Copyright

Copyright (c) 2005 by David McAdams. All rights reserved. Unpublished copyright work.

Permission is hereby granted to reproduce this document unchanged for non-commercial educational purposes only, in quantities consistent with such educational use.

About the Author

David McAdams is a candidate for a BA Mathematics Education and a BS in Mathematics at Utah Valley State College in Orem, Utah. He intends to complete his teaching credentials in December 2006.

Notes:

The unit starts with a review of fundamental geometry shapes and takes the students through creating rigorous geometric definitions. The students are taught to evaluate definitions for missing pieces and flaw. The beginning elements of proof are introduced.

This unit includes a writing assignment for each lesson. The purposes of this is to:

1. Provide a gathering activity that gets the students quickly focused on school work;
2. Familiarize the students with writing about mathematics, as opposed to just solving problems; and
3. Provide a mechanism for the teacher to evaluate such intangibles as motivation, and affect.

Established Goals: What is this?

From National Council of Teachers of Mathematics

• Build new mathematical knowledge through problem solving
• Solve problems that arise in mathematics and in other contexts
• Apply and adapt a variety of appropriate strategies to solve problems
• Monitor and reflect on the process of mathematical problem solving
  

• Recognize reasoning and proof as fundamental aspects of mathematics
• Make and investigate mathematical conjectures
• Develop and evaluate mathematical arguments and proofs
• Select and use various types of reasoning and methods of proof

• Organize and consolidate their mathematical thinking through communication
• Communicate their mathematical thinking coherently and clearly to peers, teachers, and others
• Analyze and evaluate the mathematical thinking and strategies of others;
• Use the language of mathematics to express mathematical ideas precisely.

• Recognize and use connections among mathematical ideas
• Understand how mathematical ideas interconnect and build on one another to produce a coherent whole
• Recognize and apply mathematics in contexts outside of mathematics

• Create and use representations to organize, record, and communicate mathematical ideas
• Select, apply, and translate among mathematical representations to solve problems
• Use representations to model and interpret physical, social, and mathematical phenomena

From Utah Educator Network:

Geometry Content

• 3.1.2 - write conditional statements
• 3.1.2 - write the converse statements
• 3.1.2 - write inverse statements
• 3.1.2 - determine the truth value
• 3.1.2 - write counter examples to prove a statement false, e.g. If a polygon is a quadrilateral, it is a square. Rectangle is the counter example.
• 3.1.2 - write good definitions for terms. Include symbolic representation where appropriate
• 3.1.4 - Measure angles and review classifications
• 3.1.4 - Identify angle pairs as adjacent, complementary, supplementary, linear or vertical
• 3.1.5 - Sketch, label and recognize each of the following: parallel, perpendicular and skew lines
• 3.1.6, 3.1.7 - Classify angle pairs formed by two parallel lines cut by a transversal. Prove lines parallel through angle relationships
• 3.1.12 - Classify triangles using properties and discover those properties based on classification
• 3.3.1 - Construct parallel lines and perpendicular bisectors

Algebra Content

• 2.1.2 - Use hands-on, visual, geometric patterns to predict and extend design, drawing conjectures based on inductive reasoning
• 3.1.9 - Identify and construct
  • a median using cut-out triangles and center of gravity
  • an angle bisector
  • an altitude

Process Objective

• Problem solving: draw a diagram, look for patterns, clarify understanding, "Is this true?", "What makes you think so?", check for reasonableness, guess and check, identify counter examples, consider the thinking of others. make a model or simulation, draw a picture or diagram, eliminate possibilities, extend knowledge by considering the strategies of others, propose and critique alternative approaches
• Reasoning and proof: investigate mathematical conjectures, formulate counter examples, realize that observing a pattern does not constitute proof
• Communication: class and group discussions using precise language, journal, express mathematical ideas coherently
• Connections: establish connections among mathematical expressions and physical models, use real-world applications, explore historical and multicultural contributions to math
• Representation: use a variety of visual representations and tools (protractor, compass, straight edge, manipulatives, pictures, graph paper, graphing calculators), represent patterns verbally, numerically, geometrically and algebraically

• Students will acquire rudimentary skills in writing definitions.
• Students will apply geometric reasoning in areas outside of math.
• Students will identify incomplete and invalid proofs.
• Students will apply mathematical reasoning when reading or listening.
• Students will acquire a formal knowledge of rudimentary geometric skills.
• Students will be able to identify some gross errors in proofs.