Additional Math Pages & Resources

Tuesday, November 30, 2010

How do you measure a life? Part II

Yesterday I asked if we can use elementary math to describe a person's life. I haven't come to a conclusion or answer to the question. But I learned some people insist math and science are attempts to express new things we didn't know before, in a way that everyone will understand. In contrast, poetry will express what we already know, in new ways that some will understand.

Poetry consists of an awareness of experience, expressed through meaning, sound, and rhythmic language to provoke an emotional response.

Mathematics is a science dealing with numbers, shapes, structures and change, and the relationships between these concepts.

So I wondered if a body of mathematical poetry might exist. Could it help us measure a life in more than just days lived or money accumulated?

Today I must admit that the bulk of math poetry does not. I've found plenty intended as education or entertainment, not expression of deep meaning of a person's existence. I saw memory aides - rhyming statements of how a math process works. I found plenty of problems presented to kids in a sing-song way, or limericks which pose a question to be answered or a puzzle to be solved. Here's an example:

Take five times which plus half of what,
And make the square of what you've got.
Divide by one-and-thirty square,
To get just four -- that's right, it's there.

This is not what I am looking for. After some searching, I did find tons (well, not really tons in a math sense) of serious discussion on math and how it might help us find meaning in the universe. Here's a sample:

Logic (thus math) starts with people labeling the existence of any object or phenomenon 
(God, rock, flower, etc.) by a symbol we call 1, or yes, or +, or dot, or true.  
The lack or absence of existence of an object is labeled as 0, or no, or -, or dash, or false
The symbols are used to describe objects around us as either existing or absent...

That discussion went on for many pages and my eyelids closed with a flutter! Then I found this:

If poetry is the love of carefully-chosen words and crafted phrases to convey image and idea;
  if a mathematician channels a love of pattern, quantity, and structure into carefully-chosen words and crafted phrases;
the intersection of their realms should be non-trivial.

Now we're talking! That statement led to this poem entitled An Equation for my children, by Wilmer Mills

It may be esoteric and perverse
That I consult Pythagoras to hear
A music tuning in the universe.
My interest in his math of star and sphere
Has triggered theorems too far-fetched to solve. 
They don't add up. 
But if I rack and toil
More in ether than mortal coil,
It is to comprehend how you revolve,
By formulas of orbit, ellipse, and ring.

Dear son and daughter, if I seem to range
It is to chart the numbers spiraling
Between my life and yours until the strange
And seamless beauty of equations click
Solutions for the heart's arithmetic.

Monday, November 29, 2010

How do you measure a life? Part I

Can math measure a life?

We can measure a lifespan.

How long have you lived? In my case, I can say "I'm 59 years old".

That makes sense, but it seems unreasonable; it seems nonsensical to say "I'm 21,600 days old, or 518,400 hours, or 31 million minutes, or 1,866 million seconds old".

Measuring life like this is meaningless. The larger the total of tiny units, the less the measuring means. Just as it makes sense to say "I drive 15 miles to work" but not to say "I drive 950,000 inches to work".

But regardless of units, the lifespan doesn't capture what the life was about. 

Measuring a life with math units doesn't capture its meaning, only the external evidence, such as "She earned a PhD and 94 graduate units" or "His retirement account has more than 1.5 million dollars" or "They had 4 children, 16 grandchildren and 44 great-grandchildren".

I saw this statement recently, In science one tries to tell people, in such a way as to be understood by everyone, something that no one ever knew before. In math you try to prove something that's never been proven - and tell people how you did it. But in poetry, it's the exact opposite. With poetry you want to tell people (whether they understand it or not) what we all experience.

Can we describe a life? with mathematical poems?

We'll have to look at this more tomorrow. In the meantime, try searching on mathematical poetry.

Read some. Write one.

Wednesday, November 24, 2010

The Math Tree Seedlings

Math is a language, and the ability to use it fluently is valuable. From a parent's point of view, having great math skills should result in long-term benefits for their children. As shown here:
I wish that I could promise you that money will grow on trees if you are skilled in math. But I can't promise that. I can however promise that you will amaze and astound your friends throughout life if you are confident and fluent in math. You might impress your boss, and you'll certainly be able to manage your finances.

Excel Math is like a friend with a green thumb. We simply help to plant the math; someone else nurtures and waters the seedlings as they begin to take root.

My sister is a reading specialist with many years of experience and a PhD. She always encourages parents, aunts, uncles and cousins to read to and with children. I publish math curriculum and would also like to offer my own bit of advice:

Take some time over the holidays to talk about math with your kids. You can do this while shopping for a  Christmas tree, or while doing the dishes.

How many plates did we use; how much does this pot hold; how long a nap should we take on the couch; which football games will we watch; how many floats in the parade?

How far did they fly to come see us?  How many shopping days til Christmas? What good things can we do to help those less fortunate than we are - and how do you measure the value of good works?

Water those math seedlings that have been planted this fall!

Tuesday, November 23, 2010

The Math Family Tree and its Branches

It's Thanksgiving week and lots of kids are out of school and heading for grandma's house for turkey dinner. Time to use our imaginations and consider math for the holidays.

Here's a Mathematics tree, looking for other members in his family. Let's say he represents math. What other branches of his family might show up for the big dinner on Thursday? He knows he has a cousin named Geometry (gee, I'm a tree).

