Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

Mathematics: What's the Big Idea?

# Workshop #4

## More Geometry: Quilts and Palaces

Content Guide - Beryl Jackson

Supplies Needed for Workshop #4:
Pattern blocks, graph paper, geoboards, metric tape measures, masking tape, small rectangular mirrors (2 mirrors for every pair of participants),
pencils, paper, scissors, rulers, calculators, tape, a variety of colored markers

What 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?
We will see students, at both the elementary and middle school levels, engaged in tasks which illustrate the development of geometric ideas. These ideas range from the recognition of whole shapes, to analyzing the relative properties of shapes, to investigating relationships between shapes. Ultimately, students are making conjectures about these relationships.

What issues does this workshop address?
This workshop will address how children form basic geometric concepts in mathematics. Tasks are designed to help teachers help students understand some of the underlying concepts in geometry, their connection to other strands of mathematics, and the role geometry plays in our physical world.

What teaching strategy does this workshop offer?
The teacher's and students' roles in discourse and the learning environment will all be addressed during this workshop. Each of these roles is discussed extensively in the NCTM Professional Teaching Standards.

To which NCTM Standards does this workshop relate?
Most specifically, this workshop addresses Standard 9: Geometry and Spatial Senseof the K-4 standards and Standard 12: Geometryfor teachers of grades 5-8. There is also an inherent focus on Standards 2, 3, and 4: Mathematics as Communications, Mathematics as Reasoning, and Mathematical Connections as we encourage students to use the language of geometry, understand and apply geometric properties and relationships, and link these relationships to both the real world and the world of mathematics.

### Suggested Classroom Activities

The 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.

Symmetry
Since symmetry appears so abundantly in nature - the flower, the human figure, and crystals - it is not surprising that the use of symmetry in art, architecture, and even in of the design of national flags is not by accident. Symmetry provides a naturally pleasing view of both the man-made world and the world of nature.

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.

Where's the Gold?
If the ratio of the measure of the longer side of a rectangle to the measure of the shorter side of a rectangle approximates 1.6:1 (or, more accurately, 1.6180:1), then that rectangle is said to be "golden." The rectangles formed with this special relationship were considered by the ancient Greeks to be the most aesthetically pleasing of all of the rectangular shapes.

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 Strategies

Whenever 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.

### Post-Workshop Questions

1. Despite some reform in the teaching and learning of mathematics, there is still too often major emphasis placed on arithmetic skills at the elementary level, while those skills related to geometry are viewed as less important. Discuss your view on the role of geometry in the K-8 mathematics curriculum.

2. Often, tasks are selected based on the amount of time it should take to complete them rather than on the quality of the activity itself. True geometric investigations consume larger chunks of time because they require the use of physical materials and require a higher level of thinking on the part of the students. Children need time to put the pieces together and to communicate their findings. Share your thoughts on time verses task.

3. Investigations in geometry help students develop inductive reasoning skills. When students begin a formal study of geometry, typically they experience difficulty when asked to make generalizations from their observations (inductive reasoning) or to reach logical conclusions (deductive reasoning). Identify some examples of activities related to geometry that have proven to be successful in helping students think and reason inductively and/or deductively.

### Pre-Workshop Assignment for Workshop #5

Some people say that students are not learning as much math as they need to.

• What's an example of something students coming into your class don't know as well as they need to?

• How have you seen students changing over the years with regard to their mathematics learning?

Mathematics: What's the Big Idea?