Once in a blue moon

"The Year's End"

What does that mean?  Is it over? Is it past? Are we about to see its backside? I couldn't let the end of the year go by without a posting about the parts and pieces - the context of a year.
The millenium, century, decade and year fit together. Then months, moon, days of the month, days of the week and hours, minutes and seconds.

DECADE
A year belongs to a decade. We usually say a decade starts in a year ending in zero and it ends in a year ending in 9. So a decade could be 2000-2009. This is the last year of the decade using this counting process.

We can also use the term decade to refer to any group of ten years, such as "in the last decade of his life ..."

MONTHS
This year (2009) had twelve months of various lengths totaling 365 days.  February had 28 days.

MOONS
There are 12.37 full moons in a year, to be precise. This year we have a blue moon at the end because there are 4 moons in the last quarter of 2009  (blue moons occur every 2.7 years). Here's a picture of the the third full moon of the final quarter of the year. I took this 29 days ago, facing west, at about 4 in the morning.

Here's the blue moon. It doesn't look all that blue! But it is turned around a bit. I took this shot facing east at about 6 pm December 30th. I didn't know the view of the moon flipped overnight, did you?

DAYS

A calendar year contains three hundred sixty-five days of 24 hours length. How many hours is that? No calculator is needed for this - let's use our our heads or a pencil:

So 365 x 24 =

five times 24 = 120
six times 24  = 144 plus a zero
three times 24 = 72 plus two zeros

Easy. Line these products up, then add them.

      120

    1440

    7200

for a total of ?

HOURS
8760 is the total hours in a normal year.

MINUTES
Multiply by 60 to get the minutes in a normal year = 525,600

SECONDS
Multiply by 60 again to get the seconds in a normal year = 31,536,000

HOLIDAYS

In the United States, some days are designated as Federal Holidays. Here are the holidays for 2009.
If you lived near Washington, D.C. you might also have gotten a holiday on Inauguration Day.

Thursday, January 1	New Year’s Day
Monday, January 19	Birthday of Martin Luther King, Jr.
Monday, February 16	Washington’s (or President's) Birthday
Monday, May 25	Memorial Day
Friday, July 3	Independence Day
Monday, September 7	Labor Day
Monday, October 12	Columbus Day
Wednesday, November 11	Veterans Day
Thursday, November 26	Thanksgiving Day
Friday, December 25	Christmas Day

States and local regions might have extra holidays - such as Festivus.

SPECIAL DAYS

Years contain special days - weddings, special birthdays, funerals, vacations - most of us will remember 2009 because something special happened to us or our families.

Well, here's to the end of these 31 million seconds!

A typical New Year's Eve photo from San Diego.

Christmas Eve Math

Hello and Christmas Greetings from Excel Math.

This is my hundredth post of 2009.

That equals 10 tens. Five score. Ten percent of a thousand. Eight and one-third dozen.

Centum. Century. Centa. Zentgraf. Centenaar. Quintal. Cien. Sto. Quintale. Hekaton. Hundred. Njëqind. Honerd. Sada. or מאות.

I could say a lot more about the math consequences and implications of 100, but I won't.

Today our company is starting our holiday break. We will be closed until January 4, 2010.

In compensation, I'd like to offer you a five-minute concert of seasonal music,

recorded by the US Air Force Band.

Their website says this music has been approved for

"public service broadcasting, educational activities and morale"

I think that includes us.

SOUNDS OF THE SEASON

Have a nice holiday.

Think math think math think

Henry Ford is quoted as saying,
Thinking is the hardest work there is, which is the probable reason why so few engage in it.

This quote makes sense to me. I don't think it was meant to be insulting, or ironic, or sarcastic. Thinking is hard work.

Ralph Waldo Emerson asked (and answered),
What is the hardest task in the world? To think.

Thomas Edison claimed,
There is no expedient to which a man will not go to avoid the real labour of thinking.

We are told what to think, by our parents, our friends, the press, the entertainment world, the books we read, etc.  We don't have to believe what we are told or do what they say, but either route is less work than thinking out a path for ourselves. Perhaps one reason people don't like math is because it often requires hard thinking.

Sam Ewing said,
Hard work spotlights the character of people: some turn up their sleeves, some turn up their noses, and some don't turn up at all.

