Congruence: Rigid Motions of Triangles

Congruence: Rigid Motions of Triangles Mathematics, Grade 10 Standards (Alignment) Content Standards 10.G-CO.B Understand congruence in terms of rigid motions. 10.G-CO.B.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. 10.G-CO.A Experiment with transformations in the plane. 10.G-CO.A.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. 10.G-SRT.A Understand similarity in terms of similarity transformations. 10.G-SRT.A.1 Verify experimentally the properties of dilations given by a center and a scale factor. 10.G-SRT.A.1.b The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 4. Model with mathematics. 6. Attend to precision. 7. Look for and make use of structure. DOK: 3 2015 Measured Progress. All rights reserved. Content owned by Measured Progress and licensed for use by the State of Maryland. 10_G-CO.B.6_S_11

Task Mrs. Smith has a large grid drawn on a whiteboard at the front of her classroom. She and her geometry students use an erasable marker to plot shapes on it. She plots Triangle 1, which is shown on the grid below. Part A Mrs. Smith asks Marisa to transform Triangle 1 using the rule (x, y) (x + 2, y + 3) and then draw the resulting triangle (Triangle 2) on the whiteboard. Draw Triangle 2 on the grid below. 2015 Measured Progress. All rights reserved. Content owned by Measured Progress and licensed for use by the State of Maryland. 10_G-CO.B.6_S_11

Part B Next, Mrs. Smith asks another student, James, to write a rule that would reflect a figure across the line y = 3. She then asks him to transform Triangle 2 using this rule and draw the new triangle (Triangle 3) on the whiteboard. What is a rule that reflects a figure across the line y = 3? Draw Triangle 3 on the grid below. 2015 Measured Progress. All rights reserved. Content owned by Measured Progress and licensed for use by the State of Maryland. 10_G-CO.B.6_S_11

Part C Mrs. Smith then asks a third student, Nikki, to describe a series of transformations that could be used to go from Triangle 3 to Triangle 1. What is a possible series of transformations that Nikki could describe? Part D Mrs. Smith gives a group of students a list of five different transformations of Triangle 3 to draw as follows: Transform Triangle 3 into Triangle 4 by rotating 90 counterclockwise around the point (1, 1). Transform Triangle 3 into Triangle 5 using the rule (x, y) (2x, y). Transform Triangle 3 into Triangle 6 using the rule (x, y) ( y, x 2). Transform Triangle 3 into Triangle 7 by translating 3 units right, then reflecting across the line y = x. 2015 Measured Progress. All rights reserved. Content owned by Measured Progress and licensed for use by the State of Maryland. 10_G-CO.B.6_S_11

Transform Triangle 3 into Triangle 8 using the rule 1 1 ( x, y) x, y 3. 3 3 Which of Triangles 2 through 8 are congruent to Triangle 1? Provide evidence, including why each triangle is or is not congruent to Triangle 1, to support your answers. Part E For each triangle in Part D that is not congruent to Triangle 1, Mrs. Smith asks the group to give a transformation that would make it congruent to Triangle 1. List these transformations, if any. 2015 Measured Progress. All rights reserved. Content owned by Measured Progress and licensed for use by the State of Maryland. 10_G-CO.B.6_S_11

Rubric Standard/Practice Level 1 Level 2 Level 3 Part A 10.G-CO.B.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. Attempts, given a geometric figure and a translation, to draw the transformed figure using graph paper. Given a geometric figure and a translation, draws the transformed figure using graph paper. However, reasoning and/or computation is flawed, resulting in an incorrect answer. Given a geometric figure and a translation, draws the transformed figure using graph paper. (See graph below.) 10.G-CO.A.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Part B 10.G-CO.B.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid Attempts to use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure, and to Uses geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure, and draws the Uses geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure, and draws the 2015 Measured Progress. All rights reserved. Content owned by Measured Progress and licensed for use by the State of Maryland. 10_G-CO.B.6_S_11

motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. draw the transformed figure. transformed figure. However, reasoning and/or computation is flawed, resulting in an incorrect answer. transformed figure. (The rule is: ( x, y) ( x, y 6) See the graph below.) 10.G-CO.A.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Part C 10.G-CO.A.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Attempts to specify a sequence of transformations that will carry a given figure onto another. Specifies a sequence of transformations that will carry a given figure onto another. However, reasoning and/or computation is flawed, resulting in an incorrect answer. Specifies a sequence of transformations that will carry a given figure onto another. Reflect Triangle 3 over the x-axis, then translate 2 units left and 3 units up. (There are other correct responses.) 2015 Measured Progress. All rights reserved. Content owned by Measured Progress and licensed for use by the State of Maryland. 10_G-CO.B.6_S_11

Part D 10.G-CO.B.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. 10.G-SRT.A.1.b The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Part E 10.G-CO.A.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Attempts to use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure. Attempts to describe the dilations necessary for congruence. Uses geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure. However, reasoning and/or computation is flawed, resulting in an incorrect answer. Describes the dilations necessary for congruence. However, reasoning and/or computation is flawed, resulting in an incorrect answer. Uses geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure. (Triangles 2, 3, 4, 6, and 7 are congruent to Triangle 1, because they all can be created from one another using translations, rotations, and reflections. Triangles 5 and 8 are not congruent to Triangle 1 as they are formed by dilations of Triangle 1 using a scale factor other than 1. ) Describes the dilations necessary for congruence. (Triangles 5 and 8 are not congruent to Triangle 1. Triangle 5 could be made congruent by transforming using this rule: (x, y) (0.5x, y). Triangle 8 could be made congruent by transforming using this rule: (x, y) (3x, 3y). ) 10.G-SRT.A.1.b The dilation of a line segment is longer or 2015 Measured Progress. All rights reserved. Content owned by Measured Progress and licensed for use by the State of Maryland. 10_G-CO.B.6_S_11

shorter in the ratio given by the scale factor. MP 1. Make sense of problems and persevere in solving them. MP 4. Model with mathematics. MP 6. Attend to precision. MP 7. Look for and make use of structure. Attempts to analyze the givens, constraints, relationships, or goals. Attempts to develop a solution path. Attempts to relate the problem/situation to a concept or skill that was previously learned. Attempts to model mathematics with geometrical, graphical, tabular, algebraic, or statistical representations. Attempts to give explanations. Attempts to recognize or use patterns, compositions, and/or structures in geometrical shapes or collections of figures. Shows analysis of some of the givens, constraints, relationships, and goals but with errors. Develops a solution path, but it does not work or it is not used. Relates a part of the problem/situation to a concept or skill that was previously learned, but there are gaps in the explanation. Models mathematics with geometrical, graphical, tabular, algebraic, or statistical representations but with flaws. Gives explanations but with errors. Recognizes and uses patterns, compositions, and/or structures in geometrical shapes or collections of figures but with errors. Shows correct analysis of all of the givens, constraints, relationships, and goals. Develops a solution path that works. Relates current problem/situation to a concept or skill previously learned. Correctly models mathematics with geometrical, graphical, tabular, algebraic, or statistical representations to fully describe situations. Gives clear, correct, and complete explanations. Recognizes and uses patterns, compositions, and/or structures in geometrical shapes or collections of figures. 2015 Measured Progress. All rights reserved. Content owned by Measured Progress and licensed for use by the State of Maryland. 10_G-CO.B.6_S_11