Brought to you by YouthBuild USA Teacher Fellows! Challenging Students to Discover the Pythagorean Relationship A Common Core-Aligned Lesson Plan to use in your Classroom Author Richard Singer, St. Louis YouthBuild The lesson Students will be able to determine the length of the hypotenuse of a right triangle if given the lengths of the two legs by implementing the Pythagorean Theorem. Students will be able to prove the theorem by drawing squares on all three sides of a right triangle. Students will see the relationship of the three squares of their triangle drawings, and come to the conclusion that the sum of the squares of the two legs equals the square of the hypotenuse. Objective: Having knowledge of the Pythagorean Theorem and right angles is useful and applicable in the trade of carpentry. This knowledge is also very applicable to those who go on to post-secondary training (mathematics courses, engineering majors.) Students will be able to CCSS: Math 8.G.B.7 Apply the Pythagorean theorem to determine unknown side lengths in right triangles in real world and mathematical problems in two and three dimensions. Instructional Shifts: Rigor "Pursuing conceptual understanding, procedural skill and fluency, and application all with equal intensity." Rigor includes more than how to get the answer. Rigor calls for using key concepts in a variety of contexts, including calculations related to geometrical figures. Visit https://youthbuild.org/my/education to access more classroom activity ideas 1
Here s what you do Resources needed: Dot paper (available here http://www.printablepaper.net/category/d ot for drawing triangles. Printed table with the length of two sides of right triangles listed Problem related to carpentry in which they must calculate length of hypotenuse of a right triangle on a roof truss. Rulers with centimeter scales Time: 1 hour + Instructions Academic Vocabulary: Right triangle terminology: legs, sides A, B, and C, hypotenuse, centimeter, roof truss, Pythagorean Theorem or relationship, squares and square roots, dot paper. 1) Activity: (Discuss as a large group, or use pair share and report answers to the class): a) On an overhead transparency, or computer projector, students will be shown an example of a triangle drawn on ¼ inch graph paper. They will then be shown the same triangle with the squares drawn on it. See examples.
b) After obtaining rulers and graph paper, students should use the inch side of the ruler to measure the hypotenuse and then draw their square on the hypotenuse. Be sure to explain how to make sure that two sides of the square are perpendicular to the hypotenuse. An inexpensive transparent T-square would be perfect for making sure that the sides are perpendicular to the hypotenuse. c) Students will then be given a printed table containing facts about three right triangles. The table will show the lengths of the two legs of the right triangles, but not the length of the hypotenuse. d) Students will also be given three pieces (or more if needed) of dot paper. Switching from graph to dot paper makes the lesson a bit more challenging, since the students must now measure in centimeters. The students will draw the two legs of the right triangles, one on each dot paper, and then connect the two legs by drawing the hypotenuse. (See examples at the end of this document.) Students will then draw a square on all three sides of the triangles by measuring each side and creating the square from their measurements. Students will fill in their tables with the areas of each side. 2) Wrap up: Students will be given a real-life, work-related problem involving a roof truss which contains a right triangle. Using what they learned, students will calculate the length of the hypotenuse of the right triangle on the roof truss, as follows:
Problem A: You are new on the job as an apprentice carpenter, working in a new housing subdivision. You are with your boss, working on roof trusses. Your boss gives you the below drawing and asks you to figure out the length of side C, which happens to be the hypotenuse of a right triangle. Your boss is counting on you to be accurate. Based on what you know, what is the length of side C? Examples of Triangles Drawn on Dot Paper 3) Differentiated Instruction: What are ways that you will adapt the lesson for students with different skill levels? How could you stretch the lesson for more advanced learners? How might you make the lesson more accessible for struggling/reluctant learners? Student pairs can be assigned by levels of ability; one higher level with one lower level. This would address struggling/reluctant learners because the higher level student in the paid will aid the understanding of the lower level partner. More advanced learners could be given more triangles in which they are given the length of the hypotenuse and one leg and must calculate length of the remaining leg, using an algebraic formula. Manipulative (tiles) could be used as another method of proving the theorem.
Success Tips What specific tips could you offer educators adopting this lesson? What are potential student misconceptions or struggles with the lesson? It is assumed that at this level, the students have already been exposed to right triangles and their terminology, but have not yet learned the Pythagorean theorem/relationship. Even if they have learned the theorem, it is unlikely that they have seen the actual proof by drawing squares on all three legs of a right triangle and seeing the theorem proven in a clear illustration. Students may need some guided practice to understand the measuring and drawing of the squares, and drawing the small squares inside the large squares. Because of this, the example of a drawn triangle given at the beginning of the lesson is very important. During guided instruction, do not take the pencil out of a student s hand. Explain the correction, and let the student make it. Put the objective on the board: What is the relationship among the squares of the legs and hypotenuse of a right triangle? Encourage students to answer each other s questions. Evidence of Success By the end of the lesson, students will be able to apply the Pythagorean theorem without drawing squares, and will understand the meaning of the Pythagorean theorem. Mastery will be demonstrated when students understand the relationship of the legs and hypotenuse of the right triangle. Outsiders should be able to see that the students have a deep understanding once the relationship is clear.