Students see that geometric shapes can be found in all sorts of structures as they explore the history of the Roman Empire with a focus on how engineers 2000 years ago laid the groundwork for many structures seen today. Through a short online video, brief lecture material and their own online research directed by worksheet questions, students discover how the Romans invented a structure known today as the Roman arch that enabled them to build architecture never before seen by humankind, including the amazing aqueducts. Students calculate the slope and its total drop and angle over its entire distance for an example aqueduct. Completing this lesson prepares students for the associated activity in which teams build and test model aqueducts that meet specific constraints. This lesson serves as an introduction to many other geometry—and engineering-related lessons—including statics and trusses, scale modeling, and trigonometry.

Sal introduces the famous and super important Pythagorean theorem! This lesson includes an introduction video followed by examples and practice problems involving finding the hypotenuse, finding a side, finding area of an isosceles triangle, and determining if a triangle is a right triangle. It is followed by a unit on Pythagorean theorem applications.

Khan Academy learning modules include a Community space where users can ask questions and seek help from community members. Educators should consult with their Technology administrators to determine the use of Khan Academy learning modules in their classroom. Please review materials from external sites before sharing with students.

Students explore in detail how the Romans built aqueducts using arches—and the geometry involved in doing so. Building on what they learned in the associated lesson about how innovative Roman arches enabled the creation of magnificent structures such as aqueducts, students use trigonometry to complete worksheet problem calculations to determine semicircular arch construction details using trapezoidal-shaped and cube-shaped blocks. Then student groups use hot glue and half-inch wooden cube blocks to build model aqueducts, doing all the calculations to design and build the arches necessary to support a water-carrying channel over a three-foot span. They calculate the slope of the small-sized aqueduct based on what was typical for Roman aqueducts at the time, aiming to construct the ideal slope over a specified distance in order to achieve a water flow that is not spilling over or stagnant. They test their model aqueducts with water and then reflect on their performance.

Gain a basic understanding of the Pythagorean Theorem by watching this easy to understand video tutorial. Additional resources are available as part of a paid subscription service. [12:55]

This video [4:06] provides an excellent definition of the Pythagorean Theorem. Watching and listening to the teacher and student interact help to clarify the thinking behind this concept.

This video [7:25] provides an excellent introduction to the Pythagorean Theorem. As you watch and listen to the teacher and student interact, it helps clarify the thinking behind applying this concept.

This video illustrates the concepts of square roots, slope, and the Pythagorean Theorem as they are used in designing and constructing skate park rails, ramps, and pipes. An interactive activity is also included where students are able to work through calculations of squares and square roots to solve equations concerning right triangles, as well as a printable activity.

This multimedia Learn Alberta math resource looks at the structural design of buildings and how the Pythagorean Theorem is often involved in the design process. The accompanying interactive component asks students to build a variety of right triangles as they investigate calculations of the Pythagorean Theorem. Be sure to follow the link to the printable activity included to reinforce target skills.

This resource investigates the Pythagorean Theorem. Includes animations, examples, proofs, and links to problems and activities.

Here is a site that clearly and thoroughly explains how to solve right triangles for any missing side or angle given any two pieces other sides or angles. There are example problems solved, problems for the student to attempt, and answers to the student problems. Point this site out to students who have been absent or who need additional instruction on this or many other topics.

(Nota: Esta es una traducción de un recurso educativo abierto creado por el Departamento de Educación del Estado de Nueva York (NYSED) como parte del proyecto "EngageNY" en 2013. Aunque el recurso real fue traducido por personas, la siguiente descripción se tradujo del inglés original usando Google Translate para ayudar a los usuarios potenciales a decidir si se adapta a sus necesidades y puede contener errores gramaticales o lingüísticos. La descripción original en inglés también se proporciona a continuación.)

En este módulo, los estudiantes aprenden sobre traducciones, reflexiones y rotaciones en el avión y, lo que es más importante, cómo usarlas para definir con precisión el concepto de congruencia. A lo largo del tema A, sobre las definiciones y propiedades de los movimientos rígidos básicos, los estudiantes verifican experimentalmente sus propiedades básicas y, cuando son factibles, profundicen su comprensión de estas propiedades utilizando el razonamiento. Todas las lecciones del tema B demuestran a los estudiantes la capacidad de secuenciar varias combinaciones de movimientos rígidos mientras mantienen las propiedades básicas de los movimientos rígidos individuales. Los estudiantes aprenden que la congruencia es solo una secuencia de movimientos rígidos básicos en el Tema C, y el Tema D comienza el aprendizaje del Teorema Pitagórico.

