This unit introduces students to nonproportional linear relationships by building on earlier work around similarity and slope. Then students solve systems of linear equations using graphic and algebraic methods. Students advance their understanding of lines by examining slopes in the context of data. Lastly, they use scatter plots and fitted lines to analyze numerical data.

The unit begins by revisiting different representations of proportional relationships. Students create graphs, tables, and equations in order to interpret the constant of proportionality as the rate of change of one variable with respect to the other.

Next, students analyze a relationship that is linear but not proportional. They see that the rate of change has a numerical value that is the same as the slope of the line that represents the relationship. Students also view the graph of a line in the coordinate plane as the vertical translation of a proportional relationship.

Then students consider situations represented by linear relationships with negative rates of change. They establish a way to compute the slope of a line from any two distinct points on the line. 

Next students examine systems of equations graphically and find solutions algebraically. They build on their understanding that the line representing an equation with 2 variables is made up of coordinate pairs that make the equation true. They find that the intersection of 2 lines is the point that makes both equations for the system true. Students also recognize when systems have no solution or infinite solutions based on the graphs and the slope and intercept.

Then students are introduced to scatter plots and are reminded how to interpret points on a graph using a context.  They look more closely at associations in data by informally drawing lines that model the general trend of the data. They also classify associations as positive, negative, linear, and non-linear by looking at the shape of the data in a scatter plot.

In an optional section, students look at categorical data using two-way tables and relative frequencies. They then informally look at the relative frequencies to notice whether the variables are associated or not.

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes, such as representing, interpreting, and explaining. Throughout the unit, students will benefit from routines designed to grow robust disciplinary language, both for their own sense-making and for building shared understanding with peers. Teachers can formatively assess how students are using language in these ways, particularly when students are using language to:

Represent

• Situations involving proportional relationships (Lesson 1).
• Constants of proportionality in different ways (Lesson 2).
• Linear relationships using graphs, tables, equations, and verbal descriptions (Lesson 4).
• Situations using negative slopes and slopes of zero (Lesson 8).
• Slope using expressions (Lesson 9).
• Situations by graphing lines and writing equations (Lesson 11).
• Situations involving systems of linear equations. (Lessons 13, 14, and 16).
• Data in organized ways (Lesson 18).
• Data using two-way tables, bar graphs, and segmented bar graphs (Lessons 24 and 25).
• Data using scatter plots (Lesson 28).

Interpret

• Situations involving proportional relationships (Lesson 1).
• Slopes and intercepts of linear graphs (Lesson 2).
• Situations using negative slopes and slopes of zero (Lesson 9).
• Situations involving systems of linear equations. (Lessons 13 and 14).
• Tables and scatter plots of bivariate data (Lesson 19).
• Tables, scatter plots, equations, and situations involving bivariate data (Lesson 20).
• Situations involving linear relationships (Lesson 26).

Explain

• How to graph proportional relationships (Lesson 2).
• How to use a graph to determine information about a linear situation (Lessons 4 and 5).
• How to graph linear relationships (Lesson 9).
• How to estimate using available data (Lesson 18).
• How to use tables and scatter plots to make estimates and predictions (Lesson 19).
• The meaning of slope for a situation (Lesson 20).
• How to use lines to show associations, identify outliers, and answer questions (Lesson 23).
• How to answer questions about systems of equations (Lesson 27).

In addition, students are expected to compare different representations of the same situation,  compare solutions of linear equations, describe and compare features of scatter plots, and describe graphs of systems of linear equations. Students are also asked to justify whether or not lines are good fits for a situation, justify reasoning about linear relationships, and justify correspondences between different representations. and justify associations between bivariate data. Students also have opportunities to use language to generalize about what makes a line fit a data set well and about categories for sorting scatter plots.

The table shows lessons where new terminology is first introduced, including when students are expected to understand the word or phrase receptively and when students are expected to produce the word or phrase in their own speaking or writing. Terms from the Glossary appear bolded. Teachers should continue to support students’ use of a new term in the lessons that follow where it was first introduced.

lesson	new terminology
	receptive	productive
Acc7.5.1	represent scale label	constant of proportionality
Acc7.5.2	rate of change equation
Acc7.5.4	linear relationship constant rate rate of change	slope
Acc7.5.5	vertical intercept -intercept
Acc7.5.6	initial (value or amount)	constant rate
Acc7.5.7	relate
Acc7.5.8	horizontal intercept -intercept
Acc7.5.9		rate of change vertical intercept -intercept
Acc7.5.10	constraint	horizontal line vertical line
Acc7.5.11	solution to an equation with two variables variable combination set of solutions
Acc7.5.13	ordered pair
Acc7.5.14	system of equations solution to a system of equations
Acc7.5.15	substitution	no solution (only) one solution infinitely many solutions
Acc7.5.16	algebraically
Acc7.5.17		system of equations substitution
Acc7.5.18	scatter plot
Acc7.5.20	outlier predict overpredict underpredict linear model
Acc7.5.21	positive association negative association
Acc7.5.22	linear association  nonlinear association no association fitted line
Acc7.5.23	cluster	positive association negative association linear association
Acc7.5.24	segmented bar graph relative frequency two-way (frequency) table
Acc7.5.28		scatter plot outlier cluster