In this clip the students are unsure whether the length and width of the box co-vary so she has them discuss other the relationships between quantites to help them.
In this clip the teacher leads a discussion about whether or not there are two boxes that can have the same volume.
Box Problem Summary, Clip 1, 2:40
It is important the teacher ensures the students are beginning to understand the concept of co-variation. Thus asking questions about how pairs of quantities co-vary in the box problem is a good place to do this.
Notice how the teacher goes to the various groups and asks them to report on different pairs of quantities. This is one way of recording all the answers if time is an issue with a large class.
In this instance the teacher is mostly interested in the direction of the co-variation (whether the quantity is increasing or decreasing), but given the opportunity, she does ask how it is increasing or decreasing. Because several of the relationships are linear, it is reasonable that the students should be able to be specific about how the quantities are covarying.
Notice that the teacher leaves the last group to ponder why their answer is incorrect. This can be a useful strategy because as more groups participate, the students can begin to think about why their suggestion was incorrect and fix it.
