Counting Surface Boxes*
Cuisenaire Rods look like this:
They are 3-dimensional "rods" made from wood or plastic and have various lengths (coded by their color) and their other two dimensions are fixed.

The cross sections (and thus, in particular, the two "ends") of a rod are square-shaped.

They are often used in teaching arithmetical properties.They can model some algebra, too. [See http://www.etacuisenaire.com/

Here, instead, we will use them simply a geometric objects which, in turn, we happen to choose to analyze algebraically.

Notice:

  1. A rod of length 1 has 6 flat boxes ("squares"), altogether, on its sides.

  2. A rod of length 2 has ____ boxes, altogether, on its sides.

  3. A rod of length 3 has ____ boxes, altogether, on its sides.

  4. A rod of length 4 has ____ boxes, altogether, on its sides.

  5. A rod of length 5 has ____ boxes, altogether, on its sides.
What patterns do you notice? Before jumping to answer the next question, fill this in -- knowing how to express yourself verbally and clearly and precisely is part of learning the math!

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Now answer: A rod of length N has ____ boxes, altogether, on its sides. Hmmm! Surface area corresponds to dimension D=____ and usually we think of ___________s as the kind of functions that model objects of this dimension. Why aren't we getting that here?

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* Modified from Mathematical Models and Modeling for Middle School Teachers by Cathcart/Horseman.

Created 2001 by Dr. Harel Barzilai