·       Activity Name:    Surface Area and Volume:

A Fractal Tetrahedron

 

·       Objectives:

The students will make and explore a tetrahedron fractal to find the surface area and volume for different layers.  The students will discover patterns in the increasing areas and volumes, and then work out the formulas for the geometric series.

 

·       EALR/Standards:

1.3 understand and apply concepts and procedures from geometric sense.

1.5 understand and apply concepts and procedures from algebraic sense.

2.1 investigate situations.

2.3 construct solutions.

3.1 analyze information.

3.3 draw conclusions and verify results.

4.1 gather information.

4.2 organize and interpret information.

 

·       Materials:

3 different colors of paper with 1 inch equilateral triangle grids drawn on them.

            Number of sheets of each color for each student: 4, 4, 3

Transparent tape

Scissors

Glue

 

·       Teacher Notes

o     Prerequisites for the learner:

Knowledge of geometric series.

 

o     Teacher hints for the activity:

This activity needs to be done partly in class and partly as homework if you want each student to have a completed model.  Students can do the cutting and taping at home before doing the mathematics in class.  An alternative is to have the students do the math without a model, or to have one model for everyone to share.

 

o     Introductory questions:

What is a tetrahedron?

If you take one face of a tetrahedron and connect the midpoints of each line segment, what would the new face look like?

 

o     Wrap-up questions:

How many tetrahedra will it take to make a level 3 fractal tetrahedron?

 

o     Solutions:

Introductory questions:   A four sided, 3-D model that has all equilateral triangles as the

faces.

four triangles: three corner ones and a center one that is upside down.

Wrap-up questions:      144.

 

o     Assessment suggestions:

Collect the models to check for accuracy and neatness, and collect papers showing the work done.

 

·       The Activity:

Begin by handing out the colored papers and explaining the rule for making a tetrahedron fractal.

Rules:  Each layer is made with tetrahedra that have side lengths ½ the previous layer’s side length.  A tetrahedron is attached to the center of every triangle that has a side length twice the size of the attaching tetrahedron. NOTE:  the tetrahedra are attached to triangles, NOT tetrahedra.

Have the students make the models. 

Once the students have their models, they should fill out the chart on the student page, and then answer the questions.

           

·       Extensions:

Have the students do the same project using the cube as the base figure.

Fractal Tetrahedron

 

Using your model as a guide, fill in the chart below.

 

Level

Number of Triangles

Area of 1 Triangle

Surface area of fractal Tetra.

Volume of an added Tetra.

Total Volume of Fractal

 

0                                  4                      1                                  4                        1                                1

 

1

 

2

 

3

 

4

 

5

Fractal Tetrahedron

 

Using your model as a guide, fill in the chart below.

 

Level

Number of Triangles

Area of 1 Triangle

Surface area of fractal Tetra.

Volume of an added Tetra.

Total Volume of fractal

 

0                                  4                      1                                  4                        1                                1

 

1                                  24 = 61x4         ¼                                 6 = 24(1/4)      1/8                               1.5

           

2                                  144 = 62x4       1/16                             9 = 144(1/16)  1/64                             1 7/8

 

3                                  864=63x4         1/64                             13.5                 1/512                           2 5/32

 

 

 

n                                  6nx4                 1/4n                              1.5n                        1/8n                  1 + 2(1-(3/4)n)

 

 

Patterns to notice in volume:

1:         1 + 4(1/8)                                                        = 1 + 30/21

2:         1 + 4(1/8) + 24(1/64)                                       =1 + 30/21+ 31/23

3:         1 + 4(1/8) + 24(1/64)   + 144(1/512)               =1 + 30/21+ 31/23+32/25

 

Always add 3n-1/22n-1 to the previous layer