Lesson objective: Understand that the slope is the same between any two points on a line.

Students bring prior knowledge of similar figures from 8.G.A.4. This prior knowledge is extended to slope as students use similar triangles to explain the equivalence of the slopes between all pairs of points on a nonvertical line. A conceptual challenge students may encounter is making the connection between this geometric concept and the calculation of slope.

The concept is developed through work with graphs, which allow students to gain insight into what the slope of a line tells us, and which facilitate the connection to the geometric concepts being utilized in the lesson.

This work helps students deepen their understanding of equivalence because the connection to similar triangles emphasizes the idea that, although all slope ratios may not be composed of exactly the same values, they can still be equivalent and describe the same line.

Students engage in Mathematical Practice 7 (Look for and make use of structure) as they analyze why all of the "slope triangles" drawn above/below a line are similar, and use this structure to verify that points are collinear and find additional points on a given line.

**Key vocabulary:**

- Equivalent ratio
- Hypotenuse
- Proportional
- Similar triangles
- Slope

**Special materials needed:**