8.4

8.4 is a knowledge and skills statement. These are the student expectations under it.

• 8.4.A

  use similar right triangles to develop an understanding that slope, m, given as the rate comparing the change in y-values to the change in x-values, (y2 - y1) /...

• 8.4.B

  graph proportional relationships, interpreting the unit rate as the slope of the line that models the relationship

• 8.4.C

  use data from a table or graph to determine the rate of change or slope and y-intercept in mathematical and real-world problems

01

What Students Need to Do

Students compare vertical and horizontal changes between points and calculate slope as rise divided by run. They use similar right triangles to explain why the ratio stays constant along a line. They graph unit-rate relationships and find slope and y-intercept from tables, graphs, and contexts.

02

What Mastery Looks Like

Given any two points on a line, a student gets the same slope by keeping the subtraction order consistent. The student graphs proportional relationships through the origin and interprets slope as unit rate. From a table or graph, the student identifies slope and y-intercept with correct units.

03

Common Misconceptions

Students often divide run by rise, mix subtraction orders, or count grid squares incorrectly. They may call every linear relationship proportional, even when the graph does not pass through the origin. They also confuse the y-intercept with the first table value or omit units from rates.

04

How to Assess It

Plot (1, 3), (3, 7), and (5, 11), draw two slope triangles, and find each rise-to-run ratio. State the slope, y-intercept, and whether the relationship is proportional.

1. 01

   On grid paper, use two colors to draw different right triangles along one line, then measure and compare each rise-to-run ratio.

2. 02

   Ask students to explain why y = 4x is proportional but y = 4x + 3 is not.

3. 03

   Run a card sort matching tables, graphs, equations, slopes, and y-intercepts that describe the same relationships.

4. 04

   Graph taxi cost against miles from a fare table, then interpret slope as cost per mile and the intercept as the starting fee.

• 7.4

  The student applies mathematical process standards to represent and solve problems involving proportional relationships

• 7.5

  The student applies mathematical process standards to use geometry to describe or solve problems involving proportional relationships

• 6.5

  The student applies mathematical process standards to solve problems involving proportional relationships

• 6.4

  The student applies mathematical process standards to develop an understanding of proportional relationships in problem situations