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Lesson Title: G.CO.D.13 Lesson Inscribing Figures Date: _____________ Teacher(s): ____________________ Course: CC Geometry Unit 3 Start/end times: _________________________ Lesson Standards/Objective(s): What mathematical skill(s) and understanding(s) will be developed? Which Mathematical Practices do you expect students to engage in during the lesson? G.CO.D.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. MP1: MP3: MP5: MP7: Make sense of problems and persevere in solving them. Construct viable arguments and critique the reasoning of others. Use appropriate tools strategically. Look for and make use of structure. Lesson Launch Notes: Exactly how will you use the first five minutes of the lesson? Give students paper plates, and instruct them to draw and label all of the parts of a circle that they know. Have them form small groups and get as many parts as they can. Place a paper plate under the document camera. Call on groups one at a time to send a representative to draw and label on of the parts that they had. Keep going from group to group until groups are out of ideas. Be sure that radii, chords, diameters, inscribed angles, and central angles have been added. As a class, discuss the properties of each part. Lesson Closure Notes: Exactly what summary activity, questions, and discussion will close the lesson and connect big ideas? List the questions. Provide a foreshadowing of tomorrow. Have students complete the 3-2-1 Closure. Lesson Tasks, Problems, and Activities (attach resource sheets): What specific activities, investigations, problems, questions, or tasks will students be working on during the lesson? Be sure to indicate strategic connections to appropriate mathematical practices. 1. Assign students to pairs and give each pair multiple paper plates. Have students try to construct the following shapes, inscribed in the circle formed by the plate, by folding: equilateral triangle, square, regular hexagon. As you circulate the room, be sure to ask students how they know their shapes are truly regular. 2. As a class, compare the constructions and methods used. Have pairs share which shape they thought was the most difficult to create and why. In addition, students should justify how they know each of their shapes is regular. 3. Assign each pair one of the shapes: equilateral triangle, square, and hexagon. Pairs will explore the properties of the shape they have been assigned and their relationships to the parts of a circle. Students will try to justify properties of these shapes based on the properties of circles. Equilateral Triangle Pairs – Have students explore the angles in an equilateral triangle and their relationship to the circle. They should recognize that each angle in their equilateral triangle is 60 degrees and is inscribed in the circle. The triangle divides the circle into three different arcs of equal measurement, which would be 120 degrees. This is a justification that inscribed angles are equal to one half the measurement of the arc they intercept. Square Pairs – Have students explore the diagonals of the square and their relationship to the circle. Students may easily recognize that the diagonals of the square are the diameters of the circle and must be congruent. Encourage them to also explore the angles created by the diagonals. Since the diagonals form central angles that divide the circle into four equivalent arcs, this can be used to justify that they intersect at a 90-degree angle. Conversely, if students already know that the diagonals are perpendicular, they could justify that the central angles are equivalent to the measurements of the arcs they intercept. Hexagon Pairs – Have students explore the triangles formed by the diagonals of the hexagon. Students should see that the diagonals are congruent, since they are diameters. In addition, they may be able to HCPSS Secondary Mathematics Office (v2.1); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student achievement. Portsmouth, NH: Heinemann. Lesson Title: G.CO.D.13 Lesson Inscribing Figures Course: CC Geometry Unit 3 Date: _____________ Teacher(s): ____________________ Start/end times: _________________________ explore the triangles formed based on the fact that the radii are all equivalent, or the relationship between the central angles formed by the diagonals and the arcs they intercept. 4. Once pairs have explored and discussed the relationships between the properties of their shape and the circle, regroup the class by shape and allow the larger groups to consolidate their thoughts. Encourage the larger groups to verify everything using multiple methods. Once groups have become experts in their figures, have each group present their findings to the rest of the class. Evidence of Success: What exactly do I expect students to be able to do by the end of the lesson, and how will I measure student success? That is, deliberate consideration of what performances will convince you (and any outside observer) that your students have developed a deepened and conceptual understanding. Students should be able to relate the properties of inscribed polygons to the parts of a circle, using the appropriate terminology and definitions. Notes and Nuances: Vocabulary, connections, anticipated misconceptions (and how they will be addressed), etc. Students will be exploring relationships using circle vocabulary such as radius, diameter, chord, inscribed angle, and central angle. They will make connections with properties of polygons, which may have already been explored in either Unit 2 or in middle school. Resources: What materials or resources are essential for students to successfully complete the lesson tasks or activities? Homework: Exactly what follow-up homework tasks, problems, and/or exercises will be assigned upon the completion of the lesson? Paper Plates Document Camera Have students complete various central and inscribed angle problems. Lesson Reflections: How do you know that you were effective? What questions, connected to the lesson standards/objectives and evidence of success, will you use to reflect on the effectiveness of this lesson? Can students accurately name and describe relationships between the parts of a triangle? Can students use the parts of circles to articulate justifications for properties of regular polygons inscribed within the circle? Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. HCPSS Secondary Mathematics Office (v2.1); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student achievement. Portsmouth, NH: Heinemann.