Additional Math Pages & Resources

Wednesday, March 28, 2012

What's Your Angle?

Everyone knows that an angle is the space between two lines that intersect.

Bamburgh Castle
An Angle can also refer to a member of the Germanic people that invaded England along with the Saxons and Jutes in the fifth century a.d. and merged with them to form the Anglo-Saxon peoples. Here's a photo of Bamburgh Castle, an imposing sandstone castle located on the coast at Bamburgh in Northumberland, England. In 1464, during the Wars of the Roses, Bamburgh became the first castle to succumb to cannon fire, suffering heavy damage. The ruins of the castle were restored and extended in the 18th and 19th centuries, leaving little of the earlier fortifications. Read more.

In fishing, angle means "to fish with a hook," from the Old English angel (n.) "angle, hook, fishhook," related to anga "hook," and ang- "to bend" The figurative sense of this word is first recorded from the 1580s.

In Excel Math, students learn about mathematical angles of all sizes. Here's a lesson sheet from Sixth Grade that explains the difference between adjacent, exterior, interior, and corresponding angles.
Excel Math Sixth Grade Lesson Sheet

An angle is made up of two rays or line segments that intersect or have the same endpoint. There are many geometric definitions of an angle. Here are a few types of angles:

Acute Angle = an angle that measures less than 90º
Complementary Angles = two angles whose sum equals 90º

Obtuse Angle = an angle that measures more than 90º and less than 180º
  Right Angle = a 90º angle (perpendicular lines form right angles where they cross)

        Straight Angle = a 180º angle

Supplementary Angles = two angles whose sum equals 180º

Exterior Angle = an angle on the outside of two parallel lines that are intersected by another line (r and q are exterior angles on the diagram below)
Interior Angle = an angle on the inside of two parallel lines that are intersected by another line (m and n are interior angles on the diagram)
We see angles of every shape and size in the course of our daily lives. Read about tri-angles in our previous post. It's easy to find angles at school,

at home,
at work,

and wherever we go.

See how many angles you and your students can find in each of these pictures. You can even make it a contest to see who can find the most right angles, obtuse angles, and acute angles. What's your angle?

You may also enjoy these articles: 

Monday, March 26, 2012

Malapropisms to the Max

In his hysterically funny comedy, The Rivals, Richard Brinsley Sheridan brings to life a character named "Mrs. Malaprop," who mangles the English language with panache. Through all five acts, Mrs. Malaprop gets completely mixed up in her use of words and language. Instead of saying, "contiguous countries," she describes them as "contagious countries."

At another point in the play she  gushes, "He is the very pineapple of politeness!" when she means pinnacle. She later exclaims, "Why, murder's the matter! slaughter's the matter! killing's the matter! — but he can tell you the perpendiculars" (instead of particulars), and the laughs continue throughout the play. The word "malapropism" was coined in 1949 from this comedic character who first appeared in 1775. Read more malapropisms.

My parents, with completely straight faces, would purposely include malapropisms in their daily conversation. For example, when I said I had had a feeling things would turn out in the way they actually did, my mom's response would be "you must be psychotic." So I suppose it's fitting that my husband has managed to make up a few malapropisms of his own. When he wants to spice up his meal, he'll ask for more "canine pepper" instead of cayenne. And when we're preparing condiments for dinner, he'll ask for another "mannequin" instead of a ramekin to hold the chopped veggies.

A friend of mine had a coworker who was dubbed "king of malapropisms" in their company. The friend's  name was Joe, so these became known as "Joe-isms." One of his quotes was, "money is no expense." Another favorite was, "my life passed right between my eyes." After returning from a cruise, he was asked how he liked it. To which his wife replied (imitating his love for malapropisms), "He liked the uninhibited islands the best!"

Justin Wilson (1914-2001), Cajun chef and humorist from New Orleans is known for his malapropisms, his mixed-up use of language, and his drawn-out cajun drawl, "I Gar-on-tee!" You can listen to him saying, "How ya'll are?" and view photos at One of my favorite stories told by Justin Wilson with his thick Cajun accent was his tale of visiting a southern college and seeing the boys and girls matriculating together! He was astounded. But that wasn't the worst part. They actually did it in public!

