Geometry-CCSSM Module A2 The Coordinate Plane Summary 1

Transcription

1 1 Module Overview In this inquiry module, students apply the Pythagorean Theorem to solve problems and justify solutions and solution paths for finding side lengths in right triangles. Building on their previous understanding of the coordinate plane, students apply the Pythagorean Theorem to find distances between two points. Students also use a variety of models (physical, geometric software) to make and test conjectures about the effects of rotations, reflections, and translations on shapes. Students perform and explain rotations, reflections, and translations using coordinate systems. Essential Questions What criteria are used when selecting a tool? How does change affect the measurements of geometric shapes? How do triangles help us find the distance/length? Student Focal Points 1) Applying the Pythagorean Theorem 2) Experimenting with and understanding transformations of various figures in the coordinate plane 3) Understanding congruence in terms of rigid motions 4) Describe mappings of multiple transformations in the coordinate plane Standards for Mathematical Practice Mathematically proficient students Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Standards for Mathematical Content G.CO.2 G.CO.4 G.CO.5 Sense-Making Concepts Standard(s) 8.G.8 G.CO.3 G.CO.6 Sense-Making Strategies Standard(s) Sense-Making Applications/Modeling Standard(s) 8.G.7

2 Module A2 - Scope & Sequence 2 Duration 2-3 days 1-2 days 2 days 2 days Standard(s) 8.G.7 8.G.8 Lesson 1: Pythagorean Theorem and Distance MP#7, 8 Text: Section 4.4 (The Pythagorean Theorem & the Distance Formula) Teacher Resource(s): Learnzillion: Find the Length of a Segment on the Coordinate Plan by Using the Pythagorean Theorem (online video) 8 th Grade Mathematics Unpacked Content (pages 33-35) Public Schools of North Carolina Collaborative Activity/Task: Distance Formula to Decode a Message Tools/Technology: G.CO.2 G.CO.4 G.CO.5 Dynamic Online Software (GeoGebra) Lesson 2: Translations MP#6 Text: Section 3.7 (Translations) Teacher Resource(s): High School Flip Book: Common Core State Standards for Mathematics (pages ) Collaborative Activity/Task: Translation Activity - Blockhead Tools/Technology: G.CO.2 G.CO.3 G.CO.4 G.CO.5 Students focus on translation of the plane using graph paper and function notation at this time. Lesson 3: Reflections and Symmetry MP#6, 8 Text: Section 5.7 (Reflections & Symmetry) Teacher Resource(s): High School Flip Book: Common Core State Standards for Mathematics (pages ) Symmetry (article) Collaborative Activity/Task: Reflection of a Room Error Analysis Tools/Technology: Paper-folding Dynamic Online Software (GeoGebra): GeoGebra Reflections G.CO.2 G.CO.3 Lesson 4: Rotations and Symmetry G.CO.4 MP#6, 8 G.CO.5 Text: Section 11.8 (Rotations) Teacher Resource(s): High School Flip Book: Common Core State Standards for Mathematics (pages ) Symmetry (article) Collaborative Activity/Task: Rotational Symmetry (GeoGebra Activity) Tools/Technology: Tracing paper, transparency Dynamic Online Software (GeoGebra): o GeoGebra Rotations o Rotational Symmetry

3 3 3-4 days 2 days G.CO.2 G.CO.4 G.CO.5 Lesson 5: Compositions of Transformations MP #6, 7, 8 Text: (The text does not support this standard). Teacher Resource(s): High School Flip Book: Common Core State Standards for Mathematics (pages ) Brightstorm: Geometry Compositions of Translations (YouTube video) Classzone Glide Reflections & Compositions (textbook-based resource as pdf) Collaborative Activity/Task: Composition Activity (GeoGebra Activity) Tools/Technology: Dynamic Online Software (GeoGebra):Composition Activity Module A2 Review Module A2 Test (common assessment) Total Days: days

