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Properties that are damaged by fire, water, storms, smoke, or mold require the services of a professional.  This is a job for Nate Dawson, Restoration Hero and President of Sterling Restoration.  Read on to see how he uses math to restore damaged properties back to mint condition.

Can you explain what you do for a living?

Sterling Restoration specializes in emergency repair to real property whether damaged by fire, water, storm, smoke or mold. Sterling Restoration is trusted for high quality and comprehensive cleanup, mitigation, and restoration services for both residential and commercial projects. We are a locally owned company based in Springfield, Ohio serving the Miami Valley and Central Ohio areas. We take pride in knowing that our team of professionals and extensive network of resources have the expertise to return any property to its pre-loss condition as quickly as possible.

When do you use basic math in your job?

Basic math is used in all aspects of our business including our accounting, estimating and production departments. Our accounting department uses it to calculate payroll, receivables, and payables. Our estimators use math more than anyone in our business. During the estimating process for reconstruction, we use square footage formulas (L x W) for calculating materials used, for example:  subfloor framing, roof framing , insulation, drywall, painting, etc.. We use square yard formulas (L x W/9) for calculating vinyl floors and carpet. Basic algebra formulas are used for calculating rafter lengths based on the rise and run of roof slopes.

One of our most interesting uses of basic math, and one I will focus on going forward is with water mitigation (returning a structure to dry standard). Basically, drying a wet building! Once we determine the affected area we then use a cubic footage formula (L x W x H) along with the extent of saturation to know how much dehumidification is needed. Dehumidifiers are rated based upon how many pints of water they are capable of removing from the air within a specific amount of time (AHAM Rating). Therefore, depending on the type of dehumidification used and it’s rating, we are able to determine the number of dehumidifiers we need to dry a structure within the standards of our industry (S-500 ANSI approved standard). We also use the atmospheric readings to determine whether we are creating the desired conditions required to remove water from affected materials and to determine the effectiveness of our equipment. To do this we use the temperature and relative humidity to determine specific humidity (the weight of moisture p/lbs of air) and dew point (the temperature at which water vapor will begin to condense). The formula we use to determine the number of dehumidifiers needed is as follows:

Step 1 – Determine Cubic feet (CF).

Step 2 – CF/Class Factor(a low grain refrigerant dehu has a class factor of 40 in a class two loss) = # of AHAM pints needed.

Step 3  – AHAM points needed/Dehumidifier rating = number of dehumidifiers needed.

I know! It’s starting to sound a little complicated but it is all basic math.

Do you use any technology (like calculators or computers) to help with this math? Why or why not?

Absolutely! Even though we are in the building trade we are not in the dark ages. We use the most advanced estimating system designed specifically for the insurance restoration (property repair) business. After in-putting the dimensions into a sketch type format, this system automatically calculates all the square footages, cubic footages, and linear footages. The next step is to add a specific line item. For example, when you add drywall to your estimate  it uses a current square foot price to calculate how much to charge for hanging, taping and finishing the drywall in your project. It will also calculate how many sheets of drywall, how many fasteners are needed , how much drywall tape, and how much joint compound is needed. Finally, it will calculate the material sales tax and any state sales tax on the service.

How do you think math helps you do your job better?

I do not feel it’s a matter of doing my job better. I simply could not perform my job without math! As I stated earlier, we use math in every aspect of our business. I do not feel there are too many moments throughout the day that I am not using some form of math.

What kind of math did you take in high school?

During my high school years I completed algebra and some trigonometry. If I remember correctly, that was all that was offered (yes, I graduated high school 32 years ago). Once leaving high school I furthered my math education in mechanical engineering. In my opinion, the levels of math being taught in high school today are far superior to what was then taught.

Did you like it/feel like you were good at it?

I feel like there are individuals that have an aptitude for math and those who do not. Math will obviously come easy for those who have this aptitude. I would also say that if you are good at something, the chances of enjoying it are far greater than if you are not good. Having said that, I do not believe I had this aptitude. Therefore, I had to work a little harder than others, and, at best, I was average at math. Guess where I’m going with this…no I did not like it.

