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Using Logical Reasoning to Prove Conjectures about Circles

Given conjectures about circles, the student will use deductive reasoning and counterexamples to prove or disprove the conjectures.

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Generalizing Geometric Properties of Ratios in Similar Figures

Students will investigate patterns to make conjectures about geometric relationships and apply the definition of similarity, in terms of a dilation, to identify similar figures and their proportional sides and congruent corresponding angles.

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Determining Area: Sectors of Circles

Students will use proportional reasoning to develop formulas to determine the area of sectors of circles. Students will then solve problems involving the area of sectors of circles.

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Making Conjectures About Circles and Segments

Given examples of circles and the lines that intersect them, the student will use explorations and concrete models to formulate and test conjectures about the properties and relationships among the resulting segments.

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Determining Area: Regular Polygons and Circles

The student will apply the formula for the area of regular polygons to solve problems.

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Making Conjectures About Circles and Angles

Given examples of circles and the lines that intersect them, the student will use explorations and concrete models to formulate and test conjectures about the properties of and relationships among the resulting angles.

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Solving Problems With Similar Figures

Given problem situations involving similar figures, the student will use ratios to solve the problems.

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6.08 Bonus Video: Law of Sines—The Ambiguous Case

The Law of Sines can be used to solve for sides and angles of oblique triangles. However, in some cases more than one triangle may satisfy the given conditions. We refer to this as an ambiguous case.

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Making and Verifying Conjectures about Three-Dimensional Figures

Students will explore volume conjectures and solve problems by applying the volume formulas to composite figures.

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Constructing and Justifying Statements about Geometric Figures

Students will distinguish between undefined terms, definitions, postulates, conjectures, and theorems and investigate patterns to make conjectures about geometric relationships.

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Using Counter Examples to Disprove Statements That Are False

Given statements about a geometric relationship, the student will use counter examples to disprove statements that are false.

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Using Inductive Reasoning to Formulate Conjectures

Students will practice identifying the converse, inverse, and contrapositive of conditional statements.

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Using Logical Reasoning to Prove Statements are True

Given statements about a geometric relationship, the student will distinguish between the undefined terms, definitions, postulates, conjectures, and theorems to prove the statements are true.

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Using Properties of Transformations

Given examples of mathematics in the real world, the student will use properties of transformations and their composites to describe and perform transformations of figures in a plane.

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Developing Algebraic Expressions to Represent Geometric Properties

The student will investigate patterns to make conjectures.

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Developing Algebraic Expressions to Represent Geometric Properties of Polygons

Given numerical and/or geometric patterns that represent geometric properties of polygons, the student will develop algebraic expressions that represent the geometric properties.

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Developing Algebraic Expressions to Represent Geometric Properties of Angle Relationships in Polygons

Given numerical and/or geometric patterns that represent geometric properties of angle relationships in polygons, the student will investigate patterns to make conjectures about interior and exterior angles of polygons.

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Connecting Postulates, Definitions, and Theorems

The student will distinguish the difference between undefined terms, definitions, postulates, conjectures, and theorems.

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Determining the Validity of Conditional Statements

Given a conditional statement, the student will determine its validity and the validity of the converse, inverse and contrapositive.

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Making and Verifying Conjectures about Angles

Given the relationship(s) among a set of angles, the student investigates the patterns and makes conjectures about the geometric relationships, including angles formed by parallel lines cut by a transversal.