Encuentre el resto de los recursos matemáticos de Engageny en https://archive.org/details/engageny-mathematics.

English Description:
In this module, students learn about translations, reflections, and rotations in the plane and, more importantly, how to use them to precisely define the concept of congruence. Throughout Topic A, on the definitions and properties of the basic rigid motions, students verify experimentally their basic properties and, when feasible, deepen their understanding of these properties using reasoning. All the lessons of Topic B demonstrate to students the ability to sequence various combinations of rigid motions while maintaining the basic properties of individual rigid motions. Students learn that congruence is just a sequence of basic rigid motions in Topic C, and Topic D begins the learning of Pythagorean Theorem.

Find the rest of the EngageNY Mathematics resources at https://archive.org/details/engageny-mathematics.

(Nota: Esta es una traducción de un recurso educativo abierto creado por el Departamento de Educación del Estado de Nueva York (NYSED) como parte del proyecto "EngageNY" en 2013. Aunque el recurso real fue traducido por personas, la siguiente descripción se tradujo del inglés original usando Google Translate para ayudar a los usuarios potenciales a decidir si se adapta a sus necesidades y puede contener errores gramaticales o lingüísticos. La descripción original en inglés también se proporciona a continuación.)

El módulo 7 comienza con el trabajo relacionado con el teorema de Pitágoras y los triángulos rectos. Antes de que se presenten las lecciones de este módulo a los estudiantes, es importante que las lecciones en los módulos 2 y 3 sean relacionadas con el teorema de Pitágoras se imparten (M2: Lecciones 15 y 16, M3: Lecciones 13 y 14). En los módulos 2 y 3, los estudiantes usaron el teorema de Pitágoras para determinar la longitud desconocida de un triángulo derecho. En los casos en que la longitud lateral era un entero, los estudiantes calcularon la longitud. Cuando la longitud lateral no era un entero, los estudiantes dejaron la respuesta en forma de x2 = c, donde C no era un número cuadrado perfecto. Esas soluciones se revisan y son la motivación para aprender sobre las raíces cuadradas y los números irracionales en general.

Encuentre el resto de los recursos matemáticos de Engageny en https://archive.org/details/engageny-mathematics.

English Description:
Module 7 begins with work related to the Pythagorean Theorem and right triangles.  Before the lessons of this module are presented to students, it is important that the lessons in Modules 2 and 3 related to the Pythagorean Theorem are taught (M2:  Lessons 15 and 16, M3:  Lessons 13 and 14).  In Modules 2 and 3, students used the Pythagorean Theorem to determine the unknown length of a right triangle.  In cases where the side length was an integer, students computed the length.  When the side length was not an integer, students left the answer in the form of x2=c, where c was not a perfect square number.  Those solutions are revisited and are the motivation for learning about square roots and irrational numbers in general.

Find the rest of the EngageNY Mathematics resources at https://archive.org/details/engageny-mathematics.

This resource provides in depth information on the conic sections. Click on the underlined words for definitions and diagrams.

Students connect algebra to geometric concepts with polygons as they explore the distance formula, slope criteria for parallel and perpendicular lines, and learn to calculate and justify the area and perimeter of polygons.

A concise presentation of a lesson showing how the Distance Formula is really just the Pythagorean Theorem, and how it can be used to find the distance between two sets of coordinates.

This video lesson presents a real world problem, one that a juice seller faces daily, that can be solved by using the Pythagorean theorem. [33:54]

This Flexbook is community contributed through ck12.org. It covers three lessons on the Pythagorean Theorem. 1) Introduction and Determining if the Triangle is a Right Triangle, 2) Finding the Hypotenuse, and 3) Finding a leg. It includes step by step instructions, application problems, and answers (at the end of each lesson). Ck12.org material is downloadable, editable, and accessible offline and online.

In this video lesson, students will not only learn about Pythagoras and the Pythagorean Theorem, but they will also demonstrate a proof of this theorem, use it to solve problems, and create their own real world problems to challenge the class. [57:05]