We started with the following Balance Bender. I asked them to solve it and write an algebraic solution. Most of my students figured out the correct answer, but very few could translate it to algebra.
Having students come up with algebraic expressions for each of the balances was quite a challenge. I found myself asking "What do you have on this side? How could we write that? What do you have on the other side? How would we write that? What should we do to find... ?" But we did get there and I reminded them that if they had correctly solved it in their heads, clearly they could solve these equation and they needed to figure out how to write down their thinking. I also suggested that if they were having trouble solving an equation, they could turn it into a balance picture to help them visualize.
Then I took the ropes back out. Based on how the class was going so far, I opted to work through the rate of change vs. thickness modelling as a whole class. I pulled up the table from yesterday and asked what they noticed about the equations in the red rope row.
We then did the same for the white rope row and decided that one equation was in inches, while the other was likely in mm.
I pulled up the graph I had done in preparation for this activity, one that I did not intend to use, but given the lack of good data from my class, it was helpful. We found the point where the diameter was 2.54 cm (1") and determined that each knot in the 1" rope would cause it to shorten by 21.08 cm. We could then use this information to answer the original question.
We then moved on to the main focus for today:
Amid the balloons floating around (and later confiscated), ropes being flung and blue hair dye that appeared from someone's backpack, there was little work being accomplished. I even had an observation sheet to help me keep track of what had done what (solved by graphing, solved algebraically) and who could answer my questions (How did you choose your ropes? Why does it matter? If you have a different length of the same rope, how does that change your equation?). A couple of groups did some really good work. One tried to solve graphically but the intersection of the lines was beyond their grid so they solve using Desmos. They were also able to explain the conditions under which ropes would and wouldn't work and related these to their graphs. Sadly no groups were able to test their solution out. As a side note, the other teacher had much more success with her group.
This is what it should have looked like for the thick rope and a red rope:
What I will change for next time:
1. Identify each rope with a letter. So the red ropes would be A, B and C, the thick rope would be D, the white ropes would be E and F, and so on. Students can then easily identify the rope(s) with which they work and I can easily tell if their equation is on the right track.
2. With the ropes identified I could then have a table with the equation for each rope. This could be displayed when we do systems for any students who had not completed the first part of the activity.
3. I need to consider whether doing the rate of change vs. thickness of the rope part of the activity is worth doing.
4. I may get rid of one of the smaller ropes as the two smallest ones are very close in diameter.
I still really like this activity and will reflect more on what I can do to make it run more smoothly next time.