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 2, 3:08
Notice how when a student suggests that the two quantities are inversely proportional, the teacher asks him what he means. It is very important, especially at this stage in the class, that the teacher require justifications for student statements. This will help establish justification as a norm in the classroom.
The teacher makes the move to draw both the equation and the graph on the board. This will help the students recall the equation and also help them understand what the first student meant by inversely proportional.
The students do realize it should be a straight line and the teacher makes them give an argument for why this is so, once again reinforcing that justifying one's answer will be necessary when participating in class.
She allows the students to help her describe the relationship between the two quantities which helps to keep the students involved and allows them to contribute to their own understanding.
She also uses this discussion as an opportunity to reinforce the meaning of constant rate of change.
After this clip she has a short discussion with the students about the co-variation between the length of the cutout and the width of the box. The students quickly realize it will be a very similar relationship with very similar explanations so they do not spend much time on it.
