As my devoted readers know, Totally Clueless, affectionately known as TC, has contributed many insightful comments and profound ideas for us to think about. His sobriquet belies a brilliant creative mind of course. He recently sent me a geometry problem which was motivated by his own experiences driving to work. The problem itself is accessible to advanced middle and secondary students but the result is interesting in its own right and should generate rich discussion in class. I recommend giving this as a group activity, allowing about 15 minutes for students to work on, then another 15 minutes to discuss it. Save it for an end-of-year activity or bookmark it for the future. Beyond the problem, there are important pedagogical issues here:
- How to introduce this
- Asking questions to provoke deeper thought
- Drawing conclusions and further generalizations
- Connecting this problem to other circle or geometry problems
- Maximizing student involvement
I told TC I would need some time to rework the original problem for the younger students so here goes...
Diagram for Parts I and II
Part I (middle and secondary students)
In my city, there are two circular roads "around the center" of the city, of radii 6 and 4. There are a number of radial roads that connect the two loops. Points A and B in the diagram above are at opposite ends of a diameter of the outer loop and the dashed segment is a diameter of the inner loop.
If I have to go from point A to point B on
the outer loop, I have two options:
(1) Drive along the outer loop (black arc in diagram) OR
(2) Drive radially (blue) to the inner loop, drive along the inner loop (red), and then drive radially out (blue). (Assume that there are radial roads that end at point A and point B).
Show that Option 2 is shorter than Option 1.
Part II (middle and secondary students)
Same diagram but now the radii are R and r with R > r.
Show algebraically that Option 2 is shorter.
Part III (secondary students)
To generalize even further, points A and B are distinct arbitrary points on the circle, central angle AOB has radian measure θ where θ ≤ π. OC and OD are radii of the inner loop; OA and OB are radii of the outer loop. Again the radii of the two circles are R and r, where R > r.
As before, there are two options in going from A to B:
(1) Drive along the outer loop (black arc in diagram) OR
(2) Drive radially from A to C (blue), then along the inner loop from C to D (red), then radially outward from D to B (blue).
Show that Option 2 will be shorter provided π ≥ θ > 2.
Click Read More for further discussion...
Further Comments
(1) TC's original problem was Part III. I decided to add Parts I and II to provide 'scaffolding' for students. Was this really necessary in your opinion?
(2) The results of these questions are independent of the actual radii. TC felt this was an interesting aspect of this problem and I agree. Do you think students will be surprised by this? Do we need to point this out to them? Are there other circle problems you can recall which have a similar feature?
Thanks TC for providing us with another stimulating challenge!
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7 comments:
Hi Dave,
Thanks for your comments, and for the extensions to the problem. I think I agree that the progression with definite values for the parameters, moving on to a more abstract case would cater well to students of different levels.
And, of course, one possible answer to all the variants is, to paraphrase Calvin (of Hobbes fame) is 'Given the traffic here, who knows which option is shorter!'
TC
In order the student to find themselves that this problem is independent of the radii maybe after problem 1 ask the student to play around with various radii and tell them if they can find two radii that makes option 1 is shorter than option 2 and let them guess the general result.
Excellent points, watchmath.
Giving students, particularly at the middle school level, the time to experiment with a few pairs of radii would help them to make a conjecture before turning to an algebraic solution. My guess is that many students would come to a conclusion after just a couple of trials.
I definitely could have added additional parts to this investigation, moving in smaller increments, but that's an individual teacher decision (based on the needs of the class).
Certainly, it makes sense in Part III to have them try angles like 45, 90 and 120 degrees to form the basis for a conjecture before attempting a general proof.
TC's problem is a jumping off point for us. The rest is up to the creativity of the teacher and the characteristics of the class.
Watchmath, were you at all surprised by the result? What I would want my students to come to recognize is that the conclusions are based on the difference of the radii, not the radii themselves! This is because we were not really interested in the individual distances but, rather, how the distances compared! In other words, we are really looking at the difference between the two paths.
Yes Dave, I was surprised with this counter intuitive result.
This remind me of another counter intuitive result which is also involving two circles. It was about tying the earth with rope tightly and then extend this rope by 1 meter and then the gap produced is enough to make a cat pass under the rope.
Have you seen this problem? Ok I found one blog that discuss this. Here is the link: http://www.blogcatalog.com/discuss/entry/rope-around-the-world
Nice! I have seen that problem and it's definitely worth sharing it with our students to demonstrate not only a counterintuitive idea but also the power of mathematics! Thank you for sharing that and the link...
Both problems depend on the difference of the radii. I have also seen a few SAT problems which test this same idea. I seem to recall publishing a post on something like this before but I'll have to search to find it!
For more fun with traffic problems, algebraic this time, you can demonstrate Braess' Paradox to students. The paradox demonstrates that adding a road to a travel network can increase the average travel time for the system's drivers. The Wikipedia article is a good guide.
Eric,
Fascinating article -- the example provided was very helpful. I would have probably taken that "short-cut" too! Besides, it was worth reading just to see the Nash Equilibrium concept mentioned.
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