Unit Plan  Introduction to Geometry 
Copyright
Copyright (c) 2005 by David McAdams. All rights reserved. Unpublished copyright work.
Permission is hereby granted to reproduce this document unchanged for noncommercial
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. Contact the
author at DEMcAdams@usa.net.

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

Stage 1  Desired Results 
From National Council of Teachers of Mathematics
Problem Solving
Instructional programs from prekindergarten through grade 12
should enable all students to:
 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
Reasoning and Proof
Instructional programs from prekindergarten through grade 12 should enable all students to:
 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
Communication:
Instructional programs from prekindergarten through grade 12 should enable
all students to:
 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.
Connections:
Instructional programs from prekindergarten through grade 12 should enable
all students to:
 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
Representation
Instructional programs from prekindergarten through grade 12 should enable
all students to:
 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
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 handson, visual, geometric patterns to predict and extend design, drawing conjectures based on inductive reasoning
 3.1.9  Identify and construct
 a median using cutout 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 realworld 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

Understandings: What is this?
 The world around us is full of geometry.
 The principles of geometry can be applied to everyday life.
 Mathematical definitions can be different from the definitions we know from daily life.
 Mathematical proofs allow us to understand geometry better.
 Basic geometric shapes can be combined to make more complex shapes.

Essential Questions: What is this?
 "How do I write a good definition?"
 "What are the components of a valid proof?"
 "Why can't I depend on a diagram to show proof?"
 "How do I develop a valid proof?"


Students will know/be able to: What is this?
 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.

Stage 2  Assessment Evidence 
Performance Tasks:
 Students will write a clear definition of basic geometric constructs.
 Students will demonstrate the ability to uncover an incomplete or obviously wrong proof.
 Students will identify assumptions of a geometric statement.
 Students will construct basic geometric shapes using a straight edge and a compass.
 Students will identify the properties of various geometric constructs.

Other Evidence
 Students will correctly identify valid and invalid proofs by crossing out invalid proofs.
 Students will apply geometric principles in every day life.


Stage 3  Learning Plan 
Learning Activities:
Activ Type 
Lesson Model 
Description 
Equip Revise 
Direct Instruction Model 
Points, Lines and Their Properties
This lesson is intended to introduce mathematical definitions,
properties, conjectures and theorems. As students begin to understand
these basic concepts, the teacher will expand the discussion to axioms and
definitions and introduce the concepts of conjectures and mathematical proof.
Students will begin to write definitions. The rubric for definitions
will be introduced. View lesson plan: html 
Revise Formulate 
Concept Attainment Model 
Shapes.
This lesson will introduce students to concepts of geometric shapes
and their attributes, such as convex/concave, and polygon. In the
process the students will learn how to critique their own and others’
definitions. Students will be able to identify the properties of a given shape. 
Explore 
Suchman Inquiry Model 
Triangles and their properties.
This lesson will allow the students to develop their ability to
formulate and communicate meaningful conjectures. Students will
form teams to explore various properties of triangles. 
Explore Evaluate 
WebQuest Model 
Introduction to proofs.
This lesson will give the students experience looking for mathematical
information on the Internet. The students will take the conjectures
formulated during the previous lesson and attempt to find and understand
proofs for these conjectures. The class will compare various proofs and
formulate an opinion as to what constitutes a valid and meaningful proof. 
Evaluate 
Think, Pair, Share 
Formulating a proof.
After a review of proofs formulated in the previous lesson, the
students to formulate a proof on their own. Students
will then pair up to examine each student's proof and evaluated it
for assumptions, completeness, logic, and clarity. 
Equip 
Vocabulary Acquisition Model 
Vocabulary of Geometry.
During this lesson will formalize their knowledge of the specific mathematical meanings of
various terms associated with geometry. 
Equip Explore 
Concept Development Model 
Definition, Conjecture, and Proof.
This lesson will embed the base concepts of mathematical definition,
conjecture, and proof in the student’s minds. The students will
be assigned a proof of which to write a final version. 