One amateur mathematician suggested that these are the definitive various branches of math:

1. Foundations
    -Logic & Model Theory
    -Computability Theory & Recursion Theory
    -Set Theory
    -Category Theory

2. Algebra
    -Group Theory -> Symmetry
    -Ring Theory -> Polynomials
    -Field Theory
    -Module Theory -> Linear Algebra
    -Galois Theory -> The Theory of Equations
    -Number Theory
    -Algebraic Geometry

3. Mathematical Analysis
    -Real Analysis & Measure Theory -> Calculus
    -Complex Analysis
    -Tensor & Vector Analysis
    -Differential & Integral Equations
    -Numerical Analysis
    -Functional Analysis & The Theory of Functions

4. Geometry & Topology
    -Euclidean Geometry
    -Non-Euclidean Geometry
    -Absolute Geometry
    -Metric Geometry
    -Projective Geometry
    -Affine Geometry
    -Discrete Geometry & Graph Theory
    -Differential Geometry
    -Point-Set or General Topology
    -Algebraic Topology

4. Applied Mathematics
    -Probability Theory
    -Computer Science
    -Mathematical Physics
    -Game Theory
    -Systems & Control Theory

Ok, this is a list compiled by an expert. And there are lots of big and confusing words in the list. But did you notice that he can't count?  Or doesn't know number sequence? We teach this in Kindergarten and First Grade.

Group 1, 2, 3, 4 and 4.

But that makes sense, because he left arithmetic off the list, which is what we teach in early grades of elementary school. We could do some of this list-making ourselves, I think. Here's what I come up with if I strip off all of Mr. Tree's leaves. We see these branches:

Most of these relatives will have to be introduced to you by someone else who knows them better than I do.

Monday, November 22, 2010

Backbreaking Math

I have a case study to examine today. Can we solve the problems with the elementary school math we teach to kids in our Excel Math curriculum? (Yes!)

Chris is building a planting bed in his back yard. It will keep soil from washing down from the neighbor's yard (a bit higher), and give him a place to grow vegetables. Take a look:

He bought these metal baskets called gabions, and will fill those with soil. How does he know how much soil to buy? Can he buy soil in bags from the garden center, or does he need a truck?

Chris has to fill these 8 containers:

5 large gabions = 1m x 1m x .5m = .5 cubic meters x 5 = 2.5 cubic meters
3 small gabions = 1m x .5m x .5m = .25 cubic meters x 3 = .75 cubic meters

He needs 3.25 about cubic meters of soil.

We'll estimate that Chris needs 3.5 cubic meters of soil since some will leak out the holes and wash away, or be lost in the grass. However, he has to buy soil by the cubic yard (plastic bags) or ton (determined by weighing the delivery truck), so now we need to convert from cubic meters to yards or tons.

A cubic yard of soil weighs 2000-3000 pounds or 1.0-1.5 tons.

A cubic meter = 1.3 cubic yards; a cubic yard = .76 cubic meters

3.25 cubic meters x 1.3 = 4.225 cubic yards

So we need about 4.25-4.5 cubic yards of soil.

You need 54 40-lb. bags of soil per yard, so Chris could buy 4.25 x 54 = 230 bags of soil from the garden center
4.25 x 1.0 to 1.5 tons of soil = 4.25-6.35 tons delivered by a truck

After looking at the weight and his available vehicles, Chris didn't bother with any more math, such as calculating the number of trips it would take to pick up 230 bags.

He called the soil delivery man instead. Here's the the pile of dirt which is just arriving in his back yard:

Friday, November 19, 2010

Arithmetical Words, Part V

We have come to the end of the arithmetic alphabet with R-Z today. Most of the time this blog deals with using math in real life, but without knowing math words, it's hard to communicate precisely.

We tend to say square or rectangular when we are talking about 3D objects, even when we mean rectangular prism. I was talking with my friend Chris, who said he was putting square gabions in his back yard. I knew the word gabion meant a big wire cage, but I didn't know what shape they might be. I learned gabions can be purchased as rectangular prisms, cylinders or cubes. You put them in place, then fill them with heavy material to form earthen walls.

If we want to be extra-precise, gabions aren't solid figures until they unfolded and filled. Here are some pictures of his empty ones, waiting for 2 tons of dirt. We'll do the math calculations on the dirt in another blog:

Now, on to the definitions:
  • Radius straight line from the center point of a circle to any point on the circle
  • Ray line with one endpoint
  • Rectangular Prism solid figure with 8 vertices, 12 straight edges and a total of 6 rectangular flat faces, one of which is the base
  • Rectangular Pyramid solid figure with 5 vertices, 8 straight edges, 4 triangular flat faces and 1 rectangular base
  • Reflection movement of a figure over a line that results in a mirror image; a Flip
  • Remainder amount left over when one number is divided by another
  • Right Angle angle that measures exactly 90 degrees
  • Roman Numerals number system created by ancient Romans that uses letters rather than numerals; not based on place value
  • Rotate to move or turn a figure around a point; a Turn
  • Round (1) a circular or cylindrical shape
  • Round (2) replacing an exact number with an approximate number that is more convenient to use
  • Scalene Triangle a triangle where all three sides are of different lengths
  • Similar Figures figures having the same proportions but not the same size
  • Slide when a figure moves without changing its appearance, see translation
  • Sphere 3D solid figure where all points on the surface are equidistant from the center
  • Square 2D parallelogram with 4 congruent sides and 4 congruent angles
  • Square Pyramid 3D solid figure with a total of 5 vertices, 8 straight edges, 4 triangular faces and a square base
  • Square units (1) group of squares with sides one unit in length, which are laid on top of an object to measure its area
  • Square units (2) a unit of distance when multiplied by itself becomes a measure of area 
  • Surface area the sum of the areas of all the faces of a three-dimensional figure
  • Three-Dimensional Figures 3D geometric objects with length, width and height; they are “solid”
  • Translation when a figure moves without changing its appearance, see slide.
  • Triangular Prism 3D figure with 6 vertices, 6 straight edges, 3 rectangular flat faces and 2 triangular flat faces
  • Triangular Pyramid 3D figure with 4 vertices, 6 straight edges, and 4 triangular flat faces
  • Turn to make a figure revolve around a point, Rotate
  • Two-Dimensional Figures 2D geometric objects with only length and width; they are “flat”
  • Vertex point where at least two straight lines meet (flat figures) or three straight edges meet (solid figures); plural is vertices
  • Volume measurement of the amount of space occupied by material; expressed in cubic units
  • Week period of time consisting of 7 days; not related to solar or lunar activity
  • Weight measurement that describes how heavy an object is; due to earth's gravity
  • Yard unit of length equalling 3 feet; 36 inches; .9 meters
  • Year time period of 365-366 days or 12 months; based on rotation of Earth around the sun
  • Zero Property: Addition any number added to zero has itself as the sum
  • Zero Property: Multiplication any number multiplied by zero has a product of zero

