Shifting Gears


Approximations in Cycling Michael Nakamaye


N


early 17 years ago I moved to Albuquerque. Te most important factor in this decision was the weather, propitious year-round for biking. I have been biking nearly my


entire life, for fun, transportation, and exercise. In Albuquerque, what started as an innocent 15-mile round-trip commute has now evolved into a nearly 30- mile route, including an unnecessary-but-rewarding 1,000 vertical-foot climb. When I learned that James Tanton had developed a


Math Teachers’ Circle session about biking, I watched the online video of the session with great enjoyment and fascination. Given a set of wobbly bicycle tracks that are the only clue leſt by an escaping thief, the session asks, “Which way did the thief go?” Tanton also presented a version of this session in Hawaii this past year, and so it seemed natural to build upon this in some way for my visit to Hawaii in April 2016. But where to begin? Aſter pondering the rogue thief ’s tracks for a bit,


I thought to myself, “No decent bicyclist would ever leave tracks like that!” Indeed, part of what teachers discover in Tanton’s activity is that the front wheel is traveling further than the rear wheel, which is very in- efficient. In addition, the rider is losing her momentum by swerving. An experienced cyclist mainly uses the steering to make minor adjustments in the direction of travel; the main way a cyclist turns is by leaning in the direction of the turn. Aſter asking the teachers to think about these ideas


briefly and recall some of the fun discoveries they made with Tanton, I moved the conversation in a dif- ferent direction. How does a bike work? As you pedal, the bike moves forward. What causes this forward motion? Te bike chain moves over each notch (or tooth) in the front ring near the pedals. As it does so, it also moves over one notch in the rear ring on the back wheel. So if there are x notches on the front ring and y notches on the rear ring, the bike advances by x/y


⊆


revolutions of the rear wheel each time you make a full revolution of the pedals. Te larger this fraction, the more difficult it is to pedal because you are advancing further per pedal stroke. For equivalent ratios, like 52:26 and 42:21, pedaling with 52 teeth (notches) in front and 26 in the rear will feel identical, and have the same impact, as pedaling with 42 teeth in front and 21 in the rear. A bike is like a ratio machine, giving the rider a concrete physical experience of different ratios. It turns out that there are many other ratios to consider when riding a bike, all related to the gears:


• How many teeth are there usually on the different gears? Why?


• What is a comfortable cadence (number of pedal revolutions per unit time) for most riders?


• What is the range of speeds that you can comfort- ably go on a bike?


We then moved to a different type of question: Is


there duplication between gears on a bike, i.e., two dif- ferent gears whose ratios are equivalent? (Te answer may depend on the bike, but is usually “no.”) If not, which two gears are closest? (For the bike I brought for demonstration purposes, the two closest gears were 53:23 and 39:17.) Would you be able to feel the differ- ence? (Yes, if you are working hard!)


At this stage we


“shiſted gears,” so to speak! I posed a rather abstract math- ematical question: Suppose, as a cyclist, you wish to feel √2. What are some whole number gear values that form a fraction approximating √2?


Te Math Teachers' Circle of Hawai'i (MaTCH) examines gears on a bicycle.


14 MTCircular · Summer/Autumn 2016 · American Institute of Mathematics


Page 1  |  Page 2  |  Page 3  |  Page 4  |  Page 5  |  Page 6  |  Page 7  |  Page 8  |  Page 9  |  Page 10  |  Page 11  |  Page 12  |  Page 13  |  Page 14  |  Page 15  |  Page 16  |  Page 17  |  Page 18  |  Page 19  |  Page 20