Lesson 7

Representations of Linear Relationships

Lesson Narrative

In this lesson, students develop an equation for a linear relationship by expressing regularity in repeated calculations (MP8). In an activity, students measure the volume of water in a graduated cylinder, repeatedly adding objects to the cylinder. Each additional object increases the volume of water by the same amount. They graph the relationship and interpret the initial water volume as the vertical intercept; they also interpret the slope as the rate of change, that is the amount by which the volume increases when one object is added. 

In the second activity, students explicitly formulate a procedure to compute the slope of a line from any two points that lie on the line, including two different general points. They have been doing this since the previous unit, in order to calculate and make sense of slope as well as to find an equation satisfied by all points on a line. In this unit, students have continued to graph lines, draw slope triangles, and calculate slope; this time with an emphasis on understanding the slope as a rate of change: how much vertical displacement is there per unit of horizontal displacement? This lesson continues to focus on positive slopes; in future lessons, students will start to investigate non-positive slope values.

Learning Goals

Teacher Facing

  • Create an equation that represents a linear relationship.
  • Generalize (orally and in writing) a method for calculating slope based on coordinates of two points.
  • Interpret the slope and $y$-intercept of the graph of a line in context.

Student Facing

Let’s write equations from real situations.

Required Materials

Required Preparation

Students will work in groups of 2–3.  

If doing the video presentation of spheres being added to cylinder, prepare the video for presentation.

If doing the 20 minute version of the water level task with an initial demonstration, you will need one graduated cylinder partially filled with water and 10 to 15 identical solid objects that fit into the cylinder and don’t float (marbles, dice, cubes, or hardware items such as nuts or bolts).

If doing the 45 minute version of the water level task, each group will need one graduated cylinder partially filled with water. Each group will also need about 15 identical solid objects that fit into the cylinder and don’t float. Determine a good initial water level and the approximate volume of each type of equally sized object.  

Learning Targets

Student Facing

  • I can use patterns to write a linear equation to represent a situation.
  • I can write an equation for the relationship between the total volume in a graduated cylinder and the number of objects added to the graduated cylinder.

CCSS Standards

Building On


Building Towards

Glossary Entries

  • linear relationship

    A linear relationship between two quantities means they are related like this: When one quantity changes by a certain amount, the other quantity always changes by a set amount. In a linear relationship, one quantity has a constant rate of change with respect to the other.

    The relationship is called linear because its graph is a line.

    The graph shows a relationship between number of days and number of pages read.

    When the number of days increases by 2, the number of pages read always increases by 60. The rate of change is constant, 30 pages per day, so the relationship is linear.

    Graph of line. Points plotted on line include 1 comma 70 and 4 comma 160.
  • vertical intercept

    The vertical intercept is the point where the graph of a line crosses the vertical axis.

    The vertical intercept of this line is \((0,\text-6)\) or just -6.

    A graph of a line with a vertical intercept of -6