Exploring parametrics with ships in the night
My co-teacher in Algebra II introduced me to the excellent book, Pacesetter Mathematics Precalculus through Modeling, which contains a number of wonderful investigations into mathematical modeling.
One of the problems my colleague especially liked and thought would be great for our introductions to systems is a scenario modeled off of the Andrea Doria collision where students have to determine whether two ships in the fog collide. As written, the activity is more than 10 pages of worksheets, shared below.
Being a loyal follower of Dan Meyer, I thought there had to be a better way to present this problem to students. Almost immediately after I had this thought Evan Weinberg showed me an awesome animation he created with Geogebra that animated a point moving across the screen, and I realized I could create a faux radar screen showing these two ships moving across the screen and turn them into a great 3acts problem.
To give you a sense of my students, as soon as the video was finished, 2 or three of them blurted out that you could model this situation with parametric equations. I tried to back us up with a slightly more approachable question—can we predict where the paths of the two ships cross?
Students then asked if I’d give them any more data—isn’t there some written problem statement. Nope, I said, let’s see what we can do with the video. And so we set set out to find the point of intersection for the paths of the two ships.
This venture took about 20 minutes for the class to complete, and then we were able to check the result by turning on tracking for the points in Geogebra and scrubbing the timer back and forth to see where the paths intersected. Since I was just scrubbing back and forth, the students didn’t have the time to see if the ships collided or not.
Our next step was to determine whether or not the ships actually collided. Here students took many different approaches to this problem:
- Some students wrote parametric equations for both ships. Some interesting things I noticed is that even though these students knew to write parametric equations and could write them perfectly, they didn’t know what parametric equations were, what a parameter was or how they could use their equations to tell whether they collided. One group of students found that their x equations predicted a different collision time than their y-equations, and really struggled to understand what this math was telling them.
- Other students calculated the speed of the ships along their respective paths by calculating the distance between two points using the pythagorean theorem, and then dividing that by the time it took to travel between those two points. The students then calculated the total distance the ships traveled to get to the point of intersection, and divided that by the speed to find the time each ship would arrive at the point of intersection.
- Another group entered the parametric equations into their calculator, and then searched the table of values to see if the two ships were ever at the same location at the same time.
- Some groups were pretty stuck, not knowing what to do. I suggested to one of these groups to make a table of positions of the ships at various points. They found this task easy, but tedious, and then had a much greater appreciation for the calculator group’s approach.
In the end, everyone came to the conclusion that the ships presented in this simulation didn’t hit, which allowed us to see the third act below.
Then I brought up the actual Geogebra simulation and we began to take it apart. I showed the students how easy it is to define a point that moves parametrically.
Students were also able to see that, despite all the complaints about how hard it is to make measurements from the video, their parameterization of the paths of the ships were identical matches to the equations entered into geogebra. Huzzah!
Students were also able to explain the meaning of the various numbers in the equations, for example in , the 900 gives us the starting x position of the Andrea Doria at time , while the refers to the x-velocity of the ship, moving in the negative x-direction at a speed of . And quickly thereafter they saw that one could calculate the overall speed of the ship using pythagorean theorem: .
Finally, I said ships not hitting each other is boring. How can we change the course of the Andrea Doria, leaving the Helsinki alone, so that we’ll get a collision. We saw that you could change the heading of the Andrea Doria just by making a change to its y-velocity, so that it’s parametric equations would now be:
We can set the x-equations equal to find the time they should collide
And we can then set the y-equations equal and insert the collision time to find the new y-velocity:
It was incredibly satisfying to put the new y-velocity into geogebra and see that the ships collide at 128 minutes.
- Student presentations can be hard to follow. I can see students loosing focus while their peers are trying to explain their work. This is for a lot of reasons—presenters aren’t doing the best job of explaining the big picture of their solutions, or are glossing over important details, presenters can often get lost in a sea of numbers, and we just don’t have a lot of experience yet listening to math presentations. I’d love advice on how to make these presentations more useful to the class as a whole. In physics, it seems like we are tackling simpler problems, problems are whiteboarded (as opposed to simply shared with the document camera) and students do a much better job of engaging and following along.
- What’s the proper role for vocabulary? If a student can calculate a set of parametric equations to model the position of the boat, but doesn’t know what a parameter is, is that ok?
- How do we improve at understanding what the math is telling us? : One of the biggest challenges my students seemed to face was believing that the math was telling them the ships didn’t hit. This might be because I had already told them that the real Andrea Doria did collide, but I still sense some discomfit with interpreting the solution to develop an explanation of the situation.