In Excel Math, we've created a Glossary of Math Terms to help students develop a vocabulary of math terminology. Excel Math encourages students to use words accurately as well as in proper context. Visit our website to download a free Glossary of Math Terms in English or in Spanish:

The Excel Math Glossary contains:
  • An overview of Mathematics
  • Explanations of terms such as Algebra, Arithmetic and Geometry
  • Illustrations for many of the terms
  • An indication of the lesson where each term is introduced, for every grade level where the term appears
  • Common math, punctuation and currency symbols
The Spanish glossary includes the math words in both English and Spanish, with full definitions in Spanish for parents.

Our goal with the glossary is to provide clear, short definitions of terms used throughout the Excel Math curriculum. We want students (and parents) to understand the words used in the math lessons as they learn (and review) mathematical concepts. Read more about arithmetical words.

Children are especially creative when it comes to adapting language to fit their own points of view. They are also quite adept at improvising when unsure of the correct words to use. One child learning the Pledge of Allegiance, got off to a fine start, but then came up with his own version: "I pledge allegiance to the flag of the United States of America, and to the republic of Richard Sands..." Another child, named Laurie, sang the chorus for the Battle Hymn of the Republic with great gusto, "Laurie, Laurie hallelujah," until she found out it wasn't really about her, and the word should have been glory. A student returning to school after an injury explained to his classmates, "I had a fraction in my arm and had to go to the hospital!"

Do you have some favorite malapropisms or interesting ways your students or friends have misused language? Just leave a comment to share one of your favorites.

Wednesday, March 21, 2012

Our Founding Fathers

Thomas Jefferson
On this day in history, March 21, 1790, Thomas Jefferson reported to U.S. President George Washington as the new secretary of state—the first secretary of state in this country. In 1797 Thomas Jefferson became the nation's second vice president, and just four years later he took office as the third U.S. president (1801-1809). For this reason he is often called the 1-2-3 president: first secretary of state, second vice-president, and third president. He is also well known as draftsman of the Declaration of Independence and was responsible for the Louisiana Purchase.

Both Thomas Jefferson and George Washington are featured on coins and bills. Read more about presidents (and their spouses) on coins in our previous post.

In Excel Math, we help students recognize coins, calculate cost per unit, solve word problems using mental multiplication, and calculate change using the fewest coins. Not every student will become a mathematician, but each one can learn to view math as something achievable and useful in every day life.

Many graduates of Excel Math tell us math is now their favorite subject. Here's an example of a fourth grade Excel Math Student Lesson Sheet about calculating change. (Download Excel Math worksheets from our website.) See if you can solve problems 1-6:
Fourth Grade Excel Math Lesson 26 Student Sheet
Now that we've counted our change, here are the presidents who appear on dollar bills. George Washington, the first president of the United States, has appeared on the one dollar bill since 1869:

Can you name the other statesmen who appear on dollar bills (not presidents)? The answer appears at the bottom of this post.

Continuing with the presidents who appear on bills, President Thomas Jefferson is on the two dollar bill. We don't see too many of these anymore:
President Abraham Lincoln appears on the five dollar bill:

President Andrew Jackson is on the twenty dollar bill:

President Ulysses S. Grant appears on the fifty dollar bill:

The other statesmen who appear on dollar bills (not presidents):

Alexander Hamilton appears on the ten dollar bill. Here's one that tore in half as soon as it was picked up (don't try this at home):

Benjamin Franklin appears on the 100 dollar bill:

Monday, March 19, 2012

Cricket Milestone: 100 Centuries in 100 Games

This past week, India celebrated as one of its cricket players scored 100 centuries (a century = 100 runs) in 100 international games. Sachin Tendulkar made history with this rare accomplishment last Friday. The game was against Bangladesh. His 99th century game was in March of last year. Even his millions of fans had begun to grow restless, wondering if his 100th would ever come. But come it did, much to the delight and surprise of the spectators. "His 100th century is 29 more centuries than his nearest challenger on the all-time list, Australia's Ricky Ponting," wrote 'The Telegraph'.