4 4 Content Standards Unpacking 8.G.7 (Sense-Making Applications/Modeling Standard) Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. Given real-world and mathematical problems in two and three dimensions, o Apply the Pythagorean Theorem in order to solve problems and justify solutions and solution paths for finding side lengths in right triangles within the problem contexts. Pythagorean Theorem Pythagorean Theorem, Appropriate labeling of a right triangle, (leg and hypotenuse). Solve equations involving one variable and square root, Represent real-world and mathematical contexts involving right triangles in a variety of formats (e.g., drawings, equations), Justify solutions and solution paths using conceptual understandings and vocabulary related to the Pythagorean Theorem (e.g., right angle, hypotenuse). The properties of right triangles can be used to solve problems. Pythagorean Theorem Hypotenuse Leg Right triangle Square root Square Area Notation: a 2 + b 2 = c 2 Related Standards: 8.EE.2, 8.NS.2, 7.EE.2, 6.EE.2, F-IF.8, G-SRT.6, G-SRT.8 Further Discussion & Illustrations of Standard 8.G.7:

5 5 8.G.8 (Sense-Making Concepts Standard) Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. Given real-world and mathematical problems that can be represented on a coordinate plane, o Apply the Pythagorean Theorem in order to solve problems and justify solutions and solution paths for finding side lengths (distances between points) in right triangles within the problem contexts. Pythagorean Theorem Pythagorean Theorem Operations and labeling within a coordinate system Solve equations involving one variable and square root, Represent real-world and mathematical contexts involving right triangles in a variety of formats (e.g., drawings on coordinate planes, equations), Justify solutions and solution paths using conceptual understandings and vocabulary related to the Pythagorean Theorem (right angle, hypotenuse). The properties of right triangles can be used to solve problems Theorems represent general relationships that are true for all shapes that fit certain criteria. Pythagorean Theorem Hypotenuse Leg Right triangle Square root Square Vertex Vertices Distance Notation: a 2 + b 2 = c 2 Related Standards: 8.EE.2, 8.NS.2, 6.G.3, 6.NS.8, G-GPE.1, G-GPE.2, G-GPE.4, G-GPE.5, G-GPE.7 Further Discussion & Illustrations of Standard 8.G.8:

6 6 G.CO.2 (Sense-Making Concept Standard) Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). Given undefined a variety of transformations (translations, rotations, reflections, and dilations), Represent the transformations in the plane using a variety of methods (e.g., technology, transparencies, semi-transparent mirrors (MIRAs), patty paper, compass). Describe transformations as functions that take points in the plane as inputs and give other points as outputs, explain why this satisfies the definition of a function, and adapt function notation to that of a mapping [e.g. F(x,y) F(x+a,y+b)]. Compare transformations that preserve distance and angle to those that do not. Parameters Characteristics of transformations (translations, rotations, reflections, and dilations). Methods for representing transformations. Characteristics of functions. Conventions of functions with mapping notation. Accurately perform dilations, rotations, reflections, and translations on objects in the coordinate plane with and without technology. Communicate the results of performing transformations on objects and their corresponding coordinates in the coordinate plane, including when the transformation preserves distance and angle. Use the language and notation of functions as mappings to describe transformations. Mapping one point to another through a series of transformations can be recorded as a function. Some transformations (translations, rotations, and reflections) preserve distance and angle measure, and the image is then congruent to the pre-image, while dilations preserve angle but not distance, and the pre-image is similar to the image. Distortions, such as only a horizontal stretch, preserve neither. Angle Circle Perpendicular lines Transformation Translation Reflection

7 7 Rotation Horizontal/Vertical Stretch Dilation Mapping Preservation of Distance Pre-image Image Scale Factor Corresponding Pairs of Angles Corresponding Pairs of Sides Notation: F(x,y) F(x+a,y+b) A A Related Standards: 8.G.1, 8.G.2, 8.G.4 Further Discussion & Illustrations of Standard G.CO.2:

