Adapted from the May 2012 issue of Mathematics Teacher:
A dog is running to fetch a ball thrown in the water. Point A is the dog’s starting point, point B is the location of the ball in the water, and point D can vary. Given that the dog’s rate of swimming is 1 meter per second and its rate of running is 4 meters per second, determine where point D should be located to minimize the time spent fetching the ball.
Some questions to consider here:
- In what ways has this context already been abstracted?
- Can you de-abstract (recontextualize? concretize?) the context? Describe a task that would allow students to learn about the process of abstraction rather than just encounter its result.
MaxSeptember 24, 2012 - 8:35 am -
Some things that have already been abstracted:
The dog’s speed through the water and on land have been abstracted to a constant speed.
The shoreline has been abstracted to a straight line.
The location of the ball has been abstracted to a static point, coordinated by two measurements, one along the shore line (8m) and one perpendicular to the shore line (5m).
The dog’s path through the water has been abstracted to a straight line.
The shortest time to the ball has been, if not abstracted, reformulated to be based only on the ideal point for the dog to enter the water.
The dog, ball, and points along the shore have been abstracted to points.
To get a sense of how these abstractions were actually made as modeling choices, this nifty article explains how one professor determined that his dog’s brain is somehow doing a process that gets him optimal results for time to ball: http://www.maa.org/features/elvisdog.pdf
How I would concretize the context (assuming for now that we want to keep this about dogs chasing balls into the water):
-Show a video on a Smartboard of a dog super-excited about a tennis ball along a straight shoreline. Throw the ball and pause the video immediately.
-Ask students to predict the path the dog will take to the ball, and why. Have volunteers draw their prediction on the board. Ask if anyone wants to ask a question of anyone else or revise their prediction.
-Show the dog’s actual path. Compare it to the predictions. Wonder why the dog took that path. Generate hypotheses (shortest distance, least time, first place it could see the ball was in the water, etc).
-Pick a hypothesis to test and do the math. Where would the shortest distance be? Least time? Make sure least time is one hypothesis that gets tested, even if it’s your idea — I bet the kids will buy it. Make students come up with the data we need and share the data (and how it was gathered).
-Share data about where the dog enters the water from lots of trials. Is the dog consistent? Is the point consistently the shortest path? The least time? Something else?
-Have students independently solve the same problem, more abstracted, in a different context, like the lifeguard running/paddling towards a drowning person, laying cable along a road and through a yard, etc.
Steven PetersSeptember 24, 2012 - 12:17 pm -
2. To recontextualize this problem, I would change from playing fetch with a dog to a race, since the goal is more clearly a minimum time path (the dog might be tired and take the easiest path towards the end of a fetch game). Also, if the end point is fixed, I would recommend a buoy or an anchored dock rather than a ball that could easily move with the waves. Finally, I would show a picture of the shoreline and the destination (maybe a satellite photo?).
The task can be generalized a bit by starting away from the shore, so it changes the running distance slightly. This makes the geometry a little more complicated, but it turns out that this problem is analogous to optical refraction (the path that light follows when traveling between substances with different indices of refraction). I don’t know if this is interesting to high school students, but it has been the subject of some recent research papers about robot motion planning over varied surface types.
See the following papers for more info:
Path planning by optimal-path-map construction for homogeneous-cost two-dimensional regions
Finding optimal-path maps for path planning across weighted regions
On the Optimality of Dubins Paths across Heterogeneous Terrain
GarthSeptember 24, 2012 - 1:42 pm -
I have seen this as a pipeline problem. Cost of laying pipeline on land vs. laying pipeline underwater to a well site. Do not have to deal with a stupid dog not wanting to get wet.
Jason DyerSeptember 24, 2012 - 2:07 pm -
The problem is based off a real dog fetching a real ball.
Do dogs know calculus? by Timothy J. Pennings.
Do Dogs Know Related Rates Rather than Optimization? by Pierre Perruchet and Jorge Gallego.
Bob HansenSeptember 24, 2012 - 5:46 pm -
The key here is not how the problem was abstracted by why the problem was abstracted. How will become important but “why” is the epiphany.
