipe. 
