Illustrative Math Alignment: Grade 8 Unit 8

Pythagorean Theorem and Irrational Numbers

Lesson 8: A Proof of the Pythagorean Theorem

Use the following Media4Math resources with this Illustrative Math lesson.

Title	Body	Curriculum Topic
Math Example--Coordinate Geometry--Distance Formula: Example 1	Math Example--Coordinate Geometry--Distance Formula: Example 1 Topic Geometry Description This example demonstrates the application of the distance formula to find the distance between two points on a coordinate plane. The points (3, 3) and (7, 6) are plotted on a graph, and the distance between them is calculated using the formula: √((6 - 3)2 + (7 - 3)2) = 5.	The Distance Formula
Math Example--Coordinate Geometry--Distance Formula: Example 10	Math Example--Coordinate Geometry--Distance Formula: Example 10 Topic Geometry Description This example illustrates the use of the distance formula to calculate the distance between two points on a coordinate plane. The points (1, 6) and (10, -3) are plotted on a graph, and the distance between them is determined using the formula: √((1 - 10)2 + (6 - (-3))2) = √(9^2 + 9^2) = √162 = 9√2.	The Distance Formula
Math Example--Coordinate Geometry--Distance Formula: Example 11	Math Example--Coordinate Geometry--Distance Formula: Example 11 Topic Geometry Description This example demonstrates the application of the distance formula to find the vertical distance between two points on a coordinate plane. The points (5, 6) and (5, -8) are plotted on a graph, and the distance between them is calculated using the formula: √((5 - 5)2 + (6 - (-8))2) = √(0 + (14)2) = 14.	The Distance Formula
Math Example--Coordinate Geometry--Distance Formula: Example 12	Math Example--Coordinate Geometry--Distance Formula: Example 12 Topic Geometry Description This example illustrates the use of the distance formula to calculate the distance between two points on a coordinate plane. The points (-8, -2) and (-2, 6) are plotted on a graph, and the distance between them is determined using the formula: √((-8 - (-2))2 + (-2 - 6)2) = √((-6)2 + (-8)2) = 10.	The Distance Formula
Math Example--Coordinate Geometry--Distance Formula: Example 13	Math Example--Coordinate Geometry--Distance Formula: Example 13 Topic Geometry Description This example demonstrates the application of the distance formula to find the distance between two points on a coordinate plane. The points (-10, 6) and (-1, -1) are plotted on a graph, and the distance between them is calculated using the formula: √((-10 - (-1))2 + (6 - (-1))2) = √((-9)2 + 72) = √130.	The Distance Formula
Math Example--Coordinate Geometry--Distance Formula: Example 14	Math Example--Coordinate Geometry--Distance Formula: Example 14 Topic Geometry Description This example illustrates the use of the distance formula to calculate the vertical distance between two points on a coordinate plane. The points (-5, 6) and (-5, -4) are plotted on a graph, and the distance between them is determined using the formula: √((-5 - (-5))2 + (6 - (-4))2) = √(0 + 102) = 10.	The Distance Formula
Math Example--Coordinate Geometry--Distance Formula: Example 15	Math Example--Coordinate Geometry--Distance Formula: Example 15 Topic Geometry Description This example demonstrates the application of the distance formula to find the distance between two points on a coordinate plane. The points (-6, -9) and (6, -4) are plotted on a graph, and the distance between them is calculated using the formula: √((6 - (-6))2 + (-4 - (-9))2) = √(122 + 52) = 13.	The Distance Formula
Math Example--Coordinate Geometry--Distance Formula: Example 16	Math Example--Coordinate Geometry--Distance Formula: Example 16 Topic Geometry Description This example illustrates the use of the distance formula to calculate the distance between two points on a coordinate plane. The points (-1, -2) and (9, -10) are plotted on a graph, and the distance between them is determined using the formula: √((-1 - 9)2 + (-2 - (-10))2) = √((-10)2 + 82) = 2√41.	The Distance Formula
Math Example--Coordinate Geometry--Distance Formula: Example 17	Math Example--Coordinate Geometry--Distance Formula: Example 17 Topic Geometry Description This example demonstrates the application of the distance formula to find the horizontal distance between two points on a coordinate plane. The points (-2, -7) and (3, -7) are plotted on a graph, and the distance between them is calculated using the formula: √((-2 - 3)2 + (-7 + 7)2) = √((-5)2 + 0) = 5.	The Distance Formula
Math Example--Coordinate Geometry--Distance Formula: Example 18	Math Example--Coordinate Geometry--Distance Formula: Example 18 Topic Geometry Description This example illustrates the use of the distance formula to calculate the distance between two points on a coordinate plane. The points (-4, 0) and (0, 3) are plotted on a graph, and the distance between them is determined using the formula: √((4 - 0)2 + (0 - 3)2) = √(16 + 9) = 5.	The Distance Formula
Math Example--Coordinate Geometry--Distance Formula: Example 19	Math Example--Coordinate Geometry--Distance Formula: Example 19 Topic Geometry Description This example demonstrates the application of the distance formula to find the distance between two points on a coordinate plane. The points (0, 6) and (1, 0) are plotted on a graph, and the distance between them is calculated using the formula: √((0 - 1)2 + (6 - 0)2) = √(1 + 36) = √37.	The Distance Formula