Enough pithy quotes! You've turned up, it's Christmas break and it's time for nice difficult problems to engage your brain for the rest of the holiday season. Right? Even though 5th graders are the intended audience for this series of logic problems, it doesn't mean we adults can't solve them (if we are forced to!).

The human mind prefers to be spoon-fed with the thoughts of others, but deprived of such nourishment it will, reluctantly, begin to think for itself. Agatha Christie

This is a chance to put our minds to work. Don't be afraid to write notes, cut out little horses, turn to the left or right in your chair - because these are not intended to be "in your head" puzzles.

Here's an example problem:

Johnny, Sara, Alex and Kathy are different ages.

1. Kathy is older than Sara.

2. Johnny is younger than Kathy but not the youngest.

3. Alex is the oldest.

What is the order of their ages?

(A, K, J, S)

Problem 1:

Julie, Emily, Drew and Mark have puppies named Fluffy, Molly, Randy and Cody.

1. Mark’s puppy’s name has the same first letter as Mark’s name.

2. Emily and Julie went to the movies with Randy’s owner yesterday.

3. Julie doesn’t own Cody.

Who owns each dog?

Problem 2:

Six horses ran a race: Duncan, Toby, Ella, Prancer, Caleb and Storm.

1. Prancer finished right behind Ella.

2. Toby was not first st or last.

3. Three horses came in between Storm and Caleb.

4. Duncan and Prancer finished on each side of Caleb.

5. Duncan did not finish first.

Give the finishing order, winner first.

Problem 3:

Jamie, Cory, Taylor and Sam are all employed. Their jobs are mechanic, teacher, doctor and astronaut.

1. Taylor is not the doctor or the mechanic.

2. Jamie is friends with the mechanic.

3. Cory is the astronaut.

Who has which job?


Problem 4:

Five people are in line at the grocery store: Arthur, Beth, Carrie, Diana and Eddie.

1. Beth is behind Arthur and Diana but isn’t last in line.

2. Beth and Carrie are on either side of Eddie.

3. Arthur is right behind Diana.

In what order are they standing?

Problem 5:

Carlos, Jan, Lisa and Bob are different heights.

1. Jan is not the tallest.

2. Carlos is shorter than Jan.

3. Lisa is shorter than Carlos.

Rank them in order from tallest to shortest.

Problem 6:

Five people live on 5 different floors of a 5-story apartment building. They are Dorian, Gary, Angel, Rosa and Kim.  

1. Rosa lives right below Dorian.  

2. Angel and Kim both live above Gary.  

3. Angel lives right below Rosa.

4. Kim doesn’t live on the top floor.

Starting at the ground floor, in what order do they live?

No problem can withstand the assault of sustained thinking.

Voltaire

The answers are Right Here.  Posted the day after Christmas.

Pixel Dust

I have to keep remembering the fundamental purpose of this blog. It's  Why study math? What can I do with it when I grow up?

You can count pixels. Not pixie dust, but pixels. Pixels are points of light on your screen, or dots that make up your photograph. To go on with our discussion, we need a sample picture. Here's an Oh-So-Cute-Photo of my cat Tiger with a stuffed toy.

The information contained in a picture is measured like the area of a rectangle. We use the units of pixels (i.e. so many pixels across and so many down). You may also hear pixels defined as dots per inch, or dpi.

Pixels are just one of many things you can measure in a picture. Here are information panels you get by pressing CMD-I or CNTRL-I when you are in a photo viewing program. I used 3 different programs.

This first box shows the original image data.

The left box shows the photo as it was exported from iPhoto. The right box is after I trimmed and cropped it a bit more.

The storage space required for the file has decreased from 1.4 mb to 858 kb to 444.8 kb
The pixel count has decreased from 2048x1536 to 1478x1141 to 1280x988 pixels
The dots-per-inch (dpi) stays the same in all of these tables.

How large is the original 1.4mb, 2048x1536 pixel file, IN INCHES if displayed at 72x72 dots per inch? Would it be the same size as the actual cat?

Hmmm.

2048÷72=28.444 inches
1536÷72=21.333 inches
To display that photo in its maximum original size would require a monitor 28 x 21 inches, more or less, and that's not as large as the real cat.

