Flip, Slide or Turn?

Our study of horizontal, vertical and diagonal would not be complete without recognizing that there are ways to move a shape to another orientation or position. This process is called TRANSFORMATION. There's a genre of popular children's toys called Transformers. They do flips, slides and turns. Let's see what those words mean.

We call these moves by special names, and they follow agreed-upon rules, so all of us who modify a shape can do so in a repeatable fashion. Why bother? Because mathematicians (people) like consistency.

If I ask you to create a right triangle:

• of a certain size
• with the right angle to the right side and the hypotenuse at the top
• using 1 point black lines
• fill the figure with green
• slide it 2 inches to the right
• rotate it 90 degrees

I want your work to look like the figure to the extreme right. The red dots are there to indicate a point of rotation. The line shows the "hinge" about which it is reflected

You might be doing this with pencils. No problems for you, especially if you have an eraser!

Or you might have software, like the Adobe Illustrator program I use to create Excel Math problems. If you use software, you'll be hunting around the menus frantically wondering what to do. The process to follow goes like this on my machine:

1. Create a triangle by using the polygon shape tool set to 3 sided-figure. Click and there it is!
2. Move the vertices around to get the right triangle shape you want.
3. Fill with Mint Julep Green (I can't match that green with browser-friendly Internet text colors).
4. Click and Drag (slide/move) it to the right.
5. Go to the palette (menu) and select Object/Transform/Rotate and enter 90, press ENTER

To read more about flips, slides and turns, visit our April 30, 2012 blog post and download a free math worksheet. These are the transformations we teach to elementary school kids in math class. There are others.

For example, shear slants things. Scale changes the size. Transform each lets you select whether to change angles, size in vertical or horizontal directions, etc.

Finally, although these are plane figures (on a flat surface) we can still arrange them in "vertical space" or layers. That involves some instructions regarding whether to put this shape in front, middle, back, etc.

Center of Gravity

My last few posts have dealt with concepts known as vertical, horizontal and diagonal. Because vertical is defined as the direction of the pull of gravity, it raises the question, On what does the gravity pull? Or where? We had to invent a concept called the center of gravity.

For the purposes of calculation, we define this as a point where the mass of the item is concentrated This is NOT necessarily the geometric center of the item.

For example, this photo clearly shows that the center of gravity on this car that was NOT in the center of the hoist!

Can we use math to find out what went wrong? I think we can. First we need some facts.

This is a Lotus Elise.

The car weighs 1700 lbs.
The length is 150 lbs.
The wheelbase is 90 inches.
About 32 inches of overhang on the front, and 28 on the back.
The front wheels carry 590 lbs of weight (empty gas tank).
The back wheels carry 1060 lbs of weight.

OK, that should be enough.

Here's an outline of the car, with a big black blob showing the location of the heavy engine. Two-thirds of the weight of the car is on the rear wheels when it is unladen (no people, little gas).

The red spots show where the car is lifted by a jack when you have a flat tire.
The orange in the back shows the alternative lift point when you have to raise the entire car. Notice how much farther back it is on the car, underneath the engine.

I've simplified the drawing by taking out the car and using a wedge instead. This is an estimate to show how the weight is distributed. It allows you to see the vertical red line (center of the car, and center point of the flat tire lifting points). The vertical orange line shows the center of gravity, the point about which the weight is equally balanced.

To raise this car safely, you need to position the rear lift on the orange position.

Or as the shop found out, when you take off a front wheel the car falls off the hoist.

Or a wall may fall down.

Are you Leaning, or Diagonal?

We touched on the horizontal and vertical dimensions of life yesterday. Today I'd like to discuss the diagonal. (You can see I don't have enough fancy font attributes for all these math terms.)

A diagonal is a straight line connecting two non-adjacent vertices of a polyhedron. Or as a layman might say, a slanting line across the middle to opposite corners.

When you are used to vertical and horizontal, diagonal is a bit different. It seems weaker, or less stable, or something. Shall I give you some examples?

Fabrics can be cut and sewed in a diagonal way across the warp and weft threads. This technique is called "on the bias."

That's not the same as having diagonal stripes on your shirt which is pretty rare, I think. No wonder this man's looking confused!


Then there are cutting pliers called diagonal cutters, or dikes. And finally, I found a place that recommends diagonal bookcases. You don't need bookends, because the books are already tipped over.