Did you have to learn new skills in order to do the math you use in your job? Or was it something that you could pick up using the skills you learned in school?

I had to learn how to use the math skills I had already acquired to accomplish the task at hand. For example, if you have the lengths of two sides and the angle of a triangle, you can calculate the length of the third side. It is crazy how much I use this algebraic formula; however, it took some time and experience to learn how many applications this formula has. Having said that, ninety percent of my daily tasks require math learned in high school.

Are you interested in learning more about restoration? Let me know and I will pass your information along to Nate.

A Math for Grownups follower asked me earlier this week to define algebra, and I thought that was an excellent place to start this month-long discussion. I think that most people might be surprised by what is generally found under the algebra tent. The basic definition is pretty broad:

Algebra is a branch of mathematics that uses letters and other symbols to represent numbers and quantities in formulas and equations. This system is based on a given set of axioms.

What does this mean? Well, it’s basically the step beyond arithmetic, where we only deal with numbers. Algebra allows us the flexibility of an unknown — the variable — so that we can make broader statements about situations.

Look at it this way: 8 + 3 is always 11. Always. But 8 + x depends on the value of x. This means we can pretty much substitute whatever we want for x. See? Flexibility. (Of course 8 + x has no meaning without some kind of context. But we’ll get to that later in the month.)

Algebra allows us to discover and create rules. These rules might be formulas or equations that describe a particular situation. Because of algebra, we know that the circumference of a circle is 2πr, where π is the number 3.14… and r is the radius of that circle.

Now, let’s take this definition one step further. What is the circumference of a circle with radius 1?

C = 2πr = 2π(1) = 2π

But what about the circumference of a circle with radius 2?

C = 2πr = 2π(2) = 4π

If you look closely at this, you can draw a conclusion: The larger the radius of a circle, the larger its circumference. When the radius is 1, the circumference is 2π; when the radius is twice as long, the circumference is twice as big.

This points to a critical aspect of algebra: relationships.

Algebra is a branch of mathematics that deals with general statements about the relationships between values, using numbers and variables to describe them.

The formula for the circumference of a circle is a description of the relationship between the circumference and the radius of any circle. When the radius changes, so does the circumference. When the circumference changes, so does the radius. (π is a constant, even though it is technically a Greek letter. Whenever you see π, you know you’re dealing with the number 3.14…)

So that’s it. Algebra is nothing more than a way to describe the relationships between values (numbers, measurements, etc.). In the example of circumference, we’re dealing with two branches of math. The geometry describes why the circumference is twice π times the radius. The algebra is how we describe that relationship in the form of a formula.

Without algebra, we really don’t have ways to describe many things about our lives — from geometry formulas to finding compound interest on a loan. We can fumble around and come to a conclusion, but in the end, algebra can make this process much simpler.

What do you think about these definitions of algebra? Does thinking about algebra in these ways make it a little less threatening? If so, how? Share your ideas in the comments section!

While the development of numbers continued for many, many centuries, even before the discovery or invention of zero, the Greeks were responsible for a long, long period filled with mathematical advances. By 600 B.C., a fellow named Thales of Miletus brought Babylonian mathematical discoveries to Greece, which were used to calculate distance and other measurements.

But the big player in Greece was Pythagoras. (Yes, you should recognize that name.) Born in 580 B.C. in Samos, he met old-man Thales when he was but a young lad. Perhaps Thales convinced him to travel to Egypt so that he could learn the mathematics of the Babylonians. At any rate, when Pythagoras returned from his journey, he settled in Croton (which is on the eastern coast of Italy) and this is where things get strange — at least by our modern standards.

Pythagoras established a philosophical and religious school that was made up of two societies: the akousmatikoi (hearers) and mathematikoi (learned). And while his followers look much like a cult to us, Pythagoras was in fact developing the world’s first intentional, philosophical society. Members — both men and women — were intent on living a contemplative and theoretical life, and as such divorced themselves from the culture at large, becoming completely devoted to philosophical and mathematical discovery.

