3.2. Desired Results in Coordinate Systems

"A mathematician is a machine for turning coffee into theorems."--Paul Erdös

Worth Being Familiar With

The connection between dimension in (Euclidean) space and the number of (conventional) coordinates: students should be able to state how many coordinates it takes to specify a sphere as opposed to a circle, or look at a set of coordinates in a given system and state the dimensions. Less complete responses might involve confusion over the meaning of the question, especially in higher dimensions.

The existence of different distance metrics for different spaces, which students should be able to describe if not compute, e.g. for Euclidean 2- and 3-space, for the surface of a sphere, or the minimum number of moves for a chess knight to move from one position to another. Optionally, the existence of non-Euclidean geometries and basic characteristics could be mentioned, e.g. the sum of degrees in a triangle is always greater than 180 in Riemannian geometry, and fractal dimensions through examples such as approximations to a Peano curve.

Important to Know and Do

Giving the coordinates of a point in the Cartesian and polar systems, and if time permits, in the spherical and/or cylindrical systems for 3-space as well, e.g. plotting points written in both (x, y) and (r, theta) form or finding the coordinates of given points. While Cartesian coordinates are probably long familiar to students, some may be confused by the components of the polar system and attempt to use a grid in Cartesian terms. Examples can be found in the POLAR COORDINATES notes (GIF scan) from April 22's lesson plan or HOMEWORK #4 (GIF scan) on bidirectional conversions from May 6's lesson outline.

Conversion of coordinates between systems by whatever means is expedient: some students may feel comfortable using formulas, while others may need to draw diagrams to assist themselves. I suspect some students may "eyeball" the angle when converting to polar coordinates instead of computing it, or become confused over the counterclockwise measurement. However, "eyeball" estimates are valuable in checking whether the result of a computation makes sense, and should be valued for that reason.

"Enduring" Understanding

Students should be aware of the advantages and disadvantages of different coordinate systems in various situations. Given a shape or "map" and a given task, students should be able to draw in their choice of coordinate system and explain briefly why, e.g. if a student had a "map" of a castle and was asked to "set up a coordinate system and use it give directions from the tower in the lower right to the stables, gatehouse, and kitchen," one logical response might be to use a Cartesian system centered at the tower. Students should also develop an awareness that the same underlying space can be described or represented in many ways. Multiple "maps" and tasks would probably be necessary to assess this thoroughly.

Examples can be found in HOMEWORK #1 (GIF scan of page 1, GIF scan of page 2, and GIF scan of page 3) from April 16's lesson plan, which develops "novel" coordinate systems, and GIVING DIRECTIONS (or download the RTF file), which develops different representations of the same space, i.e. the classroom.

Copyright © 2002 Yoon Ha Lee <requiescat@cityofveils.com>
Last updated on 20 May 2002