Understanding Similarity

Transcription

1 Understanding Similarity Student Probe In Quadrilateral ABCD, m A 90, m B 140, andm C 60. In Quadrilateral WXYZ, m W 90, m X 140, andm Y 60. Is Quadrilateral ABCD similar to Quadrilateral WXYZ? Explain how you know. A B C D W Answer: Yes, because the corresponding angles are congruent and the corresponding sides have a scale factor of 2. Lesson Description In this lesson students will create a 2 dilation of a square and examine the characteristics and relationships of the square and its image. The data generated will be used to develop a conceptual understanding of similarity. X 8 Rationale Understanding similarity and proportionality and learning to apply them is pivotal to students mathematical success in problem solving. Scale drawings, such as maps and building plans, rely on the proportionality of similar figures to determine actual distances and lengths. Trigonometry uses similar triangles to define the basic functions. These functions, in turn, form the basis for navigation, determining the magnitude of forces, and even music and its sound waves Y 14 Z At a Glance What: Use a dilation to understand similarity Common Core State Standard: CC.9 12.G.SRT.2 Understand similarity in terms of similarity transformations. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Matched Arkansas Standard: AR.9 12.CGT.G.5.7 (CGT.5.G.7) Draw and interpret the results of transformations and successive transformations on figures in the coordinate plane: translations, reflections, rotations (90, 180, clockwise and counterclockwise about the origin), dilations (scale factor) Mathematical Practices: Reason abstractly and quantitatively. Model with mathematics. Attend to precision. Who: Students who do are unable to identify similar figures. Grade Level: Geometry Prerequisite Vocabulary: dilation, center of dilation, similar, factor, perimeter, area Prerequisite Skills: tracing shapes, using patty paper, labeling polygons, measuring length, measuring angles Delivery Format: Individual, pairs or small groups Lesson Length: 20 minutes Materials, Resources, Technology: ruler, grid paper, patty paper. Student Worksheets: Similar Shapes (.pdf)

2 Preparation Prepare copies of Similar Shapes for each student. Provide grid paper, rulers, and protractors for each student. If using dynamic software make sure students are familiar with use of the software and have an adequate level of proficiency. Lesson The teacher says or does 1. We are going to create a 2 dilation of the square using P as the center of dilation. Expect students to say or do If students do not, then the teacher says or does Monitor students. Model, if necessary. Use patty paper to trace ABCD and point P onto grid paper. 2. Using a ruler, place one end on P and measure the distance from P to A. What is the distance? 3. Keep your ruler on line segment PA. Double this distance and make a new point. Label the new point A, the image of A. 4. Repeat this for the remaining points B, C and D. 5. Compare the shapes of the figure. What do you notice? 1 inch They have the same shape. They are both squares. 6. What do you notice? It is still a square, but the sides are longer. 7. Why do you think this is called a 2 dilaltion of the square? The measure of the sides of the new image (square) are 2 times the original image. What is 1 inch times 2? Monitor students as they work. Is it still a square, or has it changed? Measure all sides in both images. Are they both squares? Check the process used and re measure sides of both images. Monitor precision of process and correct reading of ruler.

3 The teacher says or does 8. Compare the length of the line segments of the original figure and the new image. 9. Compare the angles of the image. 10. Compare the perimeter of the original figure and the new image. 11. Compare the areas of the image. 12. What are the characteristics of similar figures? 13. Repeat, using other shapes. Expect students to say or do The lengths of the new image are 2 times as long. The angles are the same size. The perimeter is twice as much. The area is four times as much or it increased by a squared factor. Similar figures have the same shape. Corresponding angles are the same measure. Corresponding size are in the same ratio. If students do not, then the teacher says or does Be careful in making your measurements. Make certain you are comparing corresponding sides. Measure again. Make certain you are using the correct scale. Be sure you are comparing corresponding angles. What is perimeter? How did you find the perimeter? Be sure you added all sides. How do you find area? What measures did you use? Review the previous answers and process. Teacher Notes 1. Figures that have the same shape but not necessarily the same size are similar figures. To say that two figures have the same shape but not necessarily the same size is not a precise definition of similarity. Two polygons are similar if and only if corresponding angles are congruent and the corresponding sides are proportional. 2. It is important that students draw and measure precisely. 3. If precise measurement and drawing is difficult for your students, you may want to adjust the lesson. One suggestion is to begin with pre image and image already constructed. Ask students to investigate the corresponding parts by measuring and recording in a table.

4 Variations 1. The activity can be conducted using dynamic geometry software such as Geometer s Sketchpad, Geogebra, Cabri or Cabri Jr. 2. Another geometric shape, such as a triangle or rectangle may be used. Formative Assessment Triangle ABC is shown below. What would be the lengths of the sides of a 3 dilation of ABC? C 10 6 B 12 A Answer: 18, 30, 36 References Driscoll, M. (2007). Fostering Geometric Thinking. Portsmouth NH: Heinemann. Russell Gersten, P. (n.d.). RTI and Mathematics IES Practice Guide Response to Intervention in Mathematics. Retrieved 2 25, 2011, from rti4sucess. Seago, N. (2009). Learning and Teaching Geometry: Promoting an Understanding of Similarity. CA: WestEd.

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