## Description

- Overview:
- This lesson unit is intended to help teachers assess how well students are able to: work with concepts of congruency and similarity, including identifying corresponding sides and corresponding angles within and between triangles; Identify and understand the significance of a counter-example; Prove, and evaluate proofs in a geometric context.

- 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: 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: 3 Superior (1 user)

Cluster: Understand congruence in terms of rigid motions

Standard: Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

Degree of Alignment: 3 Superior (1 user)

Cluster: Understand congruence in terms of rigid motions

Standard: Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

Degree of Alignment: 3 Superior (1 user)

Cluster: Mathematical practices

Standard: Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x^2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)^2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

Degree of Alignment: 2 Strong (1 user)

# Common Core State Standards Math

Grades 9-12,Geometry: Similarity, Right Triangles, and TrigonometryCluster: Prove theorems involving similarity

Standard: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Degree of Alignment: 2 Strong (1 user)

# Common Core State Standards Math

Grades 9-12,Geometry: Similarity, Right Triangles, and TrigonometryCluster: Understand similarity in terms of similarity transformations

Standard: Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

Degree of Alignment: 2 Strong (1 user)

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: 1 Limited (1 user)

# Common Core State Standards Math

Grades 9-12,Geometry: Similarity, Right Triangles, and TrigonometryCluster: Define trigonometric ratios and solve problems involving right triangles

Standard: Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

Degree of Alignment: 0 Very Weak (1 user)

# Common Core State Standards Math

Grades 9-12,Geometry: Similarity, Right Triangles, and TrigonometryCluster: Prove theorems involving similarity

Standard: Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.

Degree of Alignment: 0 Very Weak (1 user)

Learning Domain: Geometry: Similarity, Right Triangles, and Trigonometry

Standard: Understand similarity in terms of similarity transformations

Indicator: Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

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: Geometry: Congruence

Standard: Understand congruence in terms of rigid motions

Indicator: Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Mathematical Practices

Standard: Mathematical practices

Indicator: Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 x 8 equals the well remembered 7 x 5 + 7 x 3, in preparation for learning about the distributive property. In the expression x^2 + 9x + 14, older students can see the 14 as 2 x 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 - 3(x - y)^2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Geometry: Similarity, Right Triangles, and Trigonometry

Standard: Prove theorems involving similarity

Indicator: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Geometry: Similarity, Right Triangles, and Trigonometry

Standard: Prove theorems involving similarity

Indicator: Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.

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: Congruence

Standard: Understand congruence in terms of rigid motions

Indicator: Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Geometry: Similarity, Right Triangles, and Trigonometry

Standard: Define trigonometric ratios and solve problems involving right triangles

Indicator: Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

Degree of Alignment: Not Rated (0 users)

## Evaluations

# Achieve OER

Average Score (3 Points Possible)Degree of Alignment | 1.8 (1 user) |

Quality of Explanation of the Subject Matter | 2 (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 (11)

- Mathematics
- Geometry and measures
- Angles
- CCSS
- Common Core Math
- Common Core PD
- Congruency
- Geometric Proofs
- Geometry
- ODE Learning
- Triangles

Students work with concepts of congruency and similarity, by identifying corresponding sides and corresponding angles within and between triangles. They identify and understand the significance of an example and of counter-examples in the development of proofs. They evaluate proofs within a geometric context.

Authors suggest that unit covers G-SRT 4 and G-SRT 5. However the connections are not evident in lesson design.

Lesson plans provide instructional strategies along with sample questioning strategies to build student understanding. It also provides potential student misconceptions and questions designed to build understanding.

Instructional strategies build opportunities for students to utilize mathematical practices 3 and 7. Mathematical Practice 5 was rated as a level 1 because the current lesson design does not require students to determine appropriate / strategic use of available tools for the learning process (i.e., students are given tools to use).