Lesson 14
Four Representations
Let’s contrast relationships that are and are not proportional in four different ways.
14.1: Which is the Bluest?
 Which group of blocks is the bluest?
 Order the groups of blocks from least blue to bluest.
14.2: One Scenario, Four Representations

Select two things from different lists. Make up a situation where there is a proportional relationship between quantities that involve these things.
creatures
 starfish
 centipedes
 earthworms
 dinosaurs
length
 centimeters
 cubits
 kilometers
 parsecs
time
 nanoseconds
 minutes
 years
 millennia
volume
 milliliters
 gallons
 bushels
 cubic miles
body parts
 legs
 eyes
 neurons
 digits
area
 square microns
 acres
 hides
 square lightyears
weight
 nanograms
 ounces
 deben
 metric tonnes
substance
 helium
 oobleck
 pitch
 glue
 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.

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.
 Write one or more sentences describing the relationship between the things you chose.
 Make a table with titles in each column and at least 6 pairs of numbers relating the two things.
 Graph the situation and label the axes.
 Write an equation showing the relationship and explain in your own words what each number and letter in your equation means.
 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.
Summary
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:
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.