8 8 G.CO.3 (Sense-Making Strategies Standard) Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. Given a collection of figures that include rectangles, parallelograms, trapezoids, or regular polygons, Identify which figures that have rotations or reflections that carry the figure onto itself. Perform and communicate rotations that map the object onto itself. Distinguish these transformations from those which do not carry the object back onto itself. Describe the relationship of these findings to symmetery. Characteristics of transformations (translations, rotations, reflections, and dilations). Characteristics of rectangles, parallelograms, trapezoids, and regular polygons. Accurately perform dilations, rotations, reflections, and translations on objects in the coordinate plane with and without technology. Communicate the results of performing transformations on objects and their corresponding coordinates in the coordinate plane. Mapping one point to another through a series of transformations can be recorded as a function. Since rotations and reflections preserve distance and angle measure, the image is then congruent. Reflection Linear symmetry Rotational symmetry Regular polygon Trapezoid Parallelogram Congruence Notation: Related Standards: 8.G.1, 8.G.2 Further Discussion & Illustrations of Standard G.CO.3:

9 9 G.CO.4 (Sense-Making Concept Standard) Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. Use geometric terminology (angles, circles, perpendicular lines, parallel lines, and line segments) to describe the series of steps necessary to produce a rotation, reflection, or translation. Use these descriptions to communicate precise definitions of rotation, reflection, and translation. Characteristics of transformations (translations, rotations, reflections, and dilations). Properties of mathematical definition, i.e., the smallest amount of information and properties that are enough to determine the concept. (Note: may not include all information related to concept). Accurately perform rotations, reflections, and translations on objects with and without technology. Communicate the results of performing transformations on objects. Use known and developed definitions and logical connections to develop new definitions. Geometric definitions are developed from a few undefined notions by a logical sequence of connections that lead to a precise definition. A precise definition should allow for the inclusion of all examples of the concept and require the exclusion of all non-examples. Direction of Rotation Angle of Rotation Center of Rotation Notation: Related Standards: 8.G.1, 8.G.2, F.BF.3 Further Discussion & Illustrations of Standard G.CO.4:

10 10 G.CO.5 (Sense-Making Concept Standard) Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Given a geometric figure, Produce the image of the figure under a rotation, reflection, or translation using graph paper, tracing paper, or geometry software. Describe and justify the sequence of transformations that will carry a given figure onto another. Characteristics of transformations (translations, rotations, reflections, and dilations). Techniques for producing images under transformations using graph paper, tracing paper, or geometry software. Accurately perform rotations, reflections, and translations on objects using graph paper, tracing paper, or geometry software. Communicate the results of performing transformations on objects. The same transformation may be produced using a variety of tools, but the geometric sequence of steps that describe the transformation is consistent. Any distance preserving transformation is a combination of rotations, reflections, and translations. Sequence of Transformations Notation: A A A Related Standards: 8.G.2, 8.G.3, A.REI.3, A.REI.2, A.REI.4b Further Discussion & Illustrations of Standard G.CO.5:

11 11 G.CO.6 (Sense-Making Concept Standard) Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions. Given geometric descriptions of rigid motions, Predict the effect of the rigid motion on a given figure. Produce the image of a figure under the transformation. Compare and contrast the predictions to the actual transformation. Rigid motion Characteristics of translations, rotations, and reflections including the definition of congruence. Techniques for producing images under transformations using graph paper, tracing paper, compass, or geometry software. Geometric terminology (e.g., angles, circles, perpendicular lines, parallel lines, and line segments) which describes the series of steps necessary to produce a rotation, reflection, or translation. Use geometric descriptions of rigid motions to accurately perform these transformations on objects. Communicate the results of performing transformations on objects. Any distance preserving transformation is a combination of rotations, reflections, and translations. It a series of translations, rotations, and reflections can be described that transforms one object exactly to a second object, the objects are congruent. Rigid Motion Congruent Figures Notation: Related Standards: 8.G.2 Further Discussion & Illustrations of Standard G.CO.6: