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Unit 2 Transformational Geometry Math 7 ACC Unit Goals – Stage 1 Number of Days: 24 days 10/17/16 – 11/18/16 Unit Description: Students study congruency and similarity by experimenting with dilations, rotations, reflections, and translations of geometric figures. During this unit, students are introduced to similar figures. Students will explore the relationships between interior and exterior angles of triangles and angles formed by parallel lines that are cut by a transversal. Materials: patty paper (tracing paper), calculators, pattern blocks, Geoboard dot paper*, protractors* (* Paper tool available at LBUSD Curriculum Intranet Instructional Tools Middle School Mathematical Tools) Standards for Mathematical Transfer Goals Practice Students will be able to independently use their learning to… SMP 1 Make sense of problems and • Make sense of never-before-seen problems and persevere in solving them. persevere in solving them. • Construct viable arguments and critique the reasoning of others. SMP 2 Reason abstractly and Making Meaning quantitatively. UNDERSTANDINGS ESSENTIAL QUESTIONS SMP 3 Construct viable arguments Students will understand that… Students will keep considering… and critique the reasoning of • A two-dimensional figure is congruent to another if the • How do transformations affect lines, line others. second figure can be obtained from the first figure by a segments, angles, parallel lines, and twoSMP 4 Model with mathematics. sequence of rotations, reflections, and translations. dimensional figures? SMP 5 Use appropriate tools • Similar figures have the same shape, but not • How can you use transformations to determine strategically. necessarily the same size. A two-dimensional figure is if two figures are congruent or similar? SMP 6 Attend to precision. similar to another if the second figure can be obtained • How can you find missing angle measures SMP 7 Look for and make use of from the first figure by a sequence of rotations, when parallel lines are intersected by a structure. reflections, translations, and dilations. transversal? SMP 8 Look for and express Acquisition regularity in repeated KNOWLEDGE SKILLS reasoning. Students will know… Students will be skilled at and/or be able to… • The definition of academic vocabulary • Verify the properties of rotations, reflections, and Standards for Mathematical words, such as adjacent angles, alternate translations experimentally. Content Clusters Addressed exterior angles, alternate interior angles, • Describe the effect of dilations, translations, rotations, and [a] 7.G.B Solve real‐life and complementary angles, dilation, image, line reflections on two-dimensional figures using coordinates. mathematical problems of reflection, remote interior angle, rotation, • Describe sequences of rotations, reflections, translations, involving angle measure, same-side interior angles, similar, and dilations that can verify whether two-dimensional area, surface area, and supplementary angles, transformation, figures are similar or congruent to one another. volume. translation, transversal, and vertical angles. • Use facts about supplementary, complementary, vertical, [m] 8.G.A Understand congruence • Transformations include translations, and adjacent angles in multi-step problems to write and and similarity using reflections, rotations, and dilations solve equations for unknown angles in a figure. physical models, • Complementary angles add up to 90° and • State facts about the angles created when parallel lines transparencies, or supplementary angles add up to 180°. are cut by a transversal. geometry software. • Vertical angles are congruent. • Informally argue facts about the measures of interior and exterior angles of triangles and the angle-angle criterion • The sum of the measures of the three for similarity of triangles. interior angles of a triangle is 180o. LONG BEACH UNIFIED SCHOOL DISTRICT 1 Posted 10/4/16 2016-2017 Unit 2 Transformational Geometry Math 7 ACC Assessed Grade Level Standards Standards for Mathematical Practice SMP 1 Make sense of problems and persevere in solving them. SMP 2 Reason abstractly and quantitatively. SMP 3 Construct viable arguments and critique the reasoning of others. SMP 4 Model with mathematics. SMP 5 Use appropriate tools strategically. SMP 6 Attend to precision. SMP 7 Look for and make use of structure. SMP 8 Look for and express regularity in repeated reasoning. Standards for Mathematical Content [a] 7.G.B Solve real-life and mathematical problems involving angle measure, area, surface area, and volume. 7.G.5 Use facts about supplementary, complementary, vertical, and adjacent angles in a multistep problem to write and solve simple equations for an unknown angle in a figure. [m] 8.G.A 8.G.1 8.G.2 8.G.3 8.G.4 8.G.5 Understand congruence and similarity using physical models, transparencies, or geometry software. Verify experimentally the properties of rotations, reflections, and translations: a. Lines are taken to lines, and line segments to line segments of the same length. b. Angles are taken to angles of the same measure. c. Parallel lines are taken to parallel lines. 