Coordinate systems for the plane

A way to describe the positions of points in the plane is by a coordinate system. Here is the most popular way, which was invented by the French mathematician and philosopher Rene Descartes in 1637. So a coordinate system of this sort is called a Cartesian coordinate system.

Start with two number lines (called the axes) that intersect at right angles. The scales on the number lines should have the same spacing. Call one of the lines the x axis, and the other one the y axis. The intersection point of the axes is called the origin. The coordinates of a point are two numbers, x and y. To get to the point, go to the number x on the x axis, then go out a distance y on a line parallel to the y axis.

For example, in the coordinate system at right, the x axis is horizontal and the y axis is vertical. (This is how they are usually drawn.)

Group problems

Do the set of activities on cylindrical anamorphic art from the Jan. 1998 issue of The Mathematics Teacher. The last part is individual homework.

Individual problems

1. Invent a coordinate system for the equilateral triangle graph paper you constructed in the first lesson. (Clueless? Try this interactive sketch where you can move the axes.) Draw in the axes and mark the number lines. In particular,
• You should have two number lines (let's call them the A and B axes) that intersect at an origin
• Both coordinates of all the vertices of the equilateral triangles should be integers
• No other points should have integers for both coordinates
2. Find and label the points (2,2), (1,-5), (-3,1), and (-2,-4).
3. Find and label a line (C) that is part of the triangle tessellation and is parallel to the A axis. Describe as precisely as you can the coordinates of all points on this line.
4. Find and label a line (D) that is part of the triangle tessellation and is parallel to the B axis. Describe as precisely as you can the coordinates of all points on this line.
5. There is a third set of parallel lines in the tessellation that are not parallel to either to the A or the B axis. One of these lines (E) goes through (0,0). Give a rule for the coordinates of the points on this line.
6. Find and label a line (F) that is part of the triangle tessellation and is parallel to E. Describe the coordinates of all points on this line.
7. Describe the coordinates of all points on the line through the origin that is perpendicular to the A axis. (This line is not part of the tessellation.)

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