In this unit, students will extend their understanding of proportional relationships and unit rate to constant rate of change, and use this to identify and understand linear relationships. Specifically, they will learn to recognize that linear relationships with an initial value of zero are proportional relationships, while those with a nonzero initial value are not. Students may struggle with this, as they have mostly focused on linear equations through the origin up to this point. They will use their knowledge of similar triangles to understand why the slope, the ratio of the change in the outputs to the change in the inputs, is equivalent between any two points on a given nonvertical line. Students will understand how different representations of linear relationships (graphical, numerical, symbolic, verbal) are related, and understand what information each representation provides about the relationship between the inputs and outputs.
Key Concepts:
 A constant rate of change between inputs and outputs (as the input changes by 1, the output changes in a constant way) is related to unit rate and constant of proportionality (from Grade 7, Unit 2).
 The segment created between any two points on a nonvertical line can be thought of as the hypotenuse of a right triangle. All the right triangles creating from segments from the same line are similar.
 The slope of a line gives information about how the values of x and y change together (as the value of x increases the value of y either increases or decreases, and as the value of x decreases, the value of y either increases or decreases).
Prior Knowledge Needed:
 Understand what it means for quantities to be in an proportional relationships, and that these relationships can be described using a constant of proportionality. (Grade 7, Unit 2; 7.RP.A.2)
 Lengths of corresponding sides of similar triangles are proportional. (Grade 7, Unit 13; 7.G.A.1)
Units on the Horizon:
 Patterns of association in bivariate data (Grade 8, Unit 8)
 Nonlinear functions (Grade 8, Unit 9)
 Solving linear equations (Grade 8, Unit 10)
Lessons

Lesson objective: Extend understanding of unit rate for proportional relationships to rate of change for general linear functions. Students bring prior knowledge of unit rate from 7.RP.A.2.B. This prior knowledge is extended to rate of change as student...

Lesson objective: Build on knowledge of proportional relationships to find rate of change and initial value for linear functions in various representations. This lesson helps to build fluency with calculating rate of change and identifying initial value...

Lesson objective: Apply understanding of proportional and nonproportional linear functions and their components to a situation where proportional and nonproportional data must be sorted and modeled. This lesson provides an opportunity for students to ...

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 equivale...

Lesson objective: Determine the equation of a linear function given two points by using similar triangles. This lesson helps to build procedural skill with building linear equations of the form y = mx + b using similar triangles. Graphs are predominantl...

Lesson objective: Determine the equation of a line given two points, including vertical and horizontal lines. This lesson helps to build procedural skill with slope calculation and solving for the yintercept. Algebraic equations are used here to help s...

Lesson objective: Solve problems by using the fact that the rate of change of a linear function is constant. This lesson provides an opportunity for students to apply their knowledge and understanding that the slopes between all pairs of points on a giv...

Lesson objective: Understand that the slope of a linear graph tells us how the output value changes when the input value increases by one. Students bring prior knowledge of rate of change from 8.F.B.4. This prior knowledge is extended to slope of linear...

Lesson objective: Interpret what the slope of a linear graph tells us about how two quantities change together. This lesson helps to build procedural skill with interpreting the components of linear functions, namely slope. Graphs are the focus here so ...

Lesson objective: Solve problems by interpreting information from a linear graph, specifically what the slope tells us about how the input and output values are changing together. This lesson provides an opportunity for students to apply their knowledge...