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Geometry 2016-17 ~ Unit 6 Title Properties of Circles CISD Safety Net Standards: G10B, G.12E Suggested Time Frame 4th Six Weeks Suggested Duration: 14 days Big Ideas/Enduring Understandings Guiding Questions Module 14 • Angles and segments can be used to solve real world problems. Module 14 • How can you determine the measures of central angles and inscribed angles of a circle? • How can you conclude about the angles of a quadrilateral inscribed in a circle? • What are they key theorems about tangents to a circle? • What are the relationships between the segments in circles? • What are the relationships between angles formed by lines that intersect a circle? Module 15 • Arc length and sector area of a circle can be used to solve real world problems. Module 15 • How can you justify and use the formulas for the circumference and area of a circle? • How do you find the length of an arc? • How do you find the area of a sector of a circle? • How can you write the equation of a circle if you know its radius and the coordinates of its center? Vertical Alignment Expectations TEA Vertical Alignment Chart Grades 5-8, Geometry Geometry ~ Unit 6 Updated November 30, 2016 Page 1 of 15 Geometry 2016-17 ~ Unit 6 Sample Assessment Question Coming Soon.... The resources included here provide teaching examples and/or meaningful learning experiences to address the District Curriculum. In order to address the TEKS to the proper depth and complexity, teachers are encouraged to use resources to the degree that they are congruent with the TEKS and research-based best practices. Teaching using only the suggested resources does not guarantee student mastery of all standards. Teachers must use professional judgment to select among these and/or other resources to teach the district curriculum. Some resources are protected by copyright. A username and password is required to view the copyrighted material. A portion of the District Specificity and Examples are a product of the Austin Area Math Supervisors TEKS Clarifying Documents available on the Region XI Math website. Ongoing TEKS Math Processing Skills G.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to: ● (A) apply mathematics to problems arising in everyday life, society, and the workplace; (B) use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution; Focus is on application (C) select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems; Students should assess which tool to apply rather than trying only one or all Geometry ~ Unit 6 Updated November 30, 2016 ● Page 2 of 15 Geometry 2016-17 ~ Unit 6 (D) communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate; (E) create and use representations to organize, record, and communicate mathematical ideas; ● (F) analyze mathematical relationships to connect and communicate mathematical ideas; and ● Students are expected to form conjectures based on patterns or sets of examples and non-examples (G) display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication ● Precise mathematical language is expected. Knowledge and Skills with Student Expectations ● Students should evaluate the effectiveness of representations to ensure they are communicating mathematical ideas clearly Students are expected to use appropriate mathematical vocabulary and phrasing when communicating ideas District Specificity/ Examples G.5 Logical Argument and Constructions. The student uses constructions to validate conjectures about geometric figures. G.5(A) The student is expected to investigate patterns to make conjectures about geometric relationships, Geometry ~ Unit 6 Updated November 30, 2016 G.5A Teachers should: ● Show the relationships of parallel lines cut by a transversal patterns it creates Vocabulary ● acute triangle ● adjacent triangles ● alternate exterior angles ● alternate interior angles ● altitude ● angle ● angle bisector ● central angle Suggested Resources Resources listed and categorized to indicate suggested uses. Any additional resources must be aligned with the TEKS. Book Resources: HMH Geometry Unit 6 A&M Consolidated Spring 04 Units 4-6 Redbook Geometry Section 1 Web Resources: Page 3 of 15 Geometry 2016-17 ~ Unit 6 including angles formed by parallel lines cut by a transversal, criteria required for triangle congruence, special segments of triangles, diagonals of quadrilaterals, interior and exterior angles of polygons, and special segments and angles of circles choosing from a variety of tools. ● How patterns lead to theorems Example: Pattern in interior angle sums ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Students should: ● ● Investigate patterns Discover theorems based on investigation type activities Misconceptions: Angle relationships exist with any two lines at transversal. Students must understand that lines cut by a transversal must be parallel of the angles relationships to exist. Examples: Geometry ~ Unit 6 Updated November 30, 2016 ● ● ● ● ● ● ● ● ● ● chord circle congruent corresponding angles diagonal exterior angle interior angle linear pair major arc median midsegment of a triangle minor arc obtuse triangle parallel lines perpendicular bisector polygon quadrilateral right triangle same side interior angles secant segment supplementary tangent transversal triangle Circle Theorems Formulas for Angles