The following word problem appeared in a traditional algebra textbook around. I will identify the book later. Right now I want to know your opinion of the problem, good or bad. If you dislike it, please specify why. Same if you like it.

Thanks.

## Two boys are camped at a spot where a river enters a big lake. One boy is injured so severely that every minute counts. His companion can use an outboard motorboat to get a doctor by going 3 miles down the lake and back, or by going 3 miles up the river and back. Show that even though the boy does not know the speed of either the boat or the current, he should choose the lake.

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That is a very tough problem. I don’t like it mostly because it assumes knowledge of what an outboard motor is.

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If they had said it was a motorboat without mentioning “outboard motor”, would your opinion be different? Or if it were a rowboat?

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I suppose so. But it is still a hard problem. I am stumped, and I did fine in HS Algebra and Algebra 2.

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Figured it out. But, it still feels like sort of a trick problem for young students who are not familiar with this type of setting. Vehicle problems and store price problems with increasing discounts are much more within today’s students’ ken.

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It was marked as an advanced problem, so therefore not for everyone. It isn’t a trick problem at all. Read Wayne Bishop’s comment below. It requires setting up an inequality, and expressing time for each route as a ratio of distance over rate.

This was from an algebra book, so it is not an inappropriate problem for an algebra course. Plus students may incorrectly assume the times are the same because the upstream and downstream currents are erroneously thought to cancel each other out. The math proves otherwise which is a valuable part of solving the problem–students see how math is used to prove or disprove propositions.

Anyway, I thank you for your comments on this! I’m writing an article about problems such as this so your views are helpful.

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I like the purpose of the problem – sort of an algebraic proof it seems. I try to avoid upstream/downstream problems. My students – even honors – still struggle with just consecutive integer problems and writing a linear model for a real-world situation, much less d/r/t problems involving 2 (or 3?) r’s!

I am curious though about the assumption you have to make to model the trip on the lake. Are we dealing with inequalities as well? Or maybe I have become a math zombie and am forcing a template on a problem that doesn’t require it?

We could nitpick all day about bias and sensitivity issues. It’s an old problem and could likely use some updating, but that doesn’t make it a bad problem.

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Well, if I did it correctly, I basically prove an assumption I made initially. So I suppose that as long as that assumption is true, which seems likely, then the lake is indeed a better choice.

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I sent you an email regarding the problem. Thanks for your response.

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Clever problem. Requires only basic algebra, and the knowledge that a river flows in one direction at a fixed speed (or so we assume for the duration of the trip mentioned). I don’t care for the severely injured boy factor. Could easily define the setting differently, of course.

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Assuming “show” means to prove it mathematically, it’s a great question. For a student well-versed in algebraic dirt problems (d = rt), setting up the situation algebraically is quite straightforward. A little nudge to get over the inequality step would be appropriate; students have had very little experience with inequalities at this stage so a leading question of the type, “Can you show me why this must be greater than that?” in the students’ work will probably be needed.

On the other hand, any “wave your hands” verbal argument that the student cannot support algebraically is Beautiful Sunshine.

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It’s a classic problem and one worth giving students. But this instance is worded a little less clearly than I’m happy with.

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Time is of the essence! I have no time to do math problems. I need to get that kid to the doctor!

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What a great problem! I am looking forward to giving the problem to my students.

I concur with Dr. Bishop’s assessment on where students will need assistance. It will also be difficult to get students to work in the abstract. But, they will learn to work in the abstract with problems such as the one given. In my opinion, this problem is an amazing way to end a lesson by getting students to demonstrate, mathematically, why something known to be true must be true.

The possibility of students not, initially, understanding the underlying assumptions is a positive. It creates an opportunity to discuss assumptions to be made – a necessity in many problems. (Ask a physicist.)

My only criticism is not with the problem itself but instead its location in the “advanced” section. To some, this signals that proving why a statement is true should be avoided: it’s extraneous; or it is impossible for all but the “smart” students. Rather, proving why something is true should be something students do routinely.

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