But in order to do this, they had to follow a very strict set of rules, which included vegetarianism, giving up all personal possessions and absolute secrecy. And then there are the really strange orders: do not pick up something that has fallen; do not touch a white rooster; do not look in a mirror beside the light.

That’s not all. Mysticism infused almost all the Pythagoreans did, which led to some really off-the-wall mathematical ideas, like their understanding of numbers.

  1. Nothing exists without a center, and so the circle is considered the parent of all other shapes. It was called the monad or “The First, The Essence, The Foundation, and Unity” — or according to Pythagoras, “god and the good.”
  2. The dyad was a line segment and considered to be the “door between One and Many.” It was described as audacity and anguish, illustrating the tension between the monad and something even larger.
  3. And then there’s the triad, which of course represents the number 3. Continuing in their pseudo-anthropomorphism of numbers, the triad is considered the first born, with characteristics like wisdom, peace and harmony.

I could go on. Seriously. But while the ideas of the Pythagoreans were kind of kooky, this band of deep-thinking brothers and sisters advanced mathematics in some pretty significant ways. First of all, they began classifying numbers as even and odd, prime and composite, triangular, square, perfect and irrational. Through their strange ideas of numbers, they popularized geometric constructions. They are also attributed with the discovery of the five regular solids (tetrahedron, hexahedron, octahedron, iscosahedron and dodecahedron).

But their biggest discovery is the theorem named for Pythagoras. The Pythagorean Theorem states that the in a right triangle, the square of the longest side is equal to the sum of the squares of the remaining two sides. In other words:

This is more than just a silly formula you needed to memorize in high school. Carpenters use it to be sure that they have right angles (in other words that their door frames, decks, and walls are “square”). It’s useful to find the diagonal of a television set (which is how those contraptions are measured for some reason), if you only know its length and width. And it’s the basis of a great deal of additional math discovery, like the distance formula and various area formulas.

It’s a big, honkin’ deal. And in some ways, we’re lucky it survived the secrecy of the Pythagoreans. Pythagoras wrote nothing down. (If tin foil had been invented, he might have been wearing a hat of the stuff.) But despite its closed society, this cult of nutty mathematicians and philosophers is considered one of the most important influences in all of history.

What do you remember of Pythagoras from your high school geometry class? Have you used the Pythagorean theorem in your everyday or work life? If so, how?

On Monday, Cristina Santiestevan of Outlaw Garden shared a post about the geometry of gardening, introducing us to the idea of “hexagonal spacing.” This was such a cool idea that I thought I’d explore it further. I wanted to know the math behind it. In other words, why are hexagons so darned special?

Let’s start by with the bees. In research for a magazine assignment, I’ve done some reading about bees lately, and once again, I’m in absolute awe. These little guys are the linchpins of our ecosystem in a lot of ways. Not only does their pollen-collecting insure the reproduction of a variety of plant species (and therefore the survival of critters that depend on these plants), but their colonies are efficient little factories that seem to mirror human manufacturing — from the dance the workers do to relay directions to the best pollen to the efficiency of their job descriptions.

And then there are the hives. If you think of the bees as efficient — and they are — you can deign why the hive is made up of tiny hexagons. (Remember, a hexagon is a six-sided figure.) Not wanting to waste any space whatsoever, the bees figured it out: instead of making circular cells, which leave gaps around the sides, they create a tessellation of hexagons, which leave no empty space at all.

Photo courtesy of wildxplorer

(A tessellation is the repetition of a geometric shape with no space between the figures. Think M.C. Escher or a tile floor.)