I start a problem like this, as some have already suggested, with guessing and it isn’t a reasonable answer that I am after as much as I am determining if the student understands the implications of the problem itself. And it is these implications that I use to start cluing them into the modeling process (abstraction comes much later). There is a spot you need to get to in the water (maybe your friend is drowning). Is there a quickest route to that spot? You can run faster on the shore but the shortest distance is a direct path, through the water. What is the quickest route?
There are really only three valid guesses at this point. Head towards the spot directly, run along the shore till you are directly in front of the spot and start swimming, or something in between those two extremes. This is also a good point for a discussion of why there are only three choices? We can quickly negate the other two choices, running in the opposite direction or running past C, but why are we left with only three choices? Because, thus far we only know “A”, “C”, and somewhere in between. If there is an explicit point between “A” and “C” that gives us the quickest (but not necessarily the shortest) route then we will have to solve this problem and find that point.
Another discussion point here is how we have been labeling these points and how we will use those labels as we model. I use a trick by starting with no labels, just a picture, and then show how hard it is to even talk to the problem. The student will start talking and I will go to the board and say “Here?” and the student will say “No! At right angles!”, “You mean here?”, “NO! At right angles to the shore.”, “You mean run directly away from the water?”, “NOO! #$%$!!!”
Ok, so we understand the usefulness of labeling and geometry. Next, what is guiding us in this process? Obviously distance, but if it was just distance then the shortest distance would be the solution and we all know that the shortest distance between two points is a straight line. What is it about this problem (implication) that hints that it is not as simple as drawing a straight line? This problem is about quickest not shortest and the fact that there are two speeds involved, one on land and one on water, complicates things. Some students might say “The answer wouldn’t be a straight path because why would they make it so complicated for such a simple solution?” Very good reasoning, but not proof. Maybe save that discussion for statistics.
At this point a little experimentation is in order. Using basic geometry we can at least calculate the time for the two extremes at A and C and then pick a point in between and do the same. We will find that when we turn to the water in between A and C it is quicker. But where exactly is the quickest point? And more importantly, what is going on here?
Just to do the calculations for A, C and a point in between will have been a lot of busy work. At this point we can save a lot of time by combining what we have done thus far into a single expression for time. We can apply the notion of distance and the Pythagorean theorem and arrive at a formula to calculate the time. We can take this formula (actually a function) and plot it and behold, it has a very distinctive quality, it is a curve, and it is certainly not as easily interpolated as our linear examples thus far, but it clearly has a minimum, but exactly where is it?
Now the discussion has become abstract. It isn’t about dogs or shores anymore, it isn’t even about a model of dogs and shores, it is about a function with a personality. Assuming that this is an algebra 2 class (in its later stages) then we would go on and solve for this minimum, with algebra, not calculus, using techniques like this…
This would also be a good point to mention future attractions, like calculus. Obviously, things can be modeled more easily than they can be solved.
So we have gone the full gamut, from a real context, a dog, a beach and maybe a frisbee, to a mathematical model, and then to nothing but a function, to be solved. At this point I would explain to the class that most of applied mathematics (like engineering) is in that modeling stage, but you never know exactly what form that model will take and most of the skill is being able to work with what you have and being able to refactor it and shape it towards your goal. Algebra is simply the language with which we do all of that refactoring and shaping in. There is the algebra of puzzles, like that last step, and there is the algebra of modeling.
Paul KarafiolSeptember 25, 2012 - 5:43 am -
Actually, Dan, when I taught this last year, I took a page from your book: my dog and I went to the lake, he fetched some tennis balls from the water, and I got some footage which I showed to the class. It worked really well.
James KeySeptember 25, 2012 - 6:24 am -
@6 Paul K: If you took a “page” from Dan’s “book,” you have not been paying attention! ;-)
Jason DyerSeptember 25, 2012 - 6:34 am -
Paul, do you have the video(s) up somewhere?
James KeySeptember 25, 2012 - 9:32 am -
As others have said, just show a video and let the context speak for itself.
1) Students are capable of determining that SPEED MATTERS and SPEED VARIES. They should have no trouble realizing that the dog will move more quickly on land than in water. The teacher can supply the relevant rates once the class requests that information.
