The Pythagorean Theorem and Right Triangles
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1 The Pythagorean Theorem and Right Triangles Student Probe Triangle ABC is a right triangle, with right angle C. If the length of and the length of, find the length of. Answer: the length of, since and Lesson Description Students will do a hands- on activity to verify the Pythagorean Theorem for a right triangle. They will then use the theorem to solve for the length of the hypotenuse and for the length of a leg in two additional examples. All triangles used in this lesson have sides of integral length. Students should then proceed to applying the theorem in problem situations and with triangles that do not necessarily have integral side lengths. Rationale The Pythagorean Theorem is perhaps the most At a Glance What: Pythagorean Theorem and Right Triangles Common Core Standard: CC.8.G.7 Understand and apply the Pythagorean Theorem. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real- world and mathematical problems in two and three dimensions. Matched Arkansas Standard:AR.9-2.T.G.2.4 (T.2.G.4) Apply the Pythagorean Theorem and its converse in solving practical problems Mathematical Practices: Make sense of problems and persevere in solving them. Model with mathematics. Who: Students who cannot apply the Pythagorean Theorem. Grade Level: 8 Prerequisite Vocabulary: leg, square root, hypotenuse, right angle, right triangle Prerequisite Skills: finding squares and square roots of numbers, solving numeric and algebraic equations involving squares and square roots Delivery Format: Small groups Lesson Length: 30 minutes Materials, Resources, Technology: scissors, tape or glue stick, calculators (optional) Student Worksheets: Pythagorean Theorem Puzzle important theorem in school mathematics. Equations of circles, the distance formula, many principles of trigonometry, resolution of vectors, and applications of perimeter, area, and volume are derived from it. Not only is a student s ability to correctly use and apply the Pythagorean Theorem is necessary for success in subsequent mathematics courses, many practical applications require its use. These include construction projects such as tiling or carpeting and determining if walls and corners are square or perpendicular.
2 Preparation Prepare copies of Pythagorean Theorem Puzzle for each student. Provide each student with a pair of scissors, tape or glue stick, and a calculator (optional). Lesson The teacher says or does 1. Look at the Pythagorean Theorem Puzzle. Do you see a triangle? What kind of triangle is it? How do you know it is a right triangle? 2. We are going to investigate a very important property of all right triangles, but only of right triangles. Expect students to say or do Yes. Right triangle. It has one right angle. If students do not, then the teacher says or does Point out the triangle. Review the properties of a right triangle. Review that the legs are the sides of the triangle that form the right angle. What are the lengths of the legs of the right triangle? 3, 4 3. Can you count to find the length of the hypotenuse of the right triangle? Why not? Let s find a way to determine its length. 4. How many squares do you see drawn on the grid? 5. Find the area of each of the smaller squares. No, because it is not a vertical or horizontal line segment. Count the lengths of the legs, if necessary. If students try to count the length (perhaps they will say a length of 6), use a piece of the grid to show that is not an accurate answer. 3 Make sure that students see the drawn squares, not the grid itself. 9 and 16 Review how to find the area of a square. Some students may need to count the number of unit squares in each of the squares.
3 The teacher says or does 6. Cut the squares with area 9 and area 16 away from the triangle. Cut them into individual unit squares. How many unit squares do you have? 7. Cover the big square attached to the hypotenuse with the unit squares. What do you notice? 8. Can we make a conjecture that is true for this triangle? 9. What is the length of the hypotenuse? 10. Let s see if this works for other right triangles. Suppose that a right triangle has legs of length 6 and 8. What is the length of its hypotenuse? Expect students to say or do 25, because. They fit perfectly. 5 Because. If students do not, then the teacher says or does Can you count the unit squares? Assist students as they place and secure the unit squares on the paper. Guide students from to. 10 Sketch the triangle and assist students as they label the diagram.
4 The teacher says or does Expect students to say or do If students do not, then the teacher says or does 11. This is a very important mathematical property called the Pythagorean Theorem. It states that for every right triangle ABC, Students should write this in their notebooks and/or math journals. where is the right angle,. (Note: Draw this diagram to illustrate the theorem. Emphasize that c must be the length of the hypotenuse.) 12. Let s use the Pythagorean Theorem in a different way. Suppose that a right triangle has a leg with length 12, and a hypotenuse with length 13. What is the length of the other leg? Prompt students as they draw the diagram and solve the equation. Students may use a calculator if they wish. Draw a diagram and find the length of the missing leg. 13. The Pythagorean Theorem can help us find the length of either leg or the hypotenuse of a right triangle if we know the lengths of two of the other sides.
5 Teacher Notes 1. When stating the Pythagorean Theorem as, emphasize that the hypotenuse must have length c. 2. Students may need to be reminded that either leg can be considered as having length a or b. 3. Students will need additional practice. Practice problems should include right triangles with non- integral lengths, triangles with the length of the hypotenuse missing, and triangles with the length of one leg missing. 4. The purpose of this lesson is for students to become familiar with and apply the Pythagorean Theorem correctly. Students may experience numerical and algebra difficulties that are best addressed in another lesson. 5. The Converse of the Pythagorean Theorem (If, then the triangle is a right triangle.) leads to an interesting relationship: If, then the triangle is acute. If If, then the triangle is right., then the triangle is obtuse. Variations Additional problems should include right triangles with non- integral lengths, triangles with the length of the hypotenuse missing, and triangles with the length of one leg missing. Formative Assessment Triangle ABC is a right triangle, with right angle C. If the length of, find the length of. and the length of Answer: 17
6 Resources Mathematics Preparation for Algebra. (n.d.). Retrieved 1 14, 2011, from Doing What Works: Russell Gersten, P. (n.d.). RTI and Mathematics IES Practice Guide - Response to Intervention in Mathematics. Retrieved August 16, 2011, from rti4sucess: on.pdf
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