Excel Math levels the playing field so students of all ages and abilities can learn math concepts and achieve measurable results. The unique spiraling process is an important component of Excel Math that leads to mastery and long-term competency for each student. Students regularly review concepts throughout the year while developing a solid foundation of skills. Read more about Excel Math on our website.
Now back to the sport of Cricket. Cricket is a team sport for two teams of eleven players each. A formal game of cricket can last anywhere from an afternoon to several days. Although the game play and rules are very different, the basic concept of cricket is similar to that of baseball. Teams bat in successive innings and attempt to score runs, while the opposing team fields and tries to bring an end to the batting team's innings. After each team has batted an equal number of innings (either one or two, depending on conditions chosen before the game), the team with the most runs wins.

(Note: In cricket-speak, the word "innings" is used for both the plural and the singular. "Inning" is a term used only in baseball.)

A cricket field is a roughly elliptical field of flat grass, ranging in size from about 90 to 150 meters (100-160 yards) across, bounded by an obvious fence or other marker. There is no fixed size or shape for the field, although large deviations from a low-eccentricity ellipse are discouraged. In the center of the field, and usually aligned along the long axis of the ellipse, is the pitch, a carefully prepared rectangle of closely mown and rolled grass over hard-packed earth. It is marked with white lines, called creases, like this:

The dimensions are in centimeters (divide by 2.54 for inches).

The official Test Cricket nations are currently: England, Australia, West Indies, South Africa, India, Pakistan, New Zealand, Sri Lanka, Zimbabwe, and Bangladesh.

The West Indies is actually a consortium of Caribbean countries: Barbados; Jamaica; Guyana; The Republic of Trinidad and Tobago; Antigua and Barbuda; St. Kitt's-Nevis; Dominica; St. Lucia; St. Vincent and the Grenadines; Montserrat; and Grenada, Carriacou and Petite Martinique.

Minor cricketing nations (which do not play Test cricket, but do compete for a place in the World Cup One-Day competition) include: Ireland, Kenya, Fiji, Canada, The Netherlands, USA.

In the sport of cricket, a batsman reaches his century when he scores 100 or more runs in a single inning. Sachin Tendulkar made history when he became the first batsman in history to score 100 centuries. Here's a list of those players who have scored centuries in their career, with a minimum of 22 test match centuries (some of these numbers may have gone up by the time you read this post).

Most century in test match cricket by a batsman in career.

Batsman (Country) 100s Tests 50s NO Best 0s
Sachin Tendulkar (India) 51 188 65 32 248* 14
Jacques Kallis (South Africa) 42 151 55 39 224 14
Ricky Ponting (Australia) 41 162 60 28 257 16
Rahul Dravid (India) 36 164 63 32 270 8
Sunil Gavaskar (India) 34 125 45 16 236* 12
Brian Lara (West Indies) 34 131 48 6 400* 17
Steve Waugh (Australia) 32 168 50 46 200 22
Mathew Hayden (Australia) 30 103 29 14 380 14
Mahela Jayawardene (Sri Lanka) 29 128 40 13 374 11
Don Bradman (Australia) 29 52 13 10 334 7
Kumar Sangakkara (Sri Lanka) 28 106 38 12 287 5
Allan Border (Australia) 27 156 63 44 205 11
Gary Sobers (West Indies) 26 93 30 21 365* 12
Inzamam-ul-Haq (Pakistan) 25 120 46 22 329 15
Graeme Smith (South Africa) 24 97 31 11 277 10
Shivnarine Chanderpaul (West Indies) 24 137 56 37 203* 13
Mohammad Yousuf (Pakistan) 24 90 33 12 223 11
Greg Chappell (Australia) 24 87 31 19 247* 12
Vivian Richards (West Indies) 24 121 45 12 291 10
Justin Langer (Australia) 23 105 32 10 250 11
Javed Miandad (Pakistan) 23 124 43 21 280* 6
Virender Sehwag (India) 22 96 32 6 319 15
Md Azharuddin (India) 22 99 21 9 199 5
Colin Cowdrey (England) 22 114 38 15 182 9
Wally Hammond (England) 22 85 24 16 336* 4
Geoffrey Boycott (England) 22 108 42 23 246* 10

Selected others
Those who're coming up to join this list are: Younis Khan of Pakistan with 20 centuries, Alastair Cook, Kevin Pietersen and Andrew Strauss of England and Michael Clarke of Australia with 19 centuries each, VVS Laxman of India at 17 centuries, and Ian Bell of England and Michael Hussey of Australia with 16 career centuries so far.