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. Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them. Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so. Key: [m] = major clusters; [s] = supporting clusters; [a] = additional clusters LONG BEACH UNIFIED SCHOOL DISTRICT 2016-2017 2 Posted 10/4/16 Unit 2 Transformational Geometry Math 7 ACC Evidence of Learning – Stage 2 Assessment Evidence Unit Assessment Claim 1: Students can explain and apply mathematical concepts and carry out mathematical procedures with precision and fluency. Concepts and skills that may be assessed in Claim1: 7.G.B • The student solves real-life and mathematical problems involving angle measure including problems requiring writing and solving equations. 8.G.A • The student verifies that rigid transformations preserve distance and angle measures. • The student describes sequences of rotations, reflections, translations, and dilations that can verify whether two-dimensional figures are similar or congruent to one another. • The student constructs a new figure after the original figure is dilated, rotated, reflected, or translated. • The student describes the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. Claim 2: Students can solve a range of wellposed problems in pure and applied mathematics, making productive use of knowledge and problem-solving strategies. Standard clusters that may be assessed in Claim 2: • 7.G.B • 8.G.A Claim 3: The student can clearly and precisely construct viable arguments to support their own reasoning and critique the reasoning of others. Standard clusters that may be assessed in Claim 3: • 7.G.A • 8.G.A Claim 4: The student can analyze complex, real-world scenarios and can construct and use mathematical models to interpret and solve problems. Standard clusters that may be assessed in Claim 4: • 7.G.B Other Evidence Formative Assessment Opportunities • Informal teacher observations • Checking for understanding using active participation strategies • Exit slips/summaries • Modeling Lessons (SMP 4) • Tasks • • • • • Formative Assessment Lessons (FAL) Quizzes / Chapter Tests Intranet Unit Assessment Resources Big Ideas Math Performance Tasks SBAC Interim Assessment Blocks Access Using Formative Assessment for Differentiation for suggestions. Located on the LBUSD website – “M” Mathematics – Curriculum Documents LONG BEACH UNIFIED SCHOOL DISTRICT 2016-2017 3 Posted 10/4/16 Unit 2 Transformational Geometry Math 7 ACC Learning Plan – Stage 3 Days Learning Target 1 day I will explore transformational geometry by participating in the Opening Task. I will investigate congruency by… 7-8 days Suggested Sequence of Key Learning Events and Instruction Big Ideas Math Advanced 2 Expectations (Activities and Lessons) OPENING TASK – Escher’s Artwork This Opening Task is a discussion and should take should take about 20 minutes. Display the website with Escher’s artwork and facilitate a class discussion with the discussion questions utilizing Talk Moves. This task is a gateway into the entire unit of transformational geometry. • Experimentally verifying the properties of translations, reflections, and rotations. (SMP 5) • Transforming lines, angles, parallel lines, and two-dimensional figures using translations, reflections, and rotations to verify congruence. • Describing the effect of translations, reflections, and rotations on two-dimensional figures using coordinates. • Describing a sequence of transformations that exhibits congruency, given two congruent figures. • Explaining that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of translations, reflections, and rotations. (SMP 3) • Answering questions such as… o How can you use coordinates to describe the results of a translation, reflection, or rotation? o Compare and contrast translations, reflections, and rotations. Explain why these transformations generate congruency. LONG BEACH UNIFIED SCHOOL DISTRICT 2016-2017 4 • Section 2.1 (Activity 1; Examples 1, 2, and 3) • Section 2.2 (Activities 1 and 4; Examples 1, 2, and 3) • Section 2.3 (Activities 1, 2, and 3; Examples 1, 2, and 3) • Section 2.4 (Activity 2; Examples 2, 3, 4 and 5) Curriculum Intranet Conceptual Understanding: • Escher’s Artwork Discussion Questions • M.C. Escher’s Artwork Website • Article: The Mathematical Side of M.C. Escher • What is a Tessellation? Conceptual Understanding: • Patty Paper Transformations Using Coordinates Activity • Zombie Apocalypse Lesson Plans and Game Procedural Skills and Fluency: • Reflecting a Rectangle Over a Diagonal Task • Same Size Same Shape? Task • Transformations— Shortest Sequence • Triangle Congruence with Coordinates Task Application: • Alumni Center Performance Task • Desmos: Translations • Transformations: Aaron’s Designs Task • Modeling: How Did They Make Mrs. Pac-Mac? (SMP 4) • Modeling: How Do Skytypers Write Messages? (SMP 4) Posted 10/4/16 Unit 2 Transformational Geometry Math 7 ACC Learning Plan – Stage 3 Days Learning Target I will investigate similarity by… • • • • • • • 4-5 days 2-3 days I will check my understanding of transformations by participating in the FAL. Suggested Sequence of Key Learning Events and Instruction Big