How large a print could we get if we used the 858 kb, 1478x1141 pixel file, IN INCHES if printed at 300x300 dots per inch?

Well, that's
1478÷300=4.92
1141÷300=3.80
The print would be approximately 5 inches by 4 inches.

How many pixels are on a typical screen?

It depends. Here are your choices. The shaded green area in the left top corner is the size of a NTSC TV screen. The purple shaded area is the pixel count on my largest monitor at Excel Math, a WUXGA. Yours is probably somewhere in the middle.

For our sample picture, my camera was set to record at QXGA resolution, or 2048x1536.

How many mega-pixels would that be?

3,145,728 or 3 megapixels.

If you don't understand math, you can forget about understanding photography and video. We haven't even started yet with scaling, zooming, interpolating, extrapolating, etc!

Bending Light Around Corners

Light radiates outward from its source in straight lines. That means if you are around the corner from a light, you are in darkness. It's the reason we have shade. If light poured around corners and filled up darkness like liquid fills up a container, we would have no shade.

Here's an example showing my friend Steve driving us across the desert in his 1956 Jaguar. The hat is providing shade on part of his face. The rest is in the direct path of the sunlight.

Sometimes we want light to shine around dark corners, so we have learned how it bends. And sometimes we can help it bend round corners!

RAINBOW

Light hits drops of moisture in the air and shines through those drops. But in the process, the rays of light change speed. Because it is more difficult for light to travel through water than through air, some beams slow down slightly. Light is made up of a number of different frequencies and some slow more than others. That causes the beams of light to curve. The various frequencies hit our eyes in a different way than normal, so we suddenly see the colors that make up clear light.

PRISM
A prism is a specially-cut and polished piece of glass or plastic whose purpose is to bend and separate rays of light. It's a mechanical means of making a rainbow.

OPTICAL FIBERS

If we want to transport light from one place to another, we use optical fibers. These consist of a core of plastic strands enclosed in a sheath of slightly different plastic. The light that goes into the fiber can bounce back and forth between the core and the sheath until it comes out the other end. Even if the fiber is bent, most of the light continues to travel through.

We can send light through optical fibers to illuminate a space at the other end, or we can receive light through optical fibers and see what's going on at the other end. There are lots of neat experiments that you can do to curve light. Just search Google for Bending Light and try some.

GRAVITATIONAL LENS

You'd have a hard time making this at home, but astronomers theorize that light bends when it encounters a tremendous gravitational field emanating from a star or galaxy. We think that gravity travels too. When the light and the gravity interact, light curves. This is the basis of the science fiction terms: time warp and warp speed.

Einstein and others predicted this effect, but it was not actually observed until 1979. On that car trip across Arizona and New Mexico I saw a Very Large Array of space antennas. They are hard to see in the picture, but they are scattered across the desert behind us. This is where scientists from around the world learn about Gravitational Lenses.

This NASA image shows the effect of a Gravitational Lens. You can see how the white beams of light go across a galaxy, bending around the blue (gravity field) space surrounding the planet(s).

CLOAK OF INVISIBILITY
If we can learn precisely how light is bent and reflected or refracted, scientists think we can create a way to shield or camouflage ourselves or other objects. Experiments have produced devices that can shield objects from certain types of light.

This is a popular theme in science fiction, such as H.G. Wells'  The Invisible Man, the invisibility cloak in the Harry Potter series, the alien in Predator, and the Romulan Cloaking Device in Star Trek.

You can click this link to get a scientist's take on invisibility, from Duke University.

Here are some more photos of the Jaguar - it's got curves in all the right places.

Hemispheres

In my fanciful posting yesterday, I referred to multiple dimensions in space.

Geometry has various ways of describing things that happen on a flat surface - a plane - and on curved surfaces - a sphere.

The word hemisphere means half of a sphere.

In the USA, we are used to talking about the Western Hemisphere or the Northern Hemisphere - where our country is located on the earth's surface.

There are lots of other ways to use the term hemisphere.

Chrysler has had an engine for many years, called the Hemi. It was given the name due to the shape of the combustion chambers, which resembled a hemisphere, or dome. Here is a cut-away drawing with the dome outlined in red. Although it's not an exact hemisphere, it's much more dome-like than the other designs, one of which is called the Wedge.

I recently ran across Hemisphere in an article on watch crystals. That's the glass cover over the face of your watch. On many watches, the crystal is domed, for a better appearance, greater strength, and more height over the center of the dial where the hands are. Here's a side view of two watch crystals.

Today we have all sorts of specialty technology. Of course glass companies can make tiny little domes, precisely-sized and shaped to fit my watch. But how did they do it in the old days?

Apparently way back then, they would blow a ball of glass, about a foot and a half in diameter. After it cooled, a worker would scribe small circles around the sphere with a diamond or steel cutter. Then he would press out the little circles with his thumb, and polish the edges. Voilà, a domed glass watch crystal.

Wow! Imagine the potential for breakage! And danger to your thumb!

Luckily lots of other technologies have been developed for making curved glass (including plastic alternatives that don't break easily).

Here's an older Rolex wristwatch with a domed crystal. You can see how it arches over the center of the dial.

Here are a couple side angle images, from the online glossary at the Sinn Watch Company in Frankfurt, Germany. Some Sinn watches can have either plastic or sapphire glass crystals. On the left is the domed shape, cut from a sheet of glass. On the right is a glass crystal that replicates the old domed acrylic plastic crystals.

Finally, at the bottom is the thick glass blank with the lines drawn on it to show how it will be cut out by special machines.

That's enough of that. What's the most common occurrence of the hemisphere in your life?

In my life, I think of grapefruit in the morning.

Where's the milk?

Have you ever noticed that sometimes kids (or even adults) can be insufferable?

Do you hear this sort of complaining? From a spouse, roommate or child?

The cereal is here on the table but I don't see any milk. 
And there isn't any milk in the refrigerator either! Someone drank it all. Waaah!

It might irritate you. You might get angry, or grumble to yourself. If it's your fault you might apologize.

Math can help.

In these extreme cases, I suggest you overwhelm them with your superior knowledge 


Use math.

Rather than shouting,

Get out to the barn and milk the cow!
or
Run down to the store and pick up a quart!

You can instead say, in a calm and superior tone,

The cereal is in Euclidian space. You forgot the milk resides within a Gaussian curvature. The intrinsic nature of this quadratic surface means you can't always find the milk where you expect it. 

If that's not enough, finish (in your best Twilight Zone voice) with

Sometimes you have to search in a different astral space. Remember that bookstore in the Harry Potter stories? Visible to some and not to others? Here today and gone tomorrow?

Behind the physical refrigerator door there is an alternate universe. Inside IT is the milk you are longing for...

While they are shaking their heads in confusion, you can go back to your morning paper.

Note - no milk was used or abused while preparing this blog

Third Time's A Charm

Everyone knows what a circle is. Yesterday we defined squares. Today, triangles.

A triangle is another polygon, a planar figure with three sides.

Three is the fewest sides that a polygon can have, because you can't make an enclosed shape with only one or two straight lines. Does that mean that triangles are simple? Is class over? Can we go home?

Not yet. We get to learn plenty of new words today.

Triangles are categorized by the length of their sides:

• Equilateral triangles have three sides that are of equal lengths.
  
• Iscoceles triangles have two sides that are equal in length.
• Scalene triangles have three different-length sides.

Triangles are also categorized by their angles:

• A right triangle has one angle that measures 90 degrees.
• Oblique triangles do not have a right angle.
• Acute triangles have all three angles measuring less than 90 degrees (each).
• Obtuse triangles have one angle that's greater than 90 degrees.

The sides of the triangle are called legs. The leg opposite the right angle of a right triangle has a special name - the hypotenuse.

Triangles can be similar if they have the same angles but are different sizes. Triangles may be called congruent if they are identical in every way, including size.

There are plenty of other characteristics of triangles. We don't have time to mention them all. This graphic shows some of them.

The orange lines go from a vertex to the center of the opposite side.

The blue lines originate from a vertex and intersect the opposite side at a right angle.  This is called the altitude. The point where the altitudes cross is the orthocenter.

The green lines are drawn midway along each side and intersect the sides at right angles. They cross at a single point within the triangle. These are the perpendicular bisectors (mentioned yesterday).

