## Teach The Controversy

I was walking with my wife along the River Corrib in Galway last weekend when we got into an argument that lasted the rest of the walk. I’ll present our two arguments and some illustrative video. Then I’d like you or your students to help sort us out.

Argument A: It would be much harder to swim to the other side of the river in the fast-moving water as in still water.

Argument B: It would be just as easy to swim to the other side of the river in the fast-moving water as in still water.

I hope this gets as out of hand for you and your students as it did for me and my wife.

Featured Comment

This excellent question exhibits a quality that is not found often in math curricula: it has the “specificity sweet spot”: it is specific enough for a student to answer, but non-specific enough for every kid to agree on the answer. Students making different assumptions will have different responses, thus creating a real mathematical argument.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. More here.

February 26, 2013 - 6:59 am -

I find this to be a great trig question. If we knew the speed of the water, the speed of the swimmer, and the distance across the water, we could calculate how long it would take to make the hypotenuse of a triangle.

I would argue that it would take longer to swim from one side to the other in the fast moving water, only due to the waves. This creates more of vertical movement which acts against the horizontal swimming movement, thereby making it more difficult to swim (as pertaining to my experience as a beach lifeguard for 6 years).

However, if the waves were not a factor, only the fast-moving water, then I would argue that it would take a swimmer the same amount of time to swim from one side to the other in still water and fast moving water (assuming the swimmer is not trying to compensate by swimming to the same spot). If the swimmer just swims straight–perfectly perpendicular to the water–then I would argue that the swimmer would reach the other side in the same amount of time … the only difference is where the swimmer would end up on the other side of the water.

Very thought provoking question! Thanks. I can’t wait to ask my students!

2. #### Don

February 26, 2013 - 7:32 am -

Heh, I’d say this is the sort of argument common to myself and my wife too – one where I quibble about the definition of the problem and whether success is properly defined. My wife would say it would be hard to swim across, I’d say no it’s not… you’d just end up down the river a ways.

I showed my Game Design 101 class this assortment of Spinal Tap clips on their first day to underscore this point.

3. #### Chris Shore

February 26, 2013 - 7:46 am -

Fun! As a math teacher, I say the vectors will tell you that it should be the same effort as long as you are ok finishing further down the river. Attempting to finish straight across could be exhausting. Assuming that you take the easy route, as a swim coach, I say that choppy water is harder to swim in than flat water. (We don’t live in a world of “pure” math.) P.S. If these are the arguments that you have in your marriage, you are a blessed man.

4. #### Jenny

February 26, 2013 - 8:43 am -

When you say “the other side of the river” does that mean a specific spot or just anywhere over there? Because if one wants to be directly across from where one begins, the fast moving water will make that challenging. If one is willing to be swept downstream in the process of crossing, it’s less of an issue.

That said, as a swimmer (and one who prefers sprint triathlons in swimming pools over open water) I think the choppy water makes swimming, in general, more challenging.

5. #### Matthew

February 26, 2013 - 12:28 pm -

I would argue that it does take more effort to swim in a strong current, and without taking into account waves. If you try to swim straight across the river, then you will have to make a greater effort to keep your direction straight.

And on the other hand, even if you’re okay with ending up traveling downstream by letting yourself drift with the current, the total distance travelled is increased as well as the time in the water. So even if you’re not working any harder, you still have to work longer to get to the other side as compared to the water that is still.

February 26, 2013 - 1:40 pm -

Another argument that supports the idea that it will take the same amount of time to swim across with a current, and without a current is the “bullet drop” in physics. For example, if one shoots a bullet directly parallel to the ground, the bullet will hit the ground at the exact same time as simply dropping a bullet from the same height (due to the law of gravity). In other words, the distance the bullet travels horizontally, is irrelevant to the time it takes the bullet to travel downward, vertically.

If the waves are removed, and only the the current is the variable, then I would argue that it would take the same amount of time to swim across the river to the other side, with or without the current, even considering the distance traveled from the current. This distance has no effect on reaching the distance across the river, just like the “bullet drop” example.

7. #### Scott Farrar

February 26, 2013 - 4:32 pm -

This excellent question exhibits a quality that is not found often in math curricula: it has the “specificity sweet spot”: it is specific enough for a student to answer, but non-specific enough for every kid to *agree* on the answer. Students making different assumptions will have different responses, thus creating a real mathematical argument.

Furthermore, this lack of explicitly defined assumptions means we (as students, or as a class) must define them ourselves in the course of problem solving. And isn’t that what real math is all about?

Notice how much different that is in comparison to our normal textbook questions: assumptions and answers are locked in, explicit… boring. Have you ever tried to have a class discussion around a problem and right when you feel like it may get going, one student solves it… short circuiting the whole thing.

But if we hit the specificity sweet spot, then we can actually have a lively discussion with mathematical arguments flying around the room!

As I thought about this problem myself, here are some prompts that could arise depending on the level of the class.

1) current drags you downsream, distance is longer. Harder.

2) current does the work, longer distance is negated. Not harder.

3) current affects your swimming. requires effort to stay aimed, even if its not perpendicular. Harder.

