Clarifications
Examples of Opportunities for InDepth FocusThe Pythagorean theorem is useful in practical problems, relates to gradelevel work in irrational numbers and plays an important role mathematically in coordinate geometry in high school.
General Information
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.
Test Item Specifications

Also Assessed:
 Assessment Limits :
If the triangle is part of a threedimensional figure, a graphic of the three dimensional figure must be included. Points on the coordinate grid must be where grid lines intersect.
 Calculator :
Yes
 Context :
Allowable
MAFS.8.G.2.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
Sample Test Items (4)
 Test Item #: Sample Item 1
 Question:
Triangle ABC is a right triangle. The lengths of the legs are 60 centimeters and 80 centimeters.
What is the length, in centimeters, of the hypotenuse?
 Difficulty: N/A
 Type: EE: Equation Editor
 Test Item #: Sample Item 2
 Question:
Triangle ABC is a right triangle. The length of one leg is 80 centimeters, and the hypotenuse is 120 centimeters.
What is the length, in centimeters, of the other leg?
 Difficulty: N/A
 Type: EE: Equation Editor
 Test Item #: Sample Item 3
 Question:
Two points are on the coordinate plane shown.
What is the distance between A(5,3) and B(3,5)?
 Difficulty: N/A
 Type: EE: Equation Editor
 Test Item #: Sample Item 4
 Question:
A right square pyramid is shown.
The base has a side length, b, of 30 centimeters (cm). The height, h, is 10 cm.
What is the length in centimeters, of l?
 Difficulty: N/A
 Type: EE: Equation Editor
Related Courses
Related Access Points
Related Resources
Assessments
Formative Assessments
Lesson Plans
Perspectives Video: Expert
Presentation/Slideshows
ProblemSolving Tasks
Student Center Activity
Text Resources
Tutorial
Virtual Manipulatives
STEM Lessons  Model Eliciting Activity
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.
MFAS Formative Assessments
Students are asked to determine the length of a side of a right triangle in a realworld problem.
Students are asked to determine the length of a side of a right triangle in a realworld problem.
Students are asked to determine the length of a side of a right triangle in a realworld problem.
Students are asked to determine the length of a side of a right triangle in a realworld problem.
Student Resources
Perspectives Video: Expert
Don't be a square! Learn about how even grids help archaeologists track provenience!
Type: Perspectives Video: Expert
Presentation/Slideshow
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 Midcontinent Research for Education and Learning: 1) Analyze characteristics and properties of two and threedimensional 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 realworld 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
ProblemSolving Tasks
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.
Type: ProblemSolving Task
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.
Type: ProblemSolving Task
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.
Type: ProblemSolving Task
In this problemsolving 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.
Type: ProblemSolving Task
Student Center Activity
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.
Type: Student Center Activity
Tutorial
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.
Type: Tutorial
Virtual Manipulative
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
Perspectives Video: Expert
Don't be a square! Learn about how even grids help archaeologists track provenience!
Type: Perspectives Video: Expert
ProblemSolving Tasks
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.
Type: ProblemSolving Task
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.
Type: ProblemSolving Task
The purpose of this task is for students to use the Pythagorean Theorem as a problemsolving 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).
Type: ProblemSolving Task
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.
Type: ProblemSolving Task
The purpose of this task is for students to use the Pythagorean Theorem to find the unknown sidelengths 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.
Type: ProblemSolving Task
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.
Type: ProblemSolving Task
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.
Type: ProblemSolving Task
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.
Type: ProblemSolving Task
Virtual Manipulative
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