Cube Coloring Problem

May 1994

These lesson plans are the result of the work of the teachers who have attended the Columbia Education Center's Summer Workshop. CEC is a consortium of teacher from 14 western states dedicated to improving the quality of education in the rural, western, United States, and particularly the quality of math and science Education. CEC uses Big Sky Telegraph as the hub of their telecommunications network that allows the participating teachers to stay in contact with their trainers and peers that they have met at the Workshops.

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.

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