Source: It's Only Cricket

Wednesday, March 14, 2012

The Ides of March & National Pi Day

Tomorrow is also known as the ides of March. March 15 was just another day on the calendar until Julius Caesar's assassination in 44 BC. In Shakespeare's play, The Tempest, the soothsayer warns Caesar to "beware the ides of March." So whenever March 15 rolls around, I'm reminded of Shakespeare, theater, Caesar, and phrases such as "the ides of March" that we rarely use.

In Excel Math, we teach students math concepts they will use in everyday life. Students learn the days of week, the months of year, and how to calculate dates in the future. We also teach them how to recognize a leap year.  2012 is a leap year (February had 29 days). Let's see how we can determine if a certain year is a leap year. Take a look at this lesson sheet from Fourth Grade Excel Math:
Student Lesson Sheet 124 from Fourth Grade Excel Math
From the instructions on the Lesson Sheet, can you tell which of the years shown for #2-#4 are leap years? Simply divide the year by 4. If the remainder is 0, then it is a leap year. (The exception is the year that marks the end of a century, which we're not tackling here.) The answer appears at the end of  this post.*

We don't seem to remember the Kalends of January or the Nones of May, but Shakespeare has helped us immortalize the ides of March. Ides = the 15th day of March, May, July, or October on the ancient Roman calendar and the 13th day of the other months. This was based on the phases of the moon. The ides was supposed to fall on the day of the full moon. Broadly speaking, the ides can refer to this day plus the seven days preceding it.

The Ides of March marked the beginning of the consular year from 200 BC to 153 B.C. Each year, the two consuls who were elected, began their term of office on the Ides of March.

Shakespeare's play, Julius Caesar, gave the ides of March a sense of foreboding and doom. Here's the excerpt from that play where the soothsayer warns Caesar to "beware the ides of March."
Caesar:   Who is it in the press that calls on me? I hear a tongue shriller than all the music cry "Caesar!" Speak, Caesar is turn'd to hear.
Soothsayer: Beware the ides of March. 
Caesar:   What man is that?
Brutus:   A soothsayer bids you beware the ides of March.
On that fateful day, Caesar was assassinated, and the ides of March has been a tainted date ever since. The Ides of March was the title of a 2011 movie directed by George Clooney, who played the governor of Pennsylvania. The movie took place during the frantic last days before a heavily contested Ohio presidential primary (sound familiar?), when an idealistic staffer for a new presidential candidate got a crash course on dirty politics during his stint on the campaign trail (this is where the similarity ends). Judging by the Google searches for ides following the release of this movie, quite a few people are (or were) unfamiliar with Shakespeare's famous quote.

A more common  household phrase this time of year is "march madness." In fact, the NCAA® March Madness Tournament began Tuesday night and continues tonight with the First Four. The second round starts tomorrow with Syracuse vs. University of North Carolina Asheville and the University of Kentucky vs. Western Kentucky University, among others. For an interactive bracket, visit the NCAA website. Who would you like to see win?

The madness continues throughout the month, leading up to the national semifinals, which take place on March 31. You can download an app or two, print up-to-the minute tournament brackets, stream the games live, and tweet your picks. You can even take a behind-the-scenes virtual tour, view the championship history since 1957, and play the online Bracket Challenge. How did the Romans survive without all this technology?
After 153 B.C., the Roman consuls began to take office on the Kalends of January instead of the ides of March. The kalends is the first day of the month in the Roman calendar. The word calendar comes from the Latin term kalends. Kalends, the Roman new year festival, began on January 1 and lasted until January 5.

The Romans celebrated Kalends by decorating their homes and temples with lights and greenery, exchanging gifts, fortune telling, and masquerades. Certain superstitions attached themselves to this holiday. The Romans believed bad luck would follow anyone who lent fire or iron to a neighbor during this time. Kalend’s Eve celebrations resembled our own New Year’s Eve festivities.

Speaking of festivities, today is National Pi Day! On March 14 (3.14159...) we celebrate this never-ending number with pie and ice cream, logic problems, and lots of brain teasers, such as the leap year challenge we gave you at the beginning of this post. 

