Scaling One Dimension

5 min

Teacher Prep
Setup
1–2 minutes quiet work time, followed by a whole-class discussion.

Narrative

The purpose of this Warm-up is for students to jump back into recognizing functions and determining if two quantities are functions of each other. The discussion is meant to get students using the language of functions to describe linear relationships, which continues throughout the rest of the activities.

Launch

Give students 1–2 minutes of quiet work time, and follow with a whole-class discussion.

Student Task

Here is a graph of the amount of gas burned during a trip by a tractor-trailer truck as it drives at a constant speed down a highway:

Coordinate plane, horizontal, distance traveled, miles, 0 to 240 by 40, vertical, gas burned, gallons, 0 10 100 by 10. Straight line from origin through 80 comma 10, 240 comma 30.

  1. At the end of the trip, how far did the truck drive, and how much gas did it use?
  2. If a truck traveled half this distance at the same rate, how much gas would it use?
  3. If a truck traveled double this distance at the same rate, how much gas would it use?
  4. Complete the sentence: ‾\underline{\hspace{1.5in}} is a function of ‾\underline{\hspace{1.5in}} .

 

Sample Response

  1. The truck drove 240 miles and used 30 gallons of gas.
  2. 15 gallons. Since it is a proportional relationship, if the miles are halved, then the gallons are also halved.
  3. 60 gallons. Since it is a proportional relationship, if the miles are doubled, then the gallons are also doubled.
  4. Gallons of gas burned is a function of miles traveled. The number of miles traveled is also a function of the gallons of gas burned.
Activity Synthesis (Teacher Notes)

Invite students to share their answers and their reasoning for why gas burned is a function of distance traveled. Questions to further the discussion about functions:

  • “Looking at the graph, what information do you need in order to determine how much gas was used?” (We need to know the number of miles traveled.)
  • “What is the independent variable? Dependent variable? How can you tell from the graph?” (The independent value is the distance traveled. The dependent value is the gas burned. By convention, the independent is on the xx-axis, and the dependent is on the yy-axis.)
  • “What are some ways that we can tell from the graph that the relationship between gas burned and distance traveled is proportional?” (The graph is a line that goes through the origin, and we can see a constant ratio between yy and xx in some points, like (80,10),(160,20)(80,10), (160, 20), and (240,30)(240, 30).)
  • “Is this an example of a linear function? Why or why not?” (This is a linear function because it is a proportional relationship, which can be written as y=kxy=kx. For each input value of xx, there is one and only one output value for yy.)
Anticipated Misconceptions

If students are unsure how to determine the values for half the distance or double the distance, consider asking:

  • “How did you identify the values for the first question?”
  • “How could you use what you know about proportional relationships to help you solve these questions?”
Standards
Addressing
  • 8.F.1·Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.
  • 8.F.A.1·<p>Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. <span>Function notation is not required in Grade 8.</span></p>

10 min

15 min