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 4, 2:37
It is important to establish normative ways of talking about the mathematics. In this case the situation is about using the word AT versus using the word WITH.
The graph is decreasing and concave down so the teacher explains that it can be described as decreasing AT an increasing rate, meaning that the rate that it is decreasing is increasing (i.e. it is decreasing faster and faster); or it can be described as decreasing WITH a decreasing rate since the rate of change is becoming more negative. Such distinctions are important as students continue to talk about the mathematics during the semester.
She then makes the move to ask them how they can calculate the average rate of change. The teacher reminds them of the average speed problem to facilitate a discussion about improving the accuracy by choosing a smaller interval. This is a good example of how the teacher used their prior knowledge to help them reason about the box problem.
