Interpreting Graphs of Proportional Relationships

5 min

Narrative

In this Warm-up students are given an unlabeled graph of a proportional relationship and asked to invent a situation that it could represent. This gives students an opportunity to think back to examples of proportional relationships they have encountered. After several possible contexts are shared, students label the axes of the graph, give it a title, and interpret the meaning of a point on the graph. This is an opportunity for students to attend to precision in language (MP6). During the discussion, the characteristics of a graph of a proportional relationship are reinforced.

Launch

Tell students that they will look at an unlabeled graph, and their job is to think of a situation that the graph could represent. Display the problem stem for all to see and give 1 minute of quiet think time. Ask students to give a signal when they have thought of a situation.

Invite some students to share their ideas and record the responses for all to see. The purpose of this is to provide some inspiration to students who haven't come up with anything.

Ask students how they know all of the relationships are proportional. (When one value is 0, the other is 0. The situation involves equivalent ratios. Any pair of values in the relationship has the same unit rate.)

Ask students to complete the questions.

Student Task

Here is a graph that represents a proportional relationship.

Graph of a line, x y plane, origin O. Horizontal and vertical axis both have scale 0 to 20 by 2’s. The line starts at the origin, goes through point (16 comma 12) and keeps rising.

Invent a situation that could be represented by this graph.

  1. Label the axes with the quantities in your situation.
  2. Give the graph a title.
  3. There is a point on the graph. What does it represent in your situation?

Sample Response

Answers vary. Sample response: A car is moving at a constant speed. Its speed is 34\frac34 mile per minute or its pace is 43\frac43 minutes per mile.

  1. The horizontal axis is labeled time (minutes) and the vertical axis is labeled distance (miles).
  2. Distance Traveled by a Car and How Much Time It Takes
  3. The coordinates of the point are (16,12)(16,12). In this situation, it means that the car travels 12 miles in 16 minutes.

Synthesis

Ask a few students to share their situations and other responses. After each, ask the class if they need more information to understand the situation. After a few students have shared, ask the class to think about how all the situations were different and what they had in common. What sorts of things are always true about proportional relationships? Some possible responses might be:

  • When one quantity is 0, the other is also 0.
  • There is always the same amount of one quantity for every 1 of the other quantity.
  • Context-specific considerations like constant speed, the same taste, or the same color.

Remind students that a coordinate point, (x,y)(x,y) is made up of the “xx-coordinate” and the “yy-coordinate.”

Standards
Addressing
  • 7.RP.A·Analyze proportional relationships and use them to solve real-world and mathematical problems.
  • 7.RP.A·Analyze proportional relationships and use them to solve real-world and mathematical problems.

20 min

10 min