The Pythagorean Theorem deepens studentsâ€™ understandings of two major areas of focus: area in 2dimensional and 3dimensional space, and rational/ irrational numbers. Students explore specific relationships between the squares of the legs and hypotenuse of a right triangle, and learn to find side lengths of right triangles in two and three dimensions. Furthermore, students develop operational fluency with calculating roots of perfect square numbers and estimating square roots for nonperfect squares.
Key Concepts:
 Area models can be used to demonstrate the relationship showing the sum of the squared lengths of two legs of a right triangle is the same as the square of the measure of the hypotenuse.
 If the sum of the squares of the measures of two legs of a triangle is the same as the square of the length of the third side, the triangle must be a right triangle.
 We can use the relationships shown in the Pythagorean theorem to solve problems given particular pieces of information about the problem.
Prior Knowledge Needed:
 Understand that area is additive and find areas of figures by decomposing them into nonoverlapping rectangles (Grade 3, Unit 13; 3.MD.7d)
 Identify the properties of angles in right triangles (Grade 4, Unit 10; 4.G.A.2)
 Solve problems with area of 2dimensional shapes (Grade 6, Unit 10; 6.G.1)
 Solve realworld and mathematical problems involving area, volume and surface area of two and threedimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms (Grade 7, Unit 12)
 Understand that nonperfect square roots are irrational numbers (Grade 8, Unit 4; 8.NS.A.2)
Lessons

Lesson objective: Understand that area models can be used to explain why \(a^2 + b^2 =c^2\) Students bring prior knowledge of properties of area and area models from 6.EE.A.2c, 6.G.A.1. This prior knowledge is extended to use the area of squares as stud...

Lesson objective: Rearrange area models in order to gain greater understanding of the Pythagorean Theorem. This lesson helps to build fluency with some Pythagorean Theorem proofs. Area model is used here because it highlights the equivalence of the area...

Lesson objective: Apply the method for a Pythagorean Theorem area model proof to a reallife situation. This lesson provides an opportunity for students to apply their knowledge and understanding of geometric representations of the Pythagorean Theorem t...

Lesson objective: Extend understanding of the Pythagorean Theorem to include the converse, such as if \(a^2 +b^2 = c^2\) then we have a right triangle. Students bring prior knowledge of proving the Pythagorean Theorem from 8.G.B.6. This prior knowledge ...

Lesson objective: Find Pythagorean triples and compare side lengths to similar triangles. This lesson helps to build fluency with application of the converse of the Pythagorean Theorem. This work develops students' understanding that right triangles hav...

Lesson objective: Connect work with right triangles to include right triangles contained in other shapes. This lesson provides an opportunity for students to apply their knowledge and understanding of the converse of the Pythagorean Theorem to a realli...

Lesson objective: Apply the Pythagorean Theorem to solve for missing side lengths in right triangles. Students bring prior knowledge of estimating decimal equivalents of square roots from 8.NS.A.2. This prior knowledge is extended to work with the Pytha...

Lesson objective: Find the distance between points. This lesson helps to build procedural skill with calculating unknown lengths of right triangles and distance between points in twodimensional space. Triangles and the coordinate plane are used here be...

Lesson objective: Apply the Pythagorean Theorem to solve for unknown side lengths of right triangles. This lesson provides an opportunity for students to apply their knowledge and understanding of the Pythagorean Theorem to a reallife situation. Studen...

Lesson objective: Apply the Pythagorean Theorem to distance in 3D space. This lesson provides an opportunity for students to apply their knowledge and understanding of calculating distance to a reallife situation. Students are asked to find the diagona...
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