## Applications of Geometry and Geometric Thinking - Day 1

From Dr. Hirst's Mat 5950 syllabus: The intent of this course is to introduce students to problems from the physical, biological and management sciences in which the mathematics of pre-algebra, algebra 1 and 2, pre-calculus, calculus and linear algebra are applied in the formulation and solution... -- in particular, problems in which the mathematical model has an analytical (equation or inequality) form NCTM Standards for Geometry Use Edit/Find to search for Modeling in this page

Proof as a Convincing Communication that Answers -- Why?

### Relating a Cylinder, Cone and Sphere

1. Classroom Activity Sheet

2. Valley Springs Snow Cream

3. Measuring Volumes

4. Why is the volume of a cylinder or radius r and height 2r equal to p r3?
Area of a Circle

5. Volume of a Cone Via Calculus (surface of revolution integral)

6. Volume of a Sphere (Cavalieri's Principle - If two solids have the same height and the same cross-sectional area at every level, then they have the same volume. )
Volume of a Sphere

### Toothpicks

From Geometric Systems in Architecture
Give each student 3 toothpicks and instruct each student to construct four triangles. Most students will fumble with those toothpicks and build only one triangle in a single plane. How many students will step out of the plane of the desk and build a tetrahedron with the toothpicks? The key to this activity is to step out of the 2-dimensional plane in which we compute our arithmetic and draw our geometrical figures and look at our world in a special way in order to hold the 3 sticks at an apex to form the tetrahedron while considering the table top to contain the fourth triangle.

We can use this simple exercise to make students aware of the many avenues they may explore to solve problems. Children love to manipulate their world, to build and create. We can apply this natural ability and tendency to manipulate to the geometry class by providing exercises that will make sense of the 3-dimensional figures we use in finding volume.

If we secure the apex of the tetrahedron and provide more toothpicks to build a full tetrahedron, (a small bit of modeling clay at each apex will do the trick) students can feel the strength of this collection of triangles. Man has determined that the triangle is the most stable system. Triangular braces are used to strengthen bridges, buildings, shelves, and a host of other constructions. Many students may argue that a square base or cubic system is more secure. An interesting exercise is to give half the students toothpicks and clay with instructions to build a 3-dimensional system using squares (a cubic system) while the remaining students construct a 3-dimensional system with triangles (a tetrahedral system). The system constructed with triangles will be stronger and more stable than the cubic system.

### Introduction to Geometric Models

Physical Models, Analytic Models, and Axiomatic Models of the Sphere

### Strength and Efficiency of Regular Polyhedra

Why are Bubbles Round?

The Divine Ratio

Icosahedron versus Cube

### Geodesic Domes and Buckminster Fuller

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