Learn on PengiAoPS: Introduction to Algebra (AMC 8 & 10)Chapter 8: Graphing Lines

Lesson 1: The Number Line and the Cartesian Plane

In this Grade 4 AoPS Introduction to Algebra lesson, students explore the foundations of analytic geometry by learning how to use the number line, including concepts of absolute value and magnitude, and how René Descartes extended it into the Cartesian plane. Students practice plotting points using ordered pairs and identifying x-coordinates and y-coordinates relative to the origin. This lesson from Chapter 8 sets the stage for graphing two-variable linear equations in preparation for AMC 8 and AMC 10 competition math.

Section 1

Number Line Structure and Infinite Extension

Property

A number line is a visual representation of all real numbers arranged in order, with arrows at both ends indicating that the line extends infinitely in both directions: ←⋯−3,−2,−1,0,1,2,3,⋯→\leftarrow \cdots -3, -2, -1, 0, 1, 2, 3, \cdots \rightarrow

Examples

Section 2

Magnitude and Absolute Value

Property

The magnitude of a number is another term for its absolute value, representing the distance from zero on the number line. For any real number aa, the magnitude is denoted as ∣a∣|a| and equals the absolute value.

Examples

Section 3

Cartesian Coordinate System

Property

To make a graph that includes negative values, we construct a Cartesian coordinate system. We draw two perpendicular number lines for the horizontal and vertical axes. The horizontal axis is called the xx-axis and the vertical axis is the yy-axis. The two axes divide the plane into four quadrants. The axes intersect at the origin, which has coordinates (0,0)(0, 0).

Examples

  • To plot the point (4,3)(4, 3), we start at the origin, move 4 units to the right along the x-axis, and then 3 units up. This point is in the first quadrant where both coordinates are positive.
  • The point (−5,2)(-5, 2) is located 5 units to the left of the y-axis and 2 units above the x-axis. It lies in the second quadrant.

Section 4

Quadrants and Points on the Axes

Property

The axes divide a plane into four regions, called quadrants. The quadrants are identified by Roman numerals, beginning on the upper right and proceeding counterclockwise.

Quadrant IQuadrant IIQuadrant IIIQuadrant IV
(+,+)(+, +)(−,+)(-, +)(−,−)(-, -)(+,−)(+, -)

Points on the Axes
Points with a yy-coordinate equal to 0 are on the xx-axis, and have coordinates (a,0)(a, 0).
Points with an xx-coordinate equal to 0 are on the yy-axis, and have coordinates (0,b)(0, b).

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Number Line Structure and Infinite Extension

Property

A number line is a visual representation of all real numbers arranged in order, with arrows at both ends indicating that the line extends infinitely in both directions: ←⋯−3,−2,−1,0,1,2,3,⋯→\leftarrow \cdots -3, -2, -1, 0, 1, 2, 3, \cdots \rightarrow

Examples

Section 2

Magnitude and Absolute Value

Property

The magnitude of a number is another term for its absolute value, representing the distance from zero on the number line. For any real number aa, the magnitude is denoted as ∣a∣|a| and equals the absolute value.

Examples

Section 3

Cartesian Coordinate System

Property

To make a graph that includes negative values, we construct a Cartesian coordinate system. We draw two perpendicular number lines for the horizontal and vertical axes. The horizontal axis is called the xx-axis and the vertical axis is the yy-axis. The two axes divide the plane into four quadrants. The axes intersect at the origin, which has coordinates (0,0)(0, 0).

Examples

  • To plot the point (4,3)(4, 3), we start at the origin, move 4 units to the right along the x-axis, and then 3 units up. This point is in the first quadrant where both coordinates are positive.
  • The point (−5,2)(-5, 2) is located 5 units to the left of the y-axis and 2 units above the x-axis. It lies in the second quadrant.

Section 4

Quadrants and Points on the Axes

Property

The axes divide a plane into four regions, called quadrants. The quadrants are identified by Roman numerals, beginning on the upper right and proceeding counterclockwise.

Quadrant IQuadrant IIQuadrant IIIQuadrant IV
(+,+)(+, +)(−,+)(-, +)(−,−)(-, -)(+,−)(+, -)

Points on the Axes
Points with a yy-coordinate equal to 0 are on the xx-axis, and have coordinates (a,0)(a, 0).
Points with an xx-coordinate equal to 0 are on the yy-axis, and have coordinates (0,b)(0, b).