The Algebra Problem of the Week that just finished up was about that classic playground game of trying to balance two different-sized people on a teeter totter. How many of you had an “aha!” moment on the playground one day when you realized that by sitting farther away from the fulcrum (the point where the seesaw balances on the base) means you can balance with a kid a lot heavier than you? A grown-up even? Or that you could leave your equal-sized friends dangling off the ground by sitting as far away from the fulcrum as possible?

In this PoW we asked students to work with the mathematized version of the situation. Instead of borrowing some small children and heading out to the playground to find a balancing point, we gave submitters a relationship between weights and distances from the fulcrum:

(weight 1)*(distance 1) = (weight 2)*(distance 2)

Most submitters chose to work with the equation we gave them in some fashion. Henry S. from Stony Point Elementary used guess and check to find the missing distances:

M 32*36 = 1,152 32*30 = 960 32*20 = 640 32*100 = 3,200 32*80 = 2,560 32*90 = 2,880
L 40*18 =  720 40*12 = 480 40*2 = 80 40*82 = 3,280 40*62 = 2,480 40*72 = 2,880

Other students, like Natalia N. from Sekolah Ciputra, thought of the problem as having two variables:

x=3/2+y
balance:
32x=40y
32(3/2+y)=40y
48+32y=40y
48=8y
y=6+t
x=7.5+t

I don’t know what the +t represents in Natalia’s solution. Perhaps it has to do with the fact that we don’t know how long the teeter-totter is. What do you think she might be using the t for?

Most submitters immediately thought of writing the two distances as x and x + 1.5, which impressed me because I don’t always think to ask, “how else could I write this quantity? What relationships could I use to write it in terms of another quantity?”

All of the submissions above mathematized the situation in the way we had suggested. But those weren’t the only kinds of responses. Some submitters did a lot of hard thinking about the physics involved, without getting into the mathematization, like Aspen K. from East Middle School:

They can both be balanced on it even though they don’t weigh the same. They can do this by the girl that is heavier sit on more of the end of the seesaw. So then the girl on the other end who must be lighter will sit closer to the middle. By doing that they should both be balanced out.

If I were Aspen’s teacher, I don’t think I would start my conversation with the mathematizing. Aspen is wrestling with the physics involved, and so I would head out to the playground or get out some see-saw models and start trying to balance lighter and heavier things. Once we had established the indirect relationship between weights and distances, then it would be time to start asking, “what are the quantities involved? How do they seem to be related?”

Finally, there were the students who chose to mathematize the problem in unexpected ways. These led to the solutions that were the most interesting and challenging for me, forcing me to look at the problem in totally new ways!

For example, Jeremy Z. and Steven M. from Cold Springs Elementary School used ratios to think about the problem, and never mentioned the equation with weights and distances. They may have used it to help them think of why if the weights are in a 4:5 ratio, the distances must be in a 5:4 ratio, but they didn’t explain that step of their reasoning so I’m not sure…

the ratio of the weights of them are Marnie:Lex=4:5. So the distance must be 5:4. since Marnie is sitting 1 1/2 feet further away from the middle of the seesaw then Lex. if 1 1/2 feet is 1 in difference in the ratio, then it is 90:72(inches).4*90=5*72. They both equal 360.

There were also some submissions that mathematized with some inaccuracies, or different theories about how the problem was working. Taylor H. from East Middle School focused on where to position students on a board of a given length so that one is 1.5 feet closer to the center. That was some good thinking! Several students focused on the length and how to arrange the students so they are 1.5 feet from the middle, ignoring the weight information. As Taylor worked the weight information in, I got a little confused. I think it’s neat that Taylor was trying to account for all of the quantities in the problem, and I would love to hear more from Taylor about if the weights were especially hard to think about and what they noticed about the weight information in the story:

Lets say the seesaw was 15 ft. long i subtracted 1 1/2 from that and got 13.5…. Then i had to divide that by two to get 6.75… It says that the girls weights are 32, and 40… I had to divide those by 6.75…… for Marnie it was 270.00 in. ……….. and finally for Lex it was 216.00 in. ………

Last but certainly not least, is this submission from Karrie P., also at East Middle School.

How i figured out this problem is I knew that Marine is 32 pounds and Lex is 40 pounds….. wich is a 8 pound difference. So, If Marine is 1 1/2 away from the fulcrum, than if you subtract 1/8, because there is a 8 pound differnce, from 1 1/2 it would be 1/2 away from the fulcrum for Lex. So…… over all, Marine is 1 1/2 away from the fulcrum and Lex is 1/2 away from the fulcrum.

I think it’s really neat that Karrie has the idea of reciprocal fractions (an 8 pound difference should result in a 1/8 difference in the distance) and I wonder where she got that idea. Many students wonder about the different ways to compare two quantities (should I subtract or divide? Why?) and I think that would be a neat conversation to have with Karrie (“how can you compare 32 pounds and 40 pounds? What’s the difference?”) Clearing up the fact that Marnie is 1 1/2 feet further away from the fulcrum seems like the last step to me, although it is still related to the whole idea of comparing numbers and the language we use with different types of comparisons.