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Определение возможностей

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Определение возможностей

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К концу урока ученики смогут сформулировать и применить теорему Пифагора для решения задач.

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Вопрос урока

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Вопрос урока

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Как можно использовать теорему Пифагора в реальной жизни?

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Прямая инструкция

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Прямая инструкция

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Учитель объясняет основные понятия и формулировку теоремы Пифагора. Ученики записывают в тетради.

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An Introduction to the Pythagorean Theorem | Math with Mr. J

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Практическое задание

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Практическое задание

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Разрабатывать задачи в парах с использованием теоремы Пифагора. Поделитесь решениями с классом.

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Независимая практика

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Независимая практика

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Ученики индивидуально решают задачи на применение теоремы Пифагора из предложенного списка.

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Pythagorean Theorem worksheets contain skills on right triangles, missing leg or hypotenuse, Pythagorean triple, word problems, printable charts and more.

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Дифференцированная практика

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Дифференцированная практика

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Группы учеников получают задания разного уровня сложности для применения теоремы Пифагора. Учитель помогает с трудными заданиями.

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Browse pythagorean theorem differentiated resources on Teachers Pay Teachers, a marketplace trusted by millions of teachers for original educational resources.

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Наблюдение и обсуждение

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Наблюдение и обсуждение

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Учитель наблюдает за работой учеников и организует краткие обсуждения для оценки понимания материала.

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Формативная оценка

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Формативная оценка

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Краткий опрос или тест на понимание теоремы Пифагора. Ученики отвечают на вопросы устно или письменно.

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Суммативная оценка

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Суммативная оценка

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Итоговый тест, включающий задачи на применение теоремы. Результаты оцениваются и обсуждаются.

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Ученики делятся своими выводами и отвечают на итоговые вопросы по теме теоремы Пифагора.

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Выходной билет

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Выходной билет

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Каждый ученик записывает одно главное, что он узнал на уроке, и сдает учителю.

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Use this technique to show you what students are thinking and what they have learned at the end of a lesson. Before students leave (for recess, lunch, the end of the day, their next class, or are transitioning to another subject area), they have to hand you a "ticket" filled out with an answer to a question, a solution to a problem, or a response to what they've learned.

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Дополнительная практика

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Дополнительная практика

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Решение задач на применение теоремы Пифагора для закрепления материала. Используйте дополнительные онлайн-ресурсы для самообучения.

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Use Pythagorean theorem to find right triangle side lengths (practice) | Khan Academy

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Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere.

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Проектное задание

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Проектное задание

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Создание постера или презентации на тему 'История и применение теоремы Пифагора'. Подготовьте работы с использованием программы Padlet для презентации.

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Pythagorean theorem, the well-known geometric theorem that the sum of the squares on the legs of a right triangle is equal to the square on the hypotenuse (the side opposite the right angle)-or, in familiar algebraic notation, a ² + b ² = c ². Although the theorem has long been associated with Greek mathematician-philosopher Pythagoras (c. 570-500/490 ), it is actually far older. Four Babylonian tablets from circa 1900-1600 bce indicate some knowledge of the theorem, with a very accurate calculation of the square root of 2 (the length of the hypotenuse of a right triangle with the length of both legs equal to 1) and lists of special integers known as Pythagorean triples that satisfy it (e.g., 3, 4, and 5; 3 ² + 4 ² = 5 ², 9 + 16 = 25). The theorem is mentioned in the Baudhayana Sulba-sutra of India, which was written between 800 and 400 bce. Nevertheless, the theorem came to be credited to Pythagoras. It is also proposition number 47 from Book I of Euclid's Elements.

According to the Syrian historian Iamblichus (c. 250-330 ce), Pythagoras was introduced to mathematics by Thales of Miletus and his pupil Anaximander. In any case, it is known that Pythagoras traveled to Egypt about 535 bce to further his study, was captured during an invasion in 525 bce by Cambyses II of Persia and taken to Babylon, and may possibly have visited India before returning to the Mediterranean. Pythagoras soon settled in Croton (now Crotone, Italy) and set up a school, or in modern terms a monastery ( see Pythagoreanism), where all members took strict vows of secrecy, and all new mathematical results for several centuries were attributed to his name. Thus, not only is the first proof of the theorem not known, there is also some doubt that Pythagoras himself actually proved the theorem that bears his name. Some scholars suggest that the first proof was the one shown in the figure. It was probably independently discovered in several different cultures.

Book I of the Elements ends with Euclid's famous "windmill" proof of the Pythagorean theorem. ( See Sidebar: Euclid's Windmill.) Later in Book VI of the Elements, Euclid delivers an even easier demonstration using the proposition that the areas of similar triangles are proportionate to the squares of their corresponding sides. Apparently, Euclid invented the windmill proof so that he could place the Pythagorean theorem as the capstone to Book I. He had not yet demonstrated (as he would in Book V) that line lengths can be manipulated in proportions as if they were commensurable numbers (integers or ratios of integers). The problem he faced is explained in the Sidebar: Incommensurables.

A great many different proofs and extensions of the Pythagorean theorem have been invented. Taking extensions first, Euclid himself showed in a theorem praised in antiquity that any symmetrical regular figures drawn on the sides of a right triangle satisfy the Pythagorean relationship: the figure drawn on the hypotenuse has an area equal to the sum of the areas of the figures drawn on the legs. The semicircles that define Hippocrates of Chios's lunes are examples of such an extension. ( See Sidebar: Quadrature of the Lune.)

In the Nine Chapters on the Mathematical Procedures (or Nine Chapters), compiled in the 1st century ce in China, several problems are given, along with their solutions, that involve finding the length of one of the sides of a right triangle when given the other two sides. In the Commentary of Liu Hui, from the 3rd century, Liu Hui offered a proof of the Pythagorean theorem that called for cutting up the squares on the legs of the right triangle and rearranging them ("tangram style") to correspond to the square on the hypotenuse. Although his original drawing does not survive, the next figure shows a possible reconstruction.

The Pythagorean theorem has fascinated people for nearly 4,000 years; there are now more than 300 different proofs, including ones by the Greek mathematician Pappus of Alexandria (flourished c. 320 ce), the Arab mathematician-physician Thābit ibn Qurrah (c. 836-901), the Italian artist-inventor Leonardo da Vinci (1452-1519), and even U.S. Pres. James Garfield (1831-81).

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Pythagorean theorem | Definition & History

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Pythagorean theorem, the well-known geometric theorem that the sum of the squares on the legs of a right triangle is equal to the square on the hypotenuse (the side opposite the right angle)-or, in familiar algebraic notation, a2 + b2 = c2. Although the theorem has long been associated with Greek mathematician-philosopher Pythagoras (c.