2) Show the picture/video, and ask students to represent the situation with a drawing. They will very likely produce some version of the drawing at the top of this post. If important elements are missing, the teacher can shape the discussion accordingly.
3) Point C is the point on the beach that is CLOSEST to the ball, and this point happens to lie along a line PERPENDICULAR to the shore. Students are capable of figuring both of these things out — 1) the closest point is important in the problem, and 2) the perpendicular line is used to find the relevant point.
MeghanSeptember 25, 2012 - 5:05 pm -
Weird how it’s such a particular example but so widespread. I mean I guess every topic has its ‘classic’ word problem.
But at the same time I very vividly remember hearing about this in high school and thinking ‘woah that’s so neat!’ — although I think my teacher said it was for real his actual dog.
But then still at the same time I know my kids would just DIE if they saw this problem. So much info to wade through. It’s good to remember that the stuff I thought was neat is definitely stuff that most teens aren’t so into. Or maybe my teacher really did present it in a nice way? (As it seems that Paul did with the real dog and lake and the videos, though I know for surely that my teacher didn’t bring in a video.)
RobSeptember 25, 2012 - 5:28 pm -
As a physics teacher, I feel obligated to say that this problem really shows up in optics. Light moves at different speeds in different media (air vs. glass vs. water, for example), and will bend such that it minimizes the time between the two endpoints.
With something like that, you could approach it by asking about why light bends when it hits water, and why it bends at the specific angle it does…
GregSeptember 26, 2012 - 3:04 am -
OK, does anyone have a video that shows a dog doing this? This one on YouTube comes close:
Fact is most of the time we throw the ball straight out, not on an angle like the problem indicates. That makes it a bit dodgy for me. The first question I would get from my students is “why would you do that?”. For me, that makes the pipeline laying problem better. I have also done this in the context of a car rally across the desert.
Bob HansenSeptember 30, 2012 - 9:54 am -
After reading the article “Do Dogs Know Calculus?” I tried looking at this problem again from the dog’s point of view. So your master has thrown a ball down the beach and into the water and you want to fetch it as fast as you can. To put this another way, you want to close the distance between you and the ball as quickly as possible. Since the ball is down the beach (and out in the water), at the beginning, running on land closes this distance faster. At some point though, as the angle between you, the beach and the ball widens, running along the beach does not close this distance as fast as swimming directly to the ball. Thus, the dog is simply making a judgement call and changing course when swimming directly to the ball closes this distance faster than running at an angle to it (along the beach). People do this as well when navigating crowds. Rather than plotting a direct path through the crowd we move along side it until a direct path is the faster course.
In any event, the distance DC is simply BC * Tan(ArcSin(Vw/Vb)) where Vw is the speed in water (swimming) and Vb is the speed on the beach (running). In this problem, DC = 5 * Tan(ArcSin(1/4)) = 1.29m.
Note: It doesn’t matter how far down the beach the dog starts, assuming of course that it is farther than 1.29m. The point at which it makes more sense to start swimming towards the ball depends only on how far out the ball is and the speed in water versus on land. In fact, if the dog is anywhere within that 1.29m mark, then jumping straight into the water will be the quickest course of action, and probably the one the dog would choose.
Bob HansenSeptember 30, 2012 - 7:15 pm -
I guess I should have included the step where I arrived at the minimum, it was simply looking at the speed on the beach as being a vector (Vb) and thus the rate of closing on the ball is Vb * Sin(DBC) where DBC is the angle formed by DBC. We are at the minimum when this rate is equal to the swimming speed (Vw) because running any further would close the distance to the ball slower than swimming directly towards the ball. Thus…
Vb * Sin(DBC) = Vw
Sin(DBC) = Vw/Vb
DBC = ArcSin(Vw/Vb)
Knowing DBC we can then find DC…
DC = BC * Tan(DBC)
And putting it all together…
DC = BC * Tan(ArcSin(Vw/Vb))
Cher YangNovember 3, 2012 - 10:49 am -
The biggest mistake that I see most students will be making with this type of abstract thinking is trying to understand what the problem is asking them to find. Deciding what information is useful and what is not is a big challenge for most students.
