Negative Constant of Proportionality?

Home Forums Questions about the standards 6–7 Ratios and Proportional Relationships Negative Constant of Proportionality?

This topic contains 3 replies, has 3 voices, and was last updated by  Susan Forbes 1 year, 6 months ago.

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Participant

7.RP.2a

Can the constant of proportionality ever be negative?

Our team is planning for the fall, and noticed that every example in the progression document has a positive slope in the first quadrant.

Participant

To clarify, we see at the end of the progression that we are limited to first-quadrant, positive constants of proportionality in 7th grade. But is this true in HS and college-level courses as well? We want to understand the mathematics more deeply.

Bill McCallum
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Although the standards do not say so explicitly, the examples do suggest that the constant of proportionality is always positive in Grades 6–7. As you point out, this is confirmed by the progressions document on Ratios and Proportional Relationships says on page 11 that

Proportional relationships are a major type of linear function; they are those linear functions that have a positive rate of change and take 0 to 0.

It makes sense initially to keep the constant positive, since although negative numbers have been introduced in Grade 6, most of the quantities being dealt with in proportional relationships are positive.

Once students start dealing with linear functions in Grade 8 and beyond, they become familiar with the meaning of negative slope and negative rate of change, and they can use those terms to describe functions of the form $f(x) = kx$ with $k <0$. I could go either way on whether you call that function a proportional relationship, and I don’t think it much matters which way you go. It’s probably easiest to allow the term, since that’s what people will do anyway. There’s a useful brief discussion of this question at the Math Forum.