Math Beyond Workbooks

Surface Area and Volume Discovery

Teacher-Centered ‚ A "traditional" approach to teaching about surface area and volume would probably involve the teacher writing the formulae on the board and then applying the formulae to several examples while students dutifully copied them down. The students may be able to follow the explanation without any problems, and they may be able to mimic these steps back for the teacher when given new examples. But the material is not really internalized.

Student-Centered ‚ Do not give students any formulae. Instead, try the following activity.

1. Have them sit in groups of 2 or 3 and give each group a box (it's better if none of the faces are squares).
2. Give them a centimeter ruler and a piece of paper with the surface area and volume written on it. It's best if the measurements do not contain fractions, and I'm assuming students have already learned how to calculate the area of a rectangle.
3. Their task is to figure out how to arrive at the given results. They have to come up with a procedure, and depending on their level/familiarity with algebra, they can also come up with a formula. The various groups share their results with the class.

The discussions and level of understanding generated by this activity are tremendous, and the students will remember these concepts for a long time to come. A formula is easily forgotten, but this type of understanding is not.

Next, these concepts should be related to the real world:

1. Using new boxes, tell them that the boxes are going to be used to pack sugar cubes that measure one cm on each side.
2. Tell them that their first task is to figure out how many cubes will fit in each of their boxes.
3. Tell them their second task is to calculate the number of square centimeters of cardboard they will need to make more boxes the same size.

Lots of interesting real-world packing problems can follow. What if the sugar cubes measure 1 cm by 2 cm by 3 cm? How many boxes of sugar cubes will fit into a carton measuring 50 cm by 50 cm by 1 m? Etc. The more they work with real boxes (some students have a real hard time "seeing" 3-dimensional drawings on paper), and the more they measure and answer real-world type problems, the more they will internalize the concepts.

This concept ‚ giving students a type of problem they've never done before along with the answer and asking them to work as a group to figure out how it was done ‚ can be used with nearly any new topic.