Distance on the Coordinate Plane (Grade 6)
When two points share a row (same y) or a column (same x), the distance between them is the difference of the coordinates that change, taken as the larger minus the smaller, so a distance is never negative.
Understanding distance on the coordinate plane
From (2, 3) to (7, 3), only x changes: 7 − 2 = 5 units apart, and from (7, 3) back to (2, 3) is the same 5 units.
Key Idea
When the two points sit on opposite sides of an axis, the same rule still works: from (−2, 3) to (3, 3), the larger x minus the smaller is 3 − (−2) = 5. You can see why on the grid: −2 is 2 units from the y-axis and 3 is 3 units from it, and 2 + 3 = 5. In symbols the distance is |x₂ − x₁| (or |y₂ − y₁| for a column): the absolute value of the difference, which is how far apart the two numbers are on a number line. Counting grid steps works too, but subtracting is faster and avoids miscounting.
Seeing it in action
Worked example
Distance from (2, 3) to (7, 3)?
Same y (both 3), so subtract the x's, larger minus smaller: 7 − 2 = 5 units. Starting from (7, 3) instead gives the same 5.
Same y-coordinate, so subtract x-values: 7 − 2.
Worked example 2: across an axis
Distance from (−2, 3) to (3, 3)?
Same y, so look at the x's: −2 and 3 are on opposite sides of the y-axis.
−2 is 2 units from the axis and 3 is 3 units from it: 2 + 3 = 5 units. Check with subtraction: 3 − (−2) = 5.
A common mistake, and the fix
Forgetting that zero lies between the points.
From (−2, 3) to (3, 3), 3 − 2 = 1 is wrong: the points are on opposite sides of the y-axis, so the distance is 3 − (−2) = 5. Count the grid steps once to check: 2 to reach the axis, then 3 more.
Try a few
(1, 4) to (1, 9)?
same x; 9 − 4.
(6, 0) to (0, 0)?
larger minus smaller: 6 − 0.
(3, −2) to (3, 4)?
opposite sides of the x-axis: 2 + 4, or 4 − (−2).
Coordinate Quest
A calm treasure-map game for plotting, reading, and translating points on a coordinate plane.
Try it in Numeris.
A short lesson, then practice with feedback on every answer.
Put the reasoning on paper.
Try six fraction, angle, perimeter and area questions with explained answers, for grades 4–6.
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