The same concept applies to gardening. Why waste space? As Cristina pointed out, choosing a hexagon-shaped planting scheme, you’ll get more plants in your beds.  And if you’ve got an outlaw garden, like Cristina, it’s best to make the most of your space! Here’s how:

In regular rows, you plant 6″ apart in only two directions, getting nice, even rows. But if you consider six directions, you’re replicating the hexagon, instead of a square — and as a result maximizing your space (just like the honey bees). Cristina describes it as planting on the diagonal. Or you can think of each plant at the center of the hexagon.  Then you can plant the others 6″ from the center in six directions — creating the vertices of the hexagon. (If you’ve ever looked carefully at a Grandmother’s Flower Garden quilt pattern, this idea might jump out at you. Not only is each plant the center of a hexagon, but it’s also the vertex of another hexagon.)

Drawing courtesy of Cristina Santiesteven

Did you see what I did there? Math can be described in a variety of ways! Look at the second diagram carefully, and see what jumps out at you — the hexagons or the diagonal rows?

So there you have it. We can learn a lot from a bee. And I can already think of times when this can be useful in other areas. How many more cookies can you fit on a cookie sheet, if you arrange them diagonally (or in a hexagon shape) rather than horizontal rows? What about kids desks in a classroom?

Where can you apply the hexagon to make your space more efficient? Share your ideas in the comments section!

Picasso’s Violin and Grapes (Photo courtesy of Ahisgett)

When my brother Graham was in kindergarten, he learned a little bit about Pablo Picasso.  And so my mother decided to take the whole family to a touring Picasso exhibit at the Smithsonian, which featured five or so of his paintings, including some of his most famous examples of cubism.

My brother is a man of few words, and he wasn’t any different as a little boy.  He quietly walked around the paintings, looking intently at them and being careful not to cross the red velvet ropes that kept out curious hands.  Nearby, we were all watching Graham, wondering what in the world he was thinking.

That’s when he stepped back from one of the paintings and said, “Oh, I get it.” We waited for something insightful. He pointed to the velvet ropes and said: “The paint is still wet.”

Cubism is not the easiest kind of art to understand.  But you have to admit — whether you like it or not — cubism catches the eye.

In cubism, objects are deconstructed, analyzed and reassembled — but not necessarily in their original order or size. When this is done in painting, the result is a three-dimensional object reassembled in a two-dimensional space, without regard to what can actually be seen in the real world.  So while you can’t see the back of a violin when you’re looking at the front, Picasso may depict the back and front at the same time in the same two-dimensional space.

Freaky, right?

I’ll leave it to the art experts to explain why this works.  But I can talk a bit about the

Portrait of Pablo Picasso by Juan Gris (Photo courtesy of Raxenne)

the geometry of cubism.

First, you need to know that cubism has its roots in the work of Paul Cezanne. He began playing with realism, saying he wanted to “treat nature by the cylinder, the sphere, the cone.”  In other words, he began replicating these figures as he saw them in his subjects.

Henri Matisse, Picasso, and others took Cezanne’s approach even further.  It’s not hard to recognize the cubes and angles and spheres and cones.  But it’s the flattening of three-dimensional space and disregard of symmetry that really distinguishes cubism from realism or impressionism.

Symmetry is a very common occurrence in mathematics.  From symmetric shapes to the symmetry of an equation (remember: what you do to one side of an equation, you must do to the other!), it’s fair to say that when symmetry is absent, it’s a big deal.

And the same is true for nature, the most often referenced subjects in art.  A face, a water lily, the body, a beetle — you could spend all day finding symmetry in the natural world.  Cubism turns this notion on its head.

And still, the pieces are compelling.  It’s that dissonance that draws our attention and even illustrates difficult subjects. (Picasso’s most enduring and controversial pieces is Guernica, a large painting depicting the Nazi bombing of a small Spanish town.)  The artists do this by breaking traditional rules and ignoring some mathematical truths.

Do you like cubism? Have a favorite artist? When you’ve seen cubism in the past, did you think of it mathematically? Buy the math books that will help you learn math for practical purposes, the math that you will use in your everyday life.

In my interview with painter, Samantha Hand, she mentioned something called the Rule of Thirds. I’ve heard of this, but I honestly had no idea what it was about.

