Pythagorean Theorem

Abstract

This lesson introduces and explores the Pythagorean Theorem. Three activities give students the opportunity to observe triangles, learn and use the Pythagorean Theorem and practice different ways of determining areas of triangles.

Objectives

Upon completion of this lesson, students will:

  • know the Pythagorean Theorem.
  • use the Pythagorean Theorem to find side lengths of right triangles.
  • use the Pythagorean Theorem to find areas of right triangles.
  • apply the Pythagorean Theorem to find the perimeter and area of triangles on a grid.

Standards Addressed

Grade 6

  • Geometry

    • The student solves problems (including real-world situations) using perimeter, area, or volume.

Grade 7

  • Geometry

    • The student solves problems (including real-world situations).

Grade 8

  • Geometry

    • The student solves problems (including real-world situations).

Grade 9

  • Geometry

    • The student solves problems (including real-world situations).

Grade 10

  • Geometry

    • The student solves problems (including real-world situations).

Grade 7

  • Measurement and Geometry

    • 3.0 Students know the Pythagorean theorem and deepen their understanding of plane and solid geometric shapes by constructing figures that meet given conditions and by identifying attributes of figures

Eighth Grade

  • Geometry

    • Understand and apply the Pythagorean Theorem.

Geometry

  • Similarity, Right Triangles, and Trigonometry

    • Define trigonometric ratios and solve problems involving right triangles

Grades 6-8

  • Geometry

    • Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships

Grades 9-12

  • Geometry

    • Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships

Grade 8

  • Number and Operations, Measurement, Geometry, Data Analysis and Probability, Algebra

    • COMPETENCY GOAL 3: The learner will understand and use properties and relationships in geometry.

Introductory Mathematics

  • Data Analysis and Probability

    • COMPETENCY GOAL 3: The learner will understand and use properties and relationships in geometry.

  • Geometry and Measurement

    • COMPETENCY GOAL 2: The learner will use properties and relationships in geometry and measurement concepts to solve problems.

Geometry

  • Data Analysis and Probability

    • Competency Goal 3: The learner will transform geometric figures in the coordinate plane algebraically.

Technical Mathematics I

  • Geometry and Measurement

    • Competency Goal 2: The learner will measure and apply geometric concepts to solve problems.

Technical Mathematics II

  • Geometry and Measurement

    • Competency Goal 1: The learner will use properties of geometric figures to solve problems.

7th Grade

  • Geometry

    • The student will demonstrate through the mathematical processes an understanding of proportional reasoning, tessellations, the use of geometric properties to make deductive arguments. the results of the intersection of geometric shapes in a plane, and the
    • The student will demonstrate through the mathematical processes an understanding of proportional reasoning, tessellations, the use of geometric properties to make deductive arguments. the results of the intersection of geometric shapes in a plane, and the relationships among angles formed when a transversal intersects two parallel lines.

8th Grade

  • Geometry

    • The student will demonstrate through the mathematical processes an understanding of the Pythagorean theorem; the use of ordered pairs, equations, intercepts, and intersections to locate points and lines in a coordinate plane; and the effect of a dilation
    • The student will demonstrate through the mathematical processes an understanding of the Pythagorean theorem; the use of ordered pairs, equations, intercepts, and intersections to locate points and lines in a coordinate plane; and the effect of a dilation in a coordinate plane.

Geometry

  • Geometry

    • Standard G-3: The student will demonstrate through the mathematical processes an understanding of the properties and special segments of triangles and the relationships between and among triangles.

8th Grade

  • Geometry

    • 8.10a The student will verify the Pythagorean Theorem, using diagrams, concrete materials, and measurement; and
    • 8.10 The student will

Textbooks Aligned

8th

  • Module 5 - Architects and Engineers

    • Section 3: Working with Triangles
      • Reason for Alignment: This lesson accompanies three triangle based activities: Pythagorean Explorer, Squaring the Triangle and Triangle Explorer. In this lesson, students learn how this theorem works and how to apply it. This could be reinforcement for the work in the textbook.