in Circles Page 4 of 15 Geometry 2016-17 ~ Unit 6 Angles 8 and 4 are Alternate exterior angles that are NOT congruent. VS. ● triangle congruence (SSS, SAS, AAS, HL) ● vertex ● vertical angles Angles 1 and 2 are Alternate exterior angles that ARE congruent BECAUSE the two lines are parallel. G.10 Two-dimensional and three-dimensional figures. The student uses the process skills to recognize characteristics and dimensional changes of two- and threedimensional figures. Geometry ~ Unit 6 Updated November 30, 2016 G.10B ● ● ● ● ● ● ● ● area dimensions non-proportional perimeter proportional scale factor surface area volume Page 5 of 15 Geometry 2016-17 ~ Unit 6 *CISD Safety Net* G.10(B) The student is expected to determine and describe how changes in the linear dimensions of a shape affect its perimeter, area, surface area, or volume, including proportional and non-proportional dimensional change. Big Idea: Changing a dimension affects perimeter, area (or surface area), and volume differently. ● Compare and contrast the perimeter, area, surface area, and volume of the new figure with both proportional(all dimensions) AND non-proportional (only some) dimension change. Circles - Circumference and Area Teachers should: ● Use a wide variety of examples that ask students to find the changes in perimeter, area, surface area, and volume. ● Use multiple visuals or manipulatives (i.e. Play-Doh, clay, snap cubes, geoboards, etc.) to talk about how perimeter, area, surface area, and volume change both proportionally and non-proportionally. ● Show how students could use this skill to their benefit when problem-solving (i.e. solving an easier related problem). ● Use maps, blueprints, or other real-world examples to show square-footage or other ways to think about the standard. ● Know that this is typically a low-scoring objective for students and will require more work to help them achieve. ● Connect concrete to pictorial to abstract. Students should: ● Explore through different types of hands-on activities to best understand this difficult concept. ● Use string (or another tool) to cut down a Play-Doh or clay model to understand dimensional change. ● Use a ruler (or another measurement tool) to measure dimensions and calculate perimeter, area, surface area, and volume of the new figure. ● Compare and contrast the perimeter, area, surface area, and volume of the new figure with other examples. Geometry ~ Unit 6 Updated November 30, 2016 Page 6 of 15 Geometry 2016-17 ~ Unit 6 ● Investigate to find the pattern in proportional dimensional change ( x, x^2, x^3) Misconceptions: G.12 Circles. The student ● Students think that a change in the linear dimension affects perimeter, area, surface area, and volume the same. ● Students can get confused with the non-proportional dimensional change thinking that it is proportional and thus getting the incorrect answer. uses the process skills to understand geometric relationships and apply theorems and equations about circles. G.12(A) The student is expected to apply theorems about circles, including relationships among angles, radii, chords, tangents, and secants, to solve noncontextual problems. G.12(B) The students is expected to apply the proportional relationship between the measure of an Geometry ~ Unit 6 Updated November 30, 2016 G.12A G.12B Big Idea Arc length is a piece of the circumference and the degree of rotation is a piece of the total of 360 degrees. ● angle ● arc ● arc length ● area ● center of a circle ● central angle ● chord ● circle ● circumference ● diameter ● inscribed angle ● intercepted arc ● pi ● radian measure ● radius ● ratio ● secant ● sector ● tangent Special Segments in Circles Notes Rules for Chords, Secants, and Tangents in Circles Segment Length in Circles Arc Length and Radian Measure Khan Academy - Arc Length Page 7 of 15 Geometry 2016-17 ~ Unit 6 arc length of a circle and the circumference of the circle to solve problems Teachers should show: ● Part to whole relationships ● Arc length is portion of the circle’s circumference & can be expressed in terms of π ● Teachers will need to explicitly show how the relationship is proportional. Kuta - Circumference and Arc Length Kuta - Arc Length and Sector Area Practice Example: Use examples to create a linear function that represents relationship. **Degrees Rotation = Central Angle (same thing) Student should know: ● How to use a proportion to find the arc length given a central angle ● Find the central angle given the arc length ● Find the circumference given arc length & central angle ● Similarity between all circles Sectors and Segments of Circles Khan Academy - Circle Arcs and Sectors Finding the Sector Area of a Circle Examples: Linear relationship to show proportionality Khan Academy - Radians Degree/Radian Circle Khan Academy - Unit Circle Geometry ~ Unit 6 Updated November 30, 2016 Page 8 of 15 Geometry 2016-17 ~ Unit 6 Khan Academy - Radius and Center of a Circle for an Equation in Standard Form Equation of a Circle (Standard Form) Equations of Circles Practice with Equations of Circles Application: 1. Suppose a professional baseball player swings a bat in an arc that has a radius of 48 inches and sweeps through 255° of rotation. Assuming the arc is circular, what is the distance the tip of the bat travels to the nearest inch? How many feet is this rounded to the nearest foot? 2. How far does the tip of a 14 centimeter long minute hand on a clock move in 10 minutes? 