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Pythagorean Theorem

Lesson Plan

Pythagorean Theorem

Objectives

In this lesson, students will use the Pythagorean Theorem to solve for any side of a right triangle. Students will:

  • explore where the theorem came from. [IS.2 - All Students]
  • use the theorem to solve for the hypotenuse of a right triangle.
  • use the theorem to solve for a missing leg of a right triangle.
  • apply the theorem to solve real-world application problems.

Essential Questions

  • How can you explain the relationship between congruence and similarity in both two and three dimensions?
  • How are coordinates manipulated algebraically to represent, interpret, and verify geometric relationships?

Vocabulary

  • Converse of the Pythagorean Theorem: If in a triangle, a2 + b2 = c2 and a, b, and c are the sides of the triangle, [IS.1 - All Students] then the triangle is a right triangle; if c2 > a2 + b2, then the triangle is an obtuse triangle; if c2 < a2 + b2, then the triangle is an acute triangle.
  • Hypotenuse: The side opposite the right angle in a right triangle.
  • Leg: Either one of the sides of a right triangle adjacent to the hypotenuse.
  • Pythagorean Theorem: The sum of the squares of the lengths of the legs of a right triangle is equal to the square of the length of the hypotenuse; in any right triangle where the length of one leg is a, the length of the second leg is b, and the length of the hypotenuse is c, as in: c2 = a2 + b2.
  • Pythagorean Triple: Any set of three positive integers, a, b, and c, such that a2 + b2 = c2.
  • Right Triangle: A triangle with one 90-degree angle.

Duration

120–240 minutes/24 class periods

Prerequisite Skills

Prerequisite Skills haven't been entered into the lesson plan.

Materials

Related Unit and Lesson Plans

Related Materials & Resources

The possible inclusion of commercial websites below is not an implied endorsement of their products, which are not free, and are not required for this lesson plan.

  • Cloud picture

http://www.pals.iastate.edu/carlson/main.html

  • Show students the objective of tangrams or if students have computer access, allow them to do a puzzle on the computer.

https://pbskids.org/cyberchase/games/tangram-game

  • Illustrate a set of tangrams and the length of some of the sides.

http://mathworld.wolfram.com/Tangram.html

  • Demonstrate how tangram shapes are rotated and shifted to create a “cat”.

http://demonstrations.wolfram.com/Tangram/

Formative Assessment

  • View
    • In your observations during group activities (Think-Pair-Share and the Carousel) and class discussion, evaluate the strategies students use to create their triangles. Do they inspect the shapes before assembling them? What questions do they ask their partners?
    • In the Lesson 1 Exit Ticket activity, check that students select legs and hypotenuse appropriately and use the correct operations before checking the accuracy of calculation.
    • In the Extension Activity, using the converse of the Pythagorean Theorem requires proper computation as well as the selection of the appropriate sides and hypotenuse.

Suggested Instructional Supports

  • View
    Scaffolding, Active Engagement
    W: This lesson is designed to show students where the Pythagorean Theorem originates by connecting it to something they are familiar with: puzzles. They learn what the Pythagorean Theorem is and what it is used for. Once they have used the theorem to solve some basic problems, they use the theorem to solve real-world application problems. If students are going beyond the standards, they can learn the converse of the Pythagorean Theorem and apply it to a few problems.  
    H: This lesson begins with the image of a cloud and students are asked to say what they see. This is then linked to the topic of jigsaw puzzles and then tangrams. Tangrams are similar to seeing images in clouds. Students learn about tangrams and are given the opportunity to use tangrams to replicate a picture. After they have created the picture, they are given three squares with which they have to create a triangle. Since some people believe the Pythagorean Theorem stems from the use of tangrams, it is only appropriate that students see it for themselves.  
    E: The majority of this lesson is about exploration. From when students are first given tangrams to when they see how three squares can create a triangle, students are following in the footsteps of early mathematicians. They are given the opportunity to write down their observations and thoughts, and then they are led to the Pythagorean Theorem with a graphic organizer. After students are given the theorem, they participate in a group activity in which they solve real-world application problems and present those problems to check each other’s work. 
    R: Since this lesson allows students to explore some topics on their own, walk around the classroom asking students questions and making comments to get them to rethink their observations. Students are working in pairs, which allow them to discuss the problems and to revise their answers. The carousel activity is for students to refine their problem-solving skills.  
    E: Throughout this lesson, walk around and check students’ work. Students need immediate evaluation to correct any misunderstanding of the material. The carousel activity allows students to see how other students solve problems and gives them the opportunity to evaluate themselves. Evaluate students’ exit tickets and decide if students need additional practice or if they can move forward. 
    T: This lesson is tailored to kinesthetic and visual learners. Students begin this lesson working with puzzles and manipulatives to derive an important theorem in mathematics. The graphic organizer allows students to see the Pythagorean Theorem and where it comes from. During the carousel activity, the auditory learner is given an opportunity to hear about situations involving the Pythagorean Theorem. 
    O: This lesson keeps students engaged from beginning to end. It connects the puzzles they know to puzzles from China, which leads to the main topic of the lesson. Students are given a graphic organizer, which helps them improve their note-taking skills. They learn to use their notes to solve similar problems. They begin solving problems in groups, which also helps them solve problems independently.  

