Coordinates

Applying a coordinate system to Euclidean space connects algebra and geometry, resulting in powerful methods of analysis and problem solving. Just as the number line associates numbers with locations in one dimension, a pair of perpendicular axes associates pairs of numbers with locations in two dimensions. This correspondence between numerical coordinates and geometric points allows methods from algebra to be applied to geometry and vice versa. The solution set of an equation becomes a geometric curve, making visualization a tool for doing and understanding algebra. Geometric shapes can be described by equations, making algebraic manipulation into a tool for geometric understanding, modeling and proof.

Coordinate geometry is a rich field for exploration. How does a geometric transformation such as a translation or reflection affect the coordinates of points? How is the geometric definition of a circle reflected in its equation?

Adding a third perpendicular axis associates three numbers with locations in three dimensions and extends the use of algebraic techniques to problems involving the three-dimensional world we live in.

//Connections to Shape, Quantity, Equations and Functions//. Coordinates can be used to reason about shapes. In applications, coordinate values often have units (such as meters and bushels). A one-variable equation of the form //f//(//x//) = //g//(//x//) may be solved in the coordinate plane by finding intersections of the curves //y// = //f//(//x//) and //y// = //g//(//x//).

Students understand that:
 * Core Concepts**
 * 1) Locations in the plane or in space can be specified by pairs or triples of numbers called coordinates.
 * 2) Coordinates link algebra with geometry and allow methods in one domain to solve problems in the other.
 * 3) The set of solutions to an equation in two variables forms a curve in the coordinate plane—such as a line, parabola, circle—and the solutions to systems of equations correspond to intersections of these curves.

Students can and do:
 * Core Skills**
 * 1) Translate fluently between lines in the coordinate plane and their equations.
 * 2) Identify the correspondence between parameters in common families of equations and the location and appearance of their graphs.
 * 3) Use coordinates to solve geometric problems.