Strand: GEOMETRY (8.G)

Understand congruence and similarity using physical models, transparencies, or geometry software (Standards 8.G.1-5). Understand and apply the Pythagorean Theorem and its converse (Standards 8.G.6-8). Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres (Standard 8.G.9).

Standard 8.G.7

Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

• 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.
• Applying the Pythagorean Theorem in a mathematical context
This task reads "Three right triangles surround a shaded triangle; together they form a rectangle measuring 12 units by 14 units. The figure below shows some of the dimensions but is not drawn to scale. Is the shaded triangle a right triangle? Provide a proof for your answer."
• Calculating Distance Using the Pythagorean Theorem
In this interactive students must find the distance between two points on a plane by use the Pythagorean Theorum. They then use this skill to complete an activity involving an amusement park. They create a map of a park and then figure out the distance between attractions. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Chapter 10 - Mathematical Foundation (UMSMP)
This is Chapter 10 of the Utah Middle School Math Grade 8 textbook. It provides a Mathematical Foundation for Angles, Triangles and Distance.
• Chapter 10 - Student Workbook (UMSMP)
This is Chapter 10 of the Utah Middle School Math Grade 8 student workbook. It focuses Angles, Triangles and Distance.
• Circle Sandwich
The purpose of this task is to apply knowledge about triangles, circles, and squares in order to calculate and compare two different areas.
• Glasses
This task gives students an opportunity to work with volumes of cylinders, spheres and cones.
• Grade 8 Unit 3: Geometric Applications of Exponents (Georgia Standards)
In this unit students will distinguish between rational and irrational numbers; find or estimate the square and cubed root of non-negative numbers, including 0; interpret square and cubed roots as both points of a line segment and lengths on a number line; use the properties of real numbers (commutative, associative, distributive, inverse, and identity) and the order of operations to simplify and evaluate numeric and algebraic expressions involving integer exponents, square and cubed roots; work with radical expressions and approximate them as rational numbers; solve problems involving the volume of a cylinder, cone, and sphere; determine the relationship between the hypotenuse and legs of a right triangle; use deductive reasoning to prove the Pythagorean Theorem and its converse; apply the Pythagorean Theorem to determine unknown side lengths in right triangles; determine if a triangle is a right triangle, Pythagorean triple; apply the Pythagorean Theorem to find the distance between two points in a coordinate system; and solve problems involving 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.
• IXL Game: Pythagorean theorem
This game will help eighth graders understand the pythagorean theorem via word problems. This is just one of many online games that supports the Utah Math core. Note: The IXL site requires subscription for unlimited use.
• Points from Directions
This task provides a slightly more involved use of similarity, requiring students to translate the given directions into an accurate picture, and persevere in solving a multi-step problem: They must calculate segment lengths, requiring the use of the Pythagorean theorem, and either know or derive trigonometric properties of isosceles right triangles.
• Pythagorean Explorer
This applet challenges the student to find the length of the third side of a triangle when given the two sides and the right angle.
• Pythagorean Theorem
In this lesson students will be able to use the Pythagorean Theorem to find side lengths of right triangles, the areas of right triangles, and the perimeter and areas of triangles.
• Running on the Football Field
Students need to reason as to how they can use the Pythagorean Theorem to find the distance ran by Ben Watson and Champ Bailey. The focus here should not be on who ran a greater distance, but on seeing how you can set up right triangles to apply the Pythagorean Theorem to this problem.
• Sizing up Squares
The goal of this task is for students to check that the Pythagorean Theorem holds for two specific examples. Although the work of this task does not provide a proof for the full Pythagorean Theorem, it prepares students for the area calculations they will need to make as well as the difficulty of showing that a quadrilateral in the plane is a square.
• Spiderbox
The purpose of this task is for students to work on their visualization skills and to apply the Pythagorean Theorem.
• Squaring the Triangle
Students can manipulate the sides of a triangle in this applet in order to better understand the Pythagorean Theorem.
In this task, students will create a design using rotations and reflections.
• Student Task: Circles and Squares
In this task, students must solve a problem about circles inscribed in squares
The Hopewell people were Native Americans whose culture flourished in the central Ohio Valley about 2000 years ago. They constructed earthworks using right triangles. In this task, the student will look at some of the geometrical properties of a Hopewell earthwork.
In this task, students will need to work out the actual dimensions of TV screens, which are sold according to their diagonal measurements.
• Student Task: Proofs Of The Pythagorean Theorem?
In this task, students will look at three different attempts to prove the Pythagorean theorem and determine which is the best "proof".
In this task, the student will investigate Pythagorean Triples.
During the Edo period (1603-1867) of Japanese history, geometrical puzzles were hung in the holy temples as offerings to the gods and as challenges to worshippers. Here is one such problem for students to investigate.
• The Number System (8.NS) - 8th Grade Core Guide
The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for Mathematics Grade 8 - The Number System.
• The Pythagorean Theorem and 18th-Century Cranes
A video from Annenberg Learner Learning Math shows how the Pythagorean Theorem was useful in the reconstruction of an 18th century crane. The classroom activity asks students to apply the theorem and understand its usefulness in construction and design. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• The Pythagorean Theorem: Square Areas
This lesson unit is intended to help educators 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.)
• Two Triangles' Area
This task requires the student to draw pictures of the two triangles and also make an auxiliary construction in order to calculate the areas (with the aid of the Pythagorean Theorem). Students need to know, or be able to intuitively identify, the fact that the line of symmetry of the isosceles triangle divides the base in half, and meets the base perpendicularly.

http://www.uen.org - in partnership with Utah State Board of Education (USBE) and Utah System of Higher Education (USHE).  Send questions or comments to USBE Specialist - Lindsey  Henderson and see the Mathematics - Secondary website. For general questions about Utah's Core Standards contact the Director - Jennifer  Throndsen.

These materials have been produced by and for the teachers of the State of Utah. Copies of these materials may be freely reproduced for teacher and classroom use. When distributing these materials, credit should be given to Utah State Board of Education. These materials may not be published, in whole or part, or in any other format, without the written permission of the Utah State Board of Education, 250 East 500 South, PO Box 144200, Salt Lake City, Utah 84114-4200.