Ideas Math Advanced 2 Expectations (Activities and Lessons) Extending knowledge of transformations to include • Section 2.5 dilations. (Activity 1; Examples 1, 2, and 3) Connecting scale factors to dilations. • Section 2.7 Finding missing measures in similar figures. (Activity 1; Describing the effect of dilations on twoExamples 1, 2, 3, and 5) dimensional figures using coordinates. • STEM Video: Shadow Describing a sequence of transformations that Puppets exhibits similarity, given two similar figures. Explaining that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of dilations, rotations, reflections, and translations. (SMP 3) Answering questions such as… o What is dilation? What is a scale factor? o Explain how similarity is connected to proportionality. o How does the dilation affect the length of line segments? o How can you use coordinates to find scale factors? Curriculum Intranet Conceptual Understanding: • Dilation Exploration Procedural Skills and Fluency: • Are They Similar? Task • Effects of Dilations on Lengths, Area, and Angles Task Application: • STEM Performance Task: Master Puppeteer • Soccer Complex Performance Task • Desmos: Polygraph Transformations FORMATIVE ASSESSMENT LESSON • Representing and Combining Transformations (SMP 1, 3, 5, 6, 7, 8) LONG BEACH UNIFIED SCHOOL DISTRICT 2016-2017 5 Posted 10/4/16 Unit 2 Transformational Geometry Math 7 ACC Learning Plan – Stage 3 Days Learning Target I will explore angle relationships by… • • • • • 2-3 days Suggested Sequence of Key Learning Events and Instruction Big Ideas Math Advanced 2 Expectations (Activities and Lessons) Identifying vertical angles, adjacent angles, • Section 12.1 supplementary angles, and complementary angles. (Activities 2 and 3; Examples 1, 2, and 3) Writing equations to find unknown angles in complex figures that combine adjacent, vertical, • Section 12.2 complementary, and supplementary angles. (Activities 1 and 2; Examples 1, 2, and 3) Finding unknown angle measures when parallel lines are intersected by a transversal using • Section 3.1 knowledge of corresponding angles, alternate (Activities 1 and 2; interior angles, alternate exterior angles, and Examples 1, 2, 3, and 4) same-side interior angles. Justifying angle relationships through transformations to show congruency using patty paper. (SMP 5) Answering questions such as… o Explain why two angles cannot be vertical and adjacent at the same time. o Ariana claims that two angles are complementary only if they are adjacent and make a right angle. Is Ariana correct? Justify your reasoning. o How can you use angle pairs to solve problems? o How can you find missing angle measures when parallel lines are intersected by a transversal? o How are corresponding angles, alternate interior angles, alternate exterior angles, and same-side interior angles similar and different? LONG BEACH UNIFIED SCHOOL DISTRICT 2016-2017 6 Curriculum Intranet Conceptual Understanding: • Angle Relationships Investigation with Patty Paper • Vertical Angles Dynamic Tool • Corresponding Angles Dynamic Tool Procedural Skills and Fluency: • Flipbook: Angle Relationships • Rigid Motions and Congruent Angles Task • Wanted Poster: Angles Application: • Street Intersections Task • Truss Bridge Performance Task Posted 10/4/16 Unit 2 Transformational Geometry Math 7 ACC Learning Plan – Stage 3 Days Learning Target I will apply transformations and angle relationships to triangles by … • • • • 3-4 days I will prepare for the unit assessment on transformational geometry by… Suggested Sequence of Key Learning Events and Instruction Big Ideas Math Advanced 2 Expectations (Activities and Lessons) Visually representing and explaining the Triangle • Section 3.2 Sum Theorem. (SMP 3, SMP 6) (Activities 1, 2, 3, and 4; Examples 1, 2, and 3) Finding a missing angle measure in a triangle, including the exterior angles and remote interior • Section 3.4 angles of triangles. (Activities 1 and 3; Examples 1 and 2) Determining if two triangles are similar by using the Angle-Angle (AA) Postulate or using proportional side lengths. Answering questions such as… o How can you justify the Triangle Sum Theorem? o How can you determine when two triangles are similar? o What is the Angle-Angle Similarity Postulate? o How can you determine if corresponding sides are proportional? o Are two pairs of congruent angles enough information to conclude that two triangles are similar? Explain. (SMP 3) • Incorporating the Standards for Mathematical Practice (SMPs) along with the content standards to review the unit. 1-2 days 1 day LONG BEACH UNIFIED SCHOOL DISTRICT 2016-2017 • • • • Ch. 2 Study Help (p. 68) Ch. 2 Review (p. 91 – 99) Ch. 3 Study Help (p. 116) Ch. 3 Review (p. 133 – 139) Curriculum Intranet Conceptual Understanding: • Triangle Sum Theorem PowerPoint • Triangle Sum Theorem Activity • Interior Angles of a Triangle Dynamic Tool • Similar Triangles Dynamic Tool • Triangle Exterior Angle Dynamic Tool Procedural Skills and Fluency: • What’s My Angle Task Procedural Skills and Fluency: • Intersection Performance Task Application: • Transformations Performance Task Unit Assessment Synergy: 2016-17 Math 7 ACC Unit 2 7 Posted 10/4/16