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# MAFS.8.G.2.6

Explain a proof of the Pythagorean Theorem and its converse.

### General Information

Subject Area: Mathematics
Domain-Subdomain: Geometry
Cluster: Level 2: Basic Application of Skills & Concepts
Cluster: Understand and apply the Pythagorean Theorem. (Major Cluster) -

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.

Date of Last Rating: 02/14
Status: State Board Approved
Assessed: Yes

### Test Item Specifications

• Assessment Limits :
For the converse, only perfect roots should be used.
• Calculator :

Yes

• Context :

Allowable

### Sample Test Items (3)

• Test Item #: Sample Item 2
• Question: The side lengths of a triangle are given.

3, 4, 5

Explain how you know which side will be opposite the right angle.

• Difficulty: N/A
• Type: OR: Open Response

• Test Item #: Sample Item 3
• Question:

Select three side lengths, in centimeters (cm), that can form a right triangle.

• Difficulty: N/A
• Type: MS: Multiselect

## Related Courses

This benchmark is part of these courses.
1205050: M/J Accelerated Mathematics Grade 7 (Specifically in versions: 2014 - 2015, 2015 and beyond (current))
1205070: M/J Grade 8 Pre-Algebra (Specifically in versions: 2014 - 2015, 2015 and beyond (current))
1204000: M/J Intensive Mathematics (MC) (Specifically in versions: 2014 - 2015, 2015 and beyond (current))
7812030: Access M/J Grade 8 Pre-Algebra (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MAFS.8.G.2.AP.6a: Measure the lengths of the sides of multiple right triangles to determine a relationship.

## Related Resources

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

## Formative Assessments

Pythagorean Squares:

Students are asked to explain how a pair of figures demonstrates the Pythagorean Theorem and its converse.

Type: Formative Assessment

Explaining a Proof of the Pythagorean Theorem:

Students are asked to explain the steps of a proof of the Pythagorean Theorem that uses similar triangles.

Type: Formative Assessment

Converse of the Pythagorean Theorem:

Students are asked to explain the steps of a proof of the converse of the Pythagorean Theorem.

Type: Formative Assessment

## Lesson Plans

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.

Type: Lesson Plan

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.

Type: Lesson Plan

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.)

Type: Lesson Plan

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.

Type: Lesson Plan

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.

Type: Lesson Plan

## Presentation/Slideshows

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.

Type: Presentation/Slideshow

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.

Type: Presentation/Slideshow

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.

Type: Presentation/Slideshow

Converse of the Pythagorean Theorem:

This task is for instruction purposes. Part (b) is subtle and the solution presented here uses a "dynamic" view of triangles with two side lengths fixed. This helps pave the way toward what students will see later in trigonometry but some guidance will likely be needed in order to get students started on this path.

## Text Resource

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.

Type: Text Resource

## Tutorials

Bhaskara's Proof of the Pythagorean Theorem:

This video demonstrates Bhaskara's proof of the Pythagorean Theorem.

Type: Tutorial

Pythagorean Theorem Proof Using Similar Triangles:

This video shows a proof of the Pythagorean Theorem using similar triangles.

Type: Tutorial

## Video/Audio/Animations

Pythagorean Theorem:

This video demonstrates how to prove the Pythagorean Theorem using the area of a square and its components.

Type: Video/Audio/Animation

Annotated Proof of the Pythagorean Theorem :

This resource gives an animated and then annotated proof of the Pythagorean Theorem.

Type: Video/Audio/Animation

## Virtual Manipulatives

Pythagorean Theorem Manipulatives:

This web address, from the National Library of Virtual Manipulatives, will help teachers and students validate the Pythagorean Theorem both geometrically and algebraically. It can be used interactively with the Smartboard and the Promethean Board to create a better understanding of the topic.

Type: Virtual Manipulative

Demonstrate the Pythagorean Theorem:

Representation to illustrate the Pythagorean Theorem.

Type: Virtual Manipulative

## MFAS Formative Assessments

Converse of the Pythagorean Theorem:

Students are asked to explain the steps of a proof of the converse of the Pythagorean Theorem.

Explaining a Proof of the Pythagorean Theorem:

Students are asked to explain the steps of a proof of the Pythagorean Theorem that uses similar triangles.

Pythagorean Squares:

Students are asked to explain how a pair of figures demonstrates the Pythagorean Theorem and its converse.

## Student Resources

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

## Presentation/Slideshow

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.

Type: Presentation/Slideshow

## Tutorials

Bhaskara's Proof of the Pythagorean Theorem:

This video demonstrates Bhaskara's proof of the Pythagorean Theorem.

Type: Tutorial

Pythagorean Theorem Proof Using Similar Triangles:

This video shows a proof of the Pythagorean Theorem using similar triangles.

Type: Tutorial

## Video/Audio/Animation

Annotated Proof of the Pythagorean Theorem :

This resource gives an animated and then annotated proof of the Pythagorean Theorem.

Type: Video/Audio/Animation

## Virtual Manipulative

Pythagorean Theorem Manipulatives:

This web address, from the National Library of Virtual Manipulatives, will help teachers and students validate the Pythagorean Theorem both geometrically and algebraically. It can be used interactively with the Smartboard and the Promethean Board to create a better understanding of the topic.

Type: Virtual Manipulative

## Parent Resources

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

## Virtual Manipulative

Pythagorean Theorem Manipulatives:

This web address, from the National Library of Virtual Manipulatives, will help teachers and students validate the Pythagorean Theorem both geometrically and algebraically. It can be used interactively with the Smartboard and the Promethean Board to create a better understanding of the topic.

Type: Virtual Manipulative