The Box Problem, Clip 3, 4:49

• The teacher begins by drawing a graph that resembles a downward facing parabola which the students don’t agree with due to its symmetry. After some small group discussions (which are not in this clip) the teacher leads a discussion with the class to create the graph.
• When necessary, the teacher helps the students attend to important aspects :
  • The rate of change or the slope of the graph
  • She focuses the students on changes in volume for the initial changes in the length of the cutout
  • She helps them tie this information to the concavity of the graph
• Notice that the teacher uses a box to show the students what it means for the graph to be concave down in terms of the quantities involved.
• She also points out what the graph would look like if it were in a graphing calculator and discuss what makes sense in the context of the problem.
  
• (not in this clip) She then uses the Exploring the Box with Cubes applet to demonstrate how the volume goes up, reaches a maximum, and then goes back down:
  • The boxes of unit volume in this applet make the change in volume easier for the students to see
  • This will help them conceptualize how the volume is changing as the length of the cutout is changing
  • Time permitting, the teacher can stack the boxes of unit volume into the box created