Note: This article will not be part of the book “Traditional Math”. I include it here only for your interest.
The following problem appears in an algebra 1 textbook (Dolciani, et al, 1973):
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.
In an informal survey I conducted on the internet, I asked what people thought about the problem. There were three general reactions:
1. Poorly worded, too contrived, trying too hard to be “real world”.
2. Too difficult to be included in a first-year algebra book.
3. Excellent problem requiring inequalities to prove what the problem asks
I agree that the problem is too wordy, and also agree that most first year algebra students would not be able to solve it. I note that the problem is marked as a difficult problem, and is likely used as a marker for teachers to identify exceptional students. It would be better suited for a second year algebra course in a lesson specific to proving inequalities.
Nevertheless, it is a good problem in many ways, despite its poor wording and placement in an algebra 1 textbook. To that end, one reaction in particular from Robert Craigen, a math professor at University of Manitoba, caught my attention. He expressed the opinion stated in item (3) above, but went further, saying: “A math lesson does not exist in isolation but is part of a larger structure of learning. The point is turning that problem into an important piece of connective tissue in what the learner is acquiring from the lessons.”
I agree with Professor Craigen. It is an excellent problem, not only mathematically, but for its potential to explore other connected and significant mathematical concepts. I show in this article first how the initial problem is solved, and second how the problem can be extended in subsequent classes students may have—particularly physics.
Specifically, the problem directly relates to the famous Michelson-Morley experiment about the speed of light and the attempt to prove the existence of an “ether wind” which ultimately led Einstein to important conclusions in his development of the Special Theory of Relativity.
Solving the Problem
Here is a restatement of the problem:
Given a lake that meets a river at a certain spot. At this confluence, the distance across the lake is 3 miles. There is another point 3 miles down the river. Show that it takes less time to cross the lake and back in a boat at a constant speed than the time it takes to row the boat at the same constant speed 3 miles down the river and back.
Some students find this problem difficult because neither the speed of the boat in still water nor the speed of the current are specified. In addition, they may incorrectly believe that the time should be the same for travelling across the lake as the down- and upstream trip on the river. Their reasoning is that the added speed due to the current when traveling downstream would be cancelled by the slower speed when traveling against it. The mathematics shows that this is not the case, however.
The distance for both the lake and the river in this problem is 3 miles. In the mathematics to follow, the distance is represented by D.
Let r be the speed of the boat in still water and w the speed of the river’s current, where r > w > 0.
In Dolciani, et. al. (1973), students have worked with addition of rational expressions prior to this problem, so they will be able to express mathematically the travel times of the boat for the lake and the river round trips as follows:
The time for the boat to travel down-river and back is expressed as:
It must now be shown that the time of the round trip across the lake is less than the time for the round trip on the river. We have:
Students are familiar with solving rational equations and inequalities by multiplying both sides by the lowest common denominator.
Dividing all terms by 2D, we obtain:
Since r and w are both positive and r > w, inspection shows that the left hand side must be less than the right hand side. Specifically:
The original statement in (1) can then be stated as:
which was to be proven.
Extension of the Problem
The original word problem can be extended for use in a subsequent physics course, where the original problem would be revisited. Specifically, this problem relates to the famous Michelson-Morley experiment which attempted to detect the existence of an “ether” that was thought to be the medium through which light travelled. The experiment, it was hypothesized, would also find the velocity of what was called the “ether wind”, the caused by Earth’s movement through the medium and which was believed to cause a decrease in light’s speed as it traveled against the ether wind.
Having proven that the time of the round trip on the river is greater than the trip across the lake we are now able to find how many times greater it is. We divide the time for the river trip by the lake trip time:
We can rewrite the above expression as:
This derivation employs the same approach to detect the ether wind in the Michelson-Morley experiment. The assumption was that a beam of light traveling back and forth parallel to the ether wind would result in a delay in time compared to how long it would take to make the trip without such a wind. (Of course, we now know that the outcome of the experiment was that there was no detectable delay.)
The mathematics of the Michelson-Morley uses v to represent the speed of the ether wind (substituted for w the speed of the river current). The speed of light is represented by c (substituted for r, the speed of the boat in still water.) We obtain the same result as in (2) but with the new variables:
Similarly, we can calculate the delay if the light beams are perpendicular to the direction of the ether wind. In calculating the delay of the boat traveling across the river, (that is, perpendicular to the direction of the river current) we need to consider the drift caused by the current. The boat must travel slightly sideways to arrive at the point directly opposite its starting position.
What is happening with the boat and the river current is comparable to what was thought to be happening in the Michelson-Morley experiment. The expression representing how many times greater the light will take to go across the laboratory table, perpendicular to the ether wind, than a trip with no ether wind is:
Both equations (3) and (4) are important results that were used in the Special Theory of Relativity in which fast moving objects (i.e, those moving at speeds approaching the speed of light) shrink by the following factor:
(Students taking physics will have had some experience with addition of vectors and can derive the above expression.)
The Importance of Algorithmic Reasoning and Mathematical Form
The derivation of these forms requires knowledge of and facility with algebraic expressions which in turn requires algorithmic reasoning. To follow the derivations described above, students need to be familiar with certain forms, such as:
Students also need to know how to divide a polynomial by a monomial which allows the following simplification:
The mastery of algebraic algorithms is a key skill in being able to transform symbolic expressions into forms that are most useful. Those students who take physics and see the problem revisited, will better understand that the forms in (3) and (4) make it easier to see what happens mathematically as the velocity of an object approaches the speed of light.
The importance of mathematical form is generally not obvious to students in a first year or even second year algebra course. Many students may consider algebraic algorithms and transformations tedious and ask the question “When are we ever going to use this stuff?”
Answering that question with “You’ll see in other courses or if you go into science or engineering” may seem like it is avoiding the question. Nevertheless, it is important for teachers to have the forward look that students may lack, so they can give explicit instruction to students on how to work in the abstract with problems that relate to important concepts that students will be working with in subsequent courses.
The initial algebra problem provides an opportunity for students to learn how to demonstrate mathematically that a statement is true even though it may seem that it cannot be. It is actually beneficial for students to see that their initial assumption that the time to cross the lake is the same as that of the round trip on the river is not true. To that end, the algorithmic manipulations yield significant understandings. In this case, such insights serve to better understand their application in deriving important formulas in physics.
Reference
Dolciani, Mary, and Wooton, W. (1973). In Modern Algebra: Structure and Method, Book One. p. 431. Houghton Mifflin, Boston.
See, we’ve had “rich” problems for decades.
I think this problem is a great example because it challenged my common sense intuition that the current just cancels. To understand in a more general way why this is not the case, this same problem could also be introduced again when covering topics such as linear vs. non-linear equations, convex functions, etc. Since speed is distance over time, we can also say that time is distance over speed, and therefore non-linear w.r.t. speed. And since functions of the form 1/x are convex (x>0), it follows that it is faster to travel at an average speed both ways rather than fast one way followed by slow the other way.