# Standard 2 : Understand and apply the Pythagorean Theorem. (Major Cluster)

This document was generated on CPALMS - www.cpalms.org

Clusters should not be sorted from Major to Supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work of the grade with the supporting clusters.

### General Information

Number: MAFS.8.G.2
Title: Understand and apply the Pythagorean Theorem. (Major Cluster)
Type: Cluster
Subject: Mathematics
Domain-Subdomain: Geometry

#### Related Standards

This cluster includes the following benchmarks
Code Description
MAFS.8.G.2.6: Explain a proof of the Pythagorean Theorem and its converse.
MAFS.8.G.2.7: Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

 Clarifications:Examples of Opportunities for In-Depth Focus The Pythagorean theorem is useful in practical problems, relates to grade-level work in irrational numbers and plays an important role mathematically in coordinate geometry in high school.
MAFS.8.G.2.8: Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

#### Related Access Points

This cluster includes the following access points.

#### Access Points

 Access Point Number Access Point Title MAFS.8.G.2.AP.7a: Find the hypotenuse of a two-dimensional right triangle using the Pythagorean theorem. MAFS.8.G.2.AP.6a: Measure the lengths of the sides of multiple right triangles to determine a relationship. MAFS.8.G.2.AP.8a: Apply the Pythagorean Theorem to determine lengths/distances between two points in a coordinate system by forming right triangles.

#### Related Resources

Vetted resources educators can use to teach the concepts and skills in this topic.

#### Assessments

 Name Description Sample 2 - Eighth Grade Math State Interim Assessment: This is a State Interim Assessment for eighth grade. Sample 1 - Eighth Grade Math State Interim Assessment: This is a State Interim Assessment for eighth grade.

#### Lesson Plans

 Name Description Coding with Geometry Challenge #20-22: This set of geometry challenges focuses on using Pythagorean Theorem to find missing triangle side lengths and to draw triangles. Students problem solve and think as they learn to code using block coding software.  Student will need to use their knowledge of the attributes of polygons and mathematical principals of geometry to accomplish the given challenges. The challenges start out fairly simple and move to more complex situations in which students can explore at their own pace or work as a team. Computer Science standards are seamlessly intertwined with the math standards while providing “Step it up!” and “Jump it up!” opportunities to increase rigor. Discovering and Using the Pythagorean Theorem: Students will complete a hands-on activity to discover a geometric proof of the Pythagorean Theorem, and they will use and apply the Pythagorean Theorem to solve examples and real-world situations. Pythagorean Perspective: This lesson serves as an introductory lesson to help students understand the importance of the Pythagorean Theorem and its converse. The lesson has a hands-on component that allows student to discover the properties of the theorem. There are some pictures included to help understand the process. To help acquire the Pythagorean perspective, this lesson includes worksheets that are practical for individual or cooperative learning strategies. The worksheets contain prior knowledge exercises, practice exercises and a summative assignment. After completing the lesson, the students will have the perspective needed to successfully understand the fundamentals of the Pythagorean Theorem. In Support of the Pythagorean Theorem: This lesson introduces students to the Pythagorean Theorem through a real world problem which requires application of the converse of the theorem. After students engage in hands-on practice of the theorem using color tiles to create squares, they are asked to independently solve a converse problem. Keep Calm and Hypotenuse On: This lesson covers the use of the Pythagorean theorem to find the lengths of the sides of right triangles, as well as to determine whether a triangle has one right triangle with the converse of this theorem. The lesson only deals with two-dimensional figures. As the Crow Flies: This two-day lesson teaches students to use the Pythagorean Theorem with simple right triangles on the first day, then progresses to using the theorem to find the distance between two points on a coordinate graph. The Pythagorean Theorem: Square Areas: This lesson unit is intended to help you 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 and deduce a generalizable method for finding lengths and areas (The Pythagorean Theorem.) Origami Boats - Pythagorean Theorem in the Real World: Students will create origami boats and use them to make a net drawing. The drawing will be labeled with measurements, based on the number of squares on the graph for units, such that the students will use the Pythagorean Theorem to find several of the lengths. The lesson includes a video on how to make an origami boat. This is part 2 of a lesson plan for the Pythagorean Theorem. The resource, Applying the Pythagorean ID 48973, lays the groundwork for this exercise. Applying the Pythagorean Theorem: This lesson plan is lesson 1 of two lessons. This lesson applies the Pythagorean Theorem and teaches the foundational skills required to proceed to lesson 2, Origami Boats - Pythagorean Theorem in the real world Resource ID 49055. This lesson should not be taught until the students have a knowledge of standard MAFS.8.G.2.6 Explain a proof of the Pythagorean Theorem and its converse. Bike Club Trip: In this activity the students will rank different locations for a bike club's next destination. In order to do so, the students must use Pythagorean Theorem and well as analyze data of the quantitative and qualitative type.Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom. Power of a Right Triangle: Day 1 Proving Pythagoras: In this first of three lessons on the Pythagorean Theorem students work to prove the Pythagorean theorem and verify that the theorem works. Alas, Poor Pythagoras, I Knew You Well! #2: Using different activities, students will find real life uses for the Pythagorean Theorem. A Hypotenuse is a WHAT????: Students are guided through a short history of Pythagoras and a discovery of the Pythagorean Theorem using the squaring of the sides of a right triangle.

