(Geometry > Coordinate Geometry > Distance)
I’ve seen it happen, more than once. A student working in the coordinate plane marks a diagonal segment of length units with the belief it is 1 unit in length. When I try to correct this misconception, the student stares at me blankly while I proceed through various explanations. And at the end of it all, the student is often unconvinced. This is why I love the first 4 pages of the new Distance Between Points on a Coordinate Plane document! A quick interactive visual convinces students that the horizontal or vertical distance is not the same as the diagonal distance on a gridded plane. Check it out and see what I am talking about!
The realization that the lengths are not the same creates a need for the Pythagorean Theorem. The Pythagorean Theorem is used to find those “mysterious” lengths while the distance formula is being developed informally.
The distance formula is introduced on page 10 after students work the problem by the Pythagorean Theorem. On page 11, students can choose to use either method. Once students complete the document, teachers can return to a few examples in the document and have students apply the distance formula using only coordinates if desired.