I.Title- Investigating the Pythagorean Theorem
II.Overview of the lesson- Students will do an activity to help them discover the Pythagorean theorem and then they will write journal entries on their thought process as they came up with the solution.
III.Mandated Material (Standards)- Using Geometry standard 14: Students prove the Pythagorean theorem.
IV.Time Allotted for the Lesson- 50 minutes
V.Materials Required- Scissors, glue, poster board, and squares (on graph paper) cut out of many shapes and sizes for each group of three to four students, Tri-Square Rug Game handout, journals/paper for each student to write their journal entry, and a worksheet for homework.
VI.Goals of the Lesson- For students to discover the Pythagorean theorem from a real-world situation and to see how they can use it in geometry. To learn how to read and comprehend a problem in order to discover the solution, and to learn how discussion/group work, reading and/or writing relate to solving problems in math through their journal writing.
VII.Objectives- For students to discover the Pythagorean theorem, to work on problem-solving skills, to learn reading comprehension in real-world math problems, to learn through class discussion, and to improve expressing their thoughts, feelings, and strategies through journal writing.
VIII.Skills Provided- Teacher needs to help students with the Tri-Square Rug Game if they get stuck, but let them try to do it all on their own. Also, the teacher needs to facilitate a class discussion on the game and how it relates to geometry and the real world, to hand out a worksheet with problems related to the Pythagorean theorem, and to read the journal entries to assess students' understanding of the day's activities and homework.
IX.Suggested Procedures- Teachers can begin the lesson by presenting the Tri-Square Rug Game [from Interactive Mathematics Program, Year Two] and letting the students know that they have to solve the problem (find out when the game will be fair for both players). Students are then given time to make their posters and have group discussion on what the solution is. After groups are finished with these steps, the teacher should lead a whole-class discussion about the solution (what it means and other related issues). Teachers should then give the closing time to explain the homework and then let students write in their journals (giving them suggested questions to answer to help guide their writing if they get stuck) to show how they solved the problem and what they think about the solution, etc. When students turn in their homework the next day, teachers can check homework to see if they understand how the Pythagorean theorem relates to solving equations and they can check the students' journals to understand their thinking process about their homework and the topic in general. If students are having trouble, as indicated from observation or their journal writing, teachers can then seek out these students to help them one-on-one.
X.Opening of the Lesson- At the beginning of class, I will introduce the Tri-Square Rug Game and tell the students that they need to discover when Bob and Judy will have a fair game. Students will then be able to start their discovery of the solution.
XI.Specific Activities- Students will get into groups and work on the Tri-Square Rug Game problem. They will form triangles by putting the edges of three rugs, or squares, corner to corner. They must then find the area of the squares to see if Al or Betty has a better chance of winning where Al wins if he hits the biggest rug with a dart and Betty wins if she hits either of the two smaller rugs. They will cut and paste their examples of various situations (Al is favored to win, Judy is favored, or it is a fair game) onto the poster board. After students have discussed the problem in their groups, we will have a class discussion about the work they just did and decide on a formal solution as a class. Students will then do the closing activity.
XII.Closing- Students will then draw conclusions about when there will be a fair game (which is when the rugs form a right triangle) and be able to see where the Pythagorean theorem comes from. Students will then write in their journals about what they discovered through this project: what conclusions they came to. They will also include excerpts from the class discussion and their opinions about it and also the thinking process they used to come to these conclusions and their personal opinions about the results.
XIII.Student Assessment- By reading the students' journal entries, I will be able to see who understood the lesson and who did not, so that I may spend time with certain students who were struggling to make sure that they grasp this important concept.
XIV.Extension of the Lesson- For homework, students will have a worksheet that includes problems dealing with the Pythagorean theorem. At then end of their worksheet students will write more in their journals about how the project and discussion we did in class helps them to solve the homework problems and how they can be related/used in the real world.
XV.Resources Necessary- Journals for the class, worksheet for homework and a worksheet explaining the Tri-Square Rug Game (resource: Alper, Lynne. Dan Fendel, Sherry Fraser, and Diane Resek. 1998. Interactive Mathematics Program, Year Two. Key Curriculum Press, Berkeley.).
Use the figure above to find the missing side length.
1.a = x, b = x, c = 30
2.a = 8, b = 20, c = x
3.a = x, b = 5 3, c = 10
4.If the perimeter is 30: a = 12, b = x, c = 13.
Solve the following problems. Be sure to show all of your work.
5.Find the length of the hypotenuse of a right triangle if the legs are 7cm and 24cm.
6.Find the length of the hypotenuse of an isosceles right triangle where the leg is 6in.
7.If the diagonal of a square is 5 feet, find the length of each side of the square and the area of the square.
8.Find the leg of a right triangle if the hypotenuse is 17 and one leg is 8.
9.Your little brother is pulling a toy train by a string that is 7 feet long where is arm is raised 3 feet above the ground. If he pulls the in 2 feet of the string, how far has the train moved?
10.If a window is 112in by 15in, can you fit a round mirror through it that has a 113in diameter?
11.An old tree fell to the ground as if hinged with the stump. If the stump is 12 feet high, and the top of the tree now lies 35 feet from the base, how tall was the tree before it fell?