Teacher resources and professional development across the curriculum
Teacher professional development and classroom resources across the curriculum
|Mathematics: What's the Big Idea?|
More Geometry: Quilts and PalacesContent Guide - Beryl Jackson
Supplies Needed for Workshop #4:
About the WorkshopWhat is the theme of the workshop?
No other topic in mathematics demonstrates its relationship to the real world as concretely as geometry. All around us we see geometric figures in all shapes and sizes. We walk on planes with straight or curved sidewalks, study in rectangular classrooms, shield ourselves from the rain with dome shaped objects, and model both simple and complex mathematical and scientific phenomena with geometric interpretations. We find beauty in the quilt and are in awe of the majestic structure of the palace. This workshop will focus on the mathematics in art and nature as they relate to two-dimensional geometry.
Whom do we see? What happens in the videoclips?
What issues does this workshop address?
What teaching strategy does this workshop offer?
To which NCTM Standards does this workshop relate?
Suggested Classroom ActivitiesThe Platonic Solids
A tessellation is a design that covers a flat surface without leaving gaps and without overlapping. A regular tessellation is a design whereby only one shape is used to create the tiled pattern. As a task related to the activity presented in the workshop, have your students construct models of the Platonic Solids - so named for the Greek philosopher, Plato. Plato presented a complex ideology relating these solids to the world.
There are five Platonic solids. This means that there are only five solids in which all of the faces are congruent regular polygons. The Platonic Solids are the tetrahedron (4 equilateral triangles as faces), the hexahedron (6 squares as faces), the octahedron (8 equilateral triangles as faces), the dodecahedron (12 pentagons as faces), and icosahedron (20 equilateral triangles as faces).
The solids can be constructed from paper cut-outs of squares, equilateral triangles, and regular pentagons. These cut-outs should be tabbed so that the pieces can be glued together to form the polyhedra. Another way to have students construct these solids is to use toothpicks and gumdrops (or tiny marshmallows) - the toothpicks are the edges and the gumdrops are the vertices. Still another way of building these solids is to cut straws so that they are about 2 inches in length, and then thread these straws using string to build the faces. Tie the faces (i.e. 4 equilateral triangles) together to create the solid.
After the solids have been constructed, instruct students to investigate the differences and similarities between the regular tessellations and the Platonic solids. Also, what do the tetrahedron, octahedron, and icosahedron have in common? Can students find a pattern that exists when they go from one to the other? Why are there only five of these types of solids? Middle school students may want to research Plato's ideology about these solids.
Collect and display copies of some familiar logos. These can be product logos and/or service logos. Be certain that you select logos which have some form of symmetry. However, also include a few logos that do not have any forms of symmetry. In small groups, allow students to examine several of these logos to determine if they are symmetric in any way. That is, do they have reflection symmetry (vertical, horizontal, and/or rotational)? Reconvene students in a large group and facilitate a discussion about their findings.
Next, working in pairs, instruct students to create their own logos for products of their choice. Stipulate that their logos must have symmetry and should, in some way, relate to the product selected.
Understanding how objects are symmetrical involves spatial sense. Investigations in symmetry help to develop this sense in children and provide them with opportunities to look at the world differently.
Familiarize your students with examples of these rectangles. Have them look for objects in their world which they think are "golden." Instruct them to verify their visual instincts by actually measuring/researching to find out the actual measurements of these objects. Finally, instruct them to create the ratios of the measurements of the sides to determine if the figures possess the golden proportions. This would be a good time to encourage students to use the calculator. An example of an item possessing this relation is a playing card.
If a camera is available, allow students to photograph some of the larger items they have found. Using the pictures and the actual smaller items, create a visual display. Students will be able to see (and hopefully appreciate) the attention given to the manufacturing and production of everyday items. This activity will encourage students to begin to notice their physical world and to question why things are put together as they are. Does the shape of an item entice one to purchase that item?
There are other special geometric shapes which are also designed to have the golden proportion. Have students investigate this.
Suggested StrategiesWhenever possible, bring out the relationships between geometry and measurement, number, and patterns. As illustrated with Rose Christiansen's class, it is not necessary to wait until a special time to begin building concepts with fractions. Design tasks so that there is a natural connection and flow from one strand of mathematics to another. Geometry easily fosters these relationships and connections.
Pre-Workshop Assignment for Workshop #5In preparation for Workshop #5, please answer these questions.
Some people say that students are not learning as much math as they need to.
Mathematics: What's the Big Idea?