factors tasks unit analysis

Painting Tape

I came across the following situation while shopping for paint at a local home improvement store:

Admittedly, the three varieties were not positioned like this, but this positioning does raise an interesting question.
“We can see the packages are the same height, what is that height?”

I see this question going one of two ways:
  1. The students realize that really any conceivable measurement is possible. (Barring, of course, zero and the negatives) One could make the argument that it also cannot be irrational, but this would be nit-picking. Can a roll of tape have a width of pi/6? Exactly?
  2. The students fall prey to their subconscious affinity toward the integers and begin constructing common multiples.
In fairness to the problem, both are very teachable moments, but there is nothing scarier for a teacher–under considerable time constraints–than to see a problem steer students in an alternate, but useful, direction. We know they should explore their curiosity, but can we as teachers shut-up long enough to let them?
Situation (1) leads into an explanation of unit analysis. The height can be any “x” because each roll would simply subsume the thickness of x/6, x/4, x/3 respectfully. This demonstrates great number sense. If the class immediately goes that way, I would show this picture.
Revisit the question:
“What is the height of the packaging?”
adding now,
“How thick is the individual roll in each package?”
 Ideally, students dwell on situation (2) long enough to draw out the ideas of factors, common factors, multiples, divisibility, and lowest common multiples. After which I would drop the unit analysis bomb on them anyway.
Just a thought. Yet another way that mathematics proves to be an inseparable mass despite what neatly organized curricula dictates.

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