Math Example--Coordinate Geometry--Distance Formula: Example 2	Math Example--Coordinate Geometry--Distance Formula: Example 2 Topic Geometry Description This example illustrates the use of the distance formula to calculate the distance between two points on a coordinate plane. The points (3, 7) and (9, 2) are plotted on a graph, and the distance between them is determined using the formula: √((9 - 3)2 + (2 - 7)2) = √(61).	The Distance Formula
Math Example--Coordinate Geometry--Distance Formula: Example 20	Math Example--Coordinate Geometry--Distance Formula: Example 20 Topic Geometry Description This example illustrates the use of the distance formula to calculate the horizontal distance between two points on a coordinate plane. The points (0, 0) and (8, 0) are plotted on a graph, and the distance between them is determined using the formula: √((0 - 8)2 + (0 - 0)2) = √64 = 8.	The Distance Formula
Math Example--Coordinate Geometry--Distance Formula: Example 21	Math Example--Coordinate Geometry--Distance Formula: Example 21 Topic Geometry Description This example demonstrates the application of the distance formula to find the vertical distance between two points on a coordinate plane. The points (0, 0) and (0, 6) are plotted on a graph, and the distance between them is calculated using the formula: √((0 - 0)2 + (0 - 6)2) = √(0 + (-6)2) = 6.	The Distance Formula
Math Example--Coordinate Geometry--Distance Formula: Example 3	Math Example--Coordinate Geometry--Distance Formula: Example 3 Topic Geometry Description This example demonstrates the application of the distance formula to find the distance between two points on a coordinate plane. The points (2, 4) and (9, 4) are plotted on a graph, and the distance between them is calculated using the formula: √((9 - 2)2 + (4 - 4)2) = 7.	The Distance Formula
Math Example--Coordinate Geometry--Distance Formula: Example 4	Math Example--Coordinate Geometry--Distance Formula: Example 4 Topic Geometry Description This example illustrates the use of the distance formula to calculate the vertical distance between two points on a coordinate plane. The points (5, 8) and (5, 2) are plotted on a graph, and the distance between them is determined using the formula: √((5 - 5)2 + (8 - 2)2) = 6.	The Distance Formula
Math Example--Coordinate Geometry--Distance Formula: Example 5	Math Example--Coordinate Geometry--Distance Formula: Example 5 Topic Geometry Description This example demonstrates the application of the distance formula to find the distance between two points on a coordinate plane. The points (-2, 11) and (6, 5) are plotted on a graph, and the distance between them is calculated using the formula: √((-2 - 6)2 + (11 - 5)2) = 10.	The Distance Formula
Math Example--Coordinate Geometry--Distance Formula: Example 6	Math Example--Coordinate Geometry--Distance Formula: Example 6 Topic Geometry Description This example illustrates the use of the distance formula to calculate the distance between two points on a coordinate plane. The points (-4, 8) and (6, 2) are plotted on a graph, and the distance between them is determined using the formula: √((-4 - 6)2 + (8 - 2)2) = 2 √(34).	The Distance Formula
Math Example--Coordinate Geometry--Distance Formula: Example 7	Math Example--Coordinate Geometry--Distance Formula: Example 7 Topic Geometry Description This example demonstrates the application of the distance formula to find the distance between two points on a coordinate plane. The points (-4, 3) and (2, 3) are plotted on a graph, and the distance between them is calculated using the formula: √((-4 - 2)2 + (3 - 3)2) = 6.	The Distance Formula
Math Example--Coordinate Geometry--Distance Formula: Example 8	Math Example--Coordinate Geometry--Distance Formula: Example 8 Topic Geometry Description This example illustrates the use of the distance formula to calculate the distance between two points on a coordinate plane. The points (-6, -3) and (6, 2) are plotted on a graph, and the distance between them is determined using the formula: √((-6 - 6)2 + (-3 - 2)2) = 13.	The Distance Formula
Math Example--Coordinate Geometry--Distance Formula: Example 9	Math Example--Coordinate Geometry--Distance Formula: Example 9 Topic Geometry Description This example demonstrates the application of the distance formula to find the distance between two points on a coordinate plane. The points (9, 6) and (5, -1.5) are plotted on a graph, and the distance between them is calculated using the formula: √((9 - 5)2 + (6 - (-1.5))2) = √(4^2 + 7.5^2) = √72.25 = 8.5.	The Distance Formula
Math Example--Geometric Shapes--Constructing Geometric Shapes: Example 1	Math Example--Geometric Shapes--Constructing Geometric Shapes: Example 1 Topic Geometry Description This image shows three drumsticks arranged to form a triangle. The drumsticks are placed in a closed triangular shape, with three angles marked in red. This example shows how to construct a triangle using drumsticks. The solution explains that to know it's a triangle, you need to create a closed figure with three sides and note the three interior angles.	Definition of a Quadrilateral and Geometric Constructions with Triangles
