Confusion about 8th grade Function Standards

[linseykirby]

After what we feel to be close analysis of the Function Standards at the 8th grade level, we would like some clarification on the difference between these standards and the standards that address linear relationships.

One example we have: Given a table that is a Linear Relationship, I would expect my students to make a graph that is continuous and using an infinite domain. When given a Linear Function, would I expect my students to make a graph that is infinite/continuous, continuous with limits, or discrete with limits?

We have many more questions on the topic about how functions (at the 8th grade level) differ from linear relationships. We would appreciate any and all clarification you can provide on this subject.

[Anonymous]

The situation in Baseball Cards could be a good place for students to discuss whether a continuos graph always gives a faithful picture.

[Bill McCallum]

I don’t completely understand the question, but I’ll try to say a few things that might help.

First, the phrase “linear relationship” does not occur in the standards, but the phrase “proportional relationship” does. The concept of a proportional relationship is a precursor the concept of a function. One important difference is that when you define a function you designate one of the variables as the input variable and the other as the output variable. Also, as students start to study functions, they start to think of them as objects in their own right. Later in high school they use a letter to stand for a function, and they perform various operations on functions. In Grade 8 the focus is on simply understanding a function as something that takes inputs and yields outputs. Yes, the domain is important, but truth be told the same is true with proportional relationships. So when you say “a table that is a Linear Relationship” you have to be careful. In some cases the variables may only take on whole number of values (e.g. the number of baseball cards), and then it would no more be appropriate in this case to ignore that restriction than it would be in talking about the domain of a function.

Also, it’s important to be clear about the distinction between a function and an equation that defines the function. The equation $y = 2x + 3$ can be viewed as defining a linear function, with $x$ specified as the input variable, $y$ specified as the output variable, and the equation understood as giving the value of $y$ in terms of $x$. But it can also be viewed as an equation in two variables whose solutions form a straight line; an algebraic description of a geometric object. The it is not correct to say that the equation is a function. Rather one should say that the equation defines a function, but also has other uses.

[terehi]

I also have a question that is a bit more pragmatic. I’m in the middle of reviewing an 8th grade text for my state adoption. In the earlier part of the text they do an extensive section on proportional and nonproportional “relationships”, but never mention the concept of function. Later in the text, they delve deeply into “functions” with. Definitions and such. My dilemma is that they include the earlier lessons on relationships as primary citations matched to the function standards. What are your thoughts on that? Do those citations qualify even though there is no mention of a “function” or is this an insurmountable flaw in alignment?

[Bill McCallum]

I’d have to see the text to be able to comment on it. But as a general comment I would say that there is a progression from proportional relationships to functions. At some point in that progression you introduce the concept of a function, and then you can look back and point out that a proportional relationship can be viewed as a function (in two different ways, depending on which quantity you choose as the input). I don’t know where the non-proportional relationships come in, however, that sounds a bit out of place to me.

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