Tables, Equations, and Graphs of Functions

5 min

Teacher Prep
Setup
Display the graph. 1 minute of quiet think time, followed by partner and whole-class discussions.

Narrative

The purpose of this Warm-up is to prepare students to consider which graphs do and do not represent functions, which will be useful when students practice interpreting graphs of functions or make sense of why a specific graph could not represent a function. While students may notice and wonder many things about this graph, the interpretation of the context the graph represents is the important discussion point.

As students notice and wonder, they have the opportunity to reason abstractly and quantitatively if they consider the situation that the graph represents (MP2). This Warm-up also prompts students to make sense of a problem before solving it by familiarizing themselves with a context and the mathematics that might be involved (MP1). 

Launch

Arrange students in groups of 2. Display the graph for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time and then 1 minute to discuss with their partner the things they notice and wonder.

Student Task

What do you notice? What do you wonder?

A graph in a coordinate plane.
A graph in a coordinate plane. Horizontal axis, distance from starting line in meters, vertical axis, time in seconds. The graph begins at the origin and increases steadly as it moves right until it reaches 200 comma 40.  The graph then turns and increases steadily as it moves left until it reaches the point 0 comma 74.

Sample Response

Students may notice:

  • The graph looks like a sideways V.
  • The graph doubles back on itself.
  • The graph is made of pieces that are almost, but not quite, straight.
  • Distances go between 0 and 200 meters.
  • Times go from 0 to a little more than 72 seconds.
  • The horizontal axis tells us the distance from the starting line in meters.
  • The vertical axis tells us the time in seconds. 
  • The person gets farther from the starting line and then comes back. 
  • The person was 75 meters from the starting line at about 14 seconds and 60 seconds. 
  • The person got back to the starting line in about 75 seconds. 
  • The furthest distance the person got from the start line was 200 meters. 

Students may wonder:

  • Is this about one person or more than one person?
  • Why does the graph double back?
  • Who is the graph about?
  • Why are they coming back?
  • Are they running or walking?
  • Did they take the same exact time to go out as they did to come back?
  • What is the title of this graph?
Activity Synthesis (Teacher Notes)

Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary. If possible, record the relevant reasoning on or near the graph. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and respectfully ask for clarification, point out contradicting information, or voice any disagreement.

If time allows and the situation the graph represents does not come up during the conversation, ask students to briefly discuss this idea.

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>

15 min

15 min