Type of Class:
            High School Geometry Class (all levels)

Related VA SOL:  G.9

Time Frame:
            90 minute block class

 Objectives:

  • Students will understand tessellations in relation to math and the world around them.
  • Students will know the definition and attributes of tessellations.
  • Students will be able to identify tessellations and non-tessellations, understanding why they fall into a particular category. 
  • Students will also be able to create their own tessellation.

Materials:

  • Chalk and Chalkboard
  • Tape
  • List of words or real examples (positive and negative examples)

 Procedures:

  1. Introduce the process:  Concept Attainment Lesson

Explain that you will show positive and negative examples of a concept one at a time (you will not tell them what the concept is, they will be looking to define the concept based on what you show them.)  The students’ job is to formulate a list of features that characterize the positive examples.  They will make a list of those defining features to try and come up with a definition of the concept.  As they think of new things they will add them to the list, or as things are decidedly wrong that are already on the list they will mark them off.  Mix up the examples (between positive and negative) and as a definition is being formulated ask them whether they think the example is positive or negative before you tell them.  Have them define and redefine their concept as you add new examples.

  1. Present the examples and list the attributes.Show the first example/non-example set and begin by letting the students form their definitions.  Add examples and non-examples letting the students predict their classification.
  2. Have them further develop their definition for the concept using the positive examples.
  3. Give additional examples to test whether the students truly understand their concept and definition.
  4. Discuss the process with the students, asking the questions:

  • When did you begin to understand the essential attributes?

  • Which examples were most helpful?

  • Which examples confused you?

  1. Ask for their own examples of tessellations.  Then discuss where tessellations are found in the “real world.”

 Assessment:

Each student will be asked to make a tessellation of his or her own.  They will be given the option of making it by hand, or using Geometer’s Sketchpad to help create the tessellation.  They will have 1 week to complete the project.  The following criteria is a sample that could be used to evaluate the task (total of 50 points):

  1. Neatness (15 points)

  • Project must be handed in on time (5 points)

  • Project must be done carefully (i.e., careful attention to detail) (5 points)

  • Project must be colored in some fashion (either black and white or with other colors) (5 points)

  1. Tessellation must: (15 points)

  • Be tessellated a minimum of 15 times (5 points)

  • Be a creation of your own (no regular tessellations) (10 points)

  1. Correct tessellations (using the definition to create the tessellation) (20 points)

  • There will be no gaps or spaces between the objects as they are tessellated (10 points)

  • The same object must be repeated across the plane (5 points)

  • The object must fit together with itself (5 points)

 Suggestions/Comments:

Here are some suggestions for positive and negative tessellations (negative examples can be almost anything you can think of).  You can click on some of them to find pictures of what I am suggesting.

 Positive                                                                        Negative

Flannel shirt                                                                  flower pillowcase

Checkered Towels                                                       necklace

Checkered Potholders                                                  Escher (non-tess.)

Pineapple                                                                     pear

Fish (showing scales)                                                    zebra

Snake (design or scales)                                               candle 

Escher (tess.) design                                                     fan

Honeycomb (bee’s work)                                             Kaleidoscope

Brick sidewalks                                                            Painting
           
            Shingles on houses                                                        Bugs

Ceiling tiles                                                                   Bedspread/Blanket

Linoleum                                                                      Picture of somebody

At times students may need help identifying common attributes of examples.  You may need to help them by giving hints or asking questions that will direct them toward the concept they are looking for.  This may be especially true in a standard level class, as they may be more easily frustrated not to see the concept immediately.

It is a good idea to come up with your order of examples ahead of time.  Think about what order would be most helpful for students identifying what makes a tessellation.  The better the flow of positive and negative examples, the better the students will be able to come up with clear criteria for the concept.