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 10

Geometry

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

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 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

Geometry

Data Analysis and Probability

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

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.

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 relationships among angles formed when a transversal intersects two parallel lines.

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

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 in a coordinate plane.

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

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

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: Student 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

Access to a browser

Pencil and paper

Copies of supplemental materials for the activities:

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 A^{2} + B^{2} = C^{2}, where C is the hypotenuse

right triangle

A triangle containing an angle of 90 degrees

Lesson Outline

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.

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.

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.

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.

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.