#### Perspectives Video: Expert

 Name Description Measuring a Grid for Underwater Archeology: Don't be a square! Learn about how even grids help archaeologists track provenience! Download the CPALMS Perspectives video student note taking guide.

#### Perspectives Video: Professional/Enthusiasts

 Name Description Field Sampling with the Point-centered Quarter Method: In this video, Jim Cox describes a sampling method for estimating the density of dead trees in a forest ecosystem. Download the CPALMS Perspectives video student note taking guide. What's the Distance from Here to the Middle of Nowhere?: Find out how math and technology can help you (try to) get away from civilization. Download the CPALMS Perspectives video student note taking guide.

#### Presentation/Slideshows

 Name Description The Pythagorean Theorem: Geometry’s Most Elegant Theorem: This lesson teaches students about the history of the Pythagorean theorem, along with proofs and applications. It is geared toward high school Geometry students that have completed a year of Algebra and addresses the following national standards of the National Council of Teachers of Mathematics and the Mid-continent Research for Education and Learning: 1) Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships; 2) Use visualization, spatial reasoning, and geometric modeling to solve problems; 3) Understand and apply basic and advanced properties of the concepts of geometry; and 4) Use the Pythagorean theorem and its converse and properties of special right triangles to solve mathematical and real-world problems. The video portion is about thirty minutes, and with breaks could be completed in 50 minutes. (You may consider completing over two classes, particularly if you want to allow more time for activities or do some of the enrichment material). These activities could be done individually, in pairs, or groups. I think 2 or 3 students is optimal. The materials required for the activities include scissors, tape, string and markers. A Geometric Proof of the Pythagorean Theorem: Students will see an animated presentation of the proof of the Pythagorean Theorem. This animated PowerPoint presentation uses shearing and the invariance of the area of triangles with congruent bases and heights to show a step-by-step geometric proof of the Pythagorean Theorem. Pythagoras' Theorem: This resource can be used to introduce the Pythagorean Theorem to students. It provides sketches, applets, examples and easy-to-understand visual proofs as well as an algebra proof for the theorem. It also includes interactive multiple choice practice questions on solving for a side of a right triangle, and applications involving right triangles, as well as a hands-on activity for students to do that allows them to create their own proof.

#### Student Center Activity

 Name Description Edcite: Mathematics Grade 8: Students can practice answering mathematics questions on a variety of topics. With an account, students can save their work and send it to their teacher when complete.

#### Text Resources

 Name Description Pythagoras Explained: This informational text resource is intended to support reading in the content area. The text describes a method for predicting the win-loss record for baseball teams based on runs scored and runs allowed, using the "Pythagorean Expectation" formula invented by Bill James. The text goes on to show the relationship of the prediction formula to the Pythagorean theorem, pointing out a very cool application of the theorem to the world of sports. The Pythagorean Theorem: This overview of the Pythagorean Theorem covers its purpose, equation, application, and validity. The site also provides a number of illustrations which help students visualize the theorem, and links to related resources for further understanding.

