Lesson 14: Four Representations

Let’s contrast relationships that are and are not proportional in four different ways.

14.1: Which is the Bluest?

  1. Which group of blocks is the bluest?
    Five figures of yellow and blue dots labeled A, B, C, D, and E. Figure A  has 1 yellow and 3 blue dots. Figure B has 2 yellow and 3 blue dots. Figure C has 2 yellow and 4 blue dots. Figure D has 2 yellow and 5 blue dots. Figure E has 3 yellow and 4 blue dots.
  2. Order the groups of blocks from least blue to bluest.

14.2: One Scenario, Four Representations

  1. Select two things from different lists. Make up a situation where there is a proportional relationship between quantities that involve these things.


    • starfish
    • centipedes
    • earthworms
    • dinosaurs


    • centimeters
    • cubits
    • kilometers
    • parsecs


    • nanoseconds
    • minutes
    • years
    • millennia


    • milliliters
    • gallons
    • bushels
    • cubic miles

    body parts

    • legs
    • eyes
    • neurons
    • digits


    • square microns
    • acres
    • hides
    • square light-years


    • nanograms
    • ounces
    • deben
    • metric tonnes


    • helium
    • oobleck
    • pitch
    • glue
  2. Select two other things from the lists, and make up a situation where there is a relationship between quantities that involve these things, but the relationship is not proportional.
  3. Your teacher will give you two copies of the “One Scenario, Four Representations” sheet. For each of your situations, describe the relationships in detail. If you get stuck, consider asking your teacher for a copy of the sample response.

    1. Write one or more sentences describing the relationship between the things you chose.
    2. Make a table with titles in each column and at least 6 pairs of numbers relating the two things.
    3. Graph the situation and label the axes.
    4. Write an equation showing the relationship and explain in your own words what each number and letter in your equation means.
    5. Explain how you know whether each relationship is proportional or not proportional. Give as many reasons as you can.

14.3: Make a Poster

Create a visual display of your two situations that includes all the information from the previous activity.


The constant of proportionality for a proportional relationship can often be easily identified in a graph, a table, and an equation that represents it. Here is an example of all three representations for the same relationship. The constant of proportionality is circled:

A graph in the coordinate plane and a 2-column table: A coordinate plane with the origin labeled "O". The numbers 0 through 8 appear along the x-axis and the numbers 0 through 10 appear along the y-axis. The line begins at the origin and moves upwards and to the right. It passes through the points with coordinates (1, 7/4) and (4, 7).  The first column of the table is labeled "x" and the second column is labeled "y." Row 1: 0, 0; Row 2: 1, 7/4; Row 3: 2, 7/2; Row 4: 3, 21/4; Row 5: 4, 7. The equation y = 7/4x is above the table.

On the other hand, some relationships are not proportional. If the graph of a relationship is not a straight line through the origin, if the equation cannot be expressed in the form $y = kx$, or if the table does not have a constant of proportionality that you can multiply by any number in the first column to get the associated number in the second column, then the relationship between the quantities is not a proportional relationship.

Practice Problems ▶