Student Actions

Teacher Actions

• Develop Accurate and Appropriate Procedural Fluency when using the Distance Formula and/or the Pythagorean Theorem to find the length between two points on a coordinate grid in various situations.

• Develop strategies for problem-solving when using efficient procedures to analyze the relationship between side lengths of triangles with the Pythagorean Theorem and its converse and also assess the reasonableness of solutions when categorizing triangles as right, acute, or obtuse.

• Develop the Ability to Make Conjectures, Models, and Generalizations by using models to extend patterns of relationships among triangle side lengths to develop conjectures of how the distance formula is related to the Pythagorean Theorem. 

• Facilitate mathematical discourse by engaging students in conversations that stress the use of correct vocabulary (leg, hypotenuse, point, right triangle, etc.) when describing strategies and/or solutions.

• Pose purposeful questions that enhance students’ experiences in making connections between various situations including the Pythagorean Theorem or distance formula.

• Build procedural fluency when students apply the distance formula and/or the Pythagorean Theorem to various mathematical and contextual situations. 

Key Understandings

Misconceptions 

• The Pythagorean Theorem is used to find missing side lengths of right triangles.

• The Pythagorean Theorem and its converse are used to categorize a triangle as acute, right, or obtuse:

• If the square of the longest side is equal to the sum of the squares of the other two sides, then the triangle is right.

• If the square of the longest side is greater than the sum of the squares of the other two sides, then the triangle is obtuse.

• If the square of the longest side is less than the sum of the squares of the other two sides, then the triangle is acute.

• Apply the distance formula and the Pythagorean Theorem in various contexts.

• The distance formula is derived from the Pythagorean Theorem. 

• Students cannot distinguish the longest side as the hypotenuse when using the converse of the Pythagorean Theorem for determining if triangles are acute, obtuse, or right.

• Students perform incorrect calculations when using the Pythagorean Theorem to solve problems.

• Students forget to determine if three sides could make a triangle before applying the rules of deciding which type of triangle they are working with.

• Students confuse the inequality signs when deciding if side lengths create an acute, right, or obtuse triangle by putting the sum of the squares of the lesser sides first and then the inequality sign followed by the square of the longest side.

     Example: Classify the triangle with sides of

                      8 , 7, and 12.

                    8² + 7² and 12²

                   64+49 and 144

                   113 < 144 and the student concludes incorrectly that the triangle is acute. 

  Knowledge Connections

Prior Knowledge

Leads to 

• Justify the Pythagorean theorem using measurements, diagrams, or dynamic software to solve problems in two dimensions involving right triangles. (PA.GM.1.1) 

• Use the Pythagorean theorem to find the distance between any two points in a coordinate plane. (PA.GM.1.2)

• Identify the square roots of perfect squares to 400 or, if it is not a perfect square root, locate it as an irrational number between two consecutive positive integers. (PA.N.1.4)

• Write square roots and cube roots of constants in simplest radical form. (A1.N.1.1) 

• Solve square and cube root equations with one variable, and check for extraneous solutions. (A2.A.1.5)

• Create models for situations involving trigonometry. (PC.T.2.1)

• Choose and produce an equivalent form of an expression to explain the properties of the quantity represented by the expression. (PC.T.3.2)