NY Next Generation Math GEO-G.GPE.5.a

MathGrades 9–12On the coordinate plane:

The Standard

Explore the proof for the relationship between slopes of parallel and perpendicular lines

New York State Next Generation Mathematics Learning Standards

Teacher's field guide

What Students Need to Do: Students explore a coordinate proof that parallel lines have equal slopes and nonvertical perpendicular lines have negative reciprocal slopes. They connect algebraic slope calculations to geometric direction.
What Mastery Looks Like: The student builds a convincing proof from coordinate differences or right triangles. The argument treats vertical and horizontal special cases separately.
Common Misconceptions: Students may say perpendicular slopes are merely opposites or divide coordinate differences in inconsistent order. They may assign a numerical slope to a vertical line.
How to Assess It: Give coordinates for two pairs of lines. Ask students to calculate slopes and explain why one pair is parallel and the other perpendicular.

Lesson moves

Build slope triangles on grid transparencies and rotate one triangle ninety degrees to compare ratios.
Ask why a vertical line requires a separate perpendicular-slope case.
Play slope relationship sort with line pairs, equations, and parallel, perpendicular, or neither labels.
Use mapped street segments to justify parallel lanes and perpendicular intersections with coordinates.

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