Unit Plan - Introduction to Geometry

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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.

Stage 1 - Desired Results

### From National Council of Teachers of Mathematics

Problem Solving
Instructional programs from pre-kindergarten 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 pre-kindergarten 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 pre-kindergarten 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 pre-kindergarten 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 pre-kindergarten 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 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
 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:

### 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.