Student Prerequisites

  • Arithmetic: Students must be able to:
    • add, subtract, multiply
  • Technological: Students must be able to:
    • use a calculator to square numbers
    • perform basic mouse manipulations such as point, click and drag.
    • use a browser for experimenting with the activities.

Teacher Preparation

Key Terms

area

The number of square units needed to cover a surface

perimeter

The sum of the lengths of all the sides of a polygon

Pythagorean Theorem

Used to find side lengths of right triangles, the Pythagorean Theorem states that the square of the hypotenuse is equal to the squares of the two sides, or A2 + B2 = C2, where C is the hypotenuse

right triangle

A triangle containing an angle of 90 degrees

Lesson Outline

  1. Focus and Review

    Remind students what has been learned in previous lessons that will be pertinent to this lesson and/or have them begin to think about the words and ideas of this lesson:

    • Ask students to recall information about triangles.
    • You might show students several different right triangles and then write the Pythagorean Theorem on the board and label the sides of the triangles according to the theorem.
    • Discuss what it might mean to talk about the area of a triangle.

  2. Objectives

    Let the students know what it is they will be doing and learning today. Say something like this:

    • Today, class, we will be learning about the Pythagorean Theorem. We will learn how the theorem works, and we will learn how to calculate the length of a missing side of a right triangle.
    • We are going to use the computers to learn about the Pythagorean Theorem, but please do not turn your computers on or go to this page until I ask you to. I want to show you a little about the Squaring the Triangle applet first.

  3. Teacher Input

    Explain to the students how the Squaring the Triangle applet works. You should model or demonstrate it for the students, especially if they are not familiar with how to use our computer applets.

    • Show the students two or three different sized triangles, and show them how the accompanying squares help to see how the Pythagorean Theorem works.
    • If you choose to, you may pass out the Squaring the Triangle Exploration Questions and answer the questions with students.
    • Next, open your browser to the Pythagorean Explorer in order to demonstrate this activity to the students.
    • Go through one or two examples of finding the area of a missing side of a right triangle by using the Pythagorean Theorem.

  4. Guided Practice

    Give students either a time limit or specific number of triangles to solve using the Pythagorean Explorer.

    • If your class seems to understand the process for doing this assignment, simply ask, "Can anyone tell me what you will do now?"
    • If your class seems to be having a little trouble with this process, do another example together, but let the students direct your actions:
      • Can someone describe how I would find the length of the missing side of this triangle?
      • If you choose to, you may pass out the Pythagorean Theorem Exploration Questions and have students work through the questions.

  5. Independent Practice

    • Allow the students to work on their own and to complete the worksheets, should you choose to provide them. Monitor the room for questions and to be sure that the students are on the correct web site.
    • Another option for independent practice is to have the students work in pairs (carefully chosen so that both students are of the same ability group). Have them race to find the correct areas and side lengths using the Pythagorean Explorer applet. Who ever wins gets a point. At the end of the allotted time for the game give the winning member of each pair a reward of some type.
    • For a final exercise, show students how to apply the Pythagorean Theorem in other situations using the Triangle Explorer.
    • Show several triangles using the medium and hard levels of the Triangle Explorer, and ask students if they can find the area. If needed, give them the hint that they can divide the triangle into smaller triangles with right areas.

  6. Closure

    • You may wish to bring the class back together for a discussion of the findings. Once the students have been allowed to share what they found, summarize the results of the lesson.

Alternate Outline

This lesson can be rearranged in several ways if there is only one available computer:

  • Write the Pythagorean Theorem on the board, and have students take out a piece of paper. Using the Pythagorean Explorer applet, have students write down the measurements of four or five triangles, and then give them time to find the length of the missing angle. When you are done, take up the papers and check them.
  • Have students work in groups of two or three to practice using the Pythagorean Theorem to find areas of triangles in the medium and hard levels of the Triangle Explorer. If students need a hint, show them how to divide a triangle into two triangles with right angles, then find the area of each triangle to get the whole area.