Although this question is badly worded (and I agree that giving the formula for a falling sandbag ruins the problem completely), it does not seem very different from many of your WCYDWT problems, other than that it does not come with a flashy video. If you had a video camera in a hot-air balloon, I would not at all be surprised to see a WCYDWT video of shadows.

I think your objections are not so much to pseudocontext as to presentation style.

Since I’ve begun to grasp what is meant by the term “pseudocontext”, I have pretty much agreed with the verdict of pseudocontext for all of the problems thus far presented. (Especially the dog bandana, which epitomizes everything that is wrong with pseudocontext.)

But I have to disagree with that verdict here. This problem seems legitimately perplexing to me. Despite the fact that I don’t really care much about hot air balloons and sandbags, and despite the fact that I can’t imagine anyone ever wanting this sort of information in real life, this problem still inspires me.

This problem feels complex, and the idea that we can compute the answer with basic math seems almost magical.

I agree with gasstation that this feels very much like one of the WCYDWT videos, and it inspires me in much the same way. I genuinely want to work on this problem. I don’t care much about the speed of escalators, but that doesn’t keep me from being very interested in Dan’s videos.

Mark, I don’t think this problem’s inclusion in “pseudocontext” is a condemnation on related rates as a whole. I do know people who enjoy hot air ballooning and can’t think of anyone (among that group or otherwise) who would care about the shadow’s movement.

On the other hand, change the context slightly to a (supposed) film of a UFO or Neil Armstrong’s shadow on the (supposed) moon and mix in a bit of conspiracy theory and we’ve got a slightly more interesting problem with a result that means something more to the students.

@gas/mark/gavin

I don’t care about hot air balloons at all, just as I don’t care about toasters or escalators. That’s not the point.

Where this problem fails horribly is that the shadow question is like question #382 in my mind when I hear “hot air balloon and dropping sandbag.” I’m wondering how much the balloon will rise first off, since that seems to be the reason for doing so. I’d wonder where exactly the sandbag fell. I think every physics teacher in the world has done some sort of “dropping a bomb” project. At some point I might care how fast the sandbag is falling. Only then might someone wonder if you can figure out how fast the shadow itself is going.

It’s not the balloon setup. It’s that it force feeds you the question and screams at you how to do it.

Does this fall under some arbitrary definition of pseudocontext? Arguable. (FWIW, I think it’s a partial. You wouldn’t ask the question, but you would use the method.)

What’s not arguable is that a textbook has again managed to take an interesting relationship and turn it into a mindless rote operation.

PS – You would improve it immensely simply by stripping out all the givens. Which is where the flashy presentation could be helpful.

I’ve seen loads and loads of related rates problems, but this one seems a bit above the rest, since it’s a problem that I’m actually interested in solving. At least partially because of the context. Maybe it just hits different people different ways.

It wouldn’t have to be a balloon and sandbag, but I’m not sure how else to set up this problem. If it were an object being dropped from a wall, the wall would block the shadow. What other options are there for setting up this problem?

I’m starting to think about the Uncanny Valley. http://en.wikipedia.org/wiki/Uncanny_valley

“My Favorite Orange” seems more acceptable because it is playful in its unwarranted application.

“Sandbag Shadow” tries so hard to be a true application that it becomes ludicrous. This isn’t even counting the “hint”.

@ Katie W: When I taught CPM, I remember really liking the Line Factory–and students (most of them at least) liked it too. I thought of it less as a “context” and more as a “frame”. Let me explain what I mean by that distinction… It’s kind of a matter of scale for me. Context = setting for one problem, frame = setting for large group of problems.

I actually think that one of the problems with most textbooks is the lack of any meaningful frame around the big ideas [I think IMP is a notable exception, but I have only taught 2 of the units].

Chapter 7 is not a meaningful frame. The Line Factory–while a little on the cutesy side–is a decent frame. It takes the big idea and unites all of the little concepts/skills that are a part of it under one narrative. I like how a good frame makes the connections between various concepts easier to identify.

What I didn’t like about the Line Factory was that the chapter starts out using the frame, goes off and does some other unrelated stuff for a while, and then comes back to the frame assuming that everyone remembers what it is. The second time through the unit, I wound up rewriting everything that wasn’t Line Factory into LF-material. It made for a more cohesive unit.

So pseudocontext is something that would leave prior real-world knowledge out – such as “describe what is seen from a plane when a parachute jumper leaves it.” I think we’ve all seen the video, but we are asked to believe that the jumper stays horizontally with the plane and falls directly under it. “Ignore air resistance.” Then the plane is not flying, because there’s no air resistance.
And pseudocontext would ask a question that nobody in their right mind would ask.
However, like the time to hear a splash when dropping a pebble into a well, it seems to me that this question does ask for useful information, both for the balloonists (who do drop things in case of necessity), and for the unfortunate person underneath who wants to avoid being hit with the sandbag. A video would be great. Does anyone have a balloon?

It’s my opinion, that since this is mostly a Physics concept being discussed in a Math text book, the authors are unaware of or concerned with the previous knowledge of the reader and calculus based physics since those often are concurrent classes for science and engineering majors.

Since I’m familiar with the form of classical freshman notation motion functions, I can see that the given equation defines an object under constant acceleration of gravity (g/2 being simplified to 4.9 and units implied to mean m/s), but I feel for those students not enrolled in the class that talks about how to derive this equation from scratch and understand that the given equation is essential to getting a reasonable, uniform answer related to the subject matter being discussed in the preceding chapter.

If all givens on the problem are excluded, then students would have to reconcile the fact that the balloon could have vertical or horizontal components to velocity, there might be a wind in any direction, and there might be a component of wind resistance on the sandbag as it falls perhaps giving rise to an effect of terminal velocity.

But since the problem says “The position of the sandbag is determined by s(t)” many of the above issues can be assumed to be either zero or negligible (a concept which the learned appropriate application of is in a Science class than it’s purer cousin Math).

The language is quite succinct. “The position” speaks more absolutely the concepts being implied than “the height” which could leave room for a vector of movement in the plane parallel to the ground. Suddenly I’m aware that no where in this question does it say “Assume the ground is level and flat.” Where it could be further complicated by hills, valleys, or a constant angle.

The problem also does not mention the boundaries that this equation is accurate for, t from 0 through x where s(x)=0. A clever enough student in the right classroom might be able to argue sufficiently that a correct function for the speed or position of the shadow related to time would need to be properly bound to an interval of t or else it would be a constant value of zero sometime shortly after the impact, and thus necessitate at least partial credit being taken from all other average students who answered much like the book’s key presents.

But I suspect the answer listed in the teacher’s manual implies the use of a geometric translation of sandbag height to shadow distance from impact and a derivative of that to acquire an expression of linear velocity related to time. And in which case, I’d argue much like NASA does for loss of vehicle events that a failure of imagination is acceptable for humans unexperienced at a given task.