Geometry

Explorations in Geometry

In an algebra class, sequential thinking is the dominant theme, and since in recent years algebra has been given strong emphasis, students begin a geometry course with this frame of mind. Geometry, however, involves more visual thought than algebra. These two types of thinking are not opposites, but complementary. Students normally enter a high school Geometry course underprepared in geometric thinking, and must be carefully nurtured. This nurturing normally takes place best through drawing, which in a Geometry course is called constructions.

Just as sequential and visual thinking are two common modalities, in a broader sense we have inductive vs. deductive thinking. The forte of mathematics is deduction, or logical thought, and in a Geometry course this normally takes the form of traditional two-column proofs, with Statements and Reasons. This is a higher-ordered thinking process that takes quite awhile for students to gain comfort with. I normally use constructions and guide students through inductive thinking towards the goal of deduction. After students gain familiarity with many geometric figures, they realize through experience what “must be true” in general.

In California there are 22 Geometry standards which are broken into four clusters:

• Angle Relationships, Constructions, and Lines
• Volume and Area Formulas
• Logic and Geometric Proofs
• Trigonometry

After exploring lines, triangles, quadrilaterals, other plane figures, leading into area and 3-D figures, students are then ready to handle the typical proof-style found in most Geometry courses. The content finishes up with the Pythagorean Theorem and simple trigonometry.