Thursday, November 18, 2010

Arithmetical Words, Part IV

The phrase "Mind your P's and Q's" comes to my mind today, as we get closer to the end of the alphabet and arithmetic words. Many different origins have been suggested for this slang phrase, but it seems to have always been used to encourage people to be careful.

Today we will look (carefully) at the arithmetic words beginning with P and Q
  • Parallel Lines lines that never cross, no matter how far they extend; always the same distance apart from each other
  • Parallelogram quadrilateral whose opposite sides are parallel and congruent
  • Pattern regularly repeated arrangement of letters, numbers, shapes, etc.
  • Pentagon polygon with exactly five sides
  • Percent ratio that compares a number to 100 using the % symbol
  • Percent Pie Graph a circle graph where the sum of the percentages in each section equals 100 percent
  • Perimeter distance around a closed figure
  • Permutation one of several possible orders for a series of events or items
  • Perpendicular Lines intersecting lines that form “square corners” or right angles (90 degrees) where they cross
  • Pi (π) the ratio of the circumference of a circle to its diameter; approximately equal to 3.14, or 22/7
  • Plane Figure has only length and width; Two-Dimensional or 2D Figure
  • post meridiem label for time from noon up to, but not including, midnight
  • Polygon plane figure made up of 3 or more straight lines
  • Positive Number a number greater than zero
  • Prime Factor a factor that is also a prime number
  • Prime Number a number that has itself and one as its only factors
  • Probability the likelihood that a future event will occur; expressed as a value between 0 and 1, with 0 being impossible and 1 being certain
  • Product name of the result obtained by multiplying two or more numbers together
  • Property of One any number multiplied by one has itself as the product
  • Pythagorean Theorem For any right triangle, the area of a square constructed along the triangle’s longest side (the hypotenuse) is equal to the sum of the area of squares built along the other two sides
  • Quadrilateral a polygon with 4 sides
  • Quart standard unit of measure for volume; 32 ounces; slightly more than a liter
  • Quotient number resulting from a dividend being divided by a divisor; solution to a division problem

Tuesday, November 16, 2010

Arithmetical words, Part III

This is day three of the Great Arithmetic Glossary Series, where I am trying to create the most concise definitions possible for elementary math words and concepts.

These definitions are estimates; I am rounding the meanings slightly in order to save space and words. It would be possible to carry out the definitions to many significant digits!
  • Leap Year 366-day year created by adding an extra day (29th) in February; helps compensates for uneven motion of the Earth around the sun 
  • Least Common Factor smallest factor of two or more numbers
  • Least to Greatest arrangement of numbers from lowest value to highest value
  • Length distance along a figure’s longest side from one end point to the other
  • Less Than number of smaller, or lesser, value than another number; symbol <
  • Likely Event event with a probability of greater than 0.5 but not certain or equal to 1.0
  • Line 2D straight path extending infinitely in both directions without any endpoints
  • Line Graph diagram where plotted data points form a line that (usually) shows change over time
  • Line of Symmetry line that divides a figure so each portion is a mirror image of the other
  • Line Segment portion of a line that has two endpoints
  • Lowest Common Multiple the multiple of two or more numbers that has the least value
  • Mean description of a set of values calculated by adding the values and dividing their sum by the number of items in the set; Average
  • Median description of a set of values obtained by putting the values in order from least to greatest and selecting the middle value (for an odd number of items) or by calculating the mean of the two middle values (for an even number of items)
  • Mode description of a set of values obtained by selecting the value within the set that occurs most frequently; a set may have more than one mode
  • Month unit of time containing 28-31 days; 1/12th of a year; based on lunar motion around the Earth
  • Multiple product of two whole numbers
  • Multiplicand factor being multiplied in a multiplication problem
  • Multiplier factor by which the multiplicand is multiplied in a multiplication problem
  • Negative Number number less than zero
  • Numerator portion of a fraction that is written above the line; it represents parts of a whole
  • Obtuse Angle angle that measures more than 90 degrees and less than 180 degrees
  • Octagon polygon with exactly eight sides
  • Odd Number number that cannot be divided into two equal groups; ends in 1, 3, 5, 7 or 9
  • Open Figure begins and ends at two different points
  • Order of Operations rules used to determine the sequence of performing addition, subtraction, multiplication and division in an equation
  • Order of Symmetry number of different positions to which a figure can be rotated to match itself exactly
  • Ordered Pair pair of numbers used to locate a point on a coordinate grid; horizontal (x-coordinate) is given first and vertical (y-coordinate) is next
  • Ordinal Number whole number that indicates sequential position: first, second, third, etc
  • Origin intersection of x- and y-axes on a coordinate grid; designated as a point (0, 0)
  • Outlier value in a set that is an extreme deviation from the mean value
Stay tuned for more math words tomorrow!

