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
Grade: 8
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.

Formative Assessments

Name Description
Pyramid Height:

Students are asked to determine the length of a side of a right triangle in a real-world problem.

Distance Between Two Points:

Students are asked to find the distance between two points in the coordinate plane.

Three Dimensional Diagonal:

Students are asked to determine the length of a side of a right triangle in a real-world problem.

Pythagorean Squares:

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

New Television:

Students are asked to determine the length of a side of a right triangle in a real-world problem.

How Far to School:

Students are asked to determine the length of a side of a right triangle in a real-world problem.

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.

Converse of the Pythagorean Theorem:

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

Distance on the Coordinate Plane:

Students are asked to find the distance between two points in the coordinate plane.

Coordinate Plane Triangle:

Students are asked to determine the lengths of the sides of a right triangle in the coordinate plane given the coordinates of its vertices.

Calculate Triangle Sides:

Students are asked to determine the length of each side of a right triangle in the coordinate plane given the coordinates of its vertices.

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.

Problem-Solving Tasks

Name Description
Glasses:

In this resource, students will determine the volumes of three different shaped drinking glasses. They will need prior knowledge with volume formulas for cylinders, cones, and spheres, as well as experience with equation solving, simplifying square roots, and applying the Pythagorean theorem.

A Rectangle in the Coordinate Plane:

This task provides an opportunity to apply the Pythagorean theorem to multiple triangles in order to determine the length of the hypotenuse; the converse of the Pythagorean theorem is also required in order to conclude that certain angles are right angles.

Bird and Dog Race:

The purpose of this task is for students to use the Pythagorean Theorem as a problem-solving tool to calculate the distance between two points on a grid. In this case the grid is also a map, and the street names can be viewed as defining a coordinate system (although the coordinate system is not needed to solve the problem).

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.

Is This a Rectangle?:

The goal of this task is to provide an opportunity for students to apply a wide range of ideas from geometry and algebra in order to show that a given quadrilateral is a rectangle. Creativity will be essential here as the only given information is the Cartesian coordinates of the quadrilateral's vertices. Using this information to show that the four angles are right angles will require some auxiliary constructions. Students will need ample time and, for some of the methods provided below, guidance. The reward of going through this task thoroughly should justify the effort because it provides students an opportunity to see multiple geometric and algebraic constructions unified to achieve a common purpose. The teacher may wish to have students first brainstorm for methods of showing that a quadrilateral is rectangle (before presenting them with the explicit coordinates of the rectangle for this problem): ideally, they can then divide into groups and get to work straightaway once presented with the coordinates of the quadrilateral for this problem.

Area of a Trapezoid:

The purpose of this task is for students to use the Pythagorean Theorem to find the unknown side-lengths of a trapezoid in order to determine the area. This problem will require creativity and persistence as students must decompose the given trapezoid into other polygons in order to find its area.

Applying the Pythagorean Theorem in a Mathematical Context:

Three right triangles surround a shaded triangle; together they form a rectangle measuring 12 units by 14 units. The figure used shows some of the dimensions but is not drawn to scale. Understand and apply the Pythagorean Theorem.

Areas of Geometric Shapes with the Same Perimeter:

This problem is part of a very rich tradition of problems looking to maximize the area enclosed by a shape with fixed perimeter. Only three shapes are considered here because the problem is difficult for more irregular shapes. For example, of all triangles, the one with fixed perimeter P and largest area is the equilateral triangle whose side lengths are all P3 but this is difficult to show because it is not easy to find the area of triangle in terms of the three side lengths (though Heron's formula accomplishes this). Nor is it simple to compare the area of two triangles with equal perimeter without knowing their individual areas. For quadrilaterals, a similar problem arises: showing that of all rectangles with perimeter P the one with the largest area is the square whose side lengths are P4 is a good problem which students should think about. But comparing a square to an irregularly shaped quadrilateral of equal perimeter will be difficult.

Running on the Football Field:

Students need to reason as to how they can use the Pythagorean Theorem to find the distances ran by Ben Watson and Champ Bailey. The focus here should not be on who ran a greater distance but on seeing how to set up right triangles to apply the Pythagorean Theorem to this problem. Students must use their measurement skills and make reasonable estimates to set up triangles and correctly apply the Theorem.

It's Raining!!! (Compare areas of wiped windshields):

In this problem-solving task, students are challenged to determine whether the windshield wipers on a car or a truck allow the drivers to see more area clearly. To solve this problem, students must apply the Pythagorean theorem and their ability to find area of circles and parallelograms to find the answer. Be sure to click the links in the orange bar at the top of the page for more information about the challenge. From NCTM's Figure This! Math Challenges for Families.

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.

Problem-Solving Tasks

Title Description
Glasses:

In this resource, students will determine the volumes of three different shaped drinking glasses. They will need prior knowledge with volume formulas for cylinders, cones, and spheres, as well as experience with equation solving, simplifying square roots, and applying the Pythagorean theorem.

