Overview of Investigation 1
This Investigation has four Problems. The first Problem reviews the students’ understanding of mirror or line symmetry and develops the definition of line reflection transformations. They will then test for and create such symmetries. The second Problem reviews the students’ understanding of rotational symmetry and develops the definition of rotation transformations. The third Problem develops the idea of translations and translational symmetry. The fourth Problem summarizes basic properties of shapes that are preserved by these rigid motion transformations, or transformations that preserve shape and size.
Goals of Investigation 1
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
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
Common Core Content Standards
8.G.A.1 Verify experimentally the properties of rotations, reflections, and translations. Problems 1, 2, 3, and 4
8.G.A.1a Lines are taken to lines, and line segments to line segments of the same length. Problems 1, 2, 3, and 4
8.G.A.1b Angles are taken to angles of the same measure. Problems 1, 2, 3, and 4
8.G.A.1c Parallel lines are taken to parallel lines. Problem 3 and 4
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 1.3, the students write a definition for the translation of given points. They know that a slide or translation transformation matches each point to its image. Some students will use the information to argue that if the image of two points is under translation, the distance between the points is equal to the distance between the two original points. Other students may argue that if a line connects each set of points, the two lines should have the same slope because a translation does not change the orientation of points on any figure. Working as group, the students will discuss their assumptions and develop an accurate definition.
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.1a Lines are taken to lines, and line segments to line segments of the same length. Problems 1, 2, 3, and 4
8.G.A.1b Angles are taken to angles of the same measure. Problems 1, 2, 3, and 4
8.G.A.1c Parallel lines are taken to parallel lines. Problem 3 and 4
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 1.3, the students write a definition for the translation of given points. They know that a slide or translation transformation matches each point to its image. Some students will use the information to argue that if the image of two points is under translation, the distance between the points is equal to the distance between the two original points. Other students may argue that if a line connects each set of points, the two lines should have the same slope because a translation does not change the orientation of points on any figure. Working as group, the students will discuss their assumptions and develop an accurate definition.
Students identify and record their personal experiences with the Standards for Mathematical Practice during the Mathematical Reflections at the end of the Investigation.