Arithmetical Words, Part II

I decided to devote this week's blogs to the terms we use for elementary arithmetic - in an attempt to define them clearly in as few words as possible. Read Arithmetical Words, Part I.

I should mention that some of these definitions are unique to US English. The meanings may vary in other English-speaking countries around the world. Math symbols also vary - for example, the function of the decimal "point" (as we call it) is performed by a "comma" in other countries. For a fun diversion you could visit Jenny Eather's Maths Dictionary for Kids (she is based in Australia)

You are welcome to download an illustrated, expanded version of this glossary in English or Spanish from the Excel Math website.

Our next group begins with the letters D through K
  • Day unit of time containing 24 hours; equal to one rotation of the Earth
  • Decimal (1) symbol used to separate whole numbers from smaller parts of a whole
  • Decimal (2) symbol used to separate dollar amounts from cents
  • Decimal (3) word that refers to 10
  • Decimal (4) math system based on 10 different digits
  • Decimal Number number with a decimal point; not a whole number
  • Deductive Reasoning logical process; begins with evidence and draws a conclusion; used to solve word problems
  • Denominator portion of a fraction written below the line; the total number of parts into which a whole number is divided
  • Density (1) ratio of the weight or mass of a material to its volume
  • Density (2) the number of individual items in a given space or region
  • Diagonal line segment completely inside a polygon that connects two of its non-adjoining vertices
  • Diameter line segment passing through the center of a circle, ending at either side of the circle
  • Distributive Property: Multiplication multiplying the sum of several addends gives the same result as individually multiplying the addends first, then adding the products (A + B) x C = (A x C) + (B x C)
  • Dividend quantity to be divided; beginning number from which repeated subtractions are taken
  • Divisor quantity by which a dividend is divided; amount repeatedly subtracted from a dividend
  • Edge line segment where 2 faces on a 3D figure meet; flat or curved
  • Empty Set a set that contains no items; not the same as a set containing a zero
  • Equally Likely multiple events with the same probability of occurring; probability value of 0.5
  • Equation number statement that includes an equal symbol =
  • Equilateral Triangle plane or 2D figure whose 3 sides are equal in length
  • Equivalent Fractions fractions with the same value expressed using different numbers; 4/8 is equivalent to 2/4 and 1/2
  • Estimate to quickly calculate a number that is tolerably close to the exact answer
  • Even Number number that can be divided into two equal groups; all even numbers end in 0, 2, 4, 6 or 8
  • Exterior Angle an angle on the outside of 2 parallel lines that are intersected by another line
  • Face 2D polygon (plane figure) that forms one side of a 3D figure
  • Fact Family related addition/subtraction or multiplication/division facts involving the same set of numbers
  • Factor number that divides evenly into another number.
  • Factorial product generated by multiplying a number and every positive number less than the number
  • Fahrenheit (F) temperature scale with 180 units between the freezing point (32) and the boiling point (212) of water
  • Flip change in location of a figure that results in a mirror image of the original figure; Reflection
  • Formula a mathematical statement or rule used in calculations
  • Greater Than number of higher, or larger, value than another number; symbol >
  • Greatest Common Factor the largest factor of two or more numbers
  • Greatest to Least arrangement of numbers from highest value to lowest value
  • Height vertical dimension of a 2D or 3D figure
  • Hexagon polygon with exactly six sides
  • Histogram graph where the labels for the bars are numerical intervals; used to compare data
  • Impossible Event event that will not happen; probability value of 0.0
  • Improper Fraction fraction where the numerator is greater than or equal to the denominator
  • Inequality number statement that compares two unequal expressions
  • Integer a whole number and its opposite (-2,-1,0,1,2)
  • Intercept point where a line or curve meets the x- or y-axis on a grid
  • Interest 1) fee charged by a lender to a borrower for use of money
  • Interest 2) fee a bank pays to its depositors; often a percentage of the deposit, calculated over a period of time (5% per month)
  • Interior Angle angle on the inside of two parallel lines that are intersected by another line
  • Intersecting Lines lines that cross at some point
  • Intersection of Sets a group of values or items that are common to all the sets being evaluated
  • Isosceles Triangle triangle having only 2 sides of equal length
NOTE - there are no J or K arithmetic words in our glossary!

Monday, November 15, 2010

Arithmetical words, Part I

Words, words, words, I'm so sick of words. 
I get words all day through; First from him, now from you!

This refrain from My Fair Lady has stuck in my mind for decades. Of course, as a book editor, I have to look at words all day, if not hear them. And not just any old words, but math words.

