<p>In this lesson, students will apply the Pythagorean Theorem to calculate any side of a right triangle. Students will:&nbsp;<br>- investigate where the theorem came from.&nbsp;<br>- use the theorem to get the hypotenuse of a right triangle.&nbsp;<br>- use the theorem to find a missing leg in a right triangle.&nbsp;<br>- use the theorem to solve real-world application problems.<br>&nbsp;</p>

<p>- How would you explain the relationship between congruence and similarity in two and three dimensions?&nbsp;<br>- How are coordinates algebraically transformed to express, interpret, and validate geometric relationships?<br>&nbsp;</p>

<p>- Converse of the Pythagorean Theorem: If in a triangle, <i>a</i>² + <i>b</i>² = <i>c</i>² and <i>a</i>, <i>b</i>, and <i>c</i> are the sides of the triangle, then the triangle is a right triangle; if <i>c</i>² &gt; <i>a</i>² + <i>b</i>², then the triangle is an obtuse triangle; if <i>c</i>² &lt; <i>a</i>² + <i>b</i>², then the triangle is an acute triangle.<br>- Hypotenuse: The side opposite the right angle in a right triangle.<br>- Leg: Either one of the sides of a right triangle adjacent to the hypotenuse.<br>- 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 <i>a</i>, the length of the second leg is <i>b</i>, and the length of the hypotenuse is <i>c</i>, as in: <i>c</i>² = <i>a</i>² + <i>b</i>².<br>- Pythagorean Triple: Any set of three positive integers, <i>a</i>, <i>b</i>, and <i>c</i>, such that <i>a</i>² + <i>b</i>² = <i>c</i>².<br>- Right Triangle: A triangle with one 90-degree angle.</p>

<p>- Cloud Picture handout (M-G-7-1_Cloud Picture)<br>- a jigsaw puzzle (not included; any jigsaw puzzle)<br>- Picture of a Tangram Web site handout (M-G-7-1_Picture of a Tangram Web Site)<br>- one copy per student of the Set of Tangrams handout (M-G-7-1_Set of Tangrams)<br>- handout of Tangram of a Fox (M-G-7-1_Tangram of a Fox)<br>- copies of Set of Three Squares in Inches handout (M-G-7-1_Set of Three Squares in Inches)<br>- copies of Set of Three Squares in Centimeters handout (M-G-7-1_Set of Three Squares in Centimeters)<br>- scissors<br>- rulers (with both inches and centimeters)<br>- copies of Pythagorean Theorem Graphic Organizer (M-G-7-1_Pythagorean Theorem Graphic Organizer and KEY)<br>- calculators (scientific or graphing)<br>- copies of the Pythagorean Carousel Problems (M-G-7-1_Pythagorean Carousel Problems)<br>- Extension Activity (M-G-7-1_Extension Activity and KEY)<br>- Three Squares and a Triangle Observation Page (M-G-7-1_Three Squares and a Triangle Observation Page)<br>- copies of the Lesson 1 Exit Ticket (M-G-7-1_Lesson 1 Exit Ticket and KEY)</p>

<p>- Evaluate the ways students use to create their triangles based on your observations during group activities (Think-Pair-Share and the Carousel) and class discussion. Do they inspect the shapes before they assemble them? What are the questions they ask their partners?&nbsp;<br>- Before assessing the accuracy of the computation in the Lesson 1 Exit Ticket activity, ensure that students correctly choose the legs and hypotenuse and use the correct operations.&nbsp;<br>- In the Extension Activity, using the converse of the Pythagorean Theorem requires accurate computation as well as the selection of the suitable sides and hypotenuse.<br>&nbsp;</p>

<p>Scaffolding, Active Engagement<br>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 about the Pythagorean Theorem and how it is used. After solving some basic problems with the theorem, they apply it to real-world applications. If students exceed the expectations, they can study the converse of the Pythagorean Theorem and apply it to a few problems.&nbsp;<br>H: The lesson starts with an image of a cloud and prompts students to describe what they see. This is then linked to the discussion of jigsaw puzzles and tangrams. Tangrams are similar to seeing images in clouds. Students learn about tangrams and have the opportunity to utilize them to recreate a picture. After they have completed the picture, they are given three squares with which they have to create a triangle. Because some people believe the Pythagorean Theorem is derived from the use of tangrams, it is only appropriate that students observe it for themselves.&nbsp;<br>E: This lesson focuses primarily on exploring. Students follow in the footsteps of early mathematicians from when they are given tangrams until they see how three squares can create a triangle. They are asked to write down their observations and thoughts before being guided through the Pythagorean Theorem using a graphic organizer. After learning the theorem, students participate in a group activity in which they solve real-world application problems and submit them to each other to check their work.&nbsp;<br>R: To encourage students to rethink their observations, walk around the classroom asking questions and making remarks as they explore some topics independently. Students work in pairs, allowing them to discuss problems and refine responses. The carousel activity allows students to refine their problem-solving skills.&nbsp;<br>E: During the lesson, walk around and assess students' work. Students require fast feedback to address any misunderstandings of the topic. The carousel activity helps students to observe how other students solve problems and to evaluate themselves. Evaluate students' exit tickets and decide if they require extra practice or can move forward.&nbsp;<br>T: This lesson is designed for kinesthetic and visual learners. Students begin this lesson by working with puzzles and manipulatives to develop an important mathematical theorem. The graphic organizer shows students the Pythagorean Theorem and where it comes from. During the carousel activity, the auditory learner can hear about situations involving the Pythagorean Theorem.&nbsp;<br>O: This lesson engages students from start to finish. It connects the puzzles students know to puzzles from China, leading to the main topic of the lesson. Students are given a graphic organizer to assist them improve their note-taking skills. They learn how to apply their notes to solve similar problems. They begin by solving problems in groups, which allows them to solve problems individually later.</p>

