Cartesian Coordinates

Cartesian coordinates are a system for representing points in a plane using two numbers, called the x-coordinate and the y-coordinate. This system was developed by René Descartes and is named after him. The x-coordinate represents the horizontal position of a point, while the y-coordinate represents the vertical position.

Representation

In the Cartesian coordinate system, a point is represented by an ordered pair (x, y), where x is the distance from the y-axis (measured along the horizontal axis) and y is the distance from the x-axis (measured along the vertical axis).

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Quadrants

The plane is divided into four quadrants, labeled I, II, III, and IV, which are formed by the x and y-axes. The signs of the coordinates determine in which quadrant a point lies. Points in quadrant I have both positive x and y coordinates, while points in quadrant II have negative x and positive y coordinates, and so on.

Distance Formula

The distance between two points with coordinates (x1, y1) and (x2, y2) can be calculated using the distance formula:

distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Study Guide

Here are some key topics to study when learning about Cartesian coordinates:

Understanding the concept of ordered pairs and their representation on a plane.
Identifying points in different quadrants based on their coordinates.
Calculating the distance between two points using the distance formula.
Graphing linear equations and functions using Cartesian coordinates.

Practice plotting points on a graph, identifying their coordinates, and determining in which quadrant they lie. Also, work on solving problems involving the distance between two points using the distance formula.

Understanding Cartesian coordinates is fundamental for many areas of mathematics, including geometry, algebra, and calculus. It's an essential skill for visualizing and analyzing relationships between points and shapes in a plane.

◂Math Worksheets and Study Guides Eighth Grade. Linear equations

The resources above cover the following skills:

Algebra (NCTM)

Represent and analyze mathematical situations and structures using algebraic symbols.

Explore relationships between symbolic expressions and graphs of lines, paying particular attention to the meaning of intercept and slope.

Grade 8 Curriculum Focal Points (NCTM)

Algebra: Analyzing and representing linear functions and solving linear equations and systems of linear equations

Students use linear functions, linear equations, and systems of linear equations to represent, analyze, and solve a variety of problems. They recognize a proportion (y/x = k, or y = kx) as a special case of a linear equation of the form y = mx + b, understanding that the constant of proportionality (k) is the slope and the resulting graph is a line through the origin. Students understand that the slope (m) of a line is a constant rate of change, so if the input, or x-coordinate, changes by a specific amount, a, the output, or y-coordinate, changes by the amount ma. Students translate among verbal, tabular, graphical, and algebraic representations of functions (recognizing that tabular and graphical representations are usually only partial representations), and they describe how such aspects of a function as slope and y-intercept appear in different representations. Students solve systems of two linear equations in two variables and relate the systems to pairs of lines that intersect, are parallel, or are the same line, in the plane. Students use linear equations, systems of linear equations, linear functions, and their understanding of the slope of a line to analyze situations and solve problems.

Connections to the Grade 8 Focal Points (NCTM)

Geometry: Given a line in a coordinate plane, students understand that all 'slope triangles' - triangles created by a vertical 'rise' line segment (showing the change in y), a horizontal 'run' line segment (showing the change in x), and a segment of the line itself - are similar. They also understand the relationship of these similar triangles to the constant slope of a line.