Can we consider some of A, Bs and Cs of arithmetic today? These are words anyone who studied arithmetic will have learned (although we may have forgotten a few). I'll try to present these complex words along with clear, brief definitions. But first, how about arithmetic itself:

Arithmetic study of quantities resulting from combining and separating integers, decimals and fractions;

Arithmetic branch of mathematics concerned with calculating numbers using the operations of addition, subtraction, multiplication and division in a defined order

Go ahead, test yourself and see how many you know:
  • Acute Angle angle that measures less than 90 degree
  • Adjacent Angles angles that are next to each other
  • Adjoining Sides sides that meet to form the angles of a figure
  • Alternate Exterior Angles outside angles formed when a line intersects two other parallel lines
  • Alternate Interior Angles inside angles formed when a line intersects two other parallel lines
  • ante meridiem title for time from midnight up to, but not including, noon
  • Angle two rays or line segments that intersect or have the same endpoint
  • Angle Bisector line or line segment dividing an angle into 2 congruent angles
  • Arc continuous section of a circle’s circumference 
  • Area the size of an enclosed surface, measured in square units
  • Area of a Parallelogram  = base x height; expressed in square units
  • Area of a Rectangle area = length x width; expressed in square units
  • Area of a Triangle area = 1/2 x (base x height); expressed in square units
  • Associative Property: Addition sum stays the same if grouping of addends changes
  • Associative Property: Multiplication product stays the same if grouping of factors changes
  • Average single number that describes a set of values; could be the mean, median or mode
  • Bar Graph chart where bars represent numbers and display data, such as quantities
  • Base a polygon’s side or a solid figure’s face; usually the bottom, after which the figure is named
  • Bilateral Symmetry a figure that when folded along a line of symmetry forms two halves that are mirror images
  • Celsius metric temperature scale with 100 points between freezing and boiling points of water
  • Center point in a circle an equal distance from any point on its circumference
  • Central Tendency  numerical average; center of a set of values; Mean, median or mode
  • Certain an event that will definitely happen; has a probability of 1
  • Chord line segment connecting two points on a circle’s circumference 
  • Circle closed curve with all points equidistant from a fixed point in the center
  • Circular Base a special side of a cone or cylinder that forms a closed curve.
  • Circumference the perimeter of a circle
  • Closed Figure a figure that begins and ends at one point
  • Combination a possible set of events or items
  • Commutative Property: Addition  sum remains the same if order of addends changes
  • Commutative Property: Multiplication product remains the same if order of factors changes
  • Complementary Angles two angles whose sum is 90 degrees
  • Composite Number number with more than two factors
  • Concave Polygon polygon with four or more sides having at least one internal angle greater than 180°
  • Cone 3D figure with 1 vertex, 1 curved edge, 1 circular base and 1 curved surface
  • Congruent Figures figures with identical angles and sides of equal lengths; of the same shape and size
  • Convex Polygon figure whose internal angles are all less than 180°
  • Coordinate Grid area where multiple points may be located by their horizontal and vertical distance from the origin
  • Coordinate Point location on a coordinate grid described by 2 numbers (2, -3)
  • Cost Per Unit money needed to buy a measured quantity ($1.25 per pound or dozen)
  • Cube 3D figure with 8 vertices, 12 congruent straight edges and 6 congruent square flat faces
  • Curved Edge curved line segment formed where a curved surface meets a circular base
  • Cylinder 3D figure with 2 curved edges, 2 circular bases and 1 curved surface

Friday, November 12, 2010

More than infinite, Part III

This short series of three posts were focused on how stuff accumulates in our lives (computers, closets, garages) - in fact the piles grow towards infinity if you are not careful.

NOTE: Infinity is not a Nissan luxury car (Infiniti), but a math term meaning no limit.

I've written primarily about the million+ files I have on my computers, but I have also been cleaning out the closets, cupboards, drawers and shelves around my home and office. The stuff just seems to keep accumulating. And I'm not a hoarder by any means.

Last week we had a recycling day at work, where we filled a whole pallet with wires, cables, electronic gear and computers. We probably got rid of half the total electronic stuff that sits around, unused. Note - that is NOT including the stuff our tenants accumulate in our warehouse.

As I drove in to work today I thought about the inverse of accumulation. The critical elements of our lives where we count on devices to tell us the level of de-accumulation.  For example, the self-emptying fuel tanks in our cars. We watch the gauge carefully because walking for fuel after running out is very irritating.

You can also run out of fuel on a barbecue or your household heating, but the tanks are larger (and the risk of walking reduced) so we don't have such elaborate gauges.

What else is self-emptying? I found a Greek word - kenosis - that means self-emptying in a theological sense - reducing one's interest in one's self to nothing, so one can be filled with or used by God. An interesting subject, but not math.

What other things are self-emptying? I did a Google search and found these :

  • The "full" moon, because it wanes on its own. 
  • Our checking accounts seem to work that way ... but we are the emptiers
  • Rain gauges catch rain, report it, then empty themselves
  • Patented Swim Goggles allow excess water to drain automatically
  • Trash containers in Stockholm's city square 
  • Boats with self-bailing technology; also basement sump pumps
  • The Uneedavac unit which removes dust and dirt from truck cabs and empties itself
  • De-humidifiers that remove moisture from a room, then empty themselves into a drain
  • Deleted items folders on certain computers that discard files periodically
  • Most animals are "self-emptying" including people
What is the math term for de-accumulation or self-emptying?  Subtraction

Wednesday, November 10, 2010

More than infinite, Part II

Here we are again, checking images on my Mac at work. I'm trying to get closer to infinity, (which is not a number, by the way, but a concept).

When I look for these file types, I find:

jpg files = 113,021
gif files = 2,936
png = 991
tiff = 445
bmp = 208
eps = 195,881  (my favorite) clip art
graphics total = 313,482!

