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Common Core Learning Standards GRADE 8 Mathematics GEOMETRY Common Core Learning Standards Understand congruence and similarity using physical models, transparencies, or geometry software. 8.G.1. Verify experimentally the properties of rotations, reflections, and translations: 8.G.1a. Lines are taken to lines, and line segments to line segments of the same length. Concepts Embedded Skills Transformations Translate, rotate, and reflect lines and line segments. Explain the preservation of the sides of a figure through a given transformation. Identify corresponding parts between a figure and its image using prime notation. Show that lines are taken to lines and line segments are taken to line segments. Vocabulary Rotation Reflection Translation Congruence Transformation Corresponding parts Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only. Rigorous Sample Tasks 1) Part A: Given triangle ABC below, draw the image after a transformation that preserves size and shape but NOT orientation. Scaffolded Sample Tasks 1.) Describe the given notation with its appropriate transformation. a) __________________________________________________ b) __________________________________________________ c) __________________________________________________ d) __________________________________________________ e) T(-3,5) __________________________________________________ 2.) Part B: Identify the transformation that you used to create your image. ______________________ Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only. What type of transformation is being demonstrated? _________________________________ Is orientation of triangle MATH preserved? Explain your answer. _________________________________________________________ Is the size of triangle MATH preserved? Explain your answer. _________________________________________________________ Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only. Common Core Learning Standards Understand congruence and similarity using physical models, transparencies, or geometry software. Concepts Embedded Skills Transformations Translate, rotate, and reflect geometric shapes on 8.G.1. Verify experimentally the properties of rotations, reflections, and translations: 8.G.1b. Angles are taken to angles of the same measure. a coordinate plane. Measure angles using a protractor. Identify corresponding parts between a figure and its image using prime notation. Show that angles are taken to angles of the same measure. Rigorous Sample Tasks 1) Part A: The triangle on the left was reflected across the vertical line resulting in the triangle on the right. Using what you know about line reflections, determine the value of the missing angle, x. Vocabulary Rotation Reflection Translation Congruence Properties Transformation Corresponding parts Scaffolded Sample Tasks 1) Given the clockwise rotation, label the appropriate vertices with A’, B’, and C’. Part B: The image on the right is then dilated with a scale factor of 2 . Determine the value of the angle that corresponds to 3 angle x. Justify your result. Explain why you chose the labels for each vertex. _________________________________________________________ _________________________________________________________ Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only. Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only. Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only. Common Core Learning Standards Understand congruence and similarity using physical models, transparencies, or geometry software. Concepts Embedded Skills Transformations Translate, rotate, and reflect parallel lines on a 8.G.1. Verify experimentally the properties of rotations, reflections, and translations: 8.G.1c. Parallel lines are taken to parallel lines. Rigorous Sample Tasks 1) coordinate plane. Explain that the slope and distance between two parallel lines will be preserved after a translation, rotation, or reflection of the parallel lines. Identify corresponding lines after a translation, rotation, or reflection, by using prime notation. Show that parallel lines are translated, rotated, or reflected parallel lines. Parallel lines Rotation Reflection Translation Transformation Slope/rate of change Corresponding parts Scaffolded Sample Tasks 1) What is the slope of the equation y line a Vocabulary 5 x 10 ? 6 line b Translate lines a and b, with the motion rule (x +2, y – 1). Label the appropriate images lines a’ and b’. Find the slopes of your translated images. Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only. slope of line a’ _________ slope of line b’ ___________ How do the slopes of your transformations compare with the slopes of the given parallel lines? ______________________________________________________ _____________________________________________________ 2.) Given the equation, y = 2x – 8, write the equation of a line that is parallel and has a y-intercept of -7. ______________________________ 3.) Given the lines below determine if they are parallel. Justify your result. y= 1 x 10 2 y = -2x +10 Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only. Common Core Learning Standards Understand congruence and similarity using physical models, transparencies, or geometry software. Concepts Congruent figures 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. Embedded Skills Explain the preservation of congruence when a figure is rotated, reflected, and/or translated. Describe the sequence of transformations that occurred from the original 2D figure to the image. Draw a reflection of an object. Draw a translation of an object. Draw a rotation of an object. Name corresponding parts in congruent figures. Vocabulary Transformation Reflection Rotation Translation Congruence Corresponding parts sequence SAMPLE TASKS 1) The two triangles in the picture below are congruent: a) Give a sequence of rotations, translations, and/or reflections which take b) Is it possible to show the congruence in part (a) using only translations and rotations? Explain. Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only. Common Core Learning Standards Understand congruence and similarity using physical models, transparencies, or geometry software. Concepts Embedded Skills Transformations Dilate a two-dimensional figure using coordinates. 8.G.3. Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. Rigorous Sample Tasks Describe the effect of dilating a two-dimensional figure using coordinates. Rotate a two-dimensional figure using coordinates. Describe the effect of rotating a two-dimensional figure using coordinates. Translate a two-dimensional figure using coordinates. Describe the effect of translating a twodimensional figure using coordinates. Reflect a two-dimensional figure using coordinates. Describe the effect of reflecting a two-dimensional figure using coordinates. Vocabulary Coordinate Figure Ordered pair Reflect Translate Dilate Rotate Transformation Prime Image X-axis Y-axis Scaffolded Sample Tasks Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only. Common Core Learning Standards Understand congruence and similarity using physical models, transparencies, or geometry software. Concepts Embedded Skills Similarity with Explain the preservation of similarity when a figure transformations is dilated, rotated, reflected, and/or translated. Describe the sequence of transformations that occurred from the original 2D figure to the image to show the similarity. 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. Rigorous Sample Tasks Vocabulary Rotation Reflection Translation Dilation Transformation Similarity Congruent Similar Scaffolded Sample Tasks Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only. Common Core Learning Standards Concepts Embedded Skills Understand congruence and similarity using physical models, transparencies, or geometry software. Prove/explain why the three angles of a triangle equal 180°. Prove/explain why the exterior angle theorem of a triangle. 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. Prove/explain why alternate interior angles are congruent. Prove/explain why alternate exterior angles are congruent. Rigorous Sample Tasks Prove/explain why corresponding angles are congruent. Prove/explain why angle-angle criterion works to prove similarity of two triangles. Vocabulary Triangle Similar Parallel lines Transversal Congruent Supplementary Linear pair Corresponding Vertical Alternate, exterior, interior angles Scaffolded Sample Tasks Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only. Common Core Learning Standards Understand and apply the Pythagorean Theorem. Concepts Pythagorean theorem 8.G.6. Explain a proof of the Pythagorean Theorem and its converse. Rigorous Sample Tasks Embedded Skills Explain a proof of Pythagorean theorem. (If a triangle is a right triangle, then a2 + b2 = c2) Explain a proof of the converse of Pythagorean theorem. (If a2 + b2 = c2, then a triangle is a right triangle) Identify the legs and hypotenuse of a right triangle. Solve multi-step equations. Vocabulary Pythagorean theorem Converse Proof Legs Hypotenuse Scaffolded Sample Tasks Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only. Common Core Learning Standards Understand and apply the Pythagorean Theorem. Concepts Embedded Skills Pythagorean theorem Calculate the length of a leg of a right triangle using Pythagorean theorem. Calculate the length of the hypotenuse of a right triangle using Pythagorean theorem. Calculate the diagonal of a three-dimensional figure using Pythagorean Theorem. 8.G.7. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in realworld and mathematical problems in two and three dimensions. Read and interpret a word problem involving Pythagorean Theorem. Solve word problems involving Pythagorean Theorem. Vocabulary Leg Hypotenuse Right angle Pythagorean theorem Square root Radical Diagonals Solve multi-step equations. Identify the legs and hypotenuse of a right triangle. Round to given place value. Rigorous Sample Tasks Scaffolded Sample Tasks Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only. Common Core Learning Standards Understand and apply the Pythagorean Theorem. 8.G.8. Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. Concepts Pythagorean theorem on a coordinate plane Embedded Skills Calculate the distance between two points in a coordinate plane using the Pythagorean Theorem. Plot points in a coordinate plane Solve multi-step equations Identify the legs and hypotenuse of a right triangle Solve Pythagorean Theorem Rigorous Sample Tasks Vocabulary Leg Hypotenuse Right angle Pythagorean theorem Ordered pair Coordinate plane Square root Distance formula Scaffolded Sample Tasks Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only. Common Core Learning Standards Concepts Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres. 8.G.9. Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. Embedded Skills Write and solve using the formula for the volume of a cone. Write and solve using the formula for the volume of a cylinder. Write and solve using the formula for the volume of a sphere. Solve word problems involving the volume of cones, cylinders, and spheres. Vocabulary Volume Cone Cylinder Sphere Area Base Formula Solve a multi-step equation for a missing variable. Rigorous Sample Tasks Scaffolded Sample Tasks Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.