G.2.7

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

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Formative Assessments
Formative Assessments
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CPALMS: Students are asked to determine the length of a side of a right triangle in a real-world problem.

How Far to School

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CPALMS: Students are asked to determine the length of a side of a right triangle in a real-world problem.

Three Dimensional Diagonal

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CPALMS: Students are asked to determine the length of a side of a right triangle in a real-world problem.

New Television

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CPALMS: Students are asked to determine the length of a side of a right triangle in a real-world problem.

Pyramid Height

Virtual/Hands-on Manipulatives
Virtual/Hands-on Manipulatives
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CPALMS: This resource, from NLVM, will help teachers and students validate the Pythagorean Theorem both geometrically and algebraically. It can be used interactively to better understand the topic.

Pythagorean Theorem Manipulatives

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CPALMS: This free, online calculator, will calculate the side lengths and angles for any right triangle. Students will input any valid combination of sides and/or angles and then the calculator will do the rest! It will also draw an image that can be downloaded.

Right Triangle Calculator

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Assessment Questions
Assessment Questions
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Videos of Expert Instruction
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Hook/Performance Tasks
Hook/Performance Tasks
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Illustrative Mathematics: Students use the Pythagorean Theorem to find the unknown side-lengths of a trapezoid to determine the area. They will need to be creative and persistent to decompose the trapezoid into other polygons to find area.

Area of a Trapezoid

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Illustrative Mathematics: Students work with volumes of cylinders, spheres and cones. The insight required increases as one moves across the glasses, from simple application of the formula for volume, to decomposition of volume into two, to calculating height.

Glasses

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Illustrative Mathematics: The purpose is for students to work on visualizing the two-dimensional path on the surface of the three-dimensional box and apply the Pythagorean Theorem. This task allow exploration of a concrete model to help with visualization.

Spiderbox

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Illustrative Mathematics: Very explicit shapes have been chosen to provide an opportunity to practice using knowledge of different geometric formulas for area and perimeter. The task maximizes the area enclosed by a shape with fixed perimeter.

Areas of Geometric Shapes with the Same Perimeter

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Illustrative Mathematics: Students will solve a multi-step problem which provides a more involved use of similarity to translate the given directions into an accurate picture. They calculate segment lengths and derive trig properties of isosceles right triangles.

Points from Directions

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Illustrative Mathematics: Students draw pictures two triangles and make an auxiliary construction to calculate areas. They should know, or intuitively identify, the fact that the line of symmetry of the isosceles triangle divides the base in half and meets the base perpendicularly.

Two Triangles’ Area

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Illustrative Mathematics: Students apply knowledge about triangles, circles, and squares to calculate and compare 2 different areas, which can be done several ways. Solutions use definition of congruence, Pythagorean Theorem, and symmetry considerations.

Circle Sandwich

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Illustrative Mathematics: Students will reason as to how they can use the Pythagorean Theorem to find the distance run by two people. The focus isn't who ran the greater distance but on seeing how to set up a right triangle to apply the theorem to this problem.

Running on the Football Field

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Illustrative Mathematics: Students use a drawing of a shaded triangle surrounded by three right triangles. They are given some dimensions and must use the Pythagorean Theorem to determine the hypotenuse of one triangle in order to prove whether the shaded is a right triangle.

Applying the Pythagorean Theorem in a Mathematical Context

Problem-Based Lessons and Real World Application
Real World Application
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Listen to a master quilt maker discuss the relationship between art and mathematics as she demonstrates how the Pythagorean theorem is an important part of the pattern design process. Accompanying Materials are available.

Pythagorean Theorem in Kapa Design by PBS Learning Media

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In this task, students' curiosity about how to determine who would reach a taco cart first (walking on the beach sand or on the pavement, each at a different speed and distance) is generated. They model the math with various resources to come to a conclusion.

Taco Cart – 3-Act Math Task by Dan Meyer

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Measuring a Grid for Underwater Archeology