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Unit Overview Content Area: Math Unit Title: Geometry and Transformations Unit: 6 Target Course/Grade Level: Eighth Grade Timeline: 5/6 weeks Unit Summary: Students build on their previous knowledge of measurement and geometry concepts to explore angles and lines. Students will explore angle relationships created from parallel lines. Students measure and classify angle and classify triangles. They determine the sum of their interior angles of polygons. Students use similar triangles to solve problems including height and distances. Students use the Pythagorean Theorem to solve problems with right triangles. Students will also use the coordinate grid to apply transformations to plane figures, and understand how the transformation affects its congruency and location on a coordinate plane. Primary interdisciplinary connections: Language Arts and Technology 9.1 21st-Centuries Life & Career Skills Standard 9.1 All students will demonstrate the creative, critical thinking, collaboration, and problem-solving skills needed to function successfully as both global citizens and workers in diverse ethnic and organizational cultures. Strand: A. Critical Thinking and Problem Solving B. Creativity and Innovation C. Collaboration, Teamwork and Leadership Content Statement: 9.1.8: A The ability to recognize a problem and apply critical thinking skills and problem solving skills to solve the problem is a lifelong skill that develops over time. 9.1.8: B Gathering and Evaluating knowledge and information from a variety of sources, including global perspective, fosters creativity and innovative thinking. 9.1.8: C Collaboration and team work enable individuals or groups to achieve common goals with greater efficiency. Leadership abilities develop over time through participation in group and or teams that that are engaged in challenging or competitive activities. 21st Century themes and skills: Critical Thinking and Problem Solving, Collaboration, Teamwork and Leadership, Creativity and Innovation Mathematical Practices: 8.MP.1 Make sense of problems and persevere in solving them. 8.MP.2 Reason abstractly and quantitatively. 8.MP.3 Construct viable arguments and critique the reasoning of others. 8.MP.4 Model with mathematics. 8.MP.6 Attend to precision. 8.MP.7 Look for and make use of structure. 8.MP.8 Look for and express regularity in repeated reasoning. Learning Targets Domain: Geometry Cluster: Understand congruence and similarity using physical models, transparencies or geometry software; Understand and apply the Pythagorean Theorem. Standard # Standards 8.G.1a Verify experimentally the properties of rotations, reflections, and translations. 8.G.1b a) Lines are taken to lines, and line segments of the same length. 8.G.1c b) Angles are taken of the same measure c) Parallel lines are taken to parallel 8.G.2 Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translating; given two congruent figures, describe a sequence that exhibits the congruence between them. 8.G.3 Describe the effect of dilations, translations, rotations and reflections on two dimensional figures using coordinates. 8.G.4 Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. 8.G.5 Use informational arguments to establish facts about the angle sum and exterior angle of a triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. 8.G.6 Explain a proof of the Pythagorean Theorem and its converse. 8.G.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. 8.G.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate grid system. 8.EE.2 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. 8.EE.6 Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx + b for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x,y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Develop strategies to reinforce positive attitudes and productive behaviors that impact critical thinking and problem-solving skills. 8.F.4 9.1.8.A.1 9.1.8.A.2 Implement problem-solving strategies to solve a problem in school or the community. 9.1.8.B.2 Assess data gathered to solve problems for which there are varying perspective (e.g., cross cultural, gender specific, generational, etc.) and determine how the data can best be used to design the multiple solutions. 9.1.8.C.1 Determine an individual’s responsibility for personal actions and contributions to group activities. 9.1.8.C.2 Demonstrate the use of compromise, consensus and community building strategies for carrying out different task, assignments and projects. 9.1.8.C.3 Model leadership skills during classroom and extracurricular activities. Unit Essential Questions Why is the study of lines and angles important to understanding our environment? How are angle measurements used in real-world situations? Describe the relationship between two parallel lines cut by a transversal and angles that are formed? Why is it important to understand the properties of two-dimensional figures, such as triangles and quadrilaterals? How are similar triangles, right relationships, and transformations used in the real world? How can proportional reasoning be applied to problem solving situations involving similar and right triangles? How is the Distance Formula related to the Pythagorean Theorem? How are the results of a transformation different than the original figure? How are they similar? Unit Enduring Understandings Adjacent angles share a common vertex and a common side and do not overlap. Complementary angles are two angles in which the sum of their angle measurements is 90 degrees. Supplementary angles are two angles in which the sum of their angle measurements is 180 degrees. When parallel lines are cut by a line called a transversal the following angles are created and are congruent: Vertical, Corresponding, Alternate Exterior and Alternate Interior. Triangles are classified by its side and angles measurements. A polygon is a closed figure classified by its sides. The Pythagorean Theorem describes a relationship between the legs and the hypotenuse for any right triangle. A transformation is an operation that maps an original geometric, the pre-image, onto a new figure called the image. The movements are a translation (slide), reflections (mirror image), rotation (turn), dilation (enlargement or reduction). Unit Learning Targets Students will ... Identify relationships of angles formed by two parallel lines cut by a transversal. Explore the relationship among the angles of a triangle. Identify similar polygons and find missing measures of a similar polygon. Investigate parallel lines and similar triangles. Use Pythagorean Theorem to find missing sides. Find the distance between two points on the coordinate plane. Use special right triangles to solve problems. Graph transformations on the coordinate plane. Evidence of Learning Summative Assessment Solve problems involving angles created by parallel lines cut by a transversals: vertical, alternate interior, alternate exterior and corresponding angles. Demonstrate that the sum of the angles in a triangle is 180 degrees Find measures of unknown angles and the sum of angles in polygons. Identify elements of geometric solids including faces, edges, and vertices of polyhedral. Determine missing corresponding sides of similar polygons. Use similar triangles to solve problems that include height and distance. Use the Pythagorean Theorem to find the distance between points on the coordinate plane. Graph translations, reflections, rotations, and dilations in the coordinate plane. Equipment needed: rulers, grid paper, 10x10 grids, Smart Board, white boards, calculators, Elmo Teacher Instructional Resources: Textbook (To be determined) Study Island Khan Academy Videos Formative Assessments Skill sheets Quizzes/Tests Student workbook Homework Math games Study Island Integration of Technology: Smart Board to play online games, utilize online resources, generate models with Smart Software. Kahn Academy Videos Elmo – for demonstration Study Island Technology Resources: http://www.purplemath.com http://www.khanacademy.org – Interactive 2.0 instructional and practice site. Students can view instructional videos and complete practice modules for additional practice/remediation. http://www.studyisland.com/ - Web-based instruction, practice, assessment and reporting built from NJ standards. http://www.ixl.com/math/grade-8 - IXL 8th grade online interactive activities for the students to complete http://www.aaamath.com/grade8.html - AAA math 8th grade – online interactive activities and problems for the student to complete. http://www.adaptedmind.com/Math-Worksheets.html?type=hstb – Grade level material for practice, lessons, games, etc. Opportunities for Differentiation: Decelerate: Have students create two parallel lines with masking tape on their desk and use spaghetti as the transversal to investigate the angles created. Have students look through newspapers and magazines and cut out different pictures of triangles and identify them by angles and sides. Have students make posters about triangles; include classification by angles and sides, and Pythagorean’s Theorem. Accelerated: Given a regular polygon, it is possible to draw seven diagonals from one vertex. Draw and calculate the measure of each angle. Separate in groups and have students on dot paper draw three-dimensional views of figures in the classroom and then trade drawings and have student draw the top, front and side views. Teacher Notes: