Adapted from Harrell and Fosnaugh*
Topic Area(s)
Tilings
Polygons
Congruence
Similarity
Geometric real-world situations
Objective(s)
The student will investigate geometric shapes in nature by examining the
patterns of tilings seen in nature. Students will learn how to divide triangles
an parallelograms into rep-4 tiles and rep-9 tiles.
Introductory Statement
Many geometric shapes are represented in nature. More importantly, these
shapes are repeated over and over again to build more complex patterns and
structures. For example, the honeycomb is an example of a hexagon that is
repeated many times. Geometric shapes that do this are said to tile the plane
in the same way that repeated shapes (tiles on a roof) cover a roof. You
will look at some examples some nature and then learn how some of these patterns
are actually composed of more simple, smaller repeated shapes, called rep-4
tiles and rep-9 tiles.
| Mathematics Skills | Arizona Standard |
| Describing, representing, and analyzing patterns and relationships using shapes | 3M-E2 (4-8) |
| Predict how shapes can be changed by combining or dividing them | 4M-F2 (1-3) |
| Applying geometric properties and relationships such congruence, similarity, ... to real-world situations | 4M-E2 (4-8) |
| Performing elementary transformations | 4M-E3 (4-8) |
Materials
Student recording sheets, one for each student
Snakeskin
Honeycomb
Wasp's nest
Xerox copies of: Bumblebee's eye, Onion root tip, Vertebrate striated
muscle
Focus Questions
1. What kinds of geometric shapes occur in nature?
2. Are these shapes repeated? Why would that occur?
3. How can we divide a triangle into four pieces to create a rep-4 tile?
4. How can we divide a triangle into nine pieces to create a rep-9 tile?
Teacher Preparation Notes
1. Preview and study Harell and Fosnaugh's article on Mathematics in Nature, pages 380- 384. See attached article.
2. Present Introductory Statement, Focus Questions, and briefly discuss how an equilateral triangle can be divided into four parts, a rep-4 tile.
3. Provide students with their recording sheets and let them work through the exercises. .
4. At the end of the session(s), to ensure closure, do a whole group activity
and have selected students demonstrate their responses.
What the Students Will Do
1. Listen to the Introductory Statement, Focus Questions, and presentation rep-4 tiles.
2. Examine specimens and Xeroxes of objects from nature and record shapes.
3. Complete responses to questions on response sheet about rep-4 tiles and rep-9 tiles.
4. Discuss connections between geometric shapes and those found in nature.
Discussion Questions
1. What are some common geometric shapes found in nature?
2. Compare the rep-4 tile drawings for an equilateral triangle and a right triangle.
3. Is every triangle a rep-4 tile? Why or why not?
4. What does a rep-9 tile drawing of a triangle look like?
5. For a parallelogram, what does the rep-9 tile drawing look like?
6. What is the connection between geometric shapes and those found in nature?
Extension
1. If microscopes are available, conduct the exercise with these using plant parts, etc.
2. Have students think about and discuss other examples of tilings from
nature.
Evaluation
1. Observe students' drawings and constructions.
2. Conduct discussion of students' responses to questions on their recording
sheets.
1. Geometric shapes in nature. At this station, you will find a snakeskin,
a jar of honey, and photographs of magnified objects from nature. For each
object or photograph, determine if a part of it can be duplicated and used
to "tile the plane" (i.e., repeated geometric shapes can be used to fill
in the area). For each geometric figure that will tile the plane, draw what
the tile looks like. Remember that in nature, things are not perfect. Think
of these figures as being congruent when they seem to have the same general
shape.
Table
| Object | Does this shape tile the plane? | If so, draw a tile. |
| Snakeskin | ||
| Honeycomb | ||
| Wasp's nest | ||
| Bumblebee's eye | ||
| Onion root tip | ||
| Vertebrate striated muscle |
2. Types of tilings in nature. Notice that the snakeskin appears to
be made up of parallelograms. Four copies of the same parallelogram fit together
to form a larger similar parallelogram. Four of these larger similar figures
fit together to form an even larger similar parallelogram. If we continue
this process indefinitely, we can tile the plane. See the diagram.
Because four congruent tiles are needed to create a larger similar figure,
the parallelogram is call a rep-4 tile.
Draw a triangle and determine if it is a rep-4 tile.
Draw another triangle that is different from the previous one. Determine
if this triangle is a rep-4 tile.
Do you think that every triangle is a rep-4 tile? Explain.
Mathematicians have shown that these rep-4 tiles are also rep-9 tiles. Show
that a triangle is a rep-9 tile.
Repeat this process with a parallelogram.
*Harrell, M. E., & Fosnaugh, L. S. (1997). Allium to zircon: Mathematics
and nature. Mathematics Teaching in the Middle School, 6, 380-389.