One thing a diagonal does is ADD STRENGTH to a square or rectangular structure. We teach this in some activities where we ask kids to build things with straws and string.

Here's a tandem bicycle that I built 30 years ago which successfully employed many small diagonal tubes in an effort to improve rigidity. Notice that the diagonals connect sides of the main frame rather than the exact corners. This is due to the complexity of joining the tubes in the corners.

This building by Frank Gehry is leaning, NOT a diagonal!

Vertical and Horizontal

In the last few days I have talked about

1. Counting years, using charts with rows and columns of numbers.
2. Hanging mini-blinds and measuring carefully so they fit.

Today I see that several math concepts were used but not defined - vertical (columns) and horizontal (rows). We start teaching these concepts in kindergarten so kids become familiar with terms that define our relationship with objects around us.

When we draw, we assume the left side of the paper (or screen) to be vertical, and the sweep across from left to right to be horizontal (even if we are balancing a laptop on our knees or tilting our paper at an angle). When creating a math lesson I simply draw a box with Adobe Illustrator, and it is aligned so the vertical sides do go up and down.

Look at the following two shapes. We say that one has a horizontal orientation, and the other a vertical orientation - ABCD is horizontal, and WXYZ is vertical.

OK, so much for theory, in real life how do we determine horizontal or vertical ?

We say horizontal things are at right angles to vertical things.

That's not much help! you protest.

In "the real world" we find that there are very few objects or surfaces that are exactly vertical or horizontal. In fact, since the earth is (generally-speaking) round, what we take to be flat ground may be slightly curved. How can we measure this curvature? How do we determine vertical?
 
Horizontal surfaces are always at right angles to the gravity force exerted by a local astronomical body. Earth's gravity is pulling us downward; assuming we can measure in which direction that force is strongest, we can find vertical. A right angle to that is horizontal.

We have created ways to determine these directions - we can position a stick so it casts no shadow at noon, when the sun is directly overhead. We can drip water. We can use a plumb bob (weight on a string). Even a full glass of water will indicate horizontal and vertical. These tricks have been known for thousands of years.

Now we can use various kinds of bubble levels to closely adjust items to a vertical or horizontal position. Here's a fun animated level that you can buy if you have an HTC mobile phone.

And in today's ultra-high-tech hardware stores you can buy a laser level / plumb tool that helps you in this endeavor. These shoot red lines across the room or yard, enabling you to build things correctly.

Why would you care if things are level or vertical? You might not. Here's an argument against it,

 from Austrian artist/architect Friedensreich Hundertwasser:

The flat floor is an invention of the architects. 

It fits engines – not human beings.
We do not only have eyes to see and ears to hear and noses to smell.
We also have a sense for the touch of our hands and feet.

If man is forced to walk on flat floors 

as they were planned thoughtlessly in designers’ offices, 

estranged from man’s relationship and contact to earth

a decisive part of man withers and dies. 

This has catastrophic consequences for the soul, 

the equilibrium, the well-being and the health of man.
Man’s ability to experience ceases and he becomes disabled,
mentally and organically.

An uneven and animated floor 

is a means to recover dignity of man 

which has been violated 

in our unnatural and hostile urban grid system.

The uneven floor becomes a symphony, a melody for the feet.

It brings back natural vibrations to man.
Architecture should elevate and not subdue man. 

It is good to walk on uneven floors and regain our human balance.

Measuring 101. How hard can it possibly be?

In Excel Math we teach kids about units of measure. They learn about geometric figures. It seems pretty simple to put the two together -  See the shape. Measure it.

THEORY
On a Lesson Sheet it is simple. We ask How large is this rectangle?

They answer, Five units wide and three units high. Perimeter is 16 units and area is 15 square units.

But when you move this exercise to the real world, suddenly things get a lot more complicated.

THE REAL WORLD
Let's say we want to install some mini-blinds on a window in our room. There are lots of narrow blades, all connected together by strings. They can be tilted open and closed, or pulled up and down within the window opening. Some even can be lowered to the bottom OR pulled up to the top.

It's critical to have the correct dimensions of your window openings when you order mini-blinds. It doesn't seem hard, does it? Just measure the height and width of the opening and order the blinds. But it's done incorrectly so often that mail-order suppliers offer "I measured it wrong" liability insurance!

Here's what you are supposed to do.

1. Measure the height of the opening in 3 places across the width of the window.
2. Measure the width of the window in 3 places along the height of the window.
3. Specify the blinds you want using the smallest of the 3 dimensions in each direction.

