Title: Tessellations in Nature and Rep-tiles

Audience: High School Geometry

Length of Lesson: two 50-minute periods

Sources:

http://coe.west.asu.edu/explorer/shapes/staffdevl/3.teacher.instructions.html#Student

http://illuminations.nctm.org/LessonDetail.aspx?id=L251

I. Performance or learner outcomes

The student will be able to:

Identify tessellations in nature.

Create and understand rep-tiles.

II. Overview

Students will review material from previous lesson: Introduction to Tessellations. They will explore different artifacts and pictures to see tessellations in nature. They will then learn about rep-tiles, and identify them in nature. Lastly they will apply their knowledge of rep-tiles to complete an assessment.

III.  Resources, materials and supplies needed

Snakeskin

Honeycomb

Wasp's nest

Xerox copies of:

Bumblebee's eye

Onion root tip

Vertebrate striated muscle

Construction paper

Scissors

Ruler

Protractor

Compass

Computers with Geometer’s Sketchpad and projector

IV. Supplementary materials, handouts.

Student Instructions/ Data Sheet, see Appendix A

V. Standards

TEKS

§111.34.(c) Geometric patterns: knowledge and skills and

performance descriptions. The student identifies, analyzes, and describes patterns that emerge from two- and three-dimensional geometric figures.

§111.34 (f) Similarity and the geometry of shape: knowledge and skills and performance descriptions. The student applies the concepts of similarity to justify properties of figures and solve problems.

Engage

Teacher does Teacher has students display their tessellations that they made during the previous lesson.

Probing Questions Who can remind the class of what we learned yesterday? What do you notice about the shapes used here? What regular polygons did we learn that tile the entire plane? Do we notice which regular polygons can be combined to tile the plane? What about irregular polygons? Can anyone think of a tiling that naturally occurs in nature?

Student Does We learned about tessellations Various answers Hexagons, triangles… Triangles and squares, triangles and ...... Various answers: honeycombs, etc.

Explore

Teacher does Okay we are going to get in groups of two and identify naturally occurring tessellations. You will rotate around the room to different stations and examine different things or pictures. Look at them closely and record your thoughts on the handout.

Probing Questions See attached worksheet.

Student Does

Explain

Teacher does

Probing Questions Is the artifact (or picture) a tessellation? Can you combine shapes from different artifacts to tile the plane? (The size of the shapes can be changed) What do you think is the utility of having these tessellations in nature? Why are hexagon tessellations important for bees? Think about if they used cylinders…

Student Does Yes, for example the honeycomb is a tessellation of hexagons. A student goes to the computer and attempts to make a tiling with some of the shapes observed in different artifacts. In honeycombs, hexagons share sides, therefore there is less building material (wax) required to hold more honey?

Extend

Teacher does A rep-tile is a geometric figure whose copies can fit together to form a larger similar figure. Another way that one can think of a rep-tile is as a puzzle piece, where a larger similar figure is the entire puzzle. A rep n-tile is a figure that has be property that n copies can be fitted together to create a larger similar figure

Probing Questions Can anyone think of an example of this? Do you think that any regular polygon is a rep-tile? If not, can you think of a counter example? Do any of the objects you just examine have this property? What about the honeycomb? Why do you think this is so? Can you use Geometer’s Sketchpad to demonstrate?

Student Does A square! Four squares can fit together to make a larger square. No. Students may throw out any example, are able to try to test some out in Geometer’s sketchpad. Yes, for example, the snakeskin looks like it is made up of parallelograms. 4 parallelograms can make a larger, similar parallelogram. No. Student goes to computer and using a projector, attempts to divide a large hexagon into smaller ones (all smaller ones of the same size).

Evaluate

On the overhead put the following questions:

Using a construction paper, scissors, ruler, compass and protractor prove the following properties and show your work…

1. Draw a triangle. Using construction paper cut out four copies of this triangle. Will your triangles fit together to form a larger triangle? Is the larger triangle similar to the smaller triangles? Measure the angles and sides of both the larger triangle and one of the smaller triangles to make sure the triangles are similar. Is your triangle a rep-tile? Explain.

2. Mathematicians have proved that for any natural number n greater than 1, a rep-ntile exists, for instance, a rep-2 tile, a rep-3 tile, a rep-4 tile, and so on. Draw an isosceles right triangle and show that it is a rep-2 tile.

3. Show that a 30°-60°-90° triangle is a rep-3 tile and that a right triangle whose legs measure 2 cm and 4 cm is a rep-5 tile.

Appendix A

Tilings and Polygons in Nature


Student Instructions

1. Geometric shapes in nature., 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