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G.CO.A.2 Lesson Seeds: Defining Similarity and Congruence Experiment with transformations in the plane. G.CO.A.2 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). Opening Question: Construct two figures for each of the following scenarios. a) figures that are different from each other b) figures that are almost the same as each other c) figures that are exactly the same as each other Allow students to work in groups or partners with multiple tools in order to construct each set of figures. Discussion Questions: These questions are intended to formatively assess prior knowledge and to begin student discussion of similarity versus congruence. Students should be encouraged to answer in their own words and to critique each other’s assessments. If students are unsure about an answer, have them revisit their tools from the opening question in order to make a more informed decision. 1. How do you know your figures were the same? If they are not exactly the same, how are they different? How did you interpret “almost the same”? Can you construct figures that are closer to being the same than your first pair but are still slightly different? 2. What do you think are the qualifications that make two figures exactly the same? 3. Can two figures be the same type of figure but be different from each other in multiple ways? 4. What makes two figures look like each other? 5. Can two figures look like each other but be different sizes? Can two figures be the same size but look like different figures? Which of these scenarios creates figures that look more alike? 6. How do angle measurements affect sameness? How do side lengths affect sameness? Which one has more of an effect on sameness? 7. What would it take for two figures to be proportional to each other? Other Activities: Have students construct two figures that are proportional. Instruct them to measure all sides and angles in order to make conjectures about proportional figures. Have students do the same with two identical figures, and discuss the 1:1 ratio of the second set of figures. Have students complete the graphic organizer for Similar and Congruent figures by writing information they know about each term as well as sketching examples. Have students construct multiple similar figures using appropriate tools, once they have developed a better understanding about similarity. Discuss proportionality using their figures. Have them do the same with congruent figures. Discuss which is more difficult to create as well as which has more options. Make the connection that congruence is one specific example of similarity that has a 1:1 ratio. Have students discuss transformations in relationship to similarity and congruence. Which transformations preserve congruence? Which create similar figures?