Lesson 6

Representations of Linear Relationships

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.

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.

Let’s write equations from real situations.

Rulers

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.

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.

Building On