Turns out the Rule of Thirds isn’t really about thirds, per se. Instead it’s about ninths. The idea is to divide the image into nine equal parts — something like this (Photo Credit: Lachlan via Compfight cc):

There are a couple of things to notice here. First there are exactly nine rectangles inside the one rectangle — forming a 3 x 3 grid. Second, all of the smaller rectangles are congruent, which just means they are the same size and shape. Last, each of the smaller rectangles is proportional to the larger rectangle.

What does this proportional thing mean? It’s simple, but let me explain using some numbers. Let’s say that the photo to the left measures 12 in by 6 in. (It probably doesn’t but stay with me.) From that information, we can determine the dimensions of the smaller rectangles: 12 in ÷ 3 = 4 in and 6 in ÷ 3 = 2 in. So each of the smaller rectangles is 4 in by 2 in.

If the small and large rectangles are proportional, they’ll have the same ratio. Let’s take a look:

12/6  =  2

4/2  =  2

This ratio that they have in common has a fancy name: the scale factor. (And if you know anything about drafting or making scale models, that will be familiar.)

Now before we get too far into this, let me say that Samantha — and most painters and photographers who might use the Rule of Thirds — isn’t thinking about proportion and scale factor. But this a good example of when proportions are important and intuitive.

So getting back to the Rule of Thirds — according to some research, people’s eyes are naturally drawn to where the grid lines intersect. A painter can use this information to draw viewers into the painting, especially if there are surprising elements or those that should have more emphasis. Take a look at Da Vinci’s The Last Supper.

Image courtesy of Atelier Mends

Notice how the table itself sits along the bottom horizontal line. The left vertical line crosses Judas, Peter and John, and the right vertical line crosses Thomas, James and Philip. Interestingly, the greatest tension in the piece is at these two points, while Jesus occupies the exact center of the painting with a calm demeanor.  Whatever your religious beliefs are, the story this painting tells is furthered by Da Vinci’s use of the Rule of Thirds.

In a couple of weeks, you’ll meet a photographer who probably also uses the Rule of Thirds in her work.  In the meantime, see if you can superimpose an imaginary 3 by 3 grid over your favorite paintings or photographs.  How does the Rule of Thirds draw you into the piece? How does it help you notice important or surprising details?

Have you noticed the Rule of Thirds in paintings that you love?  Share your thoughts in the comments section!

A few years ago, I got this idea that I wanted to learn how to sew.  My mother in law bought me a lightweight machine at a yard sale for $10. So I decided it was time to teach myself how to sew.  How hard could it be?

Turns it, not so much, when you have the internet at your fingertips.  With a few searches, I unearthed great Flickr tutorials for zipper pouches, blog posts with step-by-step instructions on how to make box bags and a really, really amazing month-long series of fat-quarter projects on a blog called Sew, Mama, Sew!

One of my zipper pouches made from ModGirls Sis Boom by Jennifer Paganelli

A fat quarter is a piece of 18″ x 22″ fabric.  In most cases, it’s a quarter of a yard, but not cut from one side of the width to the other.  And it’s a cheap and easy way to buy those gorgeous designer fabrics, like Amy ButlerModa and Alexander Henry.  This is a big deal, because I was quickly realizing that I’m a fabric addict.  The editors and contributors at Sew, Mama, Sew had great projects for fat quarters: purses, journal covers, pin cushions, crochet hook rolls and even fabric boxes.  I was in heaven!

When I published Math for Grownups and learned that a virtual tour would be a great way to promote the book, my first thought was that maybe — just maybe! — I could guest post on Sew, Mama, Sew.  Imagine my surprise when the editors there jumped at the chance.  I felt like I was one of the cool kids.

And today is the big day. You can read my guest post, “Nothing but Net,” which talks about how we can mentally (and physically) translate 2-dimensional figures into the 3rd dimension.  That’s what patterns are, after all.  Even if you don’t sew, this skill is a great one to have!