The Rope Stretcher
Lesson 4 of 6
Objective: SWBAT use properties of geometry and algebra to justify the statements in a proof. Students will understand the meaning of the Converse of the Pythagorean Theorem.
The warm-up prompt for this lesson asks students to write the converse of a logical statement - the statement is the Pythagorean Theorem. I ask students to follow our Team Warm-up routine, which involves sharing their responses with the other members of their cooperative learning team. I choose students at random to write the team's chosen response on the board.
Students may have a little trouble with this task, because the conclusion of the statement (and the hypothesis of the converse statement) is a statement of a relationship between the sides of a triangle in the form of an equation (MP2, MP7). I use student work on the front board to highlight a model solution, or guide the class in correcting their work.
I ask the class if they can name the theorem they have just written -- it is the Converse of the Pythagorean Theorem. I remind the class that the converse of a theorem has to be proven separately. They are not surprised when I say that we are about to do it.
I display the Agenda and Learning Targets for the lesson and briefly review them with the class.
The Rope Stretcher
With the help of a slide, I tell the class the story of the harpedonopta, or rope stretcher, who laid out the foundations of the great pyramids of ancient Egypt. I point out that the bases of the pyramids are nearly perfect squares, often several hundred feet on a side. I ask if anyone has ever tried to lay out a right angle to build a fence, lay tile, or lay out a garden. It is harder than it looks. I point out that using a carpenter's square (I have one in my classroom) is usually not sufficient, because an error of even a single degree becomes pretty large at a distance of a dozen feet or so. By the time you lay out a rectangle and return to your starting point, you are likely to find that the sