 The Visual / Auditory / Symbolic / Kinesthetic Approach to Algebra

 Dimensions A point has no length, no width, no height, no dimension at all. Mathematicians say it has 0 dimension and represent it graphically with a dot or a point on the paper. The dot is visible on the paper and therefore has some visible width or dimension, but, it is a symbolic notation rather than a factual picture. Drag the point a straight distance, a length, so as to leave a trail on the paper. That trail is a symbolic representation of a line segment - a piece of a line. It really has no width, no thickness. It has 1 dimension. It has length. Roll the line segment a straight distance in a direction at right angles to the first, a width, so as to leave a trail on the paper. That trail marks off an area. The set of points that mark the outline of that area is a rectangle. The rectangle has 2 dimensions. It has length and it has width. "Pile" the areas up, off the paper, one-on-top-of-the-other, a height, so as to fill a trail through space. That trail marks off a volume. The set of points that mark the outline of that volume is a prism. The prism has 3 dimensions. It has length and it has width and it has height.

 Units of Length, Area, and Volume

The concept of dimension is partner to the measurement of length, area, and volume. At the far right, start with the 0 dimension dot representing a point. Move left a distance marking a line of dimension 1 to record this path. Next drag the line segment up the page to record the path and mark the area, with 2 dimensions (length and width). Lastly, move the area up and off the page, sort of "stacking" the areas, to mark the 3 dimensional volume.

The basic units of length are inch, meter, foot, rod, etc.

The basic units of area are the squares of the basic units of length.

The basic units of area must incorporate two dimensions, one of length and the other of width. The basic units of area are the square inch, the square meter, the square foot, the square rod, etc.

The basic units of volume are the cubes of the basic units of length.

The basic units of volume must incorporate three dimensions -- length, width, and height. The basic units of volume are the cubic inch, the cubic meter, the cubic foot, the cubic rod, etc.

 And Other Dimensions?

What comes next dimension-wise, symbol-wise? Reconsider what was discussed earlier and follow the pattern. Think of the next dimension as a record of the next movement after moving a point a distance to mark a line segment, then dragging the line segment a width to mark an area, then "stacking" the areas a height to mark a volume. Now, take the volume (outlined by a prism or a cube) and drag it or "stack" it, or pull it. When a 4-dimensional cube is created, mathematicians call it a tesseract. This is a geometrical representation of a cube in the fourth dimension.

Mathematicians and scientist use many more than 4 dimensions. Often, one variable is used for each dimension. Often, time is called identified as the 4th dimension. With time as a 4th dimension, the snapshot becomes a movie.

Mathematicians and scientist use many more than 4 dimensions. To compute distance given coordinates, 2 coordinates or dimensions are required on a line, 4 coordinates or dimensions are required on a plane, and 6 coordinates or dimensions are required in space. See "The distance formulas used for a line, a plane, or in space" at ...80/math.htm.    www.termtiles.com, Unit 40   © 2008, A. Azzolino