[PS] Sandbags & Hot Air Balloons

[BTW: My opinion is that this isn’t pseudocontext for reasons I elaborate on in this comment.]

Calculus: Early Transcendental Functions. Larson Hostetler, Edwards. 2003. Houghton Mifflin.


Greg Hitt (a/k/a Sarcasymptote):

Et tu, Calculus?

Seriously, you are when I started to love math. Up until I met you, I trudged through year after year of pseudocontext, merely completing tasks only because they needed to be done and not out of some enjoyment or aching desire to do it. Then, you came along, and blew my freaking mind with your applicability. You were the math that worked.

And now look at you. Really, Calculus? Answer me this:

  1. What in the hell are you talking about? This is one of the most poorly worded questions I’ve ever seen. And is it common practice for someone to be so cavalier as to heave sandbags over the side from almost 200 feet in the air, all while carefully measuring both the height above the ground and the angle of elevation to the sun? I don’t know, because I don’t care about hot air balloons. I don’t know anyone who does.
  2. Why are we worried about the rate of the movement of the shadow? Is the rate that a shadow moves across the ground useful for any situation? Is there some arbitrary race between the shadow of a sandbag and a log floating down a river? Seriously, related rates should be much more applicable than this.
  3. This is textbook, er… textbook. We could pretend to think about how to model the height, but instead, let’s just skip the thinking and go right to a formula.


Moving Shadow. A sandbag is dropped from a balloon at a height of 60 meters when the angle of elevation to the sun is 30° (see figure). Find the rate at which the shadow of the sandbag is traveling along the ground when the sandbag is at a height of 35 meters. [Hint: The position of the sandbag is given by s(t) = 60 – 4.9t2.]


  1. Scan an example of pseudocontext.
  2. Email it to dan@mrmeyer.com
  3. List the textbook title, edition, and publisher.
  4. Give me your interpretation of the term “pseudocontext.”
  5. Let me know if you’d like credit (name, blog or twitter) or if you’d prefer anonymity.
I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.


  1. 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.

  2. bah ha ha, I have to disagree with gasstation. There is a context of sorts, but we’re literally chasing shadows. Which could be interesting: anyone who has had a collision with an airplane shadow wonders all sorts of interesting questions.

    But this question demands some work for worthless information. I posit that exercise #147 did not receive a lot of editorial supervision for this edition of “Early Transcendental Functions.”


  3. Mark Schwartzkopf

    October 30, 2010 - 10:51 am -

    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.

  4. I keep wondering about this pseudocontext thing. I understand what you mean by it, and my impression is that you have a lot of disdain for these kinds of problems.

    I’m teaching from CPM (College Preparatory Mathematics), however, and I can’t decide what you would make of some of their problems. For example, the algebra connections books use the idea of a “line factory” in which patrons can order lines with different characteristics. It’s obviously not even pretending to be “real world.” Does this still count as pseudo-context?

  5. 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.

  6. @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.

  7. We have a balloon race at my college every year. I am totally doing this!

    This doesn’t look like finding a make believe context just to have some word problems in the section. It looks like the author felt the need to write 147 questions or more for that section and ran out of new ideas.

    It would be a much more interesting problem to add a horizontal component to the sandbag and try to use the shadow to estimate the speed of the balloon.

  8. Mark Schwartzkopf

    October 30, 2010 - 9:40 pm -

    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?

  9. FWIW, I’ve come around to the idea that this isn’t, in fact, pseudocontext. Pseudocontext asks you to apply math that isn’t inherent to the context. There isn’t anything inherent to an orange, for instance, that would like to solving this equation. But related rates are inherent to a sandbag dropped out of a hot air balloon with the sun in the sky.

    Put another way, as I’m coming to understand pseudocontext, if there isn’t a path between your context and the math, that’s pseudocontext. The path here is rough and gravely. It isn’t a natural path. If you show your students that image and ask for questions, you’ll find ten times as many students wondering about the speed of the sandbag itself, or the force with which it’s hitting the ground, as the speed of the sandbag’s shadow. But there is a path there.