Math Example--Geometric Shapes--Constructing Geometric Shapes: Example 10	Math Example--Geometric Shapes--Constructing Geometric Shapes: Example 10 Topic Geometry Description The image shows red and yellow circular counters arranged in a grid pattern to form a square shape. The solution explains how to arrange the counters in equal numbers per layer to form a square with four clearly defined corners. This example showcases how to construct a square using these red and yellow counters.	Definition of a Quadrilateral and Geometric Constructions with Triangles
Math Example--Geometric Shapes--Constructing Geometric Shapes: Example 2	Math Example--Geometric Shapes--Constructing Geometric Shapes: Example 2 Topic Geometry Description This image shows a triangle drawn using a crayon. The triangle is closed with three sides, and the three angles are highlighted in red. This example showcases how to draw a triangle using a crayon or pencil. The solution states that to confirm it's a triangle, you need to create a closed figure with three sides and ensure it has three clearly identified angles.	Definition of a Quadrilateral and Geometric Constructions with Triangles
Math Example--Geometric Shapes--Constructing Geometric Shapes: Example 3	Math Example--Geometric Shapes--Constructing Geometric Shapes: Example 3 Topic Geometry Description This image shows a triangle drawn using paint and a brush. The triangle is closed with three sides, and the three angles are marked in red. This example showcases how to draw a triangle using paint and a brush. The solution explains that to verify it's a triangle, you must create a closed figure with three sides and make sure it has three clearly identified angles.	Definition of a Quadrilateral and Geometric Constructions with Triangles
Math Example--Geometric Shapes--Constructing Geometric Shapes: Example 4	Math Example--Geometric Shapes--Constructing Geometric Shapes: Example 4 Topic Geometry Description This image shows a triangle formed by arranging stickers in a triangular shape. The stickers form the sides of the triangle, and the angles are highlighted. This example showcases how to draw a triangle using stickers. The solution mentions that you can use stickers to form a closed figure with three sides, ensuring it has three clearly identified angles to confirm it's a triangle.	Definition of a Quadrilateral and Geometric Constructions with Triangles
Math Example--Geometric Shapes--Constructing Geometric Shapes: Example 5	Math Example--Geometric Shapes--Constructing Geometric Shapes: Example 5 Topic Geometry	Definition of a Quadrilateral and Geometric Constructions with Triangles
Math Example--Geometric Shapes--Constructing Geometric Shapes: Example 6	Math Example--Geometric Shapes--Constructing Geometric Shapes: Example 6 Topic Geometry Description The image features four drumsticks arranged in a square shape. The solution highlights how to verify that the figure is a square by checking that it has four sides and four square corners. This example showcases how to construct a square using four drumsticks. The text describes how to create a closed figure with four sides and check for square corners to confirm it's a square.	Definition of a Quadrilateral and Geometric Constructions with Triangles
Math Example--Geometric Shapes--Constructing Geometric Shapes: Example 7	Math Example--Geometric Shapes--Constructing Geometric Shapes: Example 7 Topic Geometry	Definition of a Quadrilateral and Geometric Constructions with Triangles
Math Example--Geometric Shapes--Constructing Geometric Shapes: Example 8	Math Example--Geometric Shapes--Constructing Geometric Shapes: Example 8 Topic Geometry	Definition of a Quadrilateral and Geometric Constructions with Triangles
Math Example--Geometric Shapes--Constructing Geometric Shapes: Example 9	Math Example--Geometric Shapes--Constructing Geometric Shapes: Example 9 Topic Geometry Description The image shows a set of colorful stickers with various holiday-themed designs (hearts, snowmen, reindeer, bells, etc.). The stickers are arranged to form a square shape. The solution demonstrates how to use the stickers to create a square by aligning them in a closed figure with four clear corners. This example showcases how to draw a square using these stickers.	Definition of a Quadrilateral and Geometric Constructions with Triangles
Math Example--Geometric Transformation--Translating Triangles--Example 1	Math Example--Geometric Transformation--Translating Triangles--Example 1 Topic Transformations Description A triangle on a grid is translated 4 units to the left. It shows the original triangle ABC and the translated triangle A'B'C'. Example 1: "Draw the triangle that results from the following translation: 4 units to the left." Solution: "Identify one point to translate. Then complete the triangle."	Applications of Transformations and Applications of Triangles
Math Example--Geometric Transformation--Translating Triangles--Example 10	Math Example--Geometric Transformation--Translating Triangles--Example 10 Topic Transformations Description The triangle is translated 4 units to the left and 2 units up. It displays both the initial and the translated triangles. Example 10: "Draw the triangle that results from the following translation: 4 units to the left, 2 units up." Solution: "Identify one point to translate. Then complete the triangle."	Applications of Transformations and Applications of Triangles