The red shows that a straight line can be drawn to connect the intersections of the orange, blue and green lines. This is called Euler's Line, after the mathematician.

If you draw a few circles to help describe triangles, you will find that the center of the incircle is where the triangle's angle bisectors meet. That point is called the incenter.

The circumcenter is the center of the circumcircle. It intersects the three verticles of the triangle. The circumcenter happens to land on the point where the three angle bisectors meet.

I'm stopping here because I am getting totally confused. We'll skip over calculating the area, sines and cosines, vectors, trigonometry and other aspects of triangles.

Some day we will tackle non-planar triangles - what happens when you draw a triangle over a curved surface, like the Earth - a global triangle.

Squared Off

Everyone knows what a square is. A box on paper. Could anything be simpler? 

Actually yes, there are many, complicated, math-ly ways to describe a square. Ready? Here we go...

GEOMETRY

A square is a regular quadrilateral and at the same time, a right rectangular rhombus. It's a parallelogram, and a trapezoid (or trapezium in some places).

A square is a two-dimensional (planar) object.  A polygon with four sides and four vertices (corners). All four sides are of equal length. All four angles are equal. All angles are 90 degrees (right angles).

A square's diagonals are of identical length. They divide each other and the corner angles in half, and are perpendicular to one other. That means they are perpendicular bisectors.

The diagonals are equal to the length of one side of a square multiplied by the square root of 2, or 1.414 times as long (Pythagorean Theorem).

A square's perimeter is 4 times the length of one side. Its area is equal to the product of two adjacent sides, or one of the sides squared. There is no volume to a square because it's on a planar surface.

Simple, eh?

Those are most of the definitions of a square. Now we have lots of other words that we can define more clearly. Later. Another time.

Like many of us, the square comes from a large family. In this case, it's the family of quadrilaterals (figures made up of 4 lines or having 4 sides).

Thanks to Wikipedia, here's a picture of the SQUARE FAMILY TREE.

We haven't exhausted the meaning of the word square.

CARPENTRY

A square is a tool used to mark and measure wood, so you get straight lines and accurate cuts and especially right angles. There are squares used for metalworking too.

ARCHITECTURE

A square is a open place where people and roads meet, usually in the middle of a city or town. It's also the name of one of the spaces on a chessboard.

SOCIETY

Square is a term used to infer honest and straightforward and even, as in "I'm going to square with you" or "our accounts are squared up" or "he looked him squarely in the eye".

It means substantial and sufficient - such as "three square meals a day."

A square is a term sometimes applied to a person who likes math. Someone who is not too hip or connected with modern thinking and acting. You could say they are "a square peg in a round hole."

Here's a perfect Christmas gift for a square.

You can order one here. Zazzle

(dis)Order of (un)Symmetry

Order of Symmetry is not a phrase we use very often. In fact, it's a bit of a struggle for us to get this across clearly in our math curriculum. People confuse it with a Line of Symmetry.

Rotational Symmetry is used to describe the number of times that a shape can be overlaid onto itself during a complete turn around its central point. (On the following illustrations, a corner is colored so you can watch the rotation.)

A rectangle can fit over itself twice. It has order of symmetry two. A square can be fitted over itself 4 times in a turn. It has order of symmetry 4.

This particular star can be fitted over itself 5 times in a turn. It has rotational order of symmetry 5. Not all stars are symmetrical in shape; not all have symmetry.

How many orders of symmetry does a circle have? Lots? or none? It has an infinity.

Lines of symmetry exist when you can fold a shape over a line and the folded parts lay right on top of each other. Here's one through the middle of a music player.

How and why did this come to my mind today? Gosh, it's hard to say where the ideas come from. I was reading about states and types of matter.

There are the big 2 - organic and inorganic.

And the Big 3 - gas, liquid, solid.

There once were the Big 4 - earth, water, air, fire.

But now those pesky scientists have changed their minds again - recognizing many crystalline states between liquid and solid. When some substances are magnetized or in a non-magnetic condition, or if they are conducting electricity or not - they grow crystals with different sorts of shapes and symmetries.

That's interesting and irritating! When we are classifying things we want them to remain constant under all observable conditions, not shape-shifting whenever they are subjected to outside influences!

