Teacher: Cassie Cooper Unit: The Pythagorean Theorem Lesson: Proof with Jellybeans I. Common Core State Standards 8.G.B.6—Explain a proof of the Pythagorean Theorem and its converse. II. Learning Objectives The learner will be able to construct and explain a proof of the Pythagorean Theorem using a physical model. The learner will be able to write a formula to represent the Pythagorean Theorem with 100% accuracy. III. Anticipatory Set For a warm-up, the students will be asked what they know about right triangles. They will be given around 5 minutes to brainstorm all the ideas that they have about the characteristics of right triangles. After the 5 minutes, the teacher will construct a concept map on the board. The words “right triangle” will be at the center and students will contribute ideas to construct subsequent bubbles. The teacher will build upon these ideas and focus mainly on the ideas that right triangles must have a right angle, as well as the fact that the two sides touching the right angle are called “legs” and the remaining side is the “hypotenuse.” These vocabulary words will be crucial for the lesson. IV. Purpose/Rationale “Yesterday, we talked about distance between points on a plane, as well as squares and square roots. If you struggled with the idea of square roots, you can look back at the worksheet. Today, we are going to learn about how to find the ‘flying’ distance between points on a plane. This is just talking about the direct distance between any two points. Before Discovery Education tests, we talked about the Pythagorean Theorem. Today, you’re going to actually prove this theorem and see how and why it works!” V. Input A. Task Analysis i. The students will complete a warm-up requiring them to brainstorm ideas about characteristics of right triangles. They will work independently. ii. After 5 minutes, the teacher and students will collaboratively construct a concept map of right triangles, focusing on the vocabulary needed to be successful in the lesson. iii. State the objectives and purpose of this lesson. iv. Students will get into groups of 3. The teacher will distribute the Activity Page, jellybeans, and cardboard constructions. They will then explain the directions stated at the top of the worksheet to students. v. Students will work to complete the Activity Page using the given manipulatives. vi. The teacher will be circulating the room, ensuring students are on task, assisting with questions, and asking questions for understanding. vii. viii. ix. x. xi. As students finish the Activity Page (or with 10 minutes left of class), the teacher will bring the class back together as a whole. They will ask students to find someone who was not in their group to compare their formula to. The teacher will then ask for ideas of what students concluded, writing many of them on the board. They will then write the standard formula for the Pythagorean Theorem and explain the similarities and differences between that and student responses. The teacher will then explain explicitly how the activity proves the Pythagorean Theorem. The teacher will ask students to think about how you could use side lengths to determine if a triangle is, in fact, a right triangle. As a class, they will conclude that you could see if they satisfy the Pythagorean Theorem. If they do, the triangle is right. The teacher will then pass out the homework on Pythagorean Triples. Any extra time will be used to work on this. B. Thinking Levels i. Knowledge— Students will recognize the formula of the Pythagorean Theorem. ii. Comprehension—Students will explain the Pythagorean Theorem and its’ relation to area of squares. iii. Application—Students will apply the converse of the Pythagorean Theorem to determine if a triangle is a right triangle from given leg lengths. iv. Synthesis—Students will construct a geometric proof of the Pythagorean Theorem and the formula that describes the conclusion. C. Learning Styles i. For students who are not understanding, I will work with them in their groups on how the manipulatives and hands-on aspects relate to the formula. I will also work with them to understand how the converse of the Pythagorean Theorem works and how it can be used to determine if triangles are right. ii. As an extension, there is a Bonus Challenge at the end of the Activity Page. Students will be asked to create their own model not using squares that could prove the same theorem. They can draw it and give an explanation on the back of the page. iii. Hands-on learners will be able to physically construct a proof of the Pythagorean Theorem. iv. As the teacher circulates, they can also ask students questions so that they can answer orally, as opposed to verbally. v. Visual learners will be able to see the different shapes involved in the proof of the Pythagorean Theorem, as well as see the different formulaic representations of the same concept. VI. Modeling When giving directions to the Activity Page, the teacher will demonstrate to students what they are expected to do, without the use of jellybeans. That is, they will use the cardboard construction and the Activity Page to detail what students are to do and what questions they should be able to answer at the end. The teacher will also model how to use the converse of the Pythagorean Theorem to determine if a triangle is right. They will write an example on the board before students are to do it on their homework. VII. Checking for Understanding How does this activity actually prove the Pythagorean Theorem? How is the Pythagorean Theorem related to area? Could this same activity be done with other shapes besides squares? How does the formula you wrote relate to the activity? Would this work with triangles that weren’t right triangles? Could you test this? Is the “c” square always going to be on the hypotenuse? Why or why not? VIII. Guided Practice Students will work on the activity in class, with assistance from the teacher and their group members as needed. The teacher will also demonstrate an example of using the converse of the Pythagorean Theorem to determine if a triangle is a right triangle. IX. Independent Practice Students will complete the Pythagorean Triples Homework independently. X. Closure With around 10 minutes left of the lesson, the class will come together to discuss the importance of this theorem. Questions to ask: i. How does this activity prove the Pythagorean Theorem? ii. How does the formula relate to what we did today? iii. Would this work with triangles that weren’t right triangles? iv. Is the “c” square always going to be on the hypotenuse? Why or why not? The teacher will then discuss how we could work backwards, this time determining if a triangle was a right triangle from given leg lengths. They will explain that this notion is the converse of the Pythagorean Theorem and can be used to determine if any three leg lengths are Pythagorean Triples. The teacher will reflect and evaluate the lesson after it is implemented to make any necessary changes to make the lesson more effective.