August 22, 2012 at 8:32 am #870
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.August 22, 2012 at 8:53 am #872
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.August 22, 2012 at 9:09 am #873
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.January 20, 2016 at 12:50 pm #3525
I would argue that the ratio and proportions standards when viewed in light of the overarching mathematical practices and underlying grade 5 standards 5.OA.3 and 5.G.2 would support an earlier exposure to “negative slope” relationships than grade 8. If we support the traditionally less fluent operations of subtraction and division through grade 5 exposure to descending patterns within tables, student facility with these operations can not only be remediated, but a foundation for the understanding of negative proportional relationships can be laid. In grade 6, this can then be further supported by providing situational contexts for negative slope that the students can relate to such as tracking a runner’s distance from Home Base in the problem “Running Home From Third Base”. Grade 6 student understanding of negative slope has been shown to be easily facilitated by combining this earlier introduction to descending table patterns with a subsequent physical modeling of decreasing distance over time. When this physical modeling was then combined with freeze-framed second-by-second representations of this motion on a number line students had little difficulty conceptualizing and interacting with the tabular and graphical representations of this negative slope scenario.