3. An electric winch is used to pull a boat out of the water onto a trailer. The wench winds the cable around a circular drum of diameter 5 inches. Approximately how many times will the winch have to rotate in order to roll in 5 feet of cable? Geometry ~ Unit 6 Updated November 30, 2016 Page 9 of 15 Geometry 2016-17 ~ Unit 6 G.12(C) The student is expected to apply the proportional relationship between the measure of the area of a sector of a circle and the area of the circle to solve problems. Misconceptions: ● How to manipulate pi ● Simplifying proportional ● Solving minor arc vs. major arc length ● Students can struggle with setting up proportions G.12C Big Idea: Area of a section is a piece of the area of the circle and the degree of rotation is a piece of the total of 360 degrees. Teachers should show: ● ● ● Emphasize the similarity of all circles Note that by similarity of sectors with same central angle, arc lengths are proportional to radius - Basis for introducing radian as unit of measure Not intended that it be applied to development of circular trig in this course. Area of a circle is proportional to the circle’s area and can be expressed in terms of p Students should know: ● How to solve for the area of a sector ● How to solve for the area of a circle ● Apply proportional relationships to solve for area ● Similarity between all circles Geometry ~ Unit 6 Updated November 30, 2016 Page 10 of 15 Geometry 2016-17 ~ Unit 6 Misconceptions: ● Setting up proportion ● Minor sector vs. major sector ● Simplifying proportion to solve Examples: 1. Draw a diagram of a circle with radius 8 units and a sector formed by a central angle of 30. Find the area of the sector. 2. A sprinkler head sprays water a distance of up to 20 feet from the head. Find the area of the sector covered by the spray of the sprinkler head to the nearest square foot. G.12(D) The student is expected to describe radian measure of an angle as the ratio of the length of an arc intercepted by a central angle and the radius of the circle G.12D Big Idea: Geometry ~ Unit 6 Updated November 30, 2016 Page 11 of 15 Geometry 2016-17 ~ Unit 6 Students will be using radians instead of degrees to describe an angle Teachers should show: ● Show how to write the measure of an angle in terms of radians ● Unit circle as an example Background: For this measurement, consider the unit circle (a circle of radius 1) whose center is the vertex of the angle in question. It is easy to convert between degree measurement and radian measurement. The circumference of the entire circle is 2 , so it follows that 360° equals 2 radians. Hence, 1° equals /180 radians, and 1 radian equals 180/ degrees. Most calculators can be set to use angles measured with either degrees or radians. Alternate definition of radians is sometimes given as a ratio. The radian measure of the angle is the ratio of the length of the intercepted arc to the radius of the circle. For instance, if the length of the arc is 3 and the radius of the circle is 2, then the radian measure is 1.5. The reason that this definition works is that the length of the intercepted arc is proportional to the radius of the circle. This alternate definition is more useful, however, since you can use it to relate lengths of arcs to angles. Students should know: ● Write the radian measure of an angle ● Understanding the ratio of the length of an arc intercepted by a central angle and the radius of the circle Geometry ~ Unit 6 Updated November 30, 2016 Page 12 of 15 Geometry 2016-17 ~ Unit 6 Radian measure of an angle is the length of the arc intercepted on a circle of radius 1 by an angle in standard position on a coordinate plane. Or equivalently, the radian measure of a central angle in standard position on a coordinate plane is the ratio of the intercepted arc length to the radius of the circle. Misconceptions: ● What a radian is ● How radians are different from degrees Examples: 1. Given the length of the arc l and the radius r, to find the angle at the center. (a). l = .16296, r = 12.587. (b). l = 1.3672, r = 1.2978. What is 70º in radians? To change this degree measurement to radians, we multiply as follows: To change this radian measurement to degrees, we multiply: Geometry ~ Unit 6 Updated November 30, 2016 Page 13 of 15 Geometry 2016-17 ~ Unit 6 *CISD Safety Net* G.12(E) The student is expected to show that the equation of a circle with center at the origin and radius r is x2 +y2 = r2 and determine the equation for the graph of a circle with radius r and center (h, k), (x – h)2 + (y – k)2 =r2. G.12E Big Idea Write the equation of a circle. Teachers should show: ● Expose the formula to students by explaining how r, h, and k can change the shape of the circle ● Graph the circle Students should know: ● How to write the equation for a circle ● How to find the area of the circle given the equation ● Identify parts of the equation Geometry ~ Unit 6 Updated November 30, 2016 Page 14 of 15 Geometry 2016-17 ~ Unit 6 Misconceptions: ● Students should know that circles can have a center NOT at (0, 0) ● Students can make a mistake when substituting in a negative value for h or k since the equation has x - h and y - k. Geometry ~ Unit 6 Updated November 30, 2016 Page 15 of 15