Here are some quasicrystals:

quasicrystals

It fouls up our classification system if solids can change the shape of the crystals that compose them!

Of course then we have another set of categories: animal, vegetable, mineral.

But where do slime, diatoms, lichen and fungi fit? The classification task is very complex out at the fringes! Those scientists have also given up using just those 3 categories, and have added others too confusing to explain here. As a sample,

Cellular slime molds spend most of their lives as individual unicellular yet complex microorganisms, but when a chemical signal is secreted, they assemble into a cluster that acts as one organism (protists).

These unicellular protists live on their own most of the time (perhaps in tiny singles apartment complexes) but when a signal given, they link up (like the Borg) and operate as one organism. Much like those quasicrystals, but alive. Sort of. In a science-fiction sort of way.

What is the order of symmetry for a cluster of protists? I have no idea.

At least people are fairly consistent. When your mom says "Stay still! Stop wiggling your foot!" she doesn't have to add "and stop growing new arms!".

Nigel

Could you live here?

I read an article today about the smallest condo-apartment in New York City. A married couple and two cats live in this space. I've created a floor plan with a bed and two chairs.

In Excel Math we ask students to cut out furniture shapes and arrange them on a floor plan, to use their geometry and measuring skills. In a later set of lessons, we ask them to calculate the cubic volume of a houseful of furniture. That's helpful in choosing a moving van, or deciding how much you can take to a college dorm room (this helps avoid fights with your new roommate!).

Here's the lesson content, re-arranged for the blog. Julio's dad asked him to calculate how large a truck they would need for this set of furniture.

Our math questions are these:

1. Can we fit all the furniture into the apartment?
2. If so, how high is the pile going to be?

Assumptions: there's no furniture to start with, the kitchen cabinets stay, you can't use the bathroom space, and the ceilings are 8 feet high.


I estimate that the kitchen cabinets are 2 feet by 7 feet. That's 2 x 7 x 8 = 112 cubic feet.
The main room is 10 x 15 x 8 = 1200 cubic feet. Subtract 112 and we have 1088 cubic feet.
The door still has to be opened and closed, so we have to take off about 3 x 3 x 8 or 72 cubic feet.
We'll round to 1000 cubic feet.

We only have to fit in 440 cubic feet of stuff.
The apartment will be 440/1000s full. We can simplify that to 44/100ths or 22/50, 11/25.
That's about 44%.
The depth of the pile will be 96 inches x .44 = 42 inches deep.
We will have 3 1/2 feet of stuff across every square inch of the room.

Anyone who's ever packed a moving van knows these numbers mean very little. There's always air space and you never pack everything with 100% efficiency.

Add some clothing, a few cushions and I predict the room will be filled to the ceiling!

Math R Us

People sometimes ask me, What goes into a math curriculum?

I reply, Do you mean what are the contents and how do we do it? If you've got a moment, I can tell you.

• We teach math concepts. Nearly 400 spread over 7 years. This coverage ensures our students can meet state's  standards. We cover a few things that aren't currently required, like Roman numerals.
  

• In lower grades we help students with tactile elements of math. They learn to write numbers and other math symbols, to set up problems on a page, to use counting sticks, blocks and so on.
  

• We create problems that illustrate math concepts and challenge students. There are thousands in each grade. We track them by concept, difficulty, grade level, subject matter, type of person mentioned in the problem, etc. And we provide the answers.
  

• One unique feature of Excel Math is the Checkanswer. We group 3-5 problems into sets. The sum of the answers is printed at the top of that set. Students add their own answers, then compare their sum with ours, to confirm the accuracy of their work.

• Excel Math spiraling arranges problems in an ascending path of difficulty over the years. Familiar concepts are placed in-between novel and challenging ones. This ensures no one has to learn everything the first time it's presented, and students don't forget or miss the fact that old concepts are a foundation for the new ones.

• In higher grades we provide Create A Problem pages that merge math and literacy. Students read a short story, solve problems and create a few story-based problems of their own.
  

• Activities involve students in active research - they interact with and report back on complex math issues.
  

• We give teachers ideas for classroom presentation. We offer suggestions in the Teacher Edition and we have the Projectable Lessons for putting content on the wall in front of a class.