Pi Day is a holiday celebrating the mathematical constant, π.

This year marks the twenty-fourth annual Pi Day. On March 12, 2009, the U.S. House of Representatives passed a non-binding resolution recognizing March 14, 2009, as National Pi Day. Some people also celebrate July 22 as Pi Approximation Day. Instead of the decimal equivalent to Pi, they use the fraction that represents it: 22/7. Twenty two divided by seven gives you the approximate value of Pi, hence the name of the holiday.

LaVern Christianson, a teacher  in Minnesota, even wrote some Pi Day songs for his students. One of my personal favorites is "Happy Pi Day to You." You might want to plan a Pi Day celebration for your students to help them remember this special number. They could write their own songs and poems, see how much of the number Pi they can fit in a three-inch square on a piece of paper, draw pictures of what comes to mind when they hear the word infinity, solve a brainteaser, and enjoy some Pi-related snacks.

So enjoy a piece of pie (or pi) and celebrate the day.

*The answer to Lesson Sheet 124: only #3 is evenly divisible by 4 and therefore a leap year.

Monday, March 12, 2012

Happy Birthday, Arizona—100 Years of Statehood

Happy Birthday! This year the Oreo cookie turns 100, and Arizona celebrates its centennial as part of the United States of America.

Here's a brief timeline of some of the major events from Arizona's history. For a more complete look at the past and a schedule of centennial events, visit Arizona Highways.

In Excel Math, we teach students how to read timelines and number lines so they can avoid bread lines later in life. Let's take a look at a Projectable Lesson from Excel Math First Grade. In this example, number lines are used to help students solve subtraction problems:

Excel Math First Grade Student Sheet Lesson 79
Did you figure out how to solve number 2 on the number line?

Saguaro Cactus
Now back to our timeline and some fun facts about Arizona. It became a state in 1912, and this year Arizona celebrates 100 years of statehood. Do know its nickname? How about its state flower? (the white saguaro cactus flower, the largest cactus in the United States)

The saguaro is found in the Sonoran Desert, which is in the southwestern part of Arizona. The saguaro cactus first blooms when it is 50 to 75 years old. These cacti grow very slowly at first. It takes them about ten years to grow to be just an inch tall. The Tohono O'odham people use long sticks to pick the fruit from the saguaro. They make the fruit into jelly. (Yum!) The best place to see saguaros is in Saguaro National Park near Tucson.

Arizona is also called the Grand Canyon State. The Grand Canyon is a mile deep and more than 200 miles long. At its widest point, it is 18 miles wide.

President Theodore Roosevelt called it "the one great sight which every American should see." This is a view of the Grand Canyon from the South Rim:

Grand Canyon
The canyon was formed by the Colorado River. This photo shows some of the many rock layers carved out over the years by erosion from rain, snow, and streams:
Grand Canyon

Another beautiful area of Arizona, Sedona is often called "red rock country." The red sandstone and manzanita trees are two of its signature landmarks. Sedona is surrounded by red-rock monoliths named Coffeepot, Cathedral and Thunder Mountain. At the north end of the city is the stunning Oak Creek Canyon, a breathtaking chasm.

Arizona has 20 Indian reservations that cover about one-fourth of the state. The Navajo Reservation, in the northeastern part of the state, is the largest in the country. This is a vista from an overlook on one of the Navajo Reservations located outside Grand Canyon National Park:
Navajo Reservation

Excel Math is alive and well in Arizona. The latest Arizona Mathematics Standards include the Common Core State Standards plus Arizona additions. Excel Math is fully correlated to the Common Core Standards and to the Arizona State Standards. Here's what one principal from Arizona has to say about Excel Math:
"The teachers are very pleased with the Excel Math program and with what they are able to learn about their students’ math abilities. We are very encouraged about the benefits of using this program."
—Susan, Principal from Phoenix, Arizona

With Excel Math, students learn higher-order thinking skills beyond what is required of these standards. Download the Arizona correlations (or those of your state) to see how the Excel Math Lessons, Stretches, Activities, and Exercises for each grade level correlate to those educational standards. At the bottom of each 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.

Happy Birthday, Arizona! Here's to another 100 candles on your birthday cake.