Is This a Rectangle?:

The goal of this task is to provide an opportunity for students to apply a wide range of ideas from geometry and algebra in order to show that a given quadrilateral is a rectangle. Creativity will be essential here as the only given information is the Cartesian coordinates of the quadrilateral's vertices. Using this information to show that the four angles are right angles will require some auxiliary constructions. Students will need ample time and, for some of the methods provided below, guidance. The reward of going through this task thoroughly should justify the effort because it provides students an opportunity to see multiple geometric and algebraic constructions unified to achieve a common purpose. The teacher may wish to have students first brainstorm for methods of showing that a quadrilateral is rectangle (before presenting them with the explicit coordinates of the rectangle for this problem): ideally, they can then divide into groups and get to work straightaway once presented with the coordinates of the quadrilateral for this problem.

Running on the Football Field:

Students need to reason as to how they can use the Pythagorean Theorem to find the distances ran by Ben Watson and Champ Bailey. The focus here should not be on who ran a greater distance but on seeing how to set up right triangles to apply the Pythagorean Theorem to this problem. Students must use their measurement skills and make reasonable estimates to set up triangles and correctly apply the Theorem.

It's Raining!!! (Compare areas of wiped windshields):

In this problem-solving task, students are challenged to determine whether the windshield wipers on a car or a truck allow the drivers to see more area clearly. To solve this problem, students must apply the Pythagorean theorem and their ability to find area of circles and parallelograms to find the answer. Be sure to click the links in the orange bar at the top of the page for more information about the challenge. From NCTM's Figure This! Math Challenges for Families.

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.

Problem-Solving Tasks

Title Description
Glasses:

In this resource, students will determine the volumes of three different shaped drinking glasses. They will need prior knowledge with volume formulas for cylinders, cones, and spheres, as well as experience with equation solving, simplifying square roots, and applying the Pythagorean theorem.

A Rectangle in the Coordinate Plane:

This task provides an opportunity to apply the Pythagorean theorem to multiple triangles in order to determine the length of the hypotenuse; the converse of the Pythagorean theorem is also required in order to conclude that certain angles are right angles.

Bird and Dog Race:

The purpose of this task is for students to use the Pythagorean Theorem as a problem-solving tool to calculate the distance between two points on a grid. In this case the grid is also a map, and the street names can be viewed as defining a coordinate system (although the coordinate system is not needed to solve the problem).

Is This a Rectangle?:

The goal of this task is to provide an opportunity for students to apply a wide range of ideas from geometry and algebra in order to show that a given quadrilateral is a rectangle. Creativity will be essential here as the only given information is the Cartesian coordinates of the quadrilateral's vertices. Using this information to show that the four angles are right angles will require some auxiliary constructions. Students will need ample time and, for some of the methods provided below, guidance. The reward of going through this task thoroughly should justify the effort because it provides students an opportunity to see multiple geometric and algebraic constructions unified to achieve a common purpose. The teacher may wish to have students first brainstorm for methods of showing that a quadrilateral is rectangle (before presenting them with the explicit coordinates of the rectangle for this problem): ideally, they can then divide into groups and get to work straightaway once presented with the coordinates of the quadrilateral for this problem.

Area of a Trapezoid:

The purpose of this task is for students to use the Pythagorean Theorem to find the unknown side-lengths of a trapezoid in order to determine the area. This problem will require creativity and persistence as students must decompose the given trapezoid into other polygons in order to find its area.

Applying the Pythagorean Theorem in a Mathematical Context:

Three right triangles surround a shaded triangle; together they form a rectangle measuring 12 units by 14 units. The figure used shows some of the dimensions but is not drawn to scale. Understand and apply the Pythagorean Theorem.

Areas of Geometric Shapes with the Same Perimeter:

This problem is part of a very rich tradition of problems looking to maximize the area enclosed by a shape with fixed perimeter. Only three shapes are considered here because the problem is difficult for more irregular shapes. For example, of all triangles, the one with fixed perimeter P and largest area is the equilateral triangle whose side lengths are all P3 but this is difficult to show because it is not easy to find the area of triangle in terms of the three side lengths (though Heron's formula accomplishes this). Nor is it simple to compare the area of two triangles with equal perimeter without knowing their individual areas. For quadrilaterals, a similar problem arises: showing that of all rectangles with perimeter P the one with the largest area is the square whose side lengths are P4 is a good problem which students should think about. But comparing a square to an irregularly shaped quadrilateral of equal perimeter will be difficult.

Running on the Football Field:

Students need to reason as to how they can use the Pythagorean Theorem to find the distances ran by Ben Watson and Champ Bailey. The focus here should not be on who ran a greater distance but on seeing how to set up right triangles to apply the Pythagorean Theorem to this problem. Students must use their measurement skills and make reasonable estimates to set up triangles and correctly apply the Theorem.