Geometry Quadrilaterals and Other Polygons Students have studied the special quadrilaterals in middle school and should be familiar with their definitions and some of their properties. Special quadrilaterals include parallelograms, rectangles, rhombuses, squares, trapezoids, isosceles trapezoids, and kites. Students would have investigated the properties of these special quadrilaterals using inductive reasoning. • In this chapter, students are able to investigate the properties of special quadrilaterals by using dynamic geometry software. The properties are then proven using a variety of proof formats: transformational, synthetic, analytic, and paragraph. • The first lesson in the chapter is about the angle measures in a polygon. First, students derive a formula for the sum of the interior angles and then a formula for the sum of the exterior angles. • The last four lessons are about the special quadrilaterals. • In Chapter 5, students proved triangles congruent given different hypotheses. The reasoning used in that chapter is applied to proofs involving quadrilaterals in this chapter.

UNIT 7: Quadrilaterals and Other Polygons TIME: 3 weeks UNIT NARRATIVE: This chapter starts with the properties of polygons and narrows to focus on quadrilaterals. We will study several different types of quadrilaterals: parallelograms, rhombi, rectangles, squares, kites and trapezoids. Then, we will prove that different types of quadrilaterals are parallelograms or something more specific. Drawing on their knowledge of triangle congruence from chapter 5 and 6, students now investigate the properties of special quadrilaterals. The triangle congruence criteria are used to prove properties of parallelograms. The family of quadrilaterals grows to include rectangles, rhombuses, kites and trapezoids. Students construct figures by manipulating appropriate geometric tools (compass, ruler, protractor, dynamic software, etc.) and justifying why their written instructions produce the desired figure. Properties of quadrilaterals are proven using a variety of techniques: transformational, synthetic, analytic, and paragraph.

Textbook Correlations: Big Ideas: Chapter 7

Additional Resources

ESSENTIAL QUESTIONS: 1. What is the sum of the measures of the interior angles of a polygon? 2. What are the properties of parallelograms? 3. How can you prove that a quadrilateral is a parallelogram? 4. What are the properties of the diagonals of rectangles, rhombuses, and squares? 5. What are some properties of trapezoids and kites?

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Lesson Tutorials, Game Closet, Student Journal, Skills Review Handbook, Dynamic Classroom, Interactive Lessons, Lesson Planning Tool, Puzzle Time, Practice A and B, Enrichment ACADEMIC VOCABULARY: diagonal, equilateral polygon, equiangular polygon, regular polygon, polygon, convex, interior angles exterior angles, parallelogram, rhombus, rectangle, square, trapezoid, bases, base angles, legs, isosceles trapezoid, mid-segment of a trapezoid, kite,

Geometry CLUSTER HEADING & STANDARDS: Prove Geometric theorems

MATHEMATICAL PRACTICE: All mathematical practice standards are addressed in every unit.

G.CO.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

Prove theorems involving similarity G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Apply geometric concepts in modeling situations G.MG.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). G.MG.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

MP 1 Make sense of problems and persevere when solving them. MP 2 Reason quantitatively and abstractly. MP 3 Construct viable arguments and critique the reasoning of others. MP 4 Model with mathematics. MP 5 Use appropriate tools strategically. MP 6 Attend to precision. MP 7 Look for and make use of structure. MP 8 Look for and express regularity in repeating reasoning

Learning Outcomes: By the end of this chapter, students should be able to: Use the interior angle measures of polygons. Use the exterior angle measures of polygons. Use properties to find side lengths and angles of parallelograms. Use parallelograms in the coordinate plane. Identify and verify parallelograms. Show that a quadrilateral is a parallelogram in the coordinate plane. Use properties of special parallelograms. Use properties of diagonals of special parallelograms. Use coordinate geometry to identify special types of parallelograms. Use properties of trapezoids. Use the Trapezoid Midsegment Theorem to find distances. Use properties of kites and identify quadrilaterals. End of the Unit Assessment: AC Driven DIFFERENTIATION REMEDIATION ACCELERATION ENGLISH LEARNERS SPECIAL EDUCATION Practice with algebraic proofs first to get students to apply what they already know. Tie this concept into the use of Unit 1 and 2 constructions and transformation to allow students to conceptually understand. Allow a graphic organizer to be used throughout unit with applied theorems and properties as they are developmentally discovered.

