Tuesday, November 3, 2015

Number Lines, Straight Line Spaces, and Coordinate Systems

A Number Line is widely used to represent a dense number system as an image of uniformly spaced numbers at tick marks on a straight line, think of a long ruler. Students are invited to think of the numbers as operands participating in arithmetic operations related to the geometry of the line, an analytic geometry of the straight line figure.

I want to do arithmetic, both addition-subtraction and multiplication-division, by arrangement and measurement of two line segments. The location, orientation, and scale of the second operand's segment can differ from those for a (it's our choice) first operand. The difference is determined by the segment for the first operand whose location, orientation, and scale are ours to fix. For addition-subtraction the two segments are constructed on the same Straight Line Space (SLS). For multiplication-division they can be in different SLSs.

In what follows it is necessary to distinguish numbers-as-coordinates from numbers-as-operands, from numbers as results.    ( see page 92 in Adding it Up, National Academy of Sciences, 2001 for a different approach to addition on the Number Line where this distinction is made. I got much of my understanding of arithmetic on the number line from Understanding Numbers in Elementary School Mathematics by Hung-Hsi Wu, 2010). In what follows I will try to to refer to numbers as one of: "operands", "coordinates", or "results" ( or sums or products, ideas for something better that "results"?).

I like to think of a SLS as (an idealization as of) a structure in the world, waiting there for our investigation of it with tools like coordinate systems. Of course the power of our tools increases together with our understanding of the SLS. Children can see the mathematics they develop helping them understand their world; and, conversely, come to a deeper understanding of mathematics by seeing it act in the world. 

Straight Line Space:

a collection of points densely arranged in an otherwise structureless
straight line that extends forever in both directions. No point can come to know where it is, and there is no way to distinguish one point from another.

Cartesian Coordinate Systems

A coordinate system names and locates any point by naming it with its address expressed as a coordinate (one number for a one dimensional line).
A Cartesian Coordinate System(CCS) for the SLS is the entire collection of ordered, uniformly spaced pairs of coordinate, tick-marks. It performs two functions:

  • You put the tick-mark at the point that you chose on the SLS, the CCS provides its associated coordinate.
  • You provide the coordinate, the CCS provides the tick-mark pointing at the SLS point whose coordinate it is.
Our calculation scheme requires both these CCS capabilities, so does traditional Number Line use.

Three choices are made to construct a coordinate system for the SLS. You or the calculation (but of course ultimately you) choose:
  • the point on the SLS whose coordinate will be "0", it locates the origin of the CCS at that point. This sets the CCS location.
  • the  direction away from the origin where points will have "positive" numbers for their coordinates. This sets the CCS orientation.
  • the length of a segment on the SLS, the coordinates of whose end points differ by "1".  It sets the "unit" of the coordinate system. Since the coordinates are uniformly spaced, it is sufficient to specify the unit as the distance from the origin to a point whose coordinate is "1". The unit sets the CCS scale.
Having specified a Cartesian coordinate system, any point on the SLS has a unique coordinate; inversely any coordinate designates a unique point on the SLS. The same can be said for any other coordinate system, and there is no restriction on allowed coordinate systems; since the SLS is an infinitely extended space the location of an origin is arbitrary, so is the direction from it of the half-line of points with positive coordinates, and so is the unit of length from this origin. 

To reiterate, a different one of the three coordinate system parameters

  • locates the origin of a CCS in front of us, another
  • sets a scale to accommodate the range of operand values, that interest us, and the third
  • sets a convenient orientation.   

Make it Real

We can model a CCS with a tape measure and lay out a SLS along a long, narrow, and flat slat. 
My slat SLS: 8 feet long, slat top is a 2.5 inches wide surface 26.5 inches off the floor.

The provides a section of an SLS where a tape measure can serve as a CCS. Here addition(multiplication?) will be performed by location and measurement of segments by tape measures on the slat.  

Tape measure CCS

Uniformly spaced numbers and their associated tick marks are imprinted on the tape, which is single ended and has a zero tab at the end withdrawn from the case. A Tape Measure(TM) is "handed", the commonly available ones are "right-handed":
  • intended to be worn on the user's right hip, 
  • retrieved from there with the right hand holding the case, 
  • left hand pulls the zero tab out to clip it over an edge of the object being measured, 
  • tape extended by pulling the case with the right hand , and 
  • opps! you now have your left hand to mark the measured object at the intended location on the tape. 
At least you would if your intent was to locate a point on the SLS from its coordinate. Given the limitations of a right handed TM, you might guess someone is making a "left handed" model , and you would be right. Search for one on the internet, and look at the photo:
Lower TM is right handed: its coordinates are right-side-up when the tape is extended to the right. Upper TM is left handed: its tape is the mirror image of the lower's with right-side-up coordinates when extended to the left.

  • The TM on the right(left) provides the coordinate system for points in the SLS to the right(left) of the origin.
  • A transverse slit in the slat SLS receives the zero tabs of the left and right located the CCS.
  • The numbers are the coordinates for the points their associated tick marks point to.
Numbers, tick marks, and zero tab on a left-handed tape measure. 

For the TMs shown the scale unit is one inch.


Looking down on the middle of the slat SLS. Postive coordinates to the right, negative to the left. The zero tabs of both TMs mark the origin.
The figure shows positive coordinates in their traditional position on the right. The 0 coordinate is neither positive nor negative. At this stage "positive" and "negative" are just words we use to distinguish coordinates on different sides of the origin. 
At this stage they have no other meaning.

The figure can be seen as a number line; or perhaps better as a number line under construction with the numbers and their tick marks not yet imprinted on the line

The numbers (or is it "numerals" here?) are coordinates. Our discussion of the semantics of addition says their sum should also be a coordinate; but there is no simple sense to be made of that. And what could a product be? It will turn out that sense can be made of arithmetic with segments in different coordinate systems; this will be the topic of another post    


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