prove relationships in geometric figures. using the AA Similarity Postulate and the SSS and SAS similarity Theorems Use similar triangles to solve problems Use the following resources to deepen students' conceptual understanding of mathematical content and develop their ability to apply that knowledge to problems. Domain: Cluster: Prove theorems involving similarity 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. Domain: Cluster: Prove theorems involving similarity Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Enduring Understanding(s) Polygons are similar if and only if corresponding angles are congruent and corresponding sides are proportional. Essential Question(s) How do you use proportions to find side lengths in similar polygons? Objective(s): Use proportional parts within triangles Use proportional parts with parallel lines Use the following lesson(s) first to introduce concepts/build conceptual understanding. Use the textbook resources to address procedural skill and fluency. Lesson 7.4 Parallel Lines and Proportional Parts (midsegments was previously covered in unit 2) pp. 484-492 Use the following resources to deepen students' conceptual understanding of mathematical content and develop their ability to apply that knowledge to problems. Task(s) Vocabulary Mid-segment of a triangle Activity with Discussion Use multiple representations to explore angle bisectors and proportions. See p. 492, #47
(Allow approximately 1.5 weeks for instruction, review, and assessment)t) Domain: Cluster: Define trigonometric ratios and solve problems involving right triangles 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. Enduring Understanding(s) Trigonometry can be used to measure sides and angles indirectly in right triangles. Essential Question(s) How do you find a side length or angle measure in a right triangle? Objective(s): Identify and apply side ratios in right triangles. Identify and apply side ratios in right triangles Use the following lesson(s) first to introduce concepts/build conceptual understanding. engageny Geometry Module 2, Topic D, Lesson 24 - Prove the Pythagorean Theorem Using Similarity Use the textbook resources to address procedural skill and fluency. Lesson 8.3 Special Right Triangles pp.552-559 Use the following resources to deepen students' conceptual understanding of mathematical content and develop their ability to apply that knowledge to problems. Task(s) Activity with Discussion p.559 #50 Explain how you can find the lengths of two legs of a 30-60-90 triangle in radical form if you are given the length of the hypotenuse. Domain: Cluster: Define trigonometric ratios and solve problems involving right triangles Explain and use the relationship between the sine and cosine of complementary angles. Domain: Cluster: Define trigonometric ratios and solve Enduring Understanding(s) Trigonometry can be used to measure sides and angles indirectly in right triangles. Essential Question(s) How do you find a side length or angle measure in a right triangle? Use the following lesson(s) first to introduce concepts/build conceptual understanding. engageny Geometry Module 2, Topic E Lesson 25: Incredibly Useful Ratios Lesson 26: The Definition of Sine, Cosine, and Tangent Vocabulary Trigonometry, trigonometry ratio, sine, cosine, tangent, inverse sine, inverse cosine, inverse tangent Activity with Discussion p.570 #65
to solve problems Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. Task(s) Domain: Cluster: Understand and apply theorems about circles Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Enduring Understanding(s) Relationships between angles, radii and chords will be investigated. Similarities will be applied to derive an arc length and a sector area. Essential Question(s) When lines intersect a circle, or within a circle, how do you find the measures of resulting angles, arcs, and segments? Objective(s): Identify central angles, major arcs, minor arcs, and semicircles and find their measures. Use the following lesson(s) first to introduce concepts/build conceptual understanding. . Use the textbook resources to address procedural skill and fluency. Lesson 10.2 Measuring Angles and Arcs pp.692-700 Use the following resources to deepen students' conceptual understanding of mathematical content and develop their ability to apply that knowledge to problems. Task(s) Circles and their Relationships among Central Angles, Arcs and Chords (p. 15) (pp. 46 & 52) Vocabulary Central angle, arc, minor arc, major arc, semicircle, congruent arcs, adjacent arcs, arc length Writing in Math/Discussion p.699 #62 Describe the three different types of arcs in a circle and the method for finding the measure of each one. Domain: Cluster: Understand and apply theorems about circles Enduring Understanding(s) Relationships between angles, radii and chords will be investigated. Use the following lesson(s) first to introduce concepts/build conceptual understanding. Vocabulary Inscribed angle, intercepted arc