Allow students to perform multiple step transformations. Real world connections Geogebra Technology Geometry Sketchpad Projects Write rules for finding coordinates of the images. Multiple transversals and angles

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Graphic organizers Highlighting :close” activities SIOP strategies Real-world visuals Group collaboration

Small group instruction One on one peer support Smaller size quantities Practice with manipulatives allowing for rigid motion. Review of the coordinate plane. Use of protractors and rulers to allow students to understand congruency conceptually.

Geometry Similarity This is a short chapter that revisits similarity, a concept first introduced in Chapter 4. • The first lesson introduces what it means for two polygons to be similar: corresponding sides are proportional and corresponding angles are congruent. • The next two lessons present methods for proving two triangles are similar. One of the methods involves only angles (AA), with the other two methods involving only sides (SSS) or sides and the included angle (SAS). Unlike congruency methods, the corresponding sides must be proportional versus congruent. • The last lesson presents several proportionality theorems, mostly connected to triangles. • Properties of similar triangles will be needed in the next chapter when trigonometric ratios are defined.

UNIT 8: Similarity TIME: 3 weeks UNIT NARRATIVE: In this unit, students develop the concept of similarity of triangles using corresponding angles and proportional relationships between corresponding parts. They apply concepts of similarity to solve a variety of problems involving similar triangle ratios and the relationship to scale factor. Students will use the definition of similarity in terms of similarity transformations to decide if triangles are similar. They will use transformations to explain the meaning of similar triangles while also, use the properties of similarity transformations to establish the AA criterion for similarity of two triangles. Throughout this unit, students will apply properties of similar triangles to solve problems and justify conclusions. Students will also apply properties of triangles with proportional parts to solve problems and justify their conclusions. Students will discover and prove triangles are similar using a variety of proofs. Throughout this unit, students will examine and solve real-world problems involving similar triangles. Whenever possible, allow students to use technology and concrete models to explore similar triangles.

Textbook Correlations: Big Ideas: Chapter 8

Additional Resources

ESSENTIAL QUESTIONS: 1. How are similar polygons related? 2. What can you conclude about two triangles when you know that two pairs of corresponding angles are congruent? 3. What are two ways to use corresponding sides of two triangles to determine that the triangles are similar? 4. What proportionality relationships exist in a triangle intersected by an angle bisector or by a line parallel to one of the sides?

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Lesson Tutorials, Game Closet, Student Journal, Skills Review Handbook, Dynamic Classroom, Interactive Lessons, Lesson Planning Tool, Puzzle Time, Practice A and B, Enrichment ACADEMIC VOCABULARY: similar figures, similarity, transformation, corresponding parts, slope, parallel lines, perpendicular lines, corresponding angles, ratio, proportion, proportionality

Geometry CLUSTER HEADING & STANDARDS: Understand similarity in terms of similarity transformations

MATHEMATICAL PRACTICE:

G-SRT 2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. G-SRT 3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

Prove theorems involving similarity G-SRT 4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. G-SRT 5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply geometric concepts in modeling situations G.MG.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). G.MG.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

Use coordinates to prove simple geometric theorems algebraically G.GPE.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). G.GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

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All mathematical practice standards are addressed in every unit. MP 1 Make sense of problems and persevere when solving them. MP 2 Reason quantitatively and abstractly. MP 3 Construct viable arguments and critique the reasoning of others. MP 4 Model with mathematics. MP 5 Use appropriate tools strategically. MP 6 Attend to precision. MP 7 Look for and make use of structure. MP 8 Look for and express regularity in repeating reasoning