Bob HansenNovember 3, 2012 - 11:45 am -
“Deciding what information is useful and what is not is a big challenge for most students.”
I have seen this statement, or something like it, dozens of times and never gave it much thought, till now. I suppose that the author(s) might be trying to say something else but it comes out this way. On a problem like this, I myself don’t know what information is useful and what is not. Not until I recall or contrive an applicable model that leads to a solution. Until then, all of the information might be useful. It is only at that last instant when a solution pans out that the decision is made for me as to what information is useful and what is not.
Dan MeyerNovember 3, 2012 - 12:26 pm -
If I ask you how much it would cost to wash all the windows in Seattle, you’re likely able to sort out some information that’s useful and some that isn’t, even though you’ve never seen a worked example of the same task before.
Bob HansenNovember 3, 2012 - 1:35 pm -
I was speaking of the path problem here. I agree that your problem is a “research” problem because obviously you haven’t given me enough information.
Dan MeyerNovember 4, 2012 - 2:24 pm -
Distinction without a difference.
Bob HansenNovember 4, 2012 - 4:56 pm -
When you look at it, even the Seattle problem requires that you know how to solve the problem before you know what data to go after, right? You can’t just go hunting for data without a purpose. You need to establish a model that will reach the intended goal. The probable path would be to recognize that the cost of cleaning windows is related to the area of the windows and that you need to inventory the windows in the buildings in Seattle, by size, and tally it all up, in a spreadsheet.
The dog problem is a technical problem, the kind that an engineer might solve, but the Seattle problem is much more prevalent and 9 out of 10 of your students, if they ever apply math after school, will be confronted with the Seattle problem or a version of it regularly. If you were to examine my hard drive at work you would find that the ratio of Seattle problems to dog problems to be 30 to 1. On the flip side, my programming, especially my use of SQL, more resembles the dog problem, but I am a software engineer. Someone that is not a engineer will have all Seattle problems. Someone that is an engineer will have almost all Seattle problems. Don’t tell the kids, it might depress them.:)
From a reality standpoint, the Seattle problem is much richer than the dog problem. I don’t mean from a math perspective, as far as math goes, the Seattle problem only requires arithmetic and a spreadsheet. I am talking about the rest of the model that this (simple) math ties together. You won’t be the one cleaning the windows, you will be the manager of the project, or one of the financial analysts on the project. And you wouldn’t hire a single firm to clean the windows, you will have to hire multiple. First, the task is too large. There isn’t an Acme window cleaning company sitting around with 300,000 cleaners on stand by ready for the next city to ask to have it’s windows cleaned. Secondly, for a project that large with that much ($) at stake, you wouldn’t put all of your eggs in one basket.
Even once you have established the set of primary contractors, they will subcontract a lot of the work out to more contractors. That is just how business works, especially with large projects. A large portion of the cost in a mega project is managing a mega project. The overhead is much higher because mega projects are one-offs.
In any event, the spreadsheet (actually, there will be many) will involve a lot of organization, rates, discounts, interest, and other numerical data, beyond the area of the window times a price. There will probably be an award to the primes if the job is finished ahead of schedule (time is money). There will be bonds (insurance) to purchase in case one of your window cleaners decide to drop their bucket on some poor guy’s head. There will be weekly payments for this job which is called Cash Flow. Another dimension that affects such a project.
Sadly, they don’t teach any of this anymore. It used to be covered in business math. Nothing but arithmetic and a lot of real world details. I’ll go to work tomorrow morning and review a dozen spreadsheets, involving realities having nothing to do with math, but just as deep, and then tomorrow night I will debate algebra and trig with you.:)
Maybe it doesn’t make a difference. Maybe forcing everyone through the algebra filter still gives us the best spreadsheet makers. Something in me doesn’t buy it. As an engineer I hate all that extraneous detail, but others thrive on it. I wonder how many students that thrive on that extraneous detail, give up at algebra’s and calculus’s door.
Bob HansenNovember 4, 2012 - 5:45 pm -
By the way, check out the algebra books here…
You can sign up for free to review.
I think they are pretty good and they are on the “active” side.