Unit 3: Congruence & Similarity

Transcription

1 Approximate Time Frame: 6 weeks Connections to Previous Learning: In previous grades, students made scale drawings of geometric figures and solved problems involving angle measure, surface area, and volume. Focus of this Unit: Geometric sense allows students to comprehend space and shape while exploring what can make objects similar or congruent. Students analyze the characteristics and relationships of shapes and structures, engage in logical reasoning, and use tools and techniques to determine congruence or similarity and perform transformations on the coordinate plane. Connections to Subsequent Learning: In Unit 4 students formalize their understanding of functions and can use transformations as a model. The input is the pre-image, the output is the image, and the function itself is the transformation. At the high school level, students will apply transformations to numbers, functional representations, and data. They will experiment with transformations in the plane where they will do rigid motion transformations, transformations in terms of a function, and transformations such as stretches that lead to shapes that are not similar or congruent. Desired Outcomes Standard(s): 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. Supporting Standards: 8.F.1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. 11/14/2013 3:40:24 PM Adapted from UbD framework Page 1

2 Transfer: Students will apply concepts and procedures regarding functions to represent, describe, and interpret transformations on geometric figures. Students will apply concepts of angle congruence in solving problems involving geometric figures in the plane. Ex: Prove the sum of angles in a triangle using alternate interior and alternate exterior angles. Ex: Give a function rule that takes an input of a point on the coordinate plane and gives an output of a point that is a rotation about the origin by input. of the Understandings: Students will understand that 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. If the scale factor of a dilation is greater than 1, the image resulting from the dilation is an enlargement, and if the scale factor is less than 1, the image is a reduction. A two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of transformations. Two shapes are similar if the length 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 corresponding sides. Congruent figures have the same size and shape. If the scale factor of a dilation is equal to 1, the image resulting from the dilation is congruent to the original figure. When parallel lines are cut by a transversal, corresponding angles, alternate interior angles, alternate exterior angles, and vertical angles are congruent. Essential Questions: What are transformations and what effect do they have on an object? What does the scale factor of a dilation convey? How can transformations be used to determine congruency or similarity? What angle relationships are formed by a transversal? 11/14/2013 3:40:24 PM Adapted from UbD framework Page 2

3 Mathematical Practices: (Practices to be explicitly emphasized are indicated with an *.) * 1. Make sense of problems and persevere in solving them. Students solve problems through the application of algebraic and geometric concepts. Students seek the meaning of a problem and look for efficient ways to represent and solve it. 2. Reason abstractly and quantitatively. * 3. Construct viable arguments and critique the reasoning of others. Students will justify their reasoning for choosing a particular sequence that led from the pre-image to the image. * 4. Model with mathematics. Students will manipulate figures to learn the properties of similar and congruent figures. Geometric transformations can be modeled via abstract math, such as the translation vector in translations, rotational formulas, scale factor dilations, and reflections across an axis. * 5. Use appropriate tools strategically. Students might draw pictures, use applets, or write equations to show the relationships between the angles created by a transversal. Geometric software can be incorporated into this unit especially to show transformations or a series of transformations. 6. Attend to precision. * 7. Look for and make use of structure. Students routinely seek patterns or structures to model and solve problems. Students will experimentally verify the effects of transformations and describe them in terms of congruence and similarity. 8. Look for and express regularity in repeated reasoning. Prerequisite Skills/Concepts: Students should already be able to Draw, construct, and describe geometrical figures and describe the relationships between them. (7.G.1-3) Solve real-life and mathematical problems involving angle measure, area, surface area, and volume. (7.G.4-6) Advanced Skills/Concepts: Some students may be ready to Find angle measures and patterns created by transversals with non-parallel lines. Find the vertices of the original per-image given an image and a series of transformations that had been performed. Use transformation notation including scale factor,, for a dilation yielding points, translation vectors, rotation angles, and lines of reflection. Perform rotations with centers of rotations other than, and reflections across lines other than the axes. 11/14/2013 3:40:24 PM Adapted from UbD framework Page 3

4 Knowledge: Students will know Properties of rotations, reflections and translations. (8.G.1) Exterior angle and angle sum of triangles. (8.G.5) Angles created when parallel lines are cut by a transversal. (8.G.5) Angle-angle criterion for similarity of triangles. Skills: Students will be able to Describe a series of transformations that exhibits congruence between two congruent figures. (8.G.2) Describe transformations (dilations, translations, rotations, and reflections) with words and with coordinates. Note that dilations can have centers other than. (8.G.3) Describe a series of transformations that exhibits similarity between two similar figures. (8.G.4) Find the measures of angles using transversals, the sum of angles in a triangle, the exterior angles of triangles. (8.G.5) Determine if triangles are similar using the angle-angle criterion. (8.G.5) Justify congruence or similarity of figures using a series of transformations. (8.G.2 and 8.G.4) WIDA Standard (English Language Learners): English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics. English language learners benefit from: Vocabulary of Transformational Geometry to be explicitly taught using tactile and virtual tools (ex: software tools, geometric shapes). Real world examples to reinforce the vocabulary of Geometry. Critical Terms: Transformations Pre-image Image Translation Rotation Center of Rotation Angle of rotation Reflection Line of reflection Dilations Center of Dilation Transversal Exterior angles Interior angles Academic Vocabulary: Supplemental Terms: Line segments Parallel lines Congruent (congruency) Symmetry Similarity Corresponding Scale factor 11/14/2013 3:40:24 PM Adapted from UbD framework Page 4

5 Assessment Pre-Assessments Formative Assessments Summative Assessments Self-Assessments Prior Knowledge Pre-Test Congruence and Similarity Pre-Test Transformation Matching Modeling Dilations Describing Dilations Transformation Twister Transformation Challenge Transformation Creation Good Better Best Sample Lesson Sequence 1 8.G.1,3 Effects of Transformations and Series of Transformations (model lesson). 2 8.G.2,4 Identifying Sequences of Transformations. Congruence and Similarity Post-Test Whack a Mole Activity Dilation Practice Reflection Practice Rotation Practice Translation Practice 3 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. 11/14/2013 3:40:24 PM Adapted from UbD framework Page 5