Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Golden ratio wikipedia , lookup
Lie sphere geometry wikipedia , lookup
History of geometry wikipedia , lookup
Rational trigonometry wikipedia , lookup
Euler angles wikipedia , lookup
Reuleaux triangle wikipedia , lookup
Trigonometric functions wikipedia , lookup
History of trigonometry wikipedia , lookup
Euclidean geometry wikipedia , lookup
Plainfield Public Schools Mathematics Rigorous Curriculum Design Unit Planning Organizer Grade/Course Geometry Unit of Study Unit 2 Similarity and Proof Pacing 7 weeks including 2 weeks for reteaching or enrichment Standards for Mathematical Practice Mathematical Practices that apply to the unit: MP1. MP2. MP3. MP4. MP5. MP6. MP7. MP8. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Unit Standards G.SRT.1Verify experimentally the properties of dilations given by a center and a scale factor. a. 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. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. 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. 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.CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. G.C.1 Prove that all circles are similar. “Unwrapped” Skills (students need to be able to do) “Unwrapped” Concepts DOK (students need to know) Levels FOCUS STANDARD: 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. use definition of similarity 2 explain transformations and similarity for triangles 2 FOCUS STANDARD: G.SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Use properties of similarity transformations 2 Establish ( prove) AA criterion for two triangles to be similar 3 FOCUS STANDARD: 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.CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Prove a line parallel to one side of a triangle divides the other two proportionally the Pythagorean Theorem measures of interior angles of a triangle sum to 180° base angles of isosceles triangles are congruent the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length 4 4 4 4 4 4 medians of a triangle meet at a point “Unwrapped” Skills (students need to be able to do) “Unwrapped” Concepts (students need to know) Additional Standard: DOK Levels G.C.1 Prove that all circles are similar. Prove circles are similar 4 Unit Vocabulary Terms “Unwrapped” Focus Standards Concepts Supporting Standards Concepts /Other Unit-Specific Terms Conditional Statement (all forms) Two Column Proof Postulate Similar Polygon Scale Factor Corresponding Angles Conjecture Inductive Reasoning Counter Example Complimentary Angles Supplementary Angles Geometric Mean Proportion Ratio Linear Pair Vertical Angles Essential Questions If two figures are similar, how do you find the length of a missing side? Corresponding Big Ideas Understanding Geometric Relationships In Diagrams. How do you identify a similarity transformation in a plane? Similar geometric figures can be created by transformations. All transformations create similar geometric figures. Dilations, in particular create figures that are similar, but may not be congruent. Similar geometric figures have angles that are congruent and segments that are proportional in length. Congruence is also similarity. It is just a more specifically defined similarity where the ratio of lengths is 1:1. How can the side of a triangle be partitioned into segments of a given ratio? How can this information be used to solve problems involving similar triangles? How do you write a geometric proof? How can similarity and congruence be used to solve problems and/or prove statements about or properties of triangles? The processes of proving include a variety of activities, such as developing conjectures, considering the general case, exploring with examples, looking for structural similarities across cases, and searching for counterexamples. A proof can have many different valid representational forms, including narrative, picture, diagram, twocolumn presentation, or algebraic form. Engaging Learning Experiences Engaging Scenario Student challenge Students design a star for the front of a building based on the criteria of donor. Current situation: Star must fit in a designated area. Students analyze space and determine congruent triangles. The space decreases by 50% and students must redesigns and maintains the original ratios . Student role: Students design their projects and grade others. Intended audience: Donator Product: Design of two stars. Performance Task Synopses Task 1: Students determine the shape and coordinates of the original star Complete the table comparing sides to determine if there are congruent triangles. (The trick is the congruent triangles that make up the non used space) Students calculate the 50 % of the space to use Performance Task In Detail Focus Standards: Which standard(s) (priority/supporting) will the task address? G.SRT.2, 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.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. What essential Question(s) and corresponding Big Idea(s) will this task target? Essential Questions How can the side of a triangle be partitioned into segments of a given ratio? How can this information be used to solve problems involving similar triangles? How can similarity and congruence be used to solve problems and/or prove statements about or properties of triangles? Big Ideas : Similar geometric figures can be created by transformations. All transformations create similar geometric figures. Dilations, in particular create figures that are similar, but may not be congruent. The processes of proving include a variety of activities, such as developing conjectures, considering the general case, exploring with examples, looking for structural similarities across cases, and searching for counterexamples. A proof can have many different valid representational forms, including narrative, picture, diagram, two-column presentation, or algebraic form. Which “unwrapped specific concepts and skills will this task target? Skill : use, explain, prove Concept the definition of similarity to decide if transformations are similar using similarity transformations the meaning of similarity for triangles 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 How can I differentiate the application and/or evidence to meet the varying needs of my students? Give a personal cue to begin work Give work in smaller units Provide immediate reinforces and feedback Make sure the appropriate books and materials are available Introduce the assignment in sequential steps Check for student understanding of instructions Check on progress often in the first few minutes of work Provide time suggestions for each task Provide a checklist for long detailed tasks Use technological resources Engaging Learning Experiences Engaging Scenario Advanced or Exemplary All “Goal” criteria plus: Goal Complete the original picture complete table for original picture complete second picture complete table for second picture determine theorems based on patterns Progressing Meets _2_ of the “Goal” criteria Beginning Meets fewer than _2 of the “Goal” criteria Task to be repeated after re-teaching Comments: Interdisciplinary Connections and Related Focus Standards 9.1 21st-Century Life & Career Skills 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. CCSS.ELA-Literacy.RL.9-10.1 Cite strong and thorough textual evidence to support analysis of what the text says explicitly as well as inferences drawn from the text. 21st Century Learning Skills Specific to Task those that apply for each task: ❑ Teamwork and Collaboration ❑ Initiative and Leadership ❑ Curiosity and Imagination ❑ Innovation and Creativity ❑ Critical thinking and Problem Solving ❑ Flexibility and Adaptability ❑ Effective Oral and Written Communication ❑ Accessing and Analyzing Information ❑ Other