Anyway you look at it, that's a lot of artwork. But as my clip art license insisted that there were 750,000 images on the 35 discs in the package, I must not have loaded all of them.

Going on with the census,

pdf = 31,843
html = 7495
music = 7159
movies = 511
doc = 594 + 111 (.pages)
xls = 227
applications = 189 (90% are from Apple)
xml = 180
txt = 165
presentations = 23

You should be able to tell from these statistics that I am a publisher, not an accountant, or software developer. And if I am ever going to get things cleaned up on my computers, I'll have to call in some experienced helpers to do the work:

While I am at the pool! Tomorrow we have the day off, so see you Friday.

Tuesday, November 9, 2010

More than infinite?

Infinity is a word that implies no limit. "Unbounded time, distance or quantity."
Here's the symbol for it - looks a bit like a sideways 8:   

I use the word infinity today to refer to the amount of data that we are collecting and creating on our personal computers. Lots of it is original material, but the original content is nothing relative to the copies of files that proliferate all over the web.

For example, I spent much of the data sorting, comparing, renaming and organizing photos I have taken over the past 15 years. In total, I had about 50,000 jpg images when I started, and just over 30,000 when I finished. I erased more than 60 gigabytes of data, much of it due to duplicate files that I myself have knowingly or inadvertently created.

How many duplicate files do you have on your computer(s)? Do you know?

On the Mac I have at home, I found some new features (to me) in the Finder called
  • Search For All Images = 130,119 images 
  • Search for All Movies = 256 items
  • Search for All Music = 1456 items
This is on a computer that's rarely used! But it does hold several copies of each of my photos. When I checked our family laptop, I found:
  • Search For All Images = 30,790 images 
  • Search for All Movies = 96 items
  • Search for All Music = 7508 items
I was astonished. Where are they all? Where did all the music come from? I guess from my many many CDs because I haven't purchased or downloaded more than a few dozen music files.

Is my collection large? I think not. After a bit of searching I found a guy who claimed to have 1.3 million images, managed with Picasa on his local machine.

Sheesh! I've hardly got anything by comparison ... and how many are duplicates? I don't know. But I will try to find out. In the meantime, here's an infinity pool, so-called because you can't see a limit or edge to where it ends...

Monday, November 8, 2010

American vs Italian Geometry

I saw a book called the Geometry of Pasta, and decided that it could entertain us today in the math blog. After all, geometry is clearly math. There were several very positive book reviews on Amazon, and one disgruntled review entitled "Another reason to hate geometry."

The reviewer goes on to say, "I was hoping for ... a book that would teach me about the various pastas and how to use them ... the graphic images are artsy, but useless. They are black and white ... look like art deco wallpaper or bed sheets".  

Here's a sample. What do  you think?

You can take a look at all of the pasta shapes here.

Being a writer and editor, I enjoy reading book reviews, and I think it's fair for a reader to be irritated by images of an item if they don't help connect what an item is (identity / name) to what it looks like (appearance / shape) and what you can do with it (utility / cooking).

If, like me, you grew up without Italian relatives or friends, you may not understand pasta from these pictures. I moved to the page with an alphabetical list of pasta and clicked on BUSIATI. It turns out to be a twisted worm-like shape marked 180 x 10 that's good with Pesto Genovese.  I think pesto is a green sauce and Genoa is an Italian town, but I've never had a basic pasta course, and these references baffle me.

(Yes, you are correct. I cannot successfully order pasta in an Italian restaurant.)

Remember last Friday's blog on DysLexia and DisCalculia?

Perhaps I have DisPastalia!

Let's cut through this confusion - here's what we at Excel Math consider geometry:

Identifying shapes by appearance and feel
Identifying straight and curved lines
Finding the inside and outside of a figure

Counting sides and corners
Navigating a maze

Learning terms: parallel, intersecting and perpendicular lines
Learning terms: of plane, figure, polygon, quadrilateral, parallelogram, and diagonal

Learning terms: flat and curved faces, vertices and edges
Learning terms: pentagon, hexagon, octagon and pentagon

Learning terms: rhombus and trapezoid

Recognizing 2-D figures: squares, circles, triangles and rectangles
Recognizing 2-D figures: equilateral, isosceles and scalene triangles
Recognizing 2-D figures: the parts of a circle
Recognizing 2-D figures: right, obtuse and acute angles

Recognizing lines of symmetry
Recognizing 3-D figures: sphere, cone, cylinder, cube, 
Recognizing 3-D figures: rectangular, square and triangular pyramids and prisms

Recognizing patterns
Recognizing when figures are similar or congruent

Recognizing movements: flips, turns and slides
Recognizing patterns in a sequence of figures or shading

Sorting shapes by common characteristics
Changing shapes by moving or removing lines
Drawing shapes from verbal descriptions
Creating shapes using pattern blocks
Finding simple shapes within complex patterns

Determining when figures do and do not belong in a set
Determining coordinate points
Determining if coordinate points are on a given line

Measuring line segments to the nearest half inch or half centimeter
Measuring angles
Measuring vertical or horizontal lines by subtracting X or Y coordinates

The sum of the angles for rectangles, triangles and circlesAssociating 360 degrees in a circle with 1/4, 1/2, 3/4 and full turns

Calculating area of a square and rectangle

Calculating perimeters
Calculating volume of a figure with one or more layers of cubes
Calculating the diameter, given the radius
Calculating the volume of a rectangular prism using the formula L x W x H
Calculating area and perimeter given coordinates on a coordinate grid

Calculating the area of a parallelogram
Calculating the surface area of a rectangular prism
Calculating the area of a triangle

Solving word problems involving area and perimeter

That's no doubt more than enough to make you groan or weep. I'll finish with this:  

Is the formula for the area of a pizza pie expressed as 2 π r, or is it π r²?

Friday, November 5, 2010


I saw a word for the first time today. It might have been coined about 35 years ago.


It literally means counting badly, or perhaps in a more politically-correct phrasing, difficulty with calculations - thus resulting in difficulty learning or doing math.

Here are a few things a person with dyscalculia might experience:
  • Confusing math symbols: +, −, ÷ and × and trouble with written or mental arithmetic.
  • Inability to make change, estimate price totals or balance a checkbook.
  • Trouble reading a clock and judging the passing of time, thus arriving late or early.
  • Inability to navigate or rotate a map where up is North, to face the direction of travel.
  • Inaccurately estimating the size of or distance to an object.
  • Inability to read a sequence of numbers, or transposing them (seeing 34 vs 43).
  • Trouble counting cards or scoring in sports (such as tennis, bowling).
  • Difficulty in activities requiring counting, such as dance steps.
Naturally this interested me, because this blog is about how we do those things as adults. Most people have little trouble with these maneuvers, but about 5% seem to suffer from some impairment.

Next I read about researchers in Oxford who announced that they've put electrodes on subjects' heads and passed a small electric current through the skin. In more precise terms - transcranial direct current stimulation. With a very small sample (15) group they found when the current is applied in one direction, math skills were enhanced; the other direction retarded math ability.

Here's a drawing of the brain from Wikipedia, with red indicating the area where we think math skills originate. This is where they applied the electrical current. A BBC newsman volunteered to do the same experiment - he reported that he neither felt the electrical stimulation, nor improved in his math abilities. That caused him to raise the question himself  "Was there anything in there to be stimulated?"

I think it would be safe to say there needs to be a lot more testing on this concept before we know this has any advantage or utility. And whether it is safe to try it at home. The full report will not be published until later in November.

However, I did see one article which suggested math-test-taking, over-achieving students might try it as an alternative to studying, just as over-competitive athletes take performance-enhancing drugs.

Which would you prefer? Hit the books, or throw the switch?

We could offer an optional electrical adaptor and headset with all Excel Math purchases!

Thursday, November 4, 2010

Are we math-rational?

It seems like Math should be able to help when you have to make a go/no-go decision.

Here are some examples:

  • Your car needs repairs. The garage's estimate approaches the value of the car. The car is 10 years old, runs ok, but needs $3500 worth of work on a variety of systems - do you give the car away, sell it as is, drive it until it breaks down, or fix it? How do you decide?

  • You install some new software. It doesn't work like you thought it would. Do you uninstall it, lose the value of the cash you spent (not returnable) and your time for installation - then go back to the old way? How do you determine the best option?

  • You buy a ticket to a concert. You then remember an important family event on the same date. Do you throw away the concert ticket, try to resell it, give it to a friend, or ignore the family event and go to the concert anyway, knowing you will "pay the price" for it later?

  • You buy a house in a nice area and you it. But the market is down, you owe far more than it's worth, you're struggling to make the payments and you discover you can rent a similar house down the street for much less. Do you hand the house back to the bank, do you take in a renter, or just gradually go broke?

  • You own stock in a company that is struggling. It's done well in the past but you are beginning to doubt that it will appreciate in value or pay its dividends. Do you sell? Hold? Ignore?

  • Does math help us in these situations?  In general, based on 50 years of experience, I'd say It's seldom any help.

    Why? Too many considerations in each of these decisions are not mathematical.

    I'll explain - many decisions are made in a logical manner, based on rational evaluation of the evidence. But in many other cases we are intuitive, or emotional; the options are not easily compared, and we don't have numerical criteria that can be mathematically evaluated.

    I know about car-buying decisions because I worked in automotive publishing for 25 years. No matter how much comparison information you give people in advance, they often as not will choose whatever they want on the day they go shopping. Not what they said they would buy.

    I had the chance to drive a Tesla, the electric sportscar. It was red; it was fast; it didn't handle as well as my current car. I didn't like it; I certainly didn't need it. After my drive the Tesla salesman asked me what car I thought I would buy next. Based on my whims, my shopping habits, and my needs, I said "A Ford Transit Connect."

    He was stunned - "That's $20,000! It's a mini-truck! A commercial vehicle!" That's right. Nothing at all like his product, but certainly a usable option for me and one I liked better than his fast little electric sportscar.

    But not rational. Not mathematical.

    I'm not suggesting math can't be involved in some decisions - it's just not useful in 37.78% of decisions (yes, I made that up).

    Wednesday, November 3, 2010

    Time Crawls When You Are Watching the Clock

    Yesterday I read about artist Christian Marclay's creation of a 24-hour movie, called CLOCK. He has assembled it from thousands of clips from 3000+ movies that have clocks that indicate the time of the action going on. He has been working on the project for 2 years, and has a team of 6 people just watching movies, looking for scenes with a clock or watch. Read more about it here.

    So when you walk in the gallery in London and look at the screen, some actor is up there doing their bit with a clock on the wall that matches the real time of day. Here's a typical scene that he selected:

    This CLOCK concept appealed to me, because I recently watched the 50's movie The Ladykillers and noticed the focus held on this nice watch in one scene. The bad guys were synchronizing their watches before robbing a train.

