Visualize a proof to the Pythagorean Theorem at the National Museum of Mathematics. This video focuses on modeling the concepts behind the Pythagorean Theorem with manipulatives. This video was submitted through the Innovation Math Challenge, a contest open to professional and nonprofessional producers.
Learn why the Pythagorean Theorem works for finding side lengths of right triangles. This video focuses on modeling what the Pythagorean theorem actually means and why it will find the missing side of a right triangle. This video was submitted through the Innovation Math Challenge, a contest open to professional and nonprofessional producers.
In this video, learn how to use the Pythagorean Theorem to find the distance between two points on a coordinate grid. In the accompanying classroom activity, students apply what they learned in the video as they solve similar problems and discuss solution strategies. To get the most from this lesson, students should be comfortable using the Pythagorean Theorem to find an unknown side of a right triangle, graphing points on the coordinate plane, and finding the distance between points with the same x- or y- values.
Learn why the Pythagorean theorem works for finding side lengths of right triangles. This video focuses on modeling what the Pythagorean theorem actually means and why it will find the missing side of a right triangle. This video was submitted through the Innovation Math Challenge, a contest open to professional and nonprofessional producers.
Optional Module 4, Topic E is an application of systems of linear equations (8.EE.C.8b). Specifically, a system that generates Pythagorean triples. First, students learn that a Pythagorean triple can be obtained by multiplying any known triple by a positive integer (8.G.B.7). Then, students are shown the Babylonian method for finding a triple that requires the understanding and use of a system of linear equations.
Module 7, Topic C revisits the Pythagorean Theorem and its applications, now in a context that includes the use of square roots and irrational numbers. Students learn another proof of the Pythagorean Theorem involving areas of squares off of each side of a right triangle (8.G.B.6). Another proof of the converse of the Pythagorean Theorem is presented to students, which requires an understanding of congruent triangles (8.G.B.6). With the concept of square roots firmly in place, students apply the Pythagorean Theorem to solve real-world and mathematical problems to determine an unknown side length of a right triangle and the distance between two points on the coordinate plane (8.G.B.7, 8.G.B.8).
In Module 7, Topic D, students learn that radical expressions naturally arise in geometry, such as the height of an isosceles triangle or the lateral length of a cone. The Pythagorean Theorem is applied to three-dimensional figures in Topic D as students learn some geometric applications of radicals and roots (8.G.B.7). In order for students to determine the volume of a cone or sphere, they must first apply the Pythagorean Theorem to determine the height of the cone or the radius of the sphere. Students learn that truncated cones are solids obtained by removing the top portion above a plane parallel to the base. Students know that to find the volume of a truncated cone they must access and apply their knowledge of similar figures learned in Module 3. Their work with truncated cones is an exploration of solids that is not formally assessed. In general, students solve real-world and mathematical problems in three dimensions in Topic D (8.G.C.9). For example, now that students can compute with cube roots and understand the concept of rate of change, students compute the average rate of change in the height of the water level when water is poured into a conical container at a constant rate. Students also use what they learned about the volume of cylinders, cones, and spheres to compare volumes of composite solids.
Optional Module 2, Topic D begins the learning of the Pythagorean Theorem. Students are shown the “square within a square” proof of the Pythagorean Theorem. The proof uses concepts learned in previous topics of the module, i.e., the concept of congruence and concepts related to degrees of angles. Students begin the work of finding the length of a leg or hypotenuse of a right triangle using a2 + b2 = c2. Note that this topic will not be assessed until Module 7.
It is recommended that students have some experience with the lessons in Topic D from Module 2 before beginning these lessons. In Lesson 13 of Module 3, Topic C, students are presented with a general proof that uses the Angle-Angle criterion. In Lesson 14, students are presented with a proof of the converse of the Pythagorean Theorem. Also in Lesson 14, students apply their knowledge of the Pythagorean Theorem (i.e., given a right triangle with sides a, b, c, where c is the hypotenuse, then a2 + b2 = c2) to determine unknown side lengths in right triangles. Students also use the converse of the theorem (i.e., given a triangle with lengths a, b, c, so that a2 + b2 = c2 then the triangle is a right triangle with hypotenuse c) to determine if a given triangle is in fact a right triangle.
Students will use the Quizizz response system to complete a Pythagorean Theorem and Its Converse assessment. Quizizz allows the teacher to conduct student-paced formative assessments through quizzing, collaboration, peer-led discussions, and presentation of content in a fun and engaging way for students of all ages.
