A note on linear functions - by Axelle Faughn

The following article just caught my eyes a few weeks ago, looks like we have some spontaneous uses for selfies by students for modeling with mathematics:
http://www.dailymail.co.uk/femail/article-3784114/Skills-dance-floor-Virginia-student-goes-viral-using-math-calculate-angle-dab-dance-move.html

One of my own students this semester used dabbing as a representation of a linear function for one of his selfies assignments. Not only did his submission provide me with insight into my students' world, but as the article shows, it can be taken further to perform more rigorous computations. In the article above I find it  particularly fascinating that although Anicca was able to measure all three sides of her triangle, she decided to use the tangent trigonometric ratio to calculate the angle of elevation of her dab. Linear functions are characterized by a constant rate of change: the slope. In Anicca's example, the tangent ratio represents the change in y (rise) over the change in x (run), which is precisely the slope of the associated linear function supporting the hypotenuse. Two students, two locations, same idea!

As for other representations of linear functions, here are some pictorial examples collected during recent mathematical selfies "treasure hunts" where I asked participants to illustrate these ideas:

On the left we have a good visual for connecting slope (the steps) to the linear function represented by the handrail, or its parallel line on the wall just above the steps. One should consider discussing with students what happens if steps are higher (steeper line) versus deeper, and how such decisions might be made when building a staircase. An online search provides standards for step building, but also plenty of examples or situations where standards have to be adjusted. And what might happen if the handrail and the staircase did not have the same slope (i.e. were not parallel)?













On the right, the student was concerned with the location of the line in a set of coordinates and found natural features to illustrate this concept. The height of the branch up the tree would represent the y-intercept, or initial value, of the tree branch linear function, in other words the "b" in the equation y=mx+b.




And for those of you with esthetics considerations, I thought the student who submitted the following picture did a great job at capturing the idea of embedding a quote into the picture to give it an edgy feel emphasized by the night time shot. Not only do selfies allow us to bring in more concrete and engaging representations into the classroom, they appeal to the more creative, artistically inclined students who will often come out of their way to come up with esthetically pleasing submissions.