Maryland Math 8.AT.A.2.a
The Standard
- 8.AT.A.2
- Use proportional reasoning to identify slope and represent linear relationships.
Use proportional reasoning to explain why the slope m is the same between any two points on a non-vertical line by showing that the ratios of vertical change to horizontal change between points are equivalent.
Maryland College and Career Ready Standards for Mathematics (2025) · USE PROPORTIONAL RELATIONSHIPS TO REPRESENT AND REASON ABOUT LINEAR EQUATIONS.
Teacher's field guide
What This Standard Means
What Students Need to Do
- Students find the vertical change and horizontal change between two points on a non-vertical line. They compare those ratios for different pairs of points and explain why the ratios are equal.
What Mastery Looks Like
- Given (0, 1), (2, 4), and (4, 7), a student finds slopes of 3/2 and 6/4 from different pairs. They use similar rise-and-run triangles to explain why both ratios describe the same line.
Common Misconceptions
- Some students divide horizontal change by vertical change or count squares along the slanted line as the run. Others subtract coordinates in different orders and get the wrong sign.
How to Assess It
- Give students the points (0, 1), (2, 4), and (6, 10). Ask them to find the slope using two different pairs and explain why their answers agree.
Lesson moves
Ways to Teach It
On graph paper, plot (0, 1), (2, 4), and (4, 7), then draw and label two rise-and-run triangles in different colors.
Ask: Why do triangles drawn between different pairs of points on the same line give equal rise-to-run ratios?
Give pairs of students cards showing rise and run values; have them match cards with equal slopes and check each match by division.
Draw two sections of a ramp with rises of 1 and 2 feet and runs of 4 and 8 feet; compare their slopes.
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