    Since math class is where we teach kids to tell time, using digital or analog clocks and watches, for today's blog I decided to make my own sequence of time. This is much easier than Marclay's task, as I could take still photos of clocks that are scattered around our offices. And I didn't plan to take 1440 shots (24 hr x 60 min = 1440).

    While I was doing it, our power went out! Most of the clocks run on batteries, so they're ok, but the computer doesn't do so well without electricity. So I had to go home early and finish on another machine. Here's my mini-art project:

    Tuesday, November 2, 2010


    When regular people talk about curves, they usually are thinking of people who are "traditionally-built", like Precious Ramotswe, the No.1 Lady Detective in Botswana.

    (If you don't know her, go to a bookstore or check your video sources. I promise you won't be disappointed.)

    Alternatively, they may have a winding road in mind:

    When mathematicians talk about curves, they are referring to lines that can be defined using a math formula.

    I found a website this weekend - 2D Curves - that shows 874 different types of curved lines, and explains how they are created.

    People ask me how I can find this strange stuff on the web. Well, in this case I was looking at a new Cartier watch, on which the "Durer's Folium" curve resides. The second hand travels that weird shape in the middle, rather than just spinning around an axis:

    I had no idea what a "Durer's Folium" curve was, so I went on a quest. Here's an animated illustration of what I found:

    If you understand French, you can read the MathCurve site where this image originated. They generously gave me permission to reuse this animation. (See yesterday's blog.) If you don't know French math terminology, you can still enjoy the other graphics and animations you'll find there.

    When we talk about curves in school, we often mean a "Bell Shaped Curve" which represents a distribution of test scores. The technical name for this curve is the Probability Density Function. Here's one:

    Do we teach this in Excel Math? Yes and No. We do teach how to plot lines on a grid, but we don't quite reach the level where students are doing algebraic calculations of curves.

    Monday, November 1, 2010

    How fast can you write a blog?

    I've been asked how hard it is to blog about math.

    Is it Hard? or Not Hard?

    My daily blogging can take from 30 minutes to 4 hours. Most of the time, it's not as hard as writing math curriculum!

    First I have to come up with an idea. A concept. It could be a family or group of concepts, such as fractions, ratios, division, etc. This is blog 311 for me and I haven't run out of ideas yet. It's best to think 3-4 days in advance but I seldom do that unless I am going on vacation or a business trip and have to write many blogs in one day.

    Last Friday it was "How many gallons of water have we put down this sink in the last 60 years?  (which made the drainpipe get so clogged up!)"

    Today the topic "How Fast ..." is my alternate, because the primary topic got stalled (more on this later).

    After I have an idea I start writing. Full speed, straight out. I start throwing words at the screen. This is the easiest part of the job for me. I have been writing daily at work and home, for over 40 years. I hope it shows.

    I think a lot about the illustrations, which are very important in my blogs. In many cases I take the pictures myself, so I may be wandering around our offices or the streets, looking for good subjects.

    Photo: Here's a typical photo I took with a tripod and timer. I'm wearing the cat face; my wife is the goat (not dog). This was used to show the range of colors that your eyes can distinguish. Naturally we had to get the shirts, the background set up, the photo in focus, etc. I have taken about 50,000 photos, and they're all on my hard drive:

    Clip Art: Here's an image assembled from clip art from my publishing archives. I have licensed the use of 750,000 images in order to have the raw material to make something like this - a man gazing into a crystal ball and seeing old ladies in the future. His mom, or wife perhaps?

    Mixed Media: This is a compilation of a photo taken of a blueprint and a ruler taped to my whiteboard. It  demonstrates how scale drawings work. In hindsight, it's boring.

    Here's another compilation of text and graphics to help show how fuel economy calculations vary from country to country.

    Permissions: As you can imagine, it takes time to find and construct graphics. It would be nice to just be able to think of a picture and have it appear on my page. That's sometimes possible using a source like Wikimedia that generally allows reuse, but in most cases it's considered bad form to borrow art without asking for permission. There are lots of exceptions but it is better to ask than to reuse (steal) artwork.

    (This is why I have an alternate blog topic today - I want to use some art for another topic but I need to wait for a reply to my request for permission.)

    After the blog is written I have to make sure it looks good and it's grammatically and mathematically correct. This is sometimes difficult, as I write and edit most of my own material. My wife reads it every night so there may be midnight corrections at times!

    If you are using a browser like Firefox or Safari, I have checked its appearance for you. If you read this with an RSS feed, then most of my layout work is discarded (sigh!) and you see text with art without much formatting.

    For example the headings in this post are bold and red. I do this formatting stuff last. If the post is going to contain multi-colum tables, animated GIFs, embedded YouTube videos, etc. I have to be careful and it takes much longer to make sure all the HTML coding works. Yes, there is code in this. It's a computer!

    I use Blogger to write, post and host this blog. Blogger is fairly reliable. A couple months ago Blogger started NOT working with Firefox and I was unable to determine why. Not wanting to mess up all my other inter-related accounts with Google, I switched to Safari to write all the blogs. In the last week, Firefox seems to be working again. Today I am using Safari.

    Finally I post the blog and read it. Unlike artists who perform a song then never listen to their own albums, I like to read what I write. What fun would it be to create something you don't enjoy?

    Believe it or not the "blog in the clouds" is not invulnerable. We have to back up our own work. I download the whole blog every couple months and save it on my drive and to an optical disk.

    Here I am with Janice Raymond, who created the Excel Math concept and founded the company.
    Janice and I have written virtually everything that has ever appeared in Excel Math.