<p>Ask the students: <strong>"Have you ever looked up in the sky at the clouds and seen images of animals or people?"</strong> Give students one minute to answer the question. <strong>"How about this picture? What do you see?"</strong> Show students the Cloud Picture handout (M-G-7-1_Cloud Picture). <strong>"Shapes are all around us. They give shape to objects we use every day. The top of the desk you sit at is a rectangle. The base of a lamp is a circle. Look around the classroom and tell the person next to you about the shapes you observe."</strong></p><p>While students are gazing around, remove pieces of the jigsaw puzzle and place them on the overhead projector, document camera, or whiteboard. Take out a set of tangrams and arrange them next to the jigsaw puzzle pieces. Use the Set of Tangrams worksheet (M-G-7-1_Set of Tangrams). Allow students to share their observations with the class. <strong>"How many of you have completed a jigsaw puzzle before? Today we'll be using </strong><i><strong>tangrams</strong></i><strong>, which are another type of puzzle."</strong> Point to the set on the overhead projector or document camera.<strong> "A set of tangrams is made up of seven pieces: two large right triangles, one medium right triangle, two small right triangles, one square, and one parallelogram. The goal is to use the seven pieces to create an image without overlapping them. Here's an example."</strong></p><p>If you have a computer, show students the Web page <a href="http://pbskids.org/cyberchase/games/area/tangram.html">http://pbskids.org/cyberchase/games/area/tangram.html</a>. If you don't have a computer, display the handout Picture of a Tangram Web Site (M-G-7-1_Picture of a Tangram Web Site). <strong>"Look to the upper right of the screen. What are we expected to build with these seven pieces?"</strong> You can either do the tangram on the computer in front of the class, or you can display a set on the overhead or document camera and have students assist you in moving the pieces to create the rabbit.</p><p>Distribute a set of tangrams to each student and display the Tangram of a Fox sheet (M-G-7-1_Tangram of a Fox) on the overhead projector or document camera. Give students ten minutes to try to put the fox together with their tangrams. When students have completed the fox or when time is up, distribute the Set of Three Squares in Inches sheet (M-G-7-1_Set of Three Squares in Inches) to half of the class. The red square should measure 3 inches × 3 inches, the blue square 4 inches × 4 inches, and the yellow square 5 inches × 5 inches. Distribute the Set of Three Squares in Centimeters (M-G-7-1_Set of Three Squares in Centimeters) to the remaining half of the class. The green square should measure 5 cm × 5 cm, the purple square 12 cm × 12 cm, and the orange square 13 cm × 13 cm. Students should cut the squares.</p><p><strong>Part 1: Think-Pair-Share</strong></p><p><strong>"We will use the three squares as if they were a new set of tangrams. Your goal is to create a triangle with the three squares that do not overlap. Some of you have three squares measured in inches, while others have three squares measured in centimeters." </strong>Walk around and answer questions. Students will struggle to create a triangle from the three squares, so after a few minutes, imply that the triangle will not be created of paper. It will be created by the gaps between the squares. When the students have figured it out, hand them a ruler and the Three Squares and a Triangle Observation Page (M-G-7-1_Three Squares and a Triangle Observation Page). <strong>"Write down anything that comes to mind about the tangram you just created. It could be regarding side lengths, perimeter, area, or the basic shape."</strong></p><p>After students have recorded their own findings, pair them so that one has the squares in inches and the other has the squares in centimeters. They should exchange ideas and compare and contrast. Then have the partners share with the entire class. <strong>"Did anyone notice anything about the areas of the three squares and if there is any relation among them?"</strong> Consider what students say about area and whether anyone can notice that the area of the two smaller squares combined equals the area of the larger one. If someone does, call this student as Pythagoras for the day. <strong>"The Chinese invented the game of tangrams, and experts believe a well-known formula we use today was created with tangrams. However, the credit goes to Pythagoras, a Greek philosopher and mathematician. Today we'll study about the Pythagorean Theorem and how the areas of three squares can help us calculate the side lengths of right triangles."</strong></p><p><strong>Part 2</strong></p><p>Distribute the Pythagorean Theorem Graphic Organizer (M-G-7-1_Pythagorean Theorem Graphic Organizer and KEY) and review it with the class. The bottom section contains two practice problems for students to try on their own. Once everyone has solved the problems, review the answers.</p><p><strong>Part 3: Carousel</strong></p><p>The Pythagorean Carousel handout contains five problems; if you have more than five groups, create duplicates (M-G-7-1_Pythagorean Carousel Problems). The answers are in the PowerPoint's notes section. Divide students into groups of three or four. They work on one problem at a time, then pass it clockwise (writing their work on a separate sheet of paper). When everyone is finished, if time permits, have each group present one of the problems to the class.</p><p><strong>Part 4</strong></p><p>Hand out the Lesson 1 Exit Ticket (M-G-7-1_Lesson 1 Exit Ticket and KEY) to assess students' understanding of the Pythagorean Theorem.</p><p><strong>Extension:</strong></p><p>Students who are confident can complete the Extension Activity (M-G-7-1_Extension Activity and KEY). It explains the converse of the Pythagorean Theorem and demonstrates how to determine whether a triangle is a right triangle based on its three sides.&nbsp;<br>Students that require additional practice should be paired together. Each student creates a word problem that requires the usage of the Pythagorean Theorem, which the partner solves. Partners review each other's work and, if time permits, repeat the procedure.</p>