CECmath.36


TITLE:    Cube Coloring Problem

AUTHOR:   Linda Dickerson, Redmond School District,
          Redmond, Oregon

GRADE LEVEL:   Appropriate for grades 5-12

OVERVIEW:  Investigate what happens when different sized
cubes are constructed from unit cubes, the surface areas are
painted, and the large cubes are taken apart.  How many of
the 1x1x1 unit cubes are painted on three faces, two faces,
one face, no faces?

OBJECTIVE(s):  Students will be able to:
1.   Work in groups to solve a problem
2.   Determine a pattern from the problem
3.   Write exponents fro the patterns
4.   Predict the pattern for larger cubes
5.   Graph the growth patterns
6.   Extend to algebra

RESOURCES/MATERIALS:  A large quantity of unit cubes, graph
paper, colored pencils or markers.

ACTIVITIES AND PROCEDURES:
1.   Hold up a unit cube.  Tell students this is a cube on
     its first birthday.  Ask students to describe the cube
     (eight corners, six faces, twelve edges).
2.   Ask student groups to build a 'cube' on its second
     birthday.  Ask the students to build a cube on its
     second birthday and describe it in writing.
3.   Ask students how many unit cubes it will take to build
     a cube on its third birthday, fourth, fifth...
4.   Pose this coloring problem:  The cube is ten years old.
     It is dipped into a bucket of paint.  After it dries
     the ten year old cube is taken apart into the unit
     cubes.  How many faces are painted on three faces, two
     faces, one face, no faces.
5.   Have the students chart their findings for each age
     cube up to ten and look for patterns.
6.   Have students write exponents for the number of cubes
     needed.  Expand this to the number of cubes painted on
     three faces, two faces, one face, no faces.
7.   Have students graph the findings for each dimension of
     cube up to ten and look for the graph patterns.
8.   For further extension, see NCTM ADDENDA SERIES/GRADES
     6/8.

TYING IT ALL TOGETHER:  The students will have a chance to
estimate, explore, use manipulative, predict, explain in
writing and orally.  They will note that the three painted
faces are always the corners-8 on a cube.  The cubes with
two faces painted occur on the edges between the corner and
increase by 12 each time.  The cubes with one face painted
occur as squares on the six faces of the original cube.  The
cubes with no faces painted are the cube within the cube.
This is an excellent way for students to become involved in
exploring a problem of cubic growth.