     

    IS.1 - All Students
    Show relationship of the Pythagorean Theorem in real life situations.  Show how it can be used in other subjects.  
    IS.2 - All Students
    Review and reteach solving equations based on the formative assessment of students.  
    IS.3 - All Students
    Resources should enhance the lesson.   Ensure that students understand the task from additional resources.  
    IS.4 - All Students
    Make connections between patterns and tangrams.  Use graphic organizers as appropriate.  

Instructional Procedures

  • View

    Ask students, “Have you ever looked up in the sky at the clouds and seen images of animals or people?” Give students a minute to answer the question. “What about in this picture? What do you see?” Show students the Cloud Picture handout (M-G-7-1_Cloud Picture.doc). “Shapes are all around us. They give form to things we use every day. The top of the desk you sit in is a rectangle. The base of a lamp is a circle. Look around the classroom and tell the person next to you what shapes you see.”

    While students are looking around, take pieces from the jigsaw puzzle and place them on the overhead projector, document camera, or whiteboard. Also take out a set of tangrams and place them next to the jigsaw puzzle pieces. Use the Set of Tangrams sheet (M-G-7-1_Set of Tangrams.doc). Let students share their observations with the class. “How many of you have put together a jigsaw puzzle before? Today we’re going to be using another type of puzzle called tangrams.” Point to the set on the overhead projector or document camera. “A set of tangrams consists of seven pieces: two large right triangles, one medium right triangle, two small right triangles, one square, and one parallelogram. The objective is to use the seven pieces to replicate a picture without overlapping any pieces. Here is an example.”

    If you have a computer, show students the Web site http://pbskids.org/cyberchase/games/area/tangram.html. If you do not have a computer, show the handout Picture of a Tangram Web site (M-G-7-1_Picture of a Tangram Web Site.doc). “Look at the top right of the screen. What are we supposed to make with our seven pieces?” You can either do the tangram on the computer in front of the class, or you can put a set on the overhead or document camera and have students help you move the pieces to create the rabbit.

    Hand out a set of tangrams to each student and place the Tangram of a Fox sheet (M-G-7-1_Tangram of a Fox.doc) on the overhead projector or document camera. [IS.4 - All Students] Give students about ten minutes to try to put the fox together using their tangrams. When students have successfully made the fox or when time is up, hand out the Set of Three Squares in Inches sheet (M-G-7-1_Set of Three Squares in Inches.doc) to half the class. The red square should be 3 inches x 3 inches, the blue square should be 4 inches x 4 inches, and the yellow square should be 5 inches x 5 inches. Hand out the Set of Three Squares in Centimeters (M-G-7-1_Set of Three Squares in Centimeters.doc) to the other half of the class. The green square should be 5 cm x 5 cm, the purple square should be 12 cm x 12 cm, and the orange square should be 13 cm x 13 cm. Students should cut out the squares.

    Part 1: Think-Pair-Share

    “We are going to use the three squares as though they are a new set of tangrams. Your goal is to create a triangle with the three squares with no overlap. Some of you have three squares measured in inches, and some have three squares measured in centimeters.” Walk around and answer questions. Students are going to struggle creating a triangle with the three squares, so after a few minutes give them a hint that the triangle is not going to be made of paper. It will be created by the space between the squares. When students figure it out, hand out a ruler and the Three Squares and a Triangle Observation Page (M-G-7-1_Three Squares and a Triangle Observation Page.doc). “Write down whatever comes to mind about the tangram you just created. It could be about side lengths, perimeter, area, or the shapes in general.”

    After students write down their individual findings, pair them so that one had the squares in inches, and the other had the squares in centimeters. They should share their ideas and compare and contrast. Then have the pairs share with the whole class. “Did anyone notice anything about the areas of the three squares and if there is any relation among them?” See what students say about area and if anyone is able to recognize that the area of the two smaller squares combined equals the area of the larger square. If anyone does, call this student Pythagoras for the day. “The Chinese invented the game of tangrams, and scholars believe a famous formula we use today was created using tangrams. However, a Greek philosopher and mathematician named Pythagoras is the one getting the credit. Today we are going to learn what the Pythagorean Theorem is and how the areas of three squares actually help us find side lengths of right triangles.”

    Part 2

    Hand out the Pythagorean Theorem Graphic Organizer (M-G-7-1_Pythagorean Theorem Graphic Organizer.doc and M-G-7-1_Pythagorean Theorem Graphic Organizer KEY.doc) and go through it with the class. There are two practice problems on the bottom for students to try on their own. Go over the answers once everyone has attempted the problems.

    Part 3: Carousel

    There are five problems in the Pythagorean Carousel handout, so if you have more than five groups, make duplicates (M-G-7-1_Pythagorean Carousel Problems.ppt). Answers are located in the notes section of the PowerPoint. Divide students into groups of three or four. They work one problem at a time, and when they finish, they pass the problem clockwise (writing their work on a separate sheet of paper). When everyone is done, if there is time, have each group present one of the problems to the class.

    Part 4

    Hand out the Lesson 1 Exit Ticket (M-G-7-1_Lesson 1 Exit Ticket and KEY.doc) to evaluate whether students understand the Pythagorean Theorem.

    Extension:

    • Students who are confident can work on the Extension Activity (M-G-7-1_Extension Activity and KEY.doc). It introduces the converse of the Pythagorean Theorem and illustrates how to determine if a triangle is a right triangle, given the three sides of the triangle.
    • Partner up students who may need more practice. Each student writes a word problem requiring use of the Pythagorean Theorem to solve, and the partner solves the problem. Partners check each other’s work and repeat the process if time allows.

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DRAFT 10/13/2011
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