Definitions below are written as source information for teachers to use in class room situations from grades 3 through high school. Perhaps there is a little more here than you wanted to know, but it is well worth reading. You’ll be your school’s expert on right angles! All material here is copyright of the author, James Watt. You may cite this material provided you acknowledge the author. You may also use the illustrations here to decorate your own class websites, if you like.
Simple doesn't mean stupid. It means VERY IMPORTANT. The better a child understands the simple, the easier it is to break down the complex into those well understood simple parts. Teach the simple parts well and children will master the complex. Fail to teach the simple parts well and children, and the adults they become, will fail at all things.
WHAT IS A RIGHT ANGLE?
Before we can say what a Right Angle is, we have to know the meaning of 3 important words; EXTREME, OPPOSITE and DIFFERENT.
Note; Different and Opposite do NOT mean the same thing!
EXTREME - "The edge or limit of some form."
Using the word ‘Extreme’ is to say, "There is nothing of the form past this limit". All form is bound between EXTREMES.
OPPOSITE - "One of the paired extremes of some form."
Extremes are always in pairs and these pairs are always OPPOSITE (i.e. left-right, top-bottom, etc). Opposite extremes are fundamental limits to all forms. This is because even with a simple line, you START drawing a line and then you STOP drawing it. START and STOP are the opposite extremes between which the complete simple line lies. Since complex things are made up entirely of simple parts, all complex forms are also always found between opposite extremes, only.
Incidentally, teaching children that a ‘straight line goes on forever’ is false. That is a corruption of the original Greek postulate which actually says, "A finite straight line may be produced continuously in a straight line." (...that you can extend a line - if you need to) ELEMENTS, Euclid, book I, postulate 2. In advanced mathematics, the letter ‘k’ is always assigned as the theoretical ‘stop’ or end extreme in some abstract linear series.
WHAT GOES FOR THE STRAIGHT LINE, GOES FOR THE CURVE.
"These opposite extremes are all well and good for the simple straight line", you say, "but what about the simple curve?" (A simple curve always becomes a circle.) Welllll.....
"Suppose a plane takes off on a round-the-world trip from the South Pole to the North Pole and back again. The plane flies straight North (actually a simple curve) and does not change direction. As soon as it passes the North Pole, the plane’s compass shows it is flying South. Even on simple curved lines, the extremes are there but may be hidden."
- A North-South straight line is DIFFERENT from a Northeast-Southwest straight line. Their forms are the same (straight lines) but their extremes are ‘not exactly the same’.
- A North-South line is MOST DIFFERENT from an East-West line. The two sets of opposite extremes are mutually exclusive. There is no ‘North-South’ in ‘East-West’ and vice versa.
WHAT IS A RIGHT ANGLE?
"A right angle is any intersection of two simple lines whose opposite extremes are MOST DIFFERENT ."
Simple intersecting lines of the same form but of most different opposite extremes are at a right angle to each other. A Vertical (Up-Down) line is most different from an Horizontal line (i.e. Left-Right). You will never go vertical so long as you are going horizontal. Lines whose opposite extremes are ‘most different’ to each other are ALWAYS at a right angle. Right angles are generally taught as being made of ‘simple straight lines, only’. That is not true - at all.
SIMPLE CURVED LINES CAN INTERSECT AND BE ‘MOST DIFFERENT’ TO EACH OTHER.
Take the example of the surface of the earth; Traveling directly East-West around the Equator is most different from going North-South. The relation of any latitude to any longitude is a right angle.
SIMPLE STRAIGHT AND CURVED LINES ARE ‘MOST DIFFERENT’ TO EACH OTHER.
Remember, a simple curve line ‘HAS a relation to points not in the series’ while a simple straight line ‘DOES NOT have a relation to points in the series’. These two simple types of lines are as ‘Most Different’ by form as they can possibly be to each other. The least these two forms may intersect is once (called tangent) and the most they may intersect is twice (called secant). Notice in this section how important it is that, beyond the 'most different' form character of the lines, the opposite extremes of the lines must also be 'most different'.
A PHYSICS EXAMPLE OF RIGHT ANGLE STRAIGHT AND CURVED RELATIONS.
The force of Gravity (Extremes: Up-Down from the center of the earth) is a straight line force to the 2 most different curved surface measurements of the Earth (North-South and East-West). These 3 very real ‘most different’ lines of measurements of Up-Down, North-South and East-West allow us to accurately measure and map our 3-Dimensional world and space. In geometry this is abstracted (straight-line, only) to the X, Y, Z axes. This is called Euclidean space, after the famous Greek geometer, Euclid. Euclidean geometry is still used by astronomers as the most accurate mapping geometry for space.
It is also called the Cartesian Coordinate System, after Rene Descartes. The purpose of Descartes' abstraction was to place 3 number lines (X, Y, Z measuring rulers) 'most different' to each other and numerically identify abstract locations in space. The process of abstracting is the distillation of some useful feature out from observed Nature so it may be used as a 'stand alone' logic tool or template without necessarily referring to the nature from which it originally came. In the Cartesian Coordinate System, the number values of the 'rulers' are used to describe both the form and the extremes.
SOME TEACHING NOTES ON NATURAL RIGHT ANGLES
A largely unsung part of a teacher’s job is to get children to articulate and enhance ‘what they already know’ so they may hone and expand the language tools used in conscious logical reasoning. From this type of exercise, children gain self-confidence in their native ability to ‘figure things out for themselves’. All children learn the right angle intuitively from when they are toddlers, but they don’t know this consciously. Regarding the Right Angle, a teacher should be getting children to verbalize 'what they already unconsciously know'. It is the ideal lesson in 'intellectual discovery'. Below are two citations of the easily observable evidence of this unarticulated learning.
* A baby has to learn to stand at a right angle to the ground before it can walk.
* All children are routinely drawing right angles long before they know what a right angle is. Beyond merely drawing right angles in squares and rectangles, children often draw a ‘perspective fence’ in pictures where the fence posts are slanted. Children know this looks goofy, but they can't figure out how to fix it. They haven't learned how to translate depth in a picture. The more important observation is that the child artist is laboring intensely trying to tilt the posts most differently to the rails (right angle)! This is clear evidence of children's intrinsic but inarticulate knowledge of the importance of the right angle.
The right angle is not just some mathematics theoretical 'thing'. Right angles are omnipresent in both the natural Universe and in man-made structures. The right angle is at the core of all universal form and how the universe functions'. This is why it is so important for children to understand they are not just learning some mathematical factoid that ‘will be on the test’. Here are a few aspects you can cite and/or demonstrate:
- When something is perfectly balanced, it is at a right angle to what it is balancing on.
- When something is standing on a plane at a right angle it will fall exactly the same distance in any possible direction. (Actually, if it is perfectly balanced, it won't fall at all - until it is nudged off balance. A perfectly balanced ball or piece of paper on one's finger is the result of the center line of the object being at a right angle to the finger tip).
In any future scientific math calculation students will do - all forms, concrete or abstract, whose opposite extremes are most different to each other will always be placed in and solved as right angle relations. This means, when students clearly know what a right angle really is, their ability to learn and adapt complex mathematical applications to situations continues to be understood by them as coming from UNIFIED and utterly consistent basics. The entire process of learning and using logic is considerably eased, clarified and advanced.
SOME FINAL QUICK OBSERVATIONS
While this material is written for adults as source material, teachers should have no problem picking and choosing what they want for their class. I routinely teach the majority of this material to grade school children as part of art lessons. In fact, I can't teach students proper art techniques without it.
Knowing this material is key to understanding that all the aspects of math, both numbers and form, rise from a stunning and elegant simplicity. But, as far as I am aware, this material isn’t taught in grade school, middle school or even high school. What, then, is so difficult about teaching children, "A right angle is any intersection of two simple lines whose opposite extremes are most different?" Telling children only that a right angle is "90 degrees" does not tell them what a right angle IS and therefore, is as worthless to children’s real education as it is possible to be.
If you found the above informative, it should interest you to know there is much left out of standard school math curricula which destroys or hobbles children's continuity of learning. Without the least bit of hyperbole or exaggeration I let you know, one of the most important of these omissions occurs in fundamental grade school arithmetic.
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