Picture an explorer standing at the origin $O$. If he walks $10$ meters east to point $A(10)$, or $10$ meters west to point $B(-10)$, his final positions are completely different (they are opposites of each other). Yet in terms of "energy spent" or "number of steps taken," the "intensity" of the two trips is exactly the same. This perspective, which sets aside direction and looks only at the number of steps, is the key to the "advanced journey" we are about to begin.
Observing Symmetry and Distance on the Number Line
As the opening of "Operations and Comparisons with Rational Numbers," this lesson focuses on using the number line—an intuitive tool—to make the leap from a static understanding of "numbers" to a dynamic connection with "values."
By reviewing the three essential elements of the number line, students are guided to observe the beauty of symmetry in how opposites are distributed in space. The sign determines which side of the origin we are on, while the "value" determines how far we are from the origin. Separating these two attributes is the key prerequisite for understanding absolute value calculations and the rules of addition that follow.
Numbers to the right on the number line are always greater than those to the left; and when comparing values, the absolute value describes the distance from a point to the origin—that is, the "pure value" without regard to direction.
$|10| = |-10| = 10$