WHAT COULD POSSIBLY GO WRONG?

The window might not be square. The ceiling, floor, wall trim and sills might not be horizontal. The walls might not be vertical. The window itself may be crooked in the wall. Your tape measure might slip. You might have forgotten to wear your glasses. If you have lots of windows, you might confuse the measurements when putting them onto the order form. You could read the wrong side of the tape (cm instead of inches).

When your new blinds come, you find you have problems at the top, the bottom, with the way they hang and how they pull up and down. Do you send them back, tear out the window, try to re-fabricate the trim, adjust the fit? What can you do?

POSSIBLE SOLUTIONS
The most effective solution if your measurements show your house is flaky is to give up the idea of inside mini-blinds. Put your curtains or blinds outside the window frame! That way, the fit is assured and the gaps are covered. It might not be your first choice, but it can be the best choice.

Sadly this same situation can occur with more substantial items than a mini-blind. Like the new window installed in our kitchen last year. Once they broke out the old window and put the new one in, we saw the new one was too small. Here's how it looked after it was centered and leveled in the opening. There was a lot of extra space at the top, even when lifted and leveled at the bottom.

It's true that ordering a window too large would be a bigger problem, but having this gap meant they had to cover it up with wide trim pieces. And now the glass area of our window is smaller than it needed to be. And it doesn't match the other windows.

EXPLAIN MISTAKES? OR AVOID THEM?

The foreman admitted that they didn't measure correctly. He said two people should have measured each window separately and then compared their results. If the measurements didn't match, the carpenters should measure again until the dimensions match. A single estimator had measured once, subtracted an inch, and ordered a window.

I can hear him thinking, It's not my house. As long as it fits in the hole ...

On the other hand, even when it is your own house, and you do try to be precise, it's easy to read a tape measure incorrectly. If you are doing the job alone, and trying to keep the flexy tape shoved against the far side of the window, while fumbling for your glasses, while peering at the marks on the tape, while scribbling on a scrap of paper (or the wall itself) ... you can see there's good reason for that old saying, Measure Twice, Cut Once.

A new decade, or just a new year?

Happy New Year! Happy New Decade!

I have seen various pronouncements about this being a New Year, but NOT the beginning of a New Decade. Due to various ways of counting, we can't seem to agree on whether it's a decade or not. And I still remember the wailing and gnashing of teeth that happened a decade ago when we entered a new century!

There are a few self-appointed year-numbering cops working hard to set people straight on this. But both they (and the folks ignoring them) seem to be suffering from lots of self-induced angst. Can math help?

Here's what we offer when we teach kids to count. You simply choose whether to start labeling (naming, numbering) at 0 or at one. Remember in this instance we are not counting widgets in a bin, we are numbering or labeling an arbitrarily-chosen group of moon phases, or solar rotations, or something like that...  it's far too complicated to define a year here!

In this first arrangement, each of the rows starts a new group of numbers - the twenties, the thirties, the forties, etc. That's clear. The end of the twenties is at the end of the twenties row. We drop down one row and start counting again in the thirties. The only issue here is remembering to start at zero in the first row.

In the second, more traditional way shown below, the twenties begin at the end of the teens row. I don't like that. And at the end of the twenties row, we are forced into thirty. Why have a new group of ten (a decade) starting at the end of a set? Well, one reason is that 5 comes in the middle this way, instead of 4. Another reason is that this is the way people generally count on their fingers.

 

When you sort files on a computer, the file system usually expects 0 to be first. If you named your files in the conventional way, they'll be out of order in your folder.

Computers usually sort 00, 01, 02, 03, 04, 05, 06, 07, 08, 09, 10.

And they don't keep track of place value automatically, so you need to insert leading zeros to keep your files organized. With more than 10,000 files used to create our Excel Math textbooks, we can't just number those page files any old way and expect to be able to find them later. Especially when we manage to insert a couple instances of the letter o or O in place of 0 zero!

Which do you prefer?

 

Here's my final piece of evidence. My fancy perpetual watch shows the year 2009 here, in a photo taken near midnight on New Year's Eve.

In the next shot, on New Year's Day, it shows 2010. A zero has gone and a one has appeared on the dial in the tens column. Therefore, I say It's a new decade!

But taking a note from the EPA's advice on fuel economy, your decade may vary.