Geometry Learning Outcomes: By the end of this unit, students should be able to verify through an experiment the properties of dilations given by a center and a scale factor. They should know that a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. The understanding of the dilation of a line segment is longer or shorter in the ratio based on scale factor. The students should also be able to use the definition of similarity in terms of similarity transformations. Students also use the properties of similarity transformations to establish the AA criterion. By the end of this unit, students will prove theorems about triangles, including the Pythagorean Theorem. Students should also be able to use congruence and similarity criteria for triangles to solve problems and prove relationships regarding geometric figures.

End of the Unit Assessment: AC Driven

REMEDIATION Practice with algebraic cross product including proportionality Tie this concept into the use of Unit 1, 2 and 3 to allow students to conceptually understand. Allow a graphic organizer to be used throughout unit with applied theorems and properties as they are developmentally discovered. Have students verbally explain the process prior to writing.

DIFFERENTIATION ACCELERATION ENGLISH LEARNERS Multiple transversals and angles ELD Literacy Standards Longer proof development Graphic organizers Use of converses and bi-conditions Highlighting : “cloze” activities Use of 2 unknowns within the SIOP strategies problem and prove for one and apply Real-world visuals it to prove the second. Group collaboration Number talks

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SPECIAL EDUCATION Small group/ 1x1 instruction Smaller size quantities Number Talks Practice writing and using the properties while using one step process.

Geometry Right Triangles and Trigonometry This is a fairly long chapter that introduces students to right triangle trigonometry. Students will encounter a more in-depth study of trigonometry in Algebra 2. • The first lesson on the Pythagorean Theorem will not be completely new to students who will have familiarity with this theorem from middle school. • The next two lessons use knowledge of similar triangles to investigate relationships in special right triangles (30°-60°-90° and 45°-45°-90°) as well as similar triangles that are formed when the altitude to the hypotenuse is drawn in a right triangle. Being familiar with these relationships and solving for segment lengths in triangles will be helpful in subsequent lessons. • The next three lessons present the tangent, sine, and cosine ratios. The focus of these lessons is to solve for parts of a right triangle. Many real-life applications are presented. • The last lesson of the chapter introduces the Law of Sines and the Law of Cosines so that non-right triangles can be solved.

UNIT 9: Right Triangles and Trigonometry TIME: 4 weeks UNIT NARRATIVE: In this unit students explore the relationships that exist among and between sides and angles of right triangles. Students build upon their previous knowledge of similar triangles and of the Pythagorean theorem to determine the side length ratios in special right triangles and to understand the conceptual basis for the functional ratios sine, cosine, and tangent. They explore how the values of these trigonometric functions relate in complementary angles and how to use these trigonometric ratios to solve problems. Through the work in this unit, students not only develop the skills and understanding needed for the study of many technical areas but also build a strong foundation for future study of trigonometric functions of real numbers The definitions of sine, cosine, and tangent for acute angles are founded on right triangles and similarity, and, with the Pythagorean Theorem, are fundamental in many realworld and theoretical situations. The Pythagorean Theorem is generalized to non-right triangles by the Law of Cosines. Together, the Laws of Sines and Cosines embody the triangle congruence criteria for the cases where three pieces of information suffice to completely solve a triangle. Furthermore, these laws yield two possible solutions in the ambiguous case, illustrating that Side-Side-Angle is not a congruence criterion.

