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A heated debate came up at a common core in-service I was participating in the other day over the EE cluster on proportional relationships and y = mx+ b. A number of teachers were stating that y = mx + b is proportional because after the initial point it then has a constant of proportionality. The other group of teachers said that y = mx + b is a non-proportional linear relationship and that this is only proportional when b is equal to 0. As nothing was really resolved, I thought I would bring it to the forums and see if I could bring back some clarification to the group.
I think the answer you are looking for is on Page 8 of the progression, “Ratios and Proportional Relationships,”
Where it says, “Recognizing proportional relationships. Students examine situations carefully to determine if they describe a proportional relationship…Students recognize that graphs that are not lines through the origin and tables in which there is not a constant ratio in the entries do not represent proportional relationships.”Maybe you also want to look at p. 14 of the Ratio and Proportional Relationship Progression, which describes proportional relationships in terms of ratios and begins: A proportional relationship is a collection of pairs of numbers that are in equivalent ratios.
For example, if the relationship in question is given by y = 2x + 1, it has pairs, e.g., (0, 1), (1,3), (2,5), that are not in equivalent ratios. Because of that the relationship between x and y is not a proportional relationship.
The m in y = mx + b is not always a constant of proportionality for the relationship between x and y because y = mx + b does not always represent a proportional relationship between x and y. (On the other hand, one could consider the relationship between the quantities represented by y – b and x.)
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