Unit 2 Transformational Geometry Math 8

Transcription

1 Number of Days: 24 10/16/17 11/17/17 Unit Goals Stage 1 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, dot paper, protractors Standards for Mathematical Practice SMP 1 SMP 2 SMP 3 SMP 4 SMP 5 SMP 6 SMP 7 SMP 8 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. Standards for Mathematical Content Clusters Addressed [m] 8.G.A Understand congruence and similarity using physical models, transparencies, or geometry software. Transfer Goals Students will be able to independently use their learning to Make sense of never-before-seen problems and persevere in solving them. Construct viable arguments and critique the reasoning of others. Making Meaning UNDERSTANDINGS Students will understand that Rigid transformations preserve distance and angle measures. A two-dimensional figure is congruent to another if the second figure can be obtained from the first figure by a sequence of rotations, reflections, and translations. Similar figures have the same shape, but not necessarily the same size. A two-dimensional figure is similar to another if the second figure can be obtained from the first figure by a sequence of rotations, reflections, translations, and dilations. KNOWLEDGE Students will know The definition of academic vocabulary words, such as dilation, image, line of reflection, rotation, similar, transformation, translation, and transversal. Transformations include translations, reflections, rotations, and dilations. The sum of the measures of the three interior angles of a triangle is 180 o. ESSENTIAL QUESTIONS Students will keep considering How do transformations affect lines, line segments, angles, and two-dimensional figures? How can you use transformations to determine if two figures are congruent or similar? How can you find missing angle measures when parallel lines are intersected by a transversal? Acquisition SKILLS Students will be skilled at and/or be able to Verify the properties of rotations, reflections, and translations experimentally. Describe sequences of rotations, reflections, translations, and dilations that can verify whether two-dimensional figures are similar or congruent to one another. Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. State facts about the angles created when parallel lines are cut by a transversal. Informally argue facts about the measures of interior and exterior angles of triangles and the angle-angle criterion for similarity of triangles. LONG BEACH UNIFIED SCHOOL DISTRICT 1 Posted 10/9/17

2 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. Assessed Grade Level Standards Standards for Mathematical Content [m] 8.G.A Understand congruence and similarity using physical models, transparencies, or geometry software. 8.G.1 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. 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 translations; 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 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. 8.G.5 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 2 Posted 10/9/17

3 Assessment Evidence Unit Assessment Evidence of Learning Stage 2 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: 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: 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: 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: NONE Other Evidence Formative Assessment Opportunities Informal teacher observations Checking for understanding using active participation strategies Exit slips/summaries Tasks Modeling Lessons (SMP 4) Formative Assessment Lessons (FAL) Quizzes / Chapter Tests 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 3 Posted 10/9/17

4 1 day 7-8 I will explore transformational geometry by participating in the Opening Task. I will investigate congruency by OPENING TASK Escher s Artwork This Opening Task is a discussion about Escher s artwork. Display the website with Escher s artwork and facilitate a class discussion using the discussion questions. This task is a gateway into the entire unit of transformational geometry. Experimentally verifying the properties of translations, reflections, and rotations. 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) 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. (SMP 3) o Synergy Item Bank: Item ID 51314, 69168, 33966, 36699, 47186, 50726, Section 2.1 (Activity 1; Section 2.2 (Activities 1 and 4; Section 2.3 (Activities 1 and 3; Section 2.4 (Activities 1 and 2; Examples 1, 3, 4 and 5) Escher s Artwork Discussion Questions M.C. Escher s Artwork Website The Mathematical Side of M.C. Escher What is a Tessellation? Patty Paper Transformations Using Coordinates Activity Zombie Apocalypse Lesson Plans and Game Desmos: Translations Congruent Rectangles Task Same Size Same Shape? Task Transformations Shortest Sequence Task Triangle Congruence with Coordinates Task Khan Academy: Transformations intro Alumni Center Task Aaron s Designs Task Modeling: How Did They Make Mrs. Pac-Mac? (SMP 4) Modeling: How Do Skytypers Write Messages? (SMP 4) LONG BEACH UNIFIED SCHOOL DISTRICT 4 Posted 10/9/17

5 I will investigate similarity by 4-5 Extending knowledge of transformations to include dilations. Connecting scale factors to dilations. Finding missing measures in similar figures. Describing the effect of dilations on two-dimensional figures using coordinates. 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) 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? o Synergy Item Bank: Item ID 51522, 52346, 61360, 64791, Section 2.5 (Activity 1; Section 2.7 (Activities 1 and 2; Examples 1, 2, 3, 4 and 5) STEM Video: Shadow Puppets Dilation Exploration Desmos: Polygraph Transformations Are They Similar? Task Effects of Dilations on Lengths, Area, and Angles Task Khan Academy: Dilations Khan Academy: Properties & definitions of transformations STEM Video Performance Task: Master Puppeteer Soccer Complex Task 2-3 I will check my understanding of transformations by participating in the FAL. FORMATIVE ASSESSMENT LESSON Representing and Combining Transformations (SMP 1, 3, 5, 6, 7, 8) mypd Course #7668: Implementing FAL Transformations LONG BEACH UNIFIED SCHOOL DISTRICT 5 Posted 10/9/17

6 I will explore angle relationships by 2-3 Reviewing vertical angles, adjacent angles, supplementary angles, and complementary angles. Finding unknown angle measures when parallel lines are intersected by a transversal. Justifying angle relationships through transformations to show congruency using patty paper. (SMP 5) 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? o Synergy Item Bank: Item ID 48439, 48463, 50750, 64894, What You Learned Before (p. 101) Section 3.1 (Activities 1 and 2; Examples 1, 2, 3, and 4) Angle Relationships Investigation with Patty Paper Congruence of Alternate Interior Angles Via Rotations Task Corresponding Angles Dynamic Tool Vertical Angles Dynamic Tool Angle Relationships Flipbook Rigid Motions and Congruent Angles Task Khan Academy: Angles between intersecting lines Street Intersections Task LONG BEACH UNIFIED SCHOOL DISTRICT 6 Posted 10/9/17

7 I will apply transformations and angle relationships to triangles by 3-4 Visually representing and explaining the Triangle Sum Theorem. (SMP 3, 6) Finding a missing angle measure in a triangle, including the exterior angles and remote interior angles of triangles. Determining if two triangles are similar using the angle-angle criterion. o How can you visually justify the Triangle Sum Theorem? o How can you determine when two triangles are similar? o Are two pairs of congruent angles enough information to conclude that two triangles are similar? Explain why or why not. (SMP 3) o Synergy Item Bank: Item ID 64873, 48340, Section 3.2 (Activities 1, 2, 3, and 4; Section 3.4 (Activities 1 and 3; Examples 1 and 2) Interior Angles Task Triangle Sum Theorem PowerPoint Triangle Sum Theorem Activity Interior Angles of a Triangle Dynamic Tool Similar Triangles Dynamic Tool Similar Triangles Task Triangle Exterior Angle Dynamic Tool What s My Angle Task Khan Academy: Triangle angles 1-2 I will prepare for the unit assessment on transformational geometry by Incorporating the Standards for Mathematical Practice (SMPs) along with the content standards to review the unit. Ch. 2 Study Help (p. 68) Ch. 2 Review (p ) Ch. 3 Study Help (p. 116) Ch. 3 Review (p ) Intersection Task Transformations Task 1 day Unit Assessment Students will take the Synergy Online Unit Assessment. Unit Assessment Resources (Word or PDF) can be used throughout the unit. LONG BEACH UNIFIED SCHOOL DISTRICT 7 Posted 10/9/17