The standard unit of measure for volume is the cubic unit, but we often need to fill boxes with different-sized units or packages. For example, suppose a candy factory has to package its candy in larger shipping boxes. Examine the different-sized packages of candy below. A package is defined for this activity as a solid rectangular prism whose dimensions are anything except 1 1 1.
Using just one size of package at a time, how many of each package (1-5) will fit into Box B (from Problem A1) so that the box is filled as completely as possible?
You may want to cut out the net for Box B and fold it into an open box. You can then use the box to help you visualize placing packages inside it. Close Tip
Describe your strategy for determining how many of Package 2 fit into Box B.
Notice that not all of the packages fill Box B completely so that there is no leftover space. Design the smallest box (in terms of volume) that could be used to ship all of the candy packages above. The box needs to be of a size and shape that can be completely filled by Packages 1-5 separately.
Think about the dimensions of each package and the possible dimensions of your new box. Close Tip
Is there more than one size box that can be used to ship the different candy packages? Explain why or why not.
How are the dimensions of the packages related to the dimensions of the larger shipping box?
Generalize the relationship between the dimensions of any package and the volume of any box that can be completely filled by the packages.
Video Segment Watch this segment to see how Doug and Mary solved Problem A6. They used manipulatives to represent the five packages and to figure out the dimensions of the new box.
Was your method similar or different?
If you are using a VCR, you can find this segment on the session video approximately 9 minutes and 45 seconds after the Annenberg Media logo.