At this point in the Problem String, Pam moved on to see if the next questions would help.

"So we still have 12 and nine? Okay, cool, I wonder if we could do another one. What if I said to you that we know that there's 24 hours until there's a hard freeze?"

Students had experienced the "Sticks of Gum" ratio table Problem String the day before. Because we were going to video the next day, I chose the "Sticks of Gum" Problem String because it is a really good string to get students talking and reasoning. This also had the effect of helping students move from additive thinking to multiplicative thinking. But, now students are also playing with the idea of splitting the difference, which works when the scenario has a constant rate of change, one pack has 17 sticks so 17 sticks per pack. If you know that 2 packs have 34 sticks and 4 packs have 68 sticks, you can reason that since 3 packs is exactly in the middle of 2 and 4 packs, then we can find the exact middle number of sticks between 34 and 68 to find that 3 packs have 51 sticks. This scenario can be represented by *y* = 17*x*, where *y* is the number of sticks in *x* packs.

The scenario in the orchard Problem String does not have a constant rate of change. It is not a directly proportional situation, it is an inversely proportional situation. This scenario can be represented by *y* = 120/*x*.

As students were reasoning about the amount of time it would take for 10 workers to clear the orchard, Pam chose to introduce the next problem in the string, 40 workers. Now students can consider what happens when half the number of workers show up, 10, or double the number of workers show up, 40. By keeping context at the fore, students began to develop the sense that numbers were varying inversely. Double the number of workers take half the time. Half the number of workers take double the time.

Indeed, moving on to the next problem did help students to make sense of both the multiplicative and also the inverse nature of the situation.