Kid-friendly explanation
On a grid with a horizontal line and a vertical line, the origin is the exact spot where the two lines meet. Its address is (0, 0). You can think of it as the "home base" where you start before you move right, left, up, or down. Every other point tells how far it is from this home base!
Worked examples
Example 1 (upper elementary): Finding the origin and a point
Imagine a coordinate plane, which is a grid with a horizontal x-axis and a vertical y-axis that cross in the middle.
The origin is the point where the x-axis and y-axis intersect. Its ordered pair, or coordinate address, is (0, 0).
Now look at the point (3, 2). Starting at the origin (0, 0), move 3 units to the right along the x-axis, then 2 units up along the y-axis. That is where (3, 2) is located, and you can see how the origin is the starting point for plotting the point.
Example 2 (middle/high school): A proportional relationship through the origin
Suppose a game gives 5 points for every 1 star collected. This is a proportional relationship, because the number of points is always 5 times the number of stars.
Let x be the number of stars and y be the number of points. The relationship is y = 5x.
If x = 0 stars, then y = 5(0) = 0 points, so the ordered pair is (0, 0). This means the graph of y = 5x passes through the origin.
On the coordinate plane, plot a few points like (1, 5), (2, 10), and (3, 15). Draw a straight line through these points and continue it back through (0, 0). Because this is a proportional relationship, its graph is a straight line that goes through the origin.
Going deeper
In later grades, the origin is used to describe and perform transformations, which are moves or changes to figures on the coordinate plane. For example, a rotation about the origin turns a figure around the point (0, 0), and a dilation about the origin makes a figure larger or smaller with (0, 0) as the center.
When a dilation uses the origin as the center, each point (x, y) on the original figure moves to a new point (kx, ky), where k is the scale factor. If k is greater than 1, the figure grows away from the origin. If 0 < k < 1, the figure shrinks toward the origin.
In algebra, lines that represent proportional relationships always pass through the origin and have equations of the form y = mx, where m is the slope or constant of proportionality. Lines that do not pass through the origin have equations of the form y = mx + b, where b is the y-intercept, the point where the line crosses the y-axis.
Common mistakes
- Mistake: Thinking the origin is any point with a 0 in it, such as (0, 5) or (3, 0). Correction: The origin is only the point (0, 0), where both the x-coordinate and the y-coordinate are 0.
- Mistake: Mixing up which axis is which and placing the origin at the bottom-left corner of the grid. Correction: On a standard coordinate plane, the x-axis is horizontal, the y-axis is vertical, and the origin is at their intersection in the center of the grid.
- Mistake: Believing every straight-line graph must pass through the origin. Correction: Only proportional relationships have graphs that are straight lines through the origin; many lines, such as y = 2x + 3, are straight but cross the y-axis above or below the origin.
Everyday connection
Maps and video game worlds often use something like an origin. For example, a game might treat the center of the map as (0, 0) and measure how far a character moves left, right, up, or down from that center. The idea of a starting point that everything else is measured from is just like the origin on a coordinate plane.