Paper Cup Probabilities…

This is my first entry in #MTBoS30 Today I asked my 10th grade geometry class three questions (review time!)
1. What is the probability of rolling a one (using a regular six-sided cube)?
2. If you flipped a coin, would you expect it to come up heads?
3. What is the probability that a paper cup tossed in the air will land on its side?

The ensuing discussion involved certainty. First question:
1Ss: one out of six
Several other students chimed in in agreement.
Me: who can tell me how you can be so sure?
2 Ss: because the cube has six sides, and the sides are numbered one, two, three, four, (she is ticking off on her fingers; other students were nodding in agreement and telling her what to say) five, six, and there is only one side with one!
This class is usually not this involved. I think it had to do with the fact that they really KNEW this! (Confidence is a wonderful thing!)

Second question (key word here”” “expect”)
1 Ss: yes, well, no, (???) it could be heads or tails. I mean, you could expect a heads or a tails.
Me: why can’t you expect just heads?
1 Ss: because it’s 50%. (At this point, other students begin chiming in:
“Yeah, it’s 1/2!” “A coin has two sides” and similar statements.) The question wasn’t a straightforward question about a probability fraction, so I think that caused them to not feel as confident with the answer, until one student decoded it. Think: lemmings!

It was the third question that really threw them. I held up one of those small cups, like you find in a bathroom cup dispenser. I asked them to tell me what they thought the probability would be of the cup landing on its side when tossed. The guesses ranged from 1/2 to 340/500. As we looked at the cup, the guesses got more specific. Several students noticed that the cup had a top, a bottom, and a side. The reasoning followed that there should be a 1/3 chance of landing on its side. At this point there was quite a bit of agreement. This seemed very logical (and if the strongest kids in the class said so, it must be right! Lemmings, I’m telling ya!) Multiple students jumped on the bandwagon and agreed. (No one talked about surface ratios – I figured we could tackle that later!) Then I gave each student a paper cup and asked them to create 20 trials each. I deliberately refrained from telling them instructions for tossing the cup. I just walked around and watched. Some kids tossed (across the room!), some kids tossed on their desks. Some dropped the cups on the floor. I heard disappointment as Ss complained, “it’s landing on its side every time,” how do you make it land on its top?” “There is something wrong with this cup!”

The trials were listed on the board and tallied. The students really seemed puzzled as to why the results weren’t anywhere near what they expected. They were already arguing why this was so, so I put them in groups with the instruction to:
1. Compare the actual probability from the trials to the expected.
2. Come up with some reasons for the difference.
3. Pick a spokesperson to share their ideas with the class.

Then the bell rang! Okay. The debriefing happens Monday…

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