Math Example--Geometric Transformation--Translating Triangles--Example 11	Math Example--Geometric Transformation--Translating Triangles--Example 11 Topic Transformations Description Triangle ABC is translated horizontally to the right by 4 units to form triangle A'B'C'. Example 11: The translation is described as 4 units to the right. In this topic, students explore transformations, focusing specifically on translating triangles. These examples visually demonstrate how shapes move within a coordinate plane, reinforcing understanding of shifts along axes. Translation examples assist in grasping the basic concept of shifting figures without altering their orientation or shape.	Applications of Transformations and Applications of Triangles
Math Example--Geometric Transformation--Translating Triangles--Example 12	Math Example--Geometric Transformation--Translating Triangles--Example 12 Topic Transformations Description Triangle ABC is translated vertically downward by 6 units to form triangle A'B'C'. Example 12: The translation is described as 6 units down. In this topic, students explore transformations, focusing specifically on translating triangles. These examples visually demonstrate how shapes move within a coordinate plane, reinforcing understanding of shifts along axes. Translation examples assist in grasping the basic concept of shifting figures without altering their orientation or shape.	Applications of Transformations and Applications of Triangles
Math Example--Geometric Transformation--Translating Triangles--Example 13	Math Example--Geometric Transformation--Translating Triangles--Example 13 Topic Transformations Description Triangle ABC is translated horizontally to the left by 5 units to form triangle A'B'C'. Example 13: The translation is described as 5 units to the left. In this topic, students explore transformations, focusing specifically on translating triangles. These examples visually demonstrate how shapes move within a coordinate plane, reinforcing understanding of shifts along axes. Translation examples assist in grasping the basic concept of shifting figures without altering their orientation or shape.	Applications of Transformations and Applications of Triangles
Math Example--Geometric Transformation--Translating Triangles--Example 14	Math Example--Geometric Transformation--Translating Triangles--Example 14 Topic Transformations Description Triangle ABC is translated vertically upward by 6 units to form triangle A'B'C'. Example 14: The translation is described as 6 units up. In this topic, students explore transformations, focusing specifically on translating triangles. These examples visually demonstrate how shapes move within a coordinate plane, reinforcing understanding of shifts along axes. Translation examples assist in grasping the basic concept of shifting figures without altering their orientation or shape.	Applications of Transformations and Applications of Triangles
Math Example--Geometric Transformation--Translating Triangles--Example 15	Math Example--Geometric Transformation--Translating Triangles--Example 15 Topic Transformations Description Triangle ABC is translated diagonally 5 units to the right and 5 units downward to form triangle A'B'C'. Example 15: The translation is described as 5 units to the right and 5 units down.	Applications of Transformations and Applications of Triangles
Math Example--Geometric Transformation--Translating Triangles--Example 16	Math Example--Geometric Transformation--Translating Triangles--Example 16 Topic Transformations Description Triangle ABC is translated diagonally 4 units to the right and 3 units upward to form triangle A'B'C'. Example 16: The translation is described as 4 units to the right and 3 units up. In this topic, students explore transformations, focusing specifically on translating triangles. These examples visually demonstrate how shapes move within a coordinate plane, reinforcing understanding of shifts along axes. Translation examples assist in grasping the basic concept of shifting figures without altering their orientation or shape.	Applications of Transformations and Applications of Triangles
Math Example--Geometric Transformation--Translating Triangles--Example 17	Math Example--Geometric Transformation--Translating Triangles--Example 17 Topic Transformations Description Triangle ABC is translated diagonally 4 units downward and 4 units to the right to form triangle A'B'C'. Example 17: The translation is described as 4 units down and 4 units to the right. In this topic, students explore transformations, focusing specifically on translating triangles. These examples visually demonstrate how shapes move within a coordinate plane, reinforcing understanding of shifts along axes. Translation examples assist in grasping the basic concept of shifting figures without altering their orientation or shape.	Applications of Transformations and Applications of Triangles

Illustrative Math Alignment: Grade 8 Unit 8

Pythagorean Theorem and Irrational Numbers

Lesson 8: A Proof of the Pythagorean Theorem

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