via Colossal, an art, design, and photography blog:

While waiting for a train, commuters can help themselves to square sheets of bubble wrap labelled with how long it would take to pop them.

I love this idea. The world is a better place because of it. I hestitate to bring this up, but …

the math is wrong.

It looks like the approximate times are based on *length*. Above, the ratio of side lengths is 3 to 5 to 10, or 1 to 1.67 to 3.33. Let’s assume that the small sheet does, in fact, take 3 minutes to pop, one bubble at a time. The large sheet does not have 3.33 times more bubbles; it has 3.33 times as many rows and 3.33 times as many columns. Therefore, it has 3.33^2, or 11.11, times as many bubbles. A better approximation for the large sheet would be 30 minutes. If we base the approximate times on *area*, the ratio of sides lengths would be 3 to √(5/3) to √(10/3), or 1 to 1.29 to 1.83, as shown below.

I’m thinking about how I could use this image or idea in class. Some possibilities:

**1. As-Is**

Display the photos. Ask students, “Are the times accurate?” Have students *apply* their understanding of the relationship between scale factor and area. M’eh.

**2. Hands-On**

Display the photos. In pairs, have students record how long it takes to pop a small square sheet of bubble wrap. Pose the problem, “A square sheet takes twice as long. What are its dimensions?” Have students test their predictions. In this activity, students *develop* their understanding between scale factor and area. They *poke holes* in the common misconception that when dimensions are doubled, area is doubled, too.

**3. Three-Act**

Play a video of a small square sheet of bubble wrap being popped. Include a timer. Maybe a soundtrack, too. Play the beginning of a video showing a large square sheet of bubble wrap being popped. Have students guess how long it will take. Ask, “What information would be useful?” Show the dimensions of the squares. Play the answer video.

I see this task being similar to Dan’s Penny Circle. Dan filmed himself filling a circle with 663 pennies so that the rest of us wouldn’t have to. I have a roll of bubble wrap measuring 24″ by 30′. Before I take one for the team and spend a ridiculous amount of time enjoying bubble wrap, any suggestions?

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I love popping these!! Sometimes I would twist the sheet so I can pop a bunch of them at once. I want to steal your lesson here.

I’d give each kid a square sheet (various sizes?), ruler, timer (on cell phone?). Maybe throw in a few rectangles. Then on class whiteboard, draw a 2-column chart for size and time. Ask kids to time themselves popping their sheets, record time. Ask them to measure their piece, record.

I see my kids asking, “Measure what?” And I’ll just shrug my shoulder and say, “Whatever makes sense. You’re popping the sheet, so…” I will ask that everyone measures “size” in centimeters (cuz I’m in this other country, Chris) and time in seconds. Then I’m just going to sit back and watch/listen (and pop my own sheet) to what they do. I ask that everyone share their numbers on the whiteboard.

Then I show them the BIG piece. I say, “Based on your recorded size and time of your piece, how long would it take you to pop this big one? Go, do the math!”

You mean you can’t just measure the ratio of areas the same way as you measure the ratio of linear quantities? That hardly seems fair to common “sense.” ;^)

Great find. Reminds me, too, of asking how many unit squares does it take to tile a rectangle that is, say, 4 units x 6 units, and then asking how many squares of area 1/2 square units it will take to tile the same rectangle. Bob & Ellen Kaplan posed that to a room full of Northwestern U. math club members when I went to see them present back in 2002. It was really fascinating to see how folks reacted when they thought it through. ;^) The “big” question for the afternoon was whether it was possible to tile a given rectangle with non-congruent squares. That one took us about 90 minutes to work through.

Love the idea, Chris. It didn’t occur to me to question the times listed by the store when I first saw this on Colossal. I think you have the right idea here also. It’s interesting to me that the question you ask in your “hands on” version (“what are the dimensions of the square given how long it takes?”) differs from the question you ask in the “three act” treatment (“how long will it take given the dimensions of the square?”).

My gut says the former question is a little odder and therefore a little better suited for a sequel.

If you don’t want to take this one up, I’ll definitely throw it in this month’s queue and publish it for everybody. Let me know.

Thanks, Dan. I saw the former question as a sequel as well. I hadn’t thought of it as more odd, but as more difficult/going backwards. Also, I was wondering about how the third act would play out. Whatever I brainstorm always involves measuring the side length which seems anticlimactic.

I’d love to see what you do with this. Either way, I’m going to give it a try because (a) it’s time I learned how/joined the club and (b) we’re talking about bubble wrap, man.