by Illana H. Weintraub
Unlike Algebra, Geometry is all around us – everywhere! It is physical. It is tangible.
Start talking about geometry to your children early – they won’t know at pre-school ages that it is called geometry when you talk about shapes – the ball is a "sphere", the can is a "cylinder" and the label on the can (take it off and show them) is a "rectangle." Road signs are a perfect way to practice shapes…there are triangles, squares, octagons, and kite-shapes.
Understanding geometry in our environment can grow with the child, from the simple to the complex. From pointing out the ice cubes in the ice tray and shape of the tray itself to the cross-section of an ice cream cone (if the cross-section is parallel to the ground, it forms a circle, if not then an ellipse). A hands-on experiment would be to take the cone and place the circle part on a flat surface with the point straight up. If you cut straight across the cone (parallel to the surface), you have a circle similar, but smaller, to the one on the bottom. But, if you cut the cone at an angle, you have an ellipse. This works well with clay.
Geometry in nature is a whole complex body of study. An interesting form in nature is the inside of a snail’s shell or the inside of the nautilus shell from the ocean – these are actually ever smaller right triangles with each hypotenuse becoming the leg of the next right triangle. Take a look at our logo for an example of this (MathMedia, see ad on p 64). Then, take a look at flowers. The petals of a daisy are radii from the center of the circle. The possibilities in nature are endless. A wonderful visual of these types of examples in nature are in the Disney video called "Donald Duck in Mathamagic Land" – a delightful addition to any home video library for all ages. Every time you watch it, you will discover something new.
TVs in a store are measured on the diagonal – the length or width is found using the Pythagorean Theorem (a2 + b2 = c2, where c is the diagonal of any right triangle). Billiards requires a keen innate sense of geometry with the angles and arcs necessary to predict and cause the balls to end up just the way you want them (also in the Donald Duck movie!). The immensely practical necessity of being able to calculate the area of the floor of room to know how much carpet to buy or the area of the walls of a room to know how much paint to buy – it’s all geometry. Want to buy an aquarium for a fish collection – what’s the volume of the tank? How should we cut the pizza, the pie, the cake – how many sectors and at what angle? How do we find the shortest distance between two points when driving – which route is the most efficient? For years I could not parallel park my car until someone told me to back in at a 45 degree angle – how about that?!? Every moment is a lesson – every place is a classroom!
Putting up a tent (which is a "triangular prism" with a rectangular base) requires poles perpendicular to the ground to hold up the front and back doors. A parallel bar holds up the ceiling of the tent. If a circular water sprinkler is placed in the center of the lawn, a circular area will be watered, but most lawns are rectangular – so, either the edges will not be watered or the sidewalk and building will be watered – this problem has obviously been pondered by the manufacturers of water sprinklers who have designed more efficient sprinklers that go back and forth to water a rectangular lawn.
Any kind of carpentry work involves geometry – it’s all about angles and levels (a tool to measure parallels to the ground) and circles with their radii and diameters. Obviously, an engineer building a bridge had better know geometry as well as an architect who uses geometry and physics to make sure the building will stand the test of time. Look up the "Transamerica" building in San Francisco and talk about the shapes used in that structure! Another industry dependent on geometry is theatre lighting which is all about angles and so important to some actors that they bring their own lighting designers with them. Geometry is also used to measure the width of a lake, to find the height of a tree, a building, or a mountain. In many situations, Algebra is used to find the answers to geometric situations. These concepts are usually covered in a high school geometry course.
Read any periodical and notice the geometric representation summarizing the contents of the written article. A graph is a geometric representation of data. Newspapers will use bar graphs, line graphs, picture graphs . . . lots of analytical discussion there! This being an election year means that we will be inundated with graphs on a daily basis and understanding these graphs allows us to understand the issues.
In conclusion, whether we know it or not, we live our lives through a series of geometric experiences.
(Illana H. Weintraub is the Publisher of MathMedia Educational Software, Inc.www.mathmedia.com)
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