A little bicycle history…

I started the meeting with a notice/wonder on some images from the history of the bicycle. Here are the images, with some of the things the group noticed and wondered, followed by some comments from me.

• Wheels are the same size, similar to a modern bike, but there are no tires
• There are no pedals, no chain and no gears
• Seat is between the two wheels
• Fewer spokes than there are today
• It’s like a child’s balance bike
• Did writers of the book where these images came from (Bicycle, David V. Herlihy) look into histories of the bicycle that make have happened on other continents, not just the West?
• Are these actually the first bikes?
• Do the front wheels pivot?

The first bicycles were made by carriage makers. The goal was a replacement for a horse. Even before the velocipedes, there were 4-wheeled contraptions operated by levers. They weren’t very efficient.

• There are pedals on the front wheel
• More spokes
• Seat is closer to the rear wheel
• Do you sit on the bar?

This bike got its name in comparison to the next bike, which was much more comfortable.

• Pedals attached to front wheel
• Large front wheel!
• It’s like a unicycle with a training wheel
• Seat is directly above the front wheel
• Is the purpose of the big wheel to cover more ground?
• How do you get up there?
• How do you stop?

The high wheeler was also called a penny farthing because a penny and a farthing were two different-sized coins. In 1874, David Stanton rode 106 miles from Bath to London in 8 ½ hours. Many people didn’t believe it. Do you see any issues with this bicycle?

• Back wheel is bigger
• More spokes than other bikes
• Seat too close to handlebars
• It has a chain which connects the pedals to the rear wheel – “rear-wheeled drive”
• What danger did it fix? Safety…
• Is the turning mechanism the same on all the bikes?

The “Safety” bicycle was so-called because it didn’t have the danger of throwing riders over the handlebars, which you might imagine happened with the high-wheeler.

Problem-Posing: Pedaling a bicycle

Then we looked at a video Mark and I shot for a draft 3-Act Math Task (link, link). I asked the group to write down the first question that came to mind. We watched the video a couple times.

I asked people to talk in small groups and share their questions, then we shared the questions as a group.

What Information Do You Need From Me?

I then asked what information the group needed in order to answer the first four questions. Here’s what they asked for:

• Diameter of the wheel
• Diameter of the chainring
• Diameter of the cog
• Can we look at your bike?
• The video
• A protractor
• A stopwatch

I did have the diameter of the wheel. I didn’t have the diameter of the chainring and the cog, but I had other information about it. My bike was in the room for people to use. The video was available for replay on YouTube. I didn’t have a protractor. And the video had timestamps.

1. Information about the bike: Diameter of the wheel, number of teeth on the chainring and cog, length of the crank, etc.
2. Pedaling a bike video

Problem-Solving

Small groups then had about 30 minutes to work on the problem:

Presenting Solutions

Each group presented their work at the end.

Usha shared the rough estimate of a ratio they found between revolutions of the pedals (5-6 turns) and revolutions of the back wheel (10 turns). She and Patrick started with a rough estimate and moved towards precision. They decided that the ratio of the front chainring:back cog was 1: 1 3/4. They then went to the bike to count revolutions when the bike was in different gears.

Audrey explains how she and Sophie counted chainring (pedal) revolutions and cog (rear wheel) revolutions for two different time periods in the video. The ratio between the two counts was very similar: 4/7 and 7/12. They then compared the ratios between front chainrings and back cogs for all the possible combinations on the bike. The outlined gear combinations represent the possible gears that the bike was in during the video. Both gear combinations result in the same ratio of pedal strokes to back wheel revolutions.

Audrey and Sophie also shared a question they ran into:

The ratio of chainring revolutions to cog revolutions is 7/12 and an equivalent gear combination is a 36-tooth chainring matched with a 21-tooth cog. The chainring teeth/cog teeth (36/21) has to be inverted to match the same ratio as the chainring revolutions/cog revolutions (7/12). Why wouldn’t the ratio always be chainring divided by cog, whether we’re talking about revolutions or teeth?

36/21 =1.71

21/36 = .58

7/12 = .58

This also opened an interesting question about what each part of these ratios represent. 36 is the number of teeth in the chainring. 21 is the number of teeth in the cog. What does 1.71 represent? What does .58 represent?

Greg, Kevin and Mark started off by watching the video and noticing that there was a moment when both the red and the blue fuzzies were at the bottom (6 o’clock) position of their rotations. From there they watched where the blue fuzzy (on the rear wheel) ended up with each full revolution of the pedal.

Then they moved over to the bike itself and began to repeat that same experiment with different gears, keeping track of the rear wheel in terms of degrees traveled before shifting over to keeping track of the number of revolutions similar to the 1 2/3 above. Patterns started to emerge and they saw that the smaller the cog, the more revolutions the rear wheel made for every one revolution of the pedal. They started paying attention to the number of teeth in the chainring and the rear cog. Using combinations like the 48-toothed chainring and the 12-toothed cog helped make the relationship clear, since 48 is a multiple of 12, so in that gearing, each full revolution of the pedal resulted in 4 full revolutions of the rear wheel.

Extensions

Other gear questions I handed out at the end of the meeting

One thing we started to work on at the end of the meeting was the question of how many gears Eric’s bike has. Eric shared that there are three ways to count the number of gears on a bike: possible, distinct and usable.

• There are 18 possible gears
• There are 11 distinct gears because some of the ratio combinations of the cog and chainrings are equivalent (the same number of revolutions of the rear wheel for every revolution of the pedal). There was some discussion on how different certain non-equivalent gear combinations are – is the 2.28 revolutions of the rear wheel with a 48/21 combo noticeably different to a rider from the 2.25 revolutions of the 36/16?

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In attendance: Usha, Patrick, Kevin, Audrey, Sophie, Greg, Mark & Eric

Programs Represented: NYC OACE, CUNY Start, CUNY’s Adult Literacy and HSE Program