This is an activity I did with my calculus students post AP exam. We had talked about center of mass, and we calculated it with integrals in a previous class. Then I saw this problem in my
new favorite book, so I decided to run with it. I first had the students try to balance one ruler on the desk top with the condition that it should go out as "far as it can".
Then I gave them 2 rulers and said they should line up and for the future tasks, if a ruler is on top of another ruler, it should be farther out than the one below it. So they were to play around and get the whole system to be out as far as it could be.
You can get the total out 9". There's a pattern and a reason, and I'll be mean and not give it away yet because half the fun is figuring it out. Then they got 3 rulers with the same rules.
Do you see the coolness ... almost off the table (you can get 11" with 3 rulers). Then 4 rulers.
Woot! 12.5". And just for fun, 5 rulers:
13.7" off the table edge.
Then I gave them 2 rulers and said they should line up and for the future tasks, if a ruler is on top of another ruler, it should be farther out than the one below it. So they were to play around and get the whole system to be out as far as it could be.
COOL!
ReplyDeleteWe had an extra few minutes after a review day with my calc class, and there were like a million physics textbooks, so we tried to stack them like your rulers.
http://www.prairienet.org/~pops/BookStacking.html
It didn't work so well, so I'm glad to know it works with rulers! We have a TON of those in the department.
Theoretically (certainly not in practice), you could stretch them out forever like this. If you make a sequence for the distance n rulers can stick out from the table, then
ReplyDeleted(n rulers) = d(n-1 rulers) + 6/n
So the sum to infinity is
6/1 + 6/2 + 6/3 + 6/4 + .........
As all maths teachers know, that's a divergent series - albeit only just. It might be interesting to see how quickly the class can figure that out...
Sam-
ReplyDeleteNow you have me itching to get some books to try it myself just to be contrary.
Alex-
Ooh, that series would be a great addition to the lesson. Thanks for the idea.
Ms. Cookie
That also means that at some point the ruler on top will be completely past the table. Another problem could be what's the minimum number of rulers needed to get the top one totally past the table?
ReplyDeleteWell, we got the 4th ruler past the table, and it WAS counter intuitive. I guess I could ask that question BEFORE they started experimenting:
ReplyDeleteDo you think we could balance a ruler this way and it's completely off the table?
How many rulers do you think it would take?
Ms. Cookie
love it! When I taught triangle concurrency this year, I used an activity where the kids would cut out a triangle (any triangle) out of a piece of cardboard, and then they find the centroid, and we have a competition to see who could balance their triangles (without cheating) at the centroid, on a sharpened pencil! It only works if they were extremely precise with their measurements / drawing of the medians. Randomly during class, I was able to balance mine on a pencil tip while walking around for 15 seconds! The kids were SUPER EXCITED. haha
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