4) current affects your swimming… but like a sail in the wind. Makes it easier?!?!

5) how much effort does it take to “be the sail” 100% vs. swim 100% ?

6) What mixture of “sailing” and swimming gets me across the fastest?!?!

8. #### Gaute

February 27, 2013 - 2:38 am -

Nice problem opening for a lot of discussion.

I wonder what strategy I would use to be certain that I really am working in a straight line while swimming, not fighting the current, but not getting lazy and flow with it either. Should I make a rough calculation based on estimated current and swimming speeds, to decide a priori where on the opposite bank I want to land? Will aiming for this point right from the start ensure that I swim in a “relative” straight line, or just act as a psychological help?

9. #### Santosh Zachariah

February 27, 2013 - 6:35 am -

Depends on whether either of you can approximate an airfoil shape. And whether staying afloat in fast water is harder work than swimming in still water.

10. #### David

February 27, 2013 - 8:30 am -

This could even be a stats question.

In still water, swimming at a constant rate will get you to the other side at a certain time.

A downstream current probably includes some side-to-side currents. If they’re random, then some of the time the current is increasing your velocity across the river and some of the time the current is decreasing your velocity across the river.

The best answer I have right now is that, on average, it takes the same amount of time, but that you would have a wider range of swim times over multiple attempts in a river with current.

11. #### James C.

February 27, 2013 - 12:02 pm -

On another note, I remember staring over the Niagara river upstream from the falls wondering if I could make it across before I went over the falls. I’m sure I’m not the first person to wonder that.

12. #### James Key

February 27, 2013 - 12:41 pm -

Some commenters have talked about whether you’re “aiming for the point directly across.” i.e. they speculate that it will be easier to “go with the flow” and end up at a point diagonally displaced from your point of origin, as opposed to “fighting the flow” to “stay straight.” I would wager that the latter option is more or less impossible to execute. i.e. the current is gonna carry you downstream a bit, whether you want it to or not.

Thinking about my thinking: that assumes that the swimmer is pointed in a direction that is 90 degrees to the shoreline. I guess if he/she points himself at an angle *against* the current, he/she could in fact end up at the point that point that is directly across the river. Or am I wrong??

As I think about it even more, it seems to me that the angle is really important. i.e. between the path the swimmer is attempting to move in, and the direction of flow for the river. I think this is really what determines how hard it is to swim in moving water.

13. #### Fiver

February 27, 2013 - 1:53 pm -

Methinks the debate really was: does it take longer to reach the opposite side if the water is moving? Relativity shows the answer is obvious if it doesn’t matter where on the opposite shore you end up (imagine walking the width of a train while it’s moving), but what about if you can swim x mph and you can walk y mph and the river is flowing z mph and you want to reach the point exactly opposite? Is it faster to try to swim directly across, or faster to swim perpendicular to the current and then walk back to the opposite point?

February 28, 2013 - 7:37 am -

I’m curious how we are defining “much harder” and “just as easy”? Could it be time? distance traveled? or strokes used to swim from one side or the other.
I would classify “much harder” as more strokes where distance and time are variables dependent on the current (or no current). I’d consider “just as easy” to be equivalent strokes, but that sounds weird. Furthermore, the direction of the current can be a variable during the whole trip across and who knows if the current runs parallel to the banks the entire time swimming. That said, it looks like my conclusion is Argument A, only if the current favors (runs into) the bank you leave from.
I feel like vectors are involved here and I’m demanding more information!

15. #### Richard Vahrman

February 28, 2013 - 8:51 am -

Very, very slightly longer. The river widens as it flows towards the sea.

16. #### Dan Meyer

March 1, 2013 - 1:09 am -

Unsurprisingly, both my wife and I took this comments thread as validation of our arguments.

17. #### Dan Meyer

March 1, 2013 - 1:32 pm -

Michael Pershan:

Can you guys agree on a way to settle the argument?

We’re basically negotiating on terms, same as everybody in this thread. Depending on where we put the goalposts for statistical significance, either one of us is right. She’s right that, on balance, I’d rather swim through still than choppy water. So in that sense the fast-moving is harder. But I’m right that it wouldn’t take that much longer. Particularly stacked against what our intuition tells us would happen in the fast-moving water. So we agree to disagree and go get a beer.

March 1, 2013 - 8:34 pm -

I think you and your wife should compete. Set up the tripod and press record.

19. #### Maria Droujkova

March 3, 2013 - 7:52 am -

One word: turbulence.

Real rivers that are fast are usually turbulent.

Real streams that are not turbulent aren’t any harder to swim across; you aren’t aware of the stream’s movement, and may be surprised how far it carried you (after a short swim across). Try it with an ocean current sometimes. Just make sure you survive the experience!

20. #### Chris D

March 25, 2013 - 7:23 am -

Fundamentally, I think the argument hinges on at least one participant never having tried swimming across a river flowing that fast.

Reality aside, it also depends on the shape of the river. With that kind of current, you’re pretty much guaranteed to get swept through many curves of the river, which affects the horizontal distance you have to swim.