Overview of Investigation 3
This Investigation has five Problems. The first three Problems develop coordinate rules for line reflections, translations, and rotations. The fourth Problem develops the important special property of translations and half-turns that every line is mapped onto a parallel line. The fifth Problem applies that special result to review and prove important results about angles formed when parallel lines are cut by a transversal and about the angle sum of the interior or exterior angles of any triangle.
Goals of Investigation 3
Transformations
Describe types of transformations that relate points by the motions of reflections, rotations, and translations, and methods for identifying and creating symmetric plane figures
Congruence and Similarity
Understand congruence and similarity and explore necessary and sufficient conditions for establishing congruent and similar shapes
Describe types of transformations that relate points by the motions of reflections, rotations, and translations, and methods for identifying and creating symmetric plane figures
- Recognize properties of reflection, rotation, and translation transformations
- Explore techniques for using rigid motion transformations to create symmetric designs
- Use coordinate rules for basic rigid motion transformations
Congruence and Similarity
Understand congruence and similarity and explore necessary and sufficient conditions for establishing congruent and similar shapes
- Recognize that two figures are congruent if one is derived from the other one by a sequence of reflection, rotation, and/or translation transformations
- Recognize that two figures are similar if one can be obtained from the other by a sequence of reflections, rotations, translations, and/or dilations
- Use transformations to describe a sequence that exhibits the congruence between figures
- Use transformations to explore minimum measurement conditions for establishing congruence of triangles
- Use transformations to explore minimum measurement conditions for establishing similarity of triangles
- Relate properties of angles formed by parallel lines and transversals, and the angle sum in any triangle, to properties of transformations
- Use properties of congruent and similar triangles to solve problems about shapes and measurements
Common Core Content Standards
8.G.A.1 Verify experimentally the properties of rotations, reflections, and translations. Problems 4 and 5
8.G.A.1b Angles are taken to angles of the same measure. Problem 5
8.G.A.1c Parallel lines are taken to parallel lines. Problems 4 and 5
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. Problem 5
8.G.A.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. Problems 1, 2, 3, and 4
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. Problem 5
Facilitating the Mathematical Practices
Students in Connected Mathematics classrooms display evidence of multiple Common Core Standards for Mathematical Practice every day. Here are just a few examples of when you might observe students demonstrating the Standards for Mathematical Practice during this Investigation.
Practice 1: Make sense of problems and persevere in solving them.
Students are engaged every day in solving problems and, over time, learn to persevere in solving them. To be effective, the problems embody critical concepts and skills and have the potential to engage students in making sense of mathematics. Students build understanding by reflecting, connecting, and communicating. These student-centered problem situations engage students in articulating the “knowns” in a problem situation and determining a logical solution pathway. The student-student and student-teacher dialogues help students not only to make sense of the problems, but also to persevere in finding appropriate strategies to solve them. The suggested questions in the Teacher Guides provide the metacognitive scaffolding to help students monitor and refine their problem-solving strategies.
Practice 3: Construct viable arguments and critique the reasoning of others.
In Problem 4, students use transformations to write conclusions after observing that translations and half-turns “move” lines to parallel lines. In Problem 5, they use transformations to prove that there are pairs of congruent angles in parallel lines cut by a transversal. They also use transformations to prove that the sum of the interior angles in a triangle is 180° and that the sum of the exterior angles of a triangle is 360°.
Practice 7: Look for and make use of structure.
In Problems 1–3, students look for patterns in key coordinate points after applying a transformation or a sequence of transformations. After detecting the pattern, they write coordinate rules to describe the change of all coordinates under that transformation.
Practice 8: Look for and express regularity in repeated reasoning.
In Problem 3, students use repeated reasoning to understand the coordinate rules associated with 90° and 180° rotations around the center (0,0). They also use this Practice in Problem 5. In that Problem, they rotate triangles 180° around side midpoints to observe the resulting image and figure. From the parallelogram, they can prove that the sum of the interior angles of a triangle is 180° using their results about parallel lines cut by a transversal.
Students identify and record their personal experiences with the Standards for Mathematical Practice during the Mathematical Reflections at the end of the Investigation.
8.G.A.1b Angles are taken to angles of the same measure. Problem 5
8.G.A.1c Parallel lines are taken to parallel lines. Problems 4 and 5
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. Problem 5
8.G.A.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. Problems 1, 2, 3, and 4
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. Problem 5
Facilitating the Mathematical Practices
Students in Connected Mathematics classrooms display evidence of multiple Common Core Standards for Mathematical Practice every day. Here are just a few examples of when you might observe students demonstrating the Standards for Mathematical Practice during this Investigation.
Practice 1: Make sense of problems and persevere in solving them.
Students are engaged every day in solving problems and, over time, learn to persevere in solving them. To be effective, the problems embody critical concepts and skills and have the potential to engage students in making sense of mathematics. Students build understanding by reflecting, connecting, and communicating. These student-centered problem situations engage students in articulating the “knowns” in a problem situation and determining a logical solution pathway. The student-student and student-teacher dialogues help students not only to make sense of the problems, but also to persevere in finding appropriate strategies to solve them. The suggested questions in the Teacher Guides provide the metacognitive scaffolding to help students monitor and refine their problem-solving strategies.
Practice 3: Construct viable arguments and critique the reasoning of others.
In Problem 4, students use transformations to write conclusions after observing that translations and half-turns “move” lines to parallel lines. In Problem 5, they use transformations to prove that there are pairs of congruent angles in parallel lines cut by a transversal. They also use transformations to prove that the sum of the interior angles in a triangle is 180° and that the sum of the exterior angles of a triangle is 360°.
Practice 7: Look for and make use of structure.
In Problems 1–3, students look for patterns in key coordinate points after applying a transformation or a sequence of transformations. After detecting the pattern, they write coordinate rules to describe the change of all coordinates under that transformation.
Practice 8: Look for and express regularity in repeated reasoning.
In Problem 3, students use repeated reasoning to understand the coordinate rules associated with 90° and 180° rotations around the center (0,0). They also use this Practice in Problem 5. In that Problem, they rotate triangles 180° around side midpoints to observe the resulting image and figure. From the parallelogram, they can prove that the sum of the interior angles of a triangle is 180° using their results about parallel lines cut by a transversal.
Students identify and record their personal experiences with the Standards for Mathematical Practice during the Mathematical Reflections at the end of the Investigation.