Overview of Investigation 2
This Investigation has three Problems. The first Problem introduces the concept that figures with the same size and shape can be matched in a way that shows corresponding sides, angles, and vertices. The second Problem develops the students’ understanding of the ways in which the congruence of two triangles can be confirmed by transforming one onto the other. The third Problem develops the idea that the congruence of triangles can be determined without any transformation, but by matching the measures of three strategically chosen parts of each triangle.
Goals of Investigation 2
Congruence and Similarity
Understand congruence and similarity and explore necessary and sufficient conditions for establishing congruent and similar shapes
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 1 and 2
8.G.A.1a Lines are taken to lines, and line segments to line segments of the same length. Problems 1 and 2
8.G.A.1b Angles are taken to angles of the same measure. Problems 1 and 2
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. Problems 1, 2, and 3
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 8: Look for and express regularity in repeated reasoning.
After experimenting with the transformations in Investigation 1, the students compare the original figure and its image. They look for clues to help them determine which transformation they should test for first. For example, when they see the orientation of the figure is the same, they may automatically reason that the image is under translation. If the base of the original figure is transformed to the side of the image, they may test for rotation first. Similarly, if they can picture a mirror being in between the original figure and its image, they may reason that the design must have reflectional symmetry. The patterns in the designs in Investigation 1 helped them to develop their own reasoning to describe symmetry, which also helps them test for congruency in Investigation 2.
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 and 2
8.G.A.1b Angles are taken to angles of the same measure. Problems 1 and 2
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. Problems 1, 2, and 3
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 8: Look for and express regularity in repeated reasoning.
After experimenting with the transformations in Investigation 1, the students compare the original figure and its image. They look for clues to help them determine which transformation they should test for first. For example, when they see the orientation of the figure is the same, they may automatically reason that the image is under translation. If the base of the original figure is transformed to the side of the image, they may test for rotation first. Similarly, if they can picture a mirror being in between the original figure and its image, they may reason that the design must have reflectional symmetry. The patterns in the designs in Investigation 1 helped them to develop their own reasoning to describe symmetry, which also helps them test for congruency in Investigation 2.
Students identify and record their personal experiences with the Standards for Mathematical Practice during the Mathematical Reflections at the end of the Investigation.