    Video would do a lot to inoculate the pseudocontext. For one thing, how are you going to get students wondering about speed if you don’t show something moving? I’m thinking something like this frame. But, let’s be real, if you’re contracting a hot air balloon and setting up a camera and speed gun at the exact moment the sun is 30° in the sky, you probably have the resources to film a better related rates problem than this.

    And Jason is exactly right that the givens ought to be stripped out of the problem, for a lot of different reasons, not the least of which is that you’re doing the student’s work for her when you give her that information.

  10. I like your refinement of the definition of pseudocontext and I agree this example is at least in spirit a “real problem”. What’s missing for me in most of conversations about these problems is the context in which it is posed. My first reaction to a contrived textbook-type problem like this one is: what can I do with this to make it interesting for my students and does it forward what I want them to learn? Just because its not pseudo, doesn’t make it suitable for my purposes. If it is a good idea for a problem, how can I improve it? (E.g. show a photo of a real balloon and go from there.)

  11. @ 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.

  12. The definition of pseudocontext seems to have been pretty amorphous over these weeks, so maybe we should lay out what pseudocontext is and what it is not.

    Some people have argued that this is not pseudocontext because it is interesting. A couple of points on that. Someone (I think Dan) at one point said that we shouldn’t judge a problem’s merit on whether or not someone finds it interesting, because you can always find someone who is apathetic to the math or the subject of hot air balloons or whatever. If I really think about it, I could find this an interesting exercise, but again I think that it is purely from the fact that it is a Calculus problem and involves some higher-order thinking. I also found the problem of My Favorite Orange at least entertaining, even if absurd.

    I also like Jason’s point, and this was really why I chose the problem in the first place: this is way down on the list of things to ask about when in a hot air balloon. Lack of relevancy of course does not make it pseudocontext, just a dumb problem. But I still think that there is something about creating some sort of context, no matter how ridiculous it is, and then asking a question that really doesn’t have much applicability to the construct you just created.

    I’m OK conceding that this might not be pseudocontext in the way it has been previously shown so far. I’ll admit that this is not as glaring of an example as creating an arbitrary algebra problem out of the number of jingles on a dress. However, I still think that there is something inherently wrong about this problem. The only difference between this problem and the one with the guitar is that the equation for this is derived from the position function for an object in free fall, which is, I guess, somewhat contextual, while the system of equations from the guitar example is completely arbitrary. However, no mention of free fall is made in the problem, and the student doesn’t have to use any knowledge about an object in free fall. I think that is why this problem was such a glaring example of what I thought to be pseudocontext. Look at the original statement from Boaler: “Students come to know this about math class. They know that they are entering a realm in which common sense and real-world knowledge are not needed.” It is pretty easy to do this problem without any knowledge of the real-world and the way that gravity works (oh, and, let’s totally ignore air resistance, because that is not real world, right?). It still lays out a set of parameters and allows you to plug and chug your way to the finish line. You could replace that position function with any random equation and get to some answer and you wouldn’t know that the answer didn’t make sense for how things actually behave on Earth. Again, maybe not pseudocontext, but a different manifestation of a poorly written problem. But its all a fine line, methinks.

  13. 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?

  14. >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 love this! I talked about this (less eloquently) in my post on switching to something like SBG this semester and getting away from the textbook. I framed the course as linear and quadratic topics. I framed the quadratic topics by throwing chalk over and over on the first day of this part of the course, first to get them to draw the path, then to talk about height versus time (for chalk thrown straight up).

    Today when we started talking about factoring, and a student asked when we’d use it, I was able to remind her of the chalk, and make up a problem that was sort of like the chalk, and factorable. (-16x^x+32x+48=0. I think the 32x term represents an unrealistic initial speed, and why would I start 48 feet high?)

  15. 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.

  16. Also, the problem likely fails to take into account the curvature of the Earth and the fact that light from the Sun is not perfectly parallel, something which at this low an altitude compared to the Earth’s radius and extreme distance from the Sun’s center might be argued in a Physics class as being negligible, but in a Math class might have less wiggle room since approximate answers are so often frowned upon and a reasonable expression might be determined with enough variables…

    …assuming the teacher and student are imaginative enough and care enough to apply the needed geometry to the projection.