## Description

- Overview:
- This lesson unit is intended to help teachers assess how well students are able to: use the area of right triangles to deduce the areas of other shapes; use dissection methods for finding areas; organize an investigation systematically and collect data; deduce a generalizable method for finding lengths and areas (The Pythagorean Theorem.)

- 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 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: 3 Superior (1 user)

Cluster: Understand and apply the Pythagorean Theorem

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

Degree of Alignment: 3 Superior (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: 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: 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: 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: Not Rated (0 users)

Cluster: Mathematical practices

Standard: Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.

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)

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

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)

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

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: Mathematical Practices

Standard: Mathematical practices

Indicator: Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.

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: 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

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: 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 | 3 (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 (8)

- Mathematics
- Geometry and measures
- CCSS
- Common Core Math
- Common Core PD
- Geometry
- ODE Learning
- Pythagorean Theorem

This is a lesson/assessment unit. It has many teacher resources. Detailed lesson plans, student worksheets, ppt, and suggestions on misconceptions. It is addressing 8.G.6 and 8.G.7. - proving Pythagorean Theorem and finding missing side lengths using Pythagorean Theorem. I think students will need more time understanding how to use the dot grid paper and on drawing tilted squares.