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FAYETTE COUNTY PUBLIC SCHOOLS District Curriculum Map for Mathematics: Grade 8 Topic 8G Big Idea(s) What enduring understandings are essential for application to new situations within or beyond this content? Essential Question(s) What questions will provoke and sustain student engagement while focusing learning? Enduring Standard(s) Which standards provide endurance beyond the course, leverage across multiple disciplines, and readiness for the next level? Transformations, Similarity & Congruence Transformations: Translations Reflections Rotations Dilations Similar Figures Parallel Lines Congruence Understand congruence and similarity using physical models, transparencies, or geometry software. What do transformations represent? How can I use a model to show congruencies exist when parallel lines are cut by a transversal? Enduring Understandings Coordinate geometry can be a useful tool for understanding geometric shapes and transformations. Reflections, translations, and rotations are actions that produce congruent geometric objects. A dilation is a transformation that changes the size of a figure, but not the shape. The notation used to describe a dilation includes a scale factor and a center of dilation. A dilation of scale factor k with the center of dilation at the origin may be described by the notation (kx, ky). If the scale factor of a dilation is greater than 1, the image resulting from the dilation is an enlargement. If the scale factor is less than 1, the image is a reduction. Two shapes are similar if the lengths of all the corresponding sides are proportional and all the corresponding angles are congruent. Two similar figures are related by a scale factor, which is the ratio of the lengths of the corresponding sides. Congruent figures have the same size and shape. If the scale factor of a dilation is equal to one, the image resulting from the dilation is congruent to the original figure. When parallel lines are cut by a transversal, corresponding, alternate interior and alternate exterior angles are congruent. Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. Curriculum and Instruction 2014-2015 Page 1 of 6 FAYETTE COUNTY PUBLIC SCHOOLS District Curriculum Map for Mathematics: Grade 8 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. Supporting Standard(s) Which related standards will be incorporated to support and enhance the enduring standards? Instructional Outcomes What must students learn by the end of the unit? Curriculum and Instruction Standards for Mathematical Content Geometry Understand congruence and similarity using physical models, transparencies, or geometry software. 8.G.A.1 Verify experimentally the properties of rotations, reflections, and translations: 8.G.A.1a Lines are taken to lines, and line segments to line segments of the same length. 8.G.A.1b Angles are taken to angles of the same measure. 8.G.A.1c Parallel lines are taken to parallel lines. 8.G.A.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.A.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. 8.G.A.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 twodimensional figures, describe a sequence that exhibits the similarity between them. 8.G.A.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. I can… determine if one object is a transformation of another. transform a figure using rotations, reflections, translations. determine a sequence of transformations between two congruent figures. dilate, translate, rotate, and reflect a figure in the coordinate plane. describe the effects of a dilation, reflection, rotation, or translation on the coordinates of a figure in the coordinate plane. determine the sequence of transformations between two similar objects. 2014-2015 Page 2 of 6 FAYETTE COUNTY PUBLIC SCHOOLS District Curriculum Map for Mathematics: Grade 8 calculate the angle sum of a triangle(polygon). calculate the measure of an exterior angle of a triangle(polygon). find the angles formed when parallel lines are cut by a transversal. determine if two triangles are similar. Students who demonstrate understanding can… verify experimentally the properties of rotations, reflections, and translations. understand and describe a sequence of transformations to create congruent figures. describe the effects of rotations, reflections, translations and dilations of two dimensional congruent and similar figures using coordinates. determine the angle sum and exterior angles of triangles. describe and determine angles created when parallel lines are cut by a transversal. describe similarity of triangles using the angle-angle criterion. Essential Vocabulary Performance Expectations What must students be able to do by the end of the unit to demonstrate their mastery of the instructional outcomes? Essential Vocabulary What vocabulary must students know to understand and communicate effectively about this content? adjacent angles - Two angles that share both a side and a vertex. Alternate Interior Angles - When two lines are crossed by another line (which is called a transversal), the pairs of angles on opposite sides of the transversal but inside the two lines are called alternate interior angles. corresponding angles - When two lines are crossed by a transversal, the angles in matching corners are called corresponding angles. congruent - Two plane or solid figures are congruent if one can be obtained from the other by rigid motion (a sequence of rotations, reflections, and translations). Having the same size and shape. dilation - A transformation that moves each point along the ray through the point emanating from a fixed center, and multiplies distances from the center by a common scale factor. exterior angle - An exterior angle is the angle between one side of a polygon and the extension of an adjacent side. An exterior angle of a triangle is equal to the sum of the opposite interior angles. interior angle - An angle whose sides are determined by two consecutive sides of a polygon. line of symmetry - A line across the figure such that the figure can be Curriculum and Instruction 2014-2015 Page 3 of 6 FAYETTE COUNTY PUBLIC SCHOOLS District Curriculum Map for Mathematics: Grade 8 folded along the line into matching parts; Line that divides a geometric figure into two congruent portions. parallel lines - Coplanar lines that do not intersect. perpendicular lines - A line that forms a right angle with another line or segment. reflection - A transformation resulting from a flip. rotation - A transformation in which a figure is rotated through a given angle, about a point. scale factor - A number which multiples some quantity; the ratio of any two corresponding lengths in two similar geometric figures. similar polygons - Two polygons are similar if their corresponding sides are proportional. supplementary - Two angles are supplementary if their sum is 180 degrees. transformation - A change in the position, shape, or size of a geometric figure. translation - A transformation, or change in position, resulting from a slide with no turn. transversal - A line that intersects two other lines. vertical angles - A pair of opposite angles that is formed by intersecting lines. Supporting Vocabulary alternate exterior angles angle of rotation enlargement image line of reflection reduction reflection symmetry rotational symmetry Common Core Glossary Curriculum and Instruction 2014-2015 Page 4 of 6 FAYETTE COUNTY PUBLIC SCHOOLS District Curriculum Map for Mathematics: Grade 8 http://www.corestandards.org/Math/Content/mathematicsglossary/glossary/ Curriculum and Instruction 2014-2015 Page 5 of 6 Subject and Grade Level Unit Title Summative Assessment of Learning Mathematics 8G Transformations, Similarity, and Congruence In what way will students meet the performance expectations to demonstrate mastery of the standards? Instructional Outcomes How will the instructional outcomes be sequenced into a progression of learning? Learning Activities What well-designed progression of learning tasks will intellectually engage students in challenging content? Formal Formative Assessments What is the evidence to show students have learned the lesson objective and are progressing toward mastery of the instructional outcomes? Integration Standards What standards from other disciplines will enrich the learning experiences for the students? Resources Resources What resources will be utilized to enhance student learning? Curriculum and Instruction 2014-2015 Page 6 of 6