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LMU|CMAST

Geometry Unit Descriptions

Geometry Unit Descriptions

Mathematical reasoning skills developed in previous courses provide a solid foundation for understanding proof and logic in Geometry. A study of analytical geometry is weaved throughout this course, highlighted by a study of transformations in the plane. The concept of area is also weaved throughout, with a discussion of the areas of many different geometric objects. Previous knowledge of angles and angle pairs is used to prove important results about the congruence of angles. A construction of the angle bisector is explored. Triangles, quadrilaterals, and polygons are classified and important results are proven. The triangle congruence theorems are explored in depth and used to prove two triangles congruent. The concepts of ratio and proportion are applied in the study of similar polygons. This leads to a discussion of special right triangles, and the Pythagorean theorem. Different proofs of the Pythagorean theorem are explored. The concept of a ratio is necessary for the study of the trigonometric ratios and their relationship to right triangles. Connections among shapes and their properties are made in the study of circles, including an exploration of how polygons, lines, and circles interact. Finally, the concept of shape is expanded to three dimensions, with special focus on the study of prisms, pyramids, and cylinders. The concept of function is revisited in the study of the formulas for volume and surface area. Unit Organization: Unit 1: Logic and Reasoning Unit 2: Angle and Angle Relationships Unit 3: Triangles and Triangle Congruence Unit 4: Quadrilaterals Unit 5: Polygons Unit 6: Similar Polygons Unit 7: Right Triangle Trigonometry Unit 8: Circles Unit 9: Solid Geometry Unit 10: CST Review Textbook: Geometry: Applying, Reasoning, Measuring (2004). McDougal Littell. Houghton Mifflin College Division. Ron

Larson, Laurie Boswell, Lee Stiff.

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LMU|CMAST Unit 1: Logic and Reasoning

Geometry Unit Descriptions

This unit lays the foundation for proving geometric relationships by developing the effective thinking needed to construct mathematically accurate logical arguments and proofs. Inductive reasoning is introduced as the type of thinking needed to analyze patterns and create conjectures. Counterexamples are used to prove some conjectures false. Different forms of logical statements are explored, including the converse, inverse, contrapositive, and biconditional statements. Counterexamples are used again to show that a logical statement is false. Logical statements are then used in defining basic geometric objects (i.e. points, lines, and planes), postulates (i.e. addition, etc), and properties (i.e. reflexive, symmetric, transitive). The understanding of segments is explored with constructions (i.e. copying of segments) and deductive reasoning is used to prove basic geometric statements, including those about segments. Two column proofs are introduced and a connection is made between this type of proof, and written justification of steps used in an algebra problem. Multiple types of proofs are used to prove statements about segment congruence, including two-column proofs and paragraph proofs. As an introduction to coordinate geometry, the distance formula and the midpoint are derived and organized as proofs. The definition of equidistant is given and illustrated in the coordinate plane using the distance formula. # 1A 1B 1C 1D 1E 1F 1G Learning Targets Write a conjecture from a given set of information and explain how this is inductive reasoning. Show that a statement is false by providing a counterexample. Explain what it means for a statement to be in "if-then" form, and explain the relationship between the hypothesis and conclusion. Write a negation, converse, inverse, contrapositive, and biconditional statement in words and in symbolic notation. Judge the validity of conditional statements, including the biconditional. Explain the role of proof in mathematics. Use deductive reasoning and the laws of logic to determine whether a Standard 1.0 3.0 3.0 3.0 3.0 2.0 3.0 Textbook

Pg. 6-9 # 12-33, 42-51, 68-71 Pg. 8 # 34-41; Pg. 75 #1417; Pg. 83 #24-27 Pg. 75-76 #9-13, 22-28 Pg. 76-77 # 18-24, 44-52; Pg. 91-92 #8-20 Pg. 75-76 # 14-17, 29-39; Pg. 82-83 # 13-23, 28-43 Pg. 92 #23-35

Active Practice Gallery Walk Matching

Cube T/F Strips

Puzzle Pieces 2

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LMU|CMAST

Geometry Unit Descriptions

1H 1I 1J

statement is valid. Explain the difference between inductive and deductive reasoning. State the definition of a point, line, and plane. Use a compass and protractor to construct a copy of a segment.

1.0 1.0 16.0

1K 1L

Describe the properties of segment congruence.

4.0 3.0 3.0 4.0 17.0 17.0

Pg. 92 # 21-22 Pg. 13-15 # 9-51, 6167 Pg. 38 #14-16; Pg. 105 #12-15 Pg. 105-106 #6-11, 16-18 Pg. 105 #6-11 Pg. 100 # 16-23; Pg. 101 #32

Construction

State the reflexive, symmetric, and transitive properties and use them to judge the validity of a logical statement. 1M State the addition, subtraction, multiplication, division and substitution properties, and use them to judge the validity of a logical statement. 1N Develop two column proofs or paragraph proofs involving segment congruence. 1O Use the distance formula and midpoint formula in the coordinate plane. 1P Explain the concept of equidistant, and provide examples in the coordinate plane.

Mini White Boards Find the Mistake Find the Mistake Puzzle Pieces / Gallery Walk Post-It Notes Envelop Activity

Pg. 105 #6-11

Pg. 38-39 #4-9, 17-30 Pg. 38-39 #4-9, 17-30

Essential Standards (CA): 1.0 Students demonstrate understanding by identifying and giving examples of undefined terms, axioms, theorems, and inductive and deductive reasoning. 2.0 Students write geometric proofs, including proofs by contradiction. 3.0 Students construct and judge the validity of a logical argument and give counterexamples to disprove a statement. 4.0 Students prove basic theorems involving congruence and similarity. 17.0 Students prove theorems by using coordinate geometry, including the midpoints of a line segment, the distance formula, and various forms of equations of lines and circles. Supporting Standards (CA): 16.0 Students perform basic constructions with a straightedge and compass, such as angle bisectors, perpendicular bisectors, and the line parallel to a given line through a point off the line.

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