#### Tutorials

 Name Description Bhaskara's Proof of the Pythagorean Theorem: This video demonstrates Bhaskara's proof of the Pythagorean Theorem. Pythagorean Theorem Proof Using Similar Triangles: This video shows a proof of the Pythagorean Theorem using similar triangles. Distance formula and the Pythagorean Theorem: This tutorial shows students how to find the distance between lines using the Pythagorean Theorem. This video makes a connection between the distance formula and the Pythagorean Theorem. Pythagorean Theorem: A Carpet Example: In this tutorial, you will practice finding the missing width of a carpet, given the length of one side and the diagonal of the carpet.

#### Video/Audio/Animation

 Name Description Annotated Proof of the Pythagorean Theorem : This resource gives an animated and then annotated proof of the Pythagorean Theorem.

#### Virtual Manipulatives

 Name Description Demonstrate the Pythagorean Theorem: Representation to illustrate the Pythagorean Theorem. Right Triangle Calculator: "Free online calculator will calculate the side lengths and angles for any . Just input any valid combination of sides and/or angles, and let the calculator do the rest! This free tool also draws a downloadable image of your triangle." (from Math Worksheets Go!)

#### Student Resources

Vetted resources students can use to learn the concepts and skills in this topic.

#### Perspectives Video: Expert

 Title Description Measuring a Grid for Underwater Archeology: Don't be a square! Learn about how even grids help archaeologists track provenience! Download the CPALMS Perspectives video student note taking guide.

#### Perspectives Video: Professional/Enthusiast

 Title Description What's the Distance from Here to the Middle of Nowhere?: Find out how math and technology can help you (try to) get away from civilization. Download the CPALMS Perspectives video student note taking guide.

#### Presentation/Slideshow

 Title Description The Pythagorean Theorem: Geometry’s Most Elegant Theorem: This lesson teaches students about the history of the Pythagorean theorem, along with proofs and applications. It is geared toward high school Geometry students that have completed a year of Algebra and addresses the following national standards of the National Council of Teachers of Mathematics and the Mid-continent Research for Education and Learning: 1) Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships; 2) Use visualization, spatial reasoning, and geometric modeling to solve problems; 3) Understand and apply basic and advanced properties of the concepts of geometry; and 4) Use the Pythagorean theorem and its converse and properties of special right triangles to solve mathematical and real-world problems. The video portion is about thirty minutes, and with breaks could be completed in 50 minutes. (You may consider completing over two classes, particularly if you want to allow more time for activities or do some of the enrichment material). These activities could be done individually, in pairs, or groups. I think 2 or 3 students is optimal. The materials required for the activities include scissors, tape, string and markers.

#### Student Center Activity

 Title Description Edcite: Mathematics Grade 8: Students can practice answering mathematics questions on a variety of topics. With an account, students can save their work and send it to their teacher when complete.

#### Tutorials

 Title Description Bhaskara's Proof of the Pythagorean Theorem: This video demonstrates Bhaskara's proof of the Pythagorean Theorem. Pythagorean Theorem Proof Using Similar Triangles: This video shows a proof of the Pythagorean Theorem using similar triangles. Distance formula and the Pythagorean Theorem: This tutorial shows students how to find the distance between lines using the Pythagorean Theorem. This video makes a connection between the distance formula and the Pythagorean Theorem. Pythagorean Theorem: A Carpet Example: In this tutorial, you will practice finding the missing width of a carpet, given the length of one side and the diagonal of the carpet.

#### Video/Audio/Animation

 Title Description Annotated Proof of the Pythagorean Theorem : This resource gives an animated and then annotated proof of the Pythagorean Theorem.

#### Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this topic.

#### Perspectives Video: Expert

 Title Description Measuring a Grid for Underwater Archeology: Don't be a square! Learn about how even grids help archaeologists track provenience! Download the CPALMS Perspectives video student note taking guide.

#### Perspectives Video: Professional/Enthusiast

 Title Description What's the Distance from Here to the Middle of Nowhere?: Find out how math and technology can help you (try to) get away from civilization. Download the CPALMS Perspectives video student note taking guide.