# Representing and Combining Transformations

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## Description

- Overview:
- This lesson unit is intended to help teachers assess how well students are able to: recognize and visualize transformations of 2D shapes; and translate, reflect and rotate shapes, and combine these transformations. It also aims to encourage discussion on some common misconceptions about transformations.

- Level:
- Lower Primary, Upper Primary, Middle School, High School
- Grades:
- Kindergarten, Grade 1, Grade 2, Grade 3, Grade 4, Grade 5, Grade 6, Grade 7, Grade 8, Grade 9, Grade 10, Grade 11, Grade 12
- Material Type:
- Assessment, Lesson Plan
- Provider:
- Shell Center for Mathematical Education, U.C. Berkeley
- Provider Set:
- Mathematics Assessment Project (MAP)
- Date Added:
- 04/26/2013

- License:
- Creative Commons Attribution Non-Commercial No Derivatives
- Media Format:
- Downloadable docs, Text/HTML

# Comments

## Standards

Cluster: Understand congruence and similarity using physical models, transparencies, or geometry software

Standard: Verify experimentally the properties of rotations, reflections, and translations:

Degree of Alignment: 2 Strong (1 user)

Cluster: Understand congruence and similarity using physical models, transparencies, or geometry software

Standard: 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. For example, arrange three copies of the same triangle so that the three angles appear to form a line, and give an argument in terms of transversals why this is so.

Degree of Alignment: Not Rated (0 users)

Cluster: Understand and apply the Pythagorean Theorem

Standard: Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

Degree of Alignment: Not Rated (0 users)

Cluster: Mathematical practices

Standard: Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

Degree of Alignment: Not Rated (0 users)

Cluster: Mathematical practices

Standard: Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

Degree of Alignment: Not Rated (0 users)

Cluster: Understand congruence and similarity using physical models, transparencies, or geometry software

Standard: Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.

Degree of Alignment: Not Rated (0 users)

Cluster: Mathematical practices

Standard: Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.

Degree of Alignment: Not Rated (0 users)

Cluster: Solve real-world and mathematical problems involving volume of cylinders, cones and spheres

Standard: Know the formulas for the volume of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

Degree of Alignment: Not Rated (0 users)

Standard: Lines are taken to lines, and line segments to line segments of the same length.

Degree of Alignment: Not Rated (0 users)

Standard: Angles are taken to angles of the same measure.

Degree of Alignment: Not Rated (0 users)

Standard: Parallel lines are taken to parallel lines.

Degree of Alignment: Not Rated (0 users)

Standard: 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.

Degree of Alignment: Not Rated (0 users)

Standard: Describe the effect of dilations, translations, rotations and reflections on two-dimensional figures using coordinates.

Degree of Alignment: Not Rated (0 users)

Cluster: Understand and apply the Pythagorean Theorem

Standard: Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

Degree of Alignment: Not Rated (0 users)

Cluster: Understand and apply the Pythagorean Theorem

Standard: Explain a proof of the Pythagorean Theorem and its converse.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Geometry

Standard: Understand congruence and similarity using physical models, transparencies, or geometry software

Indicator: Describe the effect of dilations, translations, rotations and reflections on two-dimensional figures using coordinates.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Geometry

Standard: Understand congruence and similarity using physical models, transparencies, or geometry software

Indicator: 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.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Geometry

Standard: Understand congruence and similarity using physical models, transparencies, or geometry software

Indicator: Verify experimentally the properties of rotations, reflections, and translations:

Degree of Alignment: Not Rated (0 users)

Learning Domain: Geometry

Indicator: 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. For example, arrange three copies of the same triangle so that the three angles appear to form a line, and give an argument in terms of transversals why this is so.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Geometry

Indicator: Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Geometry

Standard: Understand and apply the Pythagorean Theorem

Indicator: Explain a proof of the Pythagorean Theorem and its converse.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Geometry

Indicator: Parallel lines are taken to parallel lines.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Geometry

Indicator: Angles are taken to angles of the same measure.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Geometry

Indicator: Lines are taken to lines, and line segments to line segments of the same length.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Geometry

Standard: Understand and apply the Pythagorean Theorem

Indicator: Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Geometry

Standard: Solve real-world and mathematical problems involving volume of cylinders, cones and spheres

Indicator: Know the formulas for the volume of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Mathematical Practices

Standard: Mathematical practices

Indicator: Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Geometry

Standard: Understand and apply the Pythagorean Theorem

Indicator: Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Mathematical Practices

Standard: Mathematical practices

Indicator: Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and"Óif there is a flaw in an argument"Óexplain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Mathematical Practices

Standard: Mathematical practices

Indicator: Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?"ť They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

Degree of Alignment: Not Rated (0 users)

## Evaluations

# Achieve OER

Average Score (3 Points Possible)Degree of Alignment | 2 (1 user) |

Quality of Explanation of the Subject Matter | 3 (1 user) |

Utility of Materials Designed to Support Teaching | 3 (1 user) |

Quality of Assessments | N/A |

Quality of Technological Interactivity | N/A |

Quality of Instructional and Practice Exercises | N/A |

Opportunities for Deeper Learning | N/A |

# Tags (10)

- Mathematics
- Geometry and measures
- 2D
- CCSS
- Common Core Math
- Common Core PD
- Geometry
- ODE Learning
- Shapes
- Transformations

This lesson is good at connecting from one transformation to doing multiple transformations. However, it asked students to do rotations and reflections that are above the Smarter Balance Consortium's specific item specifications for 8G. In the specifications the Target Specific Attributes say that that rotations should only be done around the origin in multiplies of 90 degrees, and reflections done over the axis. In addition, it does not use the coordinate plane notation.

This is a lesson/assessment tool. It has many teacher resources like detailed lessons, student worksheets, card sets for students to match, and tips to help students get started on the task. It is addressing CCSS 8.G.1. It does not have students explore all properties of transformations, reflections, and rotations, but it does connect a series of transformations to a single transformation.