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A unit is what we count one at a time. When we count one cookie at a time then each cookie is a unit. When we count a ‘$10 note’ at a time then each ‘$10 note’ is a unit. When we count a ‘50¢ coin’ at a time then each ‘50¢ coin’ is a unit.  The word UNIT comes from a Latin word “unitus” which means “one.”

A fraction is a quantity that is part of a unit. Thus, a proper fraction is always less than a unit. If we are counting in units of a cookie then, a broken piece of a cookie is fraction of a cookie. If we are counting in units of ‘$10’ then, $5 is a fraction of ‘$10’. If we are counting in units of ‘50¢’ then, ‘10¢’ is fraction of ‘50¢’. The word FRACTION comes from a Latin word “fractere” which means, “a broken piece.”

Common fractions are ‘half’, ‘quarter’, etc. It must be stated that a fraction appears relative to the unit. Thus, if we are counting in units of ‘$10’ then, $5 is HALF of ‘$10’. Thus, we see large numbers being fractions of still larger numbers. Here are some exercises in this subject for the kindergarten level.

LEVEL K3: UNITS & FRACTIONS

Once again, a unit is what we count one at a time. Thus, if we are counting ‘half a cookie’ at a time then ‘half a cookie’ is the unit. This can appear very confusing unless we go strictly by the definitions of UNIT and FRACTION.

• UNIT: what we count one at a time.
• FRACTION: a part of a unit.

A unit can be arbitrary. A fraction is always relative to the unit. Fractions address ways to represent quantities, which cannot be represented by whole numbers.

We are always looking at a fraction of the universe.

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Spatial sense is having the sense of direction, distance and location with respect to one’s environment. Orientation is getting adjusted to that environment.

To get oriented one needs to spot the relative locations of various things in one’s environment. To spot a location one only needs to know the direction it is in and its distance from one.

A direction is the line along which attention may be directed. There is infinity of directions radiating out from one’s location. The main directions are FRONT, BACK, ABOVE, BELOW, LEFT and RIGHT. The directions of LEFT and RIGHT are difficult for a child to recognize until he or she reaches Kindergarten.

A distance is the separation between two locations. There is infinity of different distances in any one direction. The distances may be identified roughly as NEAR or FAR.

A position of an object tells us how it is located in relation to other objects, such as, IN, OUT, ON, UNDER, MIDDLE, and NEXT TO. A child may learn these positions as part of learning the language, but may also be taught as part of mathematics.

Spatial locations combine into shape. A very common shape is rectangle that most doors and windows have. Other shapes are triangles, circles, etc. These shapes may be drawn on a plane surface. They are two dimensional because they have length and breadth, or width and height.

Most objects are three dimensional because they have a third dimension of thickness or depth. Examples of simple objects are cubes, spheres, cylinders, cones, etc.

Shapes and objects are symmetrical when one half is the mirror image of the other half. Most bodies are symmetrical. Objects may slide, flip or spin in space.

Our body is an object that exists in space. Therefore, orientation and spatial sense is an important subject for a child to become familiar with. Here are some exercises in this subject for the kindergarten level.

LEVEL K1: ORIENTATION & SPATIAL SENSE

“Orientation & Spatial Sense” forms the foundation of the subject of GEOMETRY. It introduces the elements of space and how these elements may relate to the student.

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A counting number is based on the idea of a unit. A unit could be a goat, a house, a cookie or anything.

A UNIT is what we count one at a time. A COUNTING NUMBER is how many units we have counted.

The term “fraction” comes from the concept of a “broken piece.” A unit may be broken into pieces and these pieces may then be counted. For example, we may break a cookie into 4 equal parts. Each part is called a quarter. We may then count 3 of those parts as three-quarters of the cookie.

A FRACTION is part of a unit, which, in its turn, may act as a smaller unit. Thus, both numbers and fractions are based on the idea of counting.

The Brotherhood established by Pythagoras believed that by understanding the relationships between numbers they could uncover the spiritual secrets of the universe and bring themselves closer to the gods. Today that basic search continues in terms of finding that one equation that would explain all universal phenomena.

In particular the Brotherhood focused on the study of rational numbers as described in the essay Numbers & Consciousness. Rational numbers depend on the idea of ratio. A ratio tells you how many times a number is to another number in terms of the same unit. If Johnny is 10 years old and his father is 40 years old, then his father is four times as old as Johnny. The ratio of father’s age to Johnny’s age is 4 to 1. If Johnny’s mother is 30 years old, then his mother is three times as old as Johnny. The ratio of mother’s age to Johnny’s age is 3 to 1.

Since the mother is 3 times Johnny’s age, and the father is 4 time’s Johnny’s age, the ratio of his mother’s age to his father’s age may be expressed as 3 to 4 using Johnny’s age as the common measure or common “unit”.

To summarize, if a and b represent counting numbers then a/b represents a rational number. Here both a and b are multiples of some indivisible common unit.

A RATIONAL NUMBER is a number that can be expressed exactly by a ratio of two counting numbers based on some indivisible common unit.

It seemed at that point in time that rational numbers represented all possible numbers that could ever exist.  A unit could be broken into smaller and smaller units making it possible to represent any quantity as a ratio of its mutiples. Therefore, it came as a big surprise when numbers, such as √2, were discovered that could not be written down as a ratio based on some indivisible common unit. It meant that no small enough common unit could be found for such numbers. The idea of an ultimate indivisible unit came under intense doubt.

It was a discovery so illogical that it was rejected outright by Pythagoras. The following is the earliest proof available: (see Irrational number – Wikipedia, the free encyclopedia).

The first proof of the existence of irrational numbers is usually attributed to a Pythagorean (possibly Hippasus of Metapontum.) The Pythagorean method at that time would have claimed that there must be some sufficiently small, indivisible unit that could fit evenly into the hypotenuse just as well as into the arms of an isoscles right angle triangle.  However, Hippasus, in the 5th century BC, was able to deduce that there was in fact no common unit of measure, and that the assertion of such an existence was in fact a contradiction. He demonstrated clearly that there may exist numbers, such as √2, that cannot be expressed as ratios of two counting numbers. Hence, they are not based on any unit that can be counted. It is said that Pythagoras was so enraged that he ordered Hippasus to be drowned.

These are irrational numbers. They defy the sanctity of the idea of a permanent indivisible unit. If you attempt to express an irrational number as a decimal you end up with a number that continues forever with no regular or consistent pattern. There can be two rational numbers that are infinitesimally close to each other, and yet there can be infinity of  irrational numbers betweem them. There is no limit to how small the difference between two numbers can be.

There was no going around this new consciousness that could not be disproven even when Hippasus was drowned. Today, the most famous irrational number is π (pi), which represents the ratio of the circumference of a circle to its diameter.

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