Textbook Correlations: Big Ideas: Chapter 9

Additional Resources

ESSENTIAL QUESTIONS: 1. How can you prove the Pythagorean Theorem? 2. What is a Pythagorean triple? 3. What is the relationship among the side lengths of 45°- 45°- 90° triangles? 30°60°- 90° triangles? 4. How are altitudes and geometric means of right triangles related? 5. How is a right triangle used to find the tangent of an acute angle? Is there a unique right triangle that must be used? Draft

Lesson Tutorials, Game Closet, Student Journal, Skills Review Handbook, Dynamic Classroom, Interactive Lessons, Lesson Planning Tool, Puzzle Time, Practice A and B, Enrichment ACADEMIC VOCABULARY: Product Property of Square Roots, Cross Products Property, Pythagorean triple, right triangle, legs of a right triangle, hypotenuse, isosceles triangle, geometric mean, altitude of a triangle, similar figures, trigonometric ratio, tangent, angle of elevation, sine, cosine, angle of depression, inverse tangent, inverse sine, inverse cosine, solve a right triangle, Law of Sines, Law of Cosines

Geometry 6. How is a right triangle used to find the sine and cosine of an acute angle? Is there a unique right triangle that must be used? 7. When you know the lengths of the sides of a right triangle, how can you find the measures of the two acute angles? 8. What are the Law of Sines and the Law of Cosines?

CLUSTER HEADING & STANDARDS: Prove theorems involving similarity

MATHEMATICAL PRACTICE: All mathematical practice standards are addressed in every unit.

G-SRT 4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. G-SRT 5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Define trigonometric ratios and solve problems involving right triangles. G.SRT.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. G.SRT.7 Explain and use the relationship between the sine and cosine of complementary angles. G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.* Apply geometric concepts in modeling situations G.MG.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). G.MG.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

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MP.1. Make sense of problems and persevere in solving them. MP.2. Reason abstractly and quantitatively. MP.3. Construct a viable argument and critique the reasoning of others. MP.4. Model with mathematics. MP.5. Use appropriate tools strategically. MP.6. Attend to precision. MP.7. Look for and make use of structure. MP.8. Look for and express regularity.

Geometry California Standards: G.SRT.8.1 Students know the definitions of the basic trigonometric functions defined by the angles of a right triangle. They also know and are able to use elementary relationships between them. For example, tan(x) = sin(x)/cos(x), (sin(x)) 2 + (cos(x)) 2 = 1.

Learning Outcomes: By the end of this unit, students should be able to: The relationships among the lengths of the legs and the hypotenuse are the same for all right triangles with acute angles of 30 degrees and 60 degrees and can be derived by halving an equilateral triangle using an altitude. The relationships among the lengths of the legs and hypotenuse are the same for all right triangles with two acute angles of 45 degree angles and can be derived by halving a square along the diagonal. Similar right triangles produce trigonometric ratios. Trigonometric ratios are dependent only on angle measure. Trigonometric ratios can be used to solve application problems involving right triangles. Define trigonometric ratios and solve problems involving right triangles. Apply trigonometry to general triangles. Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. Recognize when the Law of Sines or Law of Cosines can be applied to a problem and solve problems in context using them. Prove Law of Sine and Cosine. Apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).

End of the Unit Assessment: AC Driven

REMEDIATION Revisit solving multiple step equations to find a given unknown. Revisit Algebra Common Core Standards involving ratios, Pythagorean theorem, expressions, and equations. Visual graphic organizers with labeled characteristics of triangles and angles. Small group instruction Partner activities for collaboration. Real-world situations

DIFFERENTIATION ACCELERATION Multiple transversals and angles Longer proof development Use of converses and bi-conditions Use of 2 unknowns within the problem and prove for one and apply it to prove the second. Real-world applications Technology and the use of Geogebra Presentations Interactive Projects

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ENGLISH LEARNERS

ELD Literacy Standards Graphic organizers Highlighting : “cloze” activities SIOP strategies Real-world visuals Group collaboration Number talks

SPECIAL EDUCATION Small group/ 1x1 instruction Smaller size quantities Number Talks Practice writing and using the properties while using one step process. Revisit solving multiple step equations to find a given unknown. Revisit Algebra Common Core Standards involving ratios, Pythagorean theorem, expressions, and equations. Visual graphic organizers with labeled characteristics of triangles and angles.