Pyramid of Cubes
Tools and materials: Lots of wooden cubes (the teacher
should know approximately how many). The cubes can have edges 1 cm, 1/2 in, 3/4
in or 1 in long. TI-34 II calculators or similar ones.
Task:
Students work in groups. Each
group sits around a table.
Instructions:
You are going to build a
pyramid from wooden cubes. You can get at most c cubes (e.g. c=160). The
pyramid has a base measuring n by n; the second layer has (n-1) by (n-1) cubes.
This pattern continues until one block crowns your pyramid. Before you start
building the pyramid, you have to compute the exact number of blocks you are
going to use, and take this number of blocks from the supply table. After you
build the pyramid, compute its surface area (the base doesn't count) either in
square inches or in square centimeters.
Example of a solution:
Computation of the number of
blocks b, as a function of the length of the edge of the base n. (The number of
blocks that are available is c.)
We will use the TI-34 II calculator.
Calculator variables:
A holds
the length of the edge n.
B holds
the number of blocks b in the whole pyramid.
Define
OP1=A+1→A
increases
n;
OP2=B+A2→B
computes b;
Enter:
0→A[=]
0→B[=]
Repeat
OP1 write the
value of n;
OP2 write the
value of b;
until b>c
Table of values
n: b:
1
1
2
5
3 14
... ...
Computation of visible surface area
Roof area
= n*n
"Do you see it?" (Look from the top!).
See how you can slide all the cubes to a corner, and when
you look from the top the roof area is simply n*n.
Side
area = 4*n + 4*(n-1) +....+ 4*2+4*1
= 4*(n+ (n-1) + ... +2+1)
= 4*(n*(n+1)/2)
= 2*n*(n+1)
(You may
derive this formula or compute its value with a calculator.)
Therefore:
The surface area = n*n +
2*n*(n+1).
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