• Ancillary materials support the main components of our curriculum. These include Glossary, Manipulatives, Mental Math, Training Videos, and In-Service presentations. The Scope and Sequence in the Teacher Edition outlines how concepts are presented in that grade.
  

• We solicit reviews by school boards, curriculum supervisors, teachers and parents. California's formal social content review ensures we fairly portray cultural and racial diversity in our problems and artwork; we mention the societal contributions of minorities, males and females; we show people in various positive contributing roles to help students view school constructively; and we refrain from using brand names, products or corporate logos.

In addition to what you'd definitely consider math (addition, geometry, etc.), we cover related elements that educators have decided should be taught in a math class. These include skills like:

• reading maps
• giving directions (to a locale, for a process)
• telling time, using calendars
  
• understanding coins, currency, money and foreign exchange
  
• strategies for responsible purchasing, saving, taxes, etc.
• dealing with probability (wagering, random chance, statistics)

Other than that, math curriculum is a piece of cake. It's all we do. No literacy, no history, no geography, no physical education - except as these fit within our math problems.

Math R Us.

Dash it all, Holmes!

THE ORTHOGRAPHY OF HYPHENATION

Translated into English, that means the proper way to put hyphens in words. You know, the little line in between syllables.

A hyphenation algorithm is a set of rules (used by people or a computer program) that decide at which points a word can be split using a hyphen. For example, a hyphenation algorithm might decide how the word impeachment can be broken.

impeach-ment 
or
im-peachment 
but not
impe-achment

This is handy for line justification and line-wrapping, which is a cosmetic process applied to a paragraph to keep the right margin smooth. Notice the next paragraph's appearance.

The rules of word-breaking are very complex and there are many exceptions (much to the irritation of editors). In the simplest terms - you should only break multi-syllable words, between the syllables. Of course it gets much more complicated when you care about how the line fits in a paragraph, or on a page, etc. One of the most important rules is NOT to break words across columns or page breaks.

An important math rule that overrides hyphenation dictionaries says you never separate math characters, currency symbols, etc. from the numbers preceding or following them.

As you can see here, $100
is not the same as $-
100.

JUSTIFICATION

Here's the Excel Math "house rule" on hyphenation and justification for Lesson Sheets:  DON'T. EVER.

Please don't confuse the hyphenation algorithm with the rules for inserting hyphens when you create compound words, such as mother-of-pearl.

Since computers have taken over most hyphenation tasks, there are new computer-variant characters, such as the soft hyphen or non-breaking hyphen. These trigger special instructions in the hyphenation algorithm.

Strictly speaking, the hyphen is NOT the same as the minus sign (which is longer than a hypen), or any of the dashes.

UNICODIFICATION

When a publisher or its computer system must distinguish between these characters (which seem so very much alike), we use special instruction tags known as Unicode.  

Unicode provides a unique number for every character, no matter what the computer platform, no matter what the application program, no matter what the human language being represented.

Unicode is shown in this form: a U, a + and four digits:

HYPHENIC VARIATIONS
hyphen U+2010
non-breaking hyphen U+2211
minus U+2212
figure dash U+2012 (also used as minus sign)
en dash U+2013
em dash U+2014
horizontal bar U+2015 (longer than em dash)
macron U+00AF (short, high overline)
overline U+203E (over other characters)
underscore U+005F (forces words apart visually while keeping them joined)
underline U+0332 (under other characters)
short stroke overlay U+0335 (through other characters, with inter-character gaps)
continuous strikethrough U+0336 (through other characters with no gaps)
diagonal stroke overlay U+0337 (through other characters, diagonally)
soft (invisible) hyphen  U+00AD (invisible but indicates a preferred word break point)

I'm going to stop here with the special characters. Isn't this a pain in the neck really fun stuff?

If you want more UNICODE, you can go to the website.

MATHEMATICATIONS
Math meanings can be dramatically changed by hyphen placement. An example? Try this:

• three-hundred-year-old trees means some trees that are 300 years old.
• three hundred-year-old trees means 3 trees that are 100 years old each.
  
• three hundred year-old trees means 300 trees that are 1 year old each.

Writing word problems about big old trees is a tricky business!