Word problems are challenging, no matter the age or the
mathematical level of the problem solver.
I can almost hear the collective groans of students when I even say word problems, and for calculus students
related rates problems produce some of the loudest groans. For most word problems, the actual process of
calculating a solution is not the hard part.
The hard part is setting up the problem to get to the point where
calculations can happen.
Related rates is one of the categories of mathematical selfies I always include in
a Calculus I project. Like many
categories of word problems in textbooks, related rates problems fall into a
few basic types among which are changes in two attributes of a geometric shape such
as volume and diameter, and change in length of shadows and speed of an object’s
movement. After a good bit of practice,
students can usually learn to solve the standard types of problems, but they
often don’t see any relevance of those problems to themselves. Given that there are typical problems that
students are asked to solve, it isn’t surprising that for this part of the
project, students notice or recreate those very problems. What is interesting is how those problems
become more personal and how the creation of the mathematical selfie requires that they experience the relationships
in a dynamic way. Textbook illustrations
for these problems are obviously static, and so the dynamic relationships
between the rates of change within a problem can be hard for students to grasp.
I am always interested to discover what
types of examples students see, and how they make this process more relevant to
themselves. Here are examples of some of the standard problems and how students
translated them.
1. Geometric attributes
One typical problem for this type of related rates is
pouring some substance at a given rate into a pile. There is usually a diagram in an industrial
setting often with a conveyor belt and a pile of sand. That’s probably not going to mean a lot to
most students. Making cookies, however,
does allow a student to see this exact process in a dynamic way. It was interesting that two students
submitted very similar photos, which probably means that cookies are an
important part of college life! Photo 1
describes the process in a way that indicates the rates of change involved but
doesn’t include the units. Photo 2 notes the all-important units of the rates
of change, which is always a big step in student understanding. I am particularly intrigued by the text on
the two photos and how it reveals a difference in student understanding. In photo 1, the student has recreated the
diagram of the standard problem, but the two rates described actually represent
the same attribute of volume--the change in volume of the sugar being sifted
and the change in volume of the sugar in the pile. This says to me that while the mathematical selfie recreates the
diagram of a related rates problem, the text indicates a lack of understanding
of the nature of related rates as opposed to a single rate of change. Photo 2, on the other hand, shows a deeper
understanding when it relates two different attributes, rate of change of
volume and rate of change of height. The
author even attempts to write the description like a typical textbook question.
2. Shadows and motion
This type of problem is
usually described with a spot light, a person’s shadow, and motion of either
the light or the person. I often get
questions from students of why there is a spot light pointed at a wall, or why
a person would be walking between the light and the wall. This classic question leads to a great use
of similar triangles or trig ratios in the solution, but it is very
contrived. A student, however, provides
a pertinent example of this very process in photo 3. Music and concerts are an integral part of
student life. I am really impressed that
a student saw the related rates happening in this environment. I also think the
problem that the student implies in the description could be a very challenging
related rates problem because we can’t really expect the musician to walk in a
line on stage, and even if we limit the problem to just one of the shadows that
would be cast by these lights, it would still be very challenging given a
particular pathway of the musician’s movement.
There are several other typical types of related rates
problems, but I’ll stop with these two for this time. The next time I teach
Calculus I, I plan to use these mathematical
selfies as the basis for some typical related rates problem examples that
students may find more relevant, although I will need to put several
constraints on the shadows problem or risk a word problem mutiny!
