Here's a great comment about my previous post from Dan Meyer:

Oh
man, I love this. Love the vacuum-sealed bags. Love the alternative
definition for "flipped." Can you help me understand how the question is
even solvable with the given information, though? I can calculate the
average weight of a pre- and post-1982 penny but it seems like a random
bag of pennies would be impossible to sort out.

I took some screenshots to show what I hoped my students would come up with the first time I did this experiment as an inquiry-based experience. They need to know the mass of a pre-82 and post-82 penny. Since those can differ slightly from penny to penny, the average is probably best (and isn't that a great conversation to have because this has probably not occurred to them).

Then they have to find the average mass of a penny in their particular bag. And since the bags are all different, this will differ from group to group.

Then the use some algebra (and they groan when I say "and you suggested to your teacher in 8th grade that you'd never need this stuff!") and solve for x.

When you do this math for the pennies on the balance in the 10 random pennies picture, you find there are 3 pre-82 pennies and 7 post-82 pennies. And again, that was what I hoped I would see. I didn't anticipate that there would be so many great questions to ask and so many different ways kids tackle this.

Questions:

- Based on the mass, which type of penny do you guess is more abundant in your sample? Why does your answer make sense (or not make sense)?
- What are some sources of error that are built in to this experiment?
- How else could we solve this problem?

And that's where kids have really wow-ed me. Some do it this way. Some use guess and check. The good estimators take a guess, based on the average, and guess and check up or down from their guess if they weren't exactly right. This year one student tried something I had never seen (and I wish I would have copied his paper before I returned it). He assumed he had 10 older pennies and created an equation where x = the number of newer pennies so he could mathematically adjust the mass down based on how many of the newer pennies in the bag. When he explained it to me and his lab partners, he seemed totally mystified that this is NOT the way we were planning to do it.

Then, when I am teaching these calculations in the context of isotopes, a student notices this relationship:

And he suggests that we don't even need to do the math; we can just bank on the digits past the decimal point being the percent. So I ask, "Does that always work?" What happens if the masses of the isotopes are more than one unit away from each other? Like these:

And he says, "That's easy. We divide the digits past the decimal by 2." So we named a postulate after him because he created a rule. The next day, on the quiz, he came to my desk and asked if he had to show his work for the problem or could he just use his postulate? What's the use, I asked him, of having a postulate if we can't use it?

Now if only I would have used ClassKick while we did this activity, then the students could have seen each other's solutions and heard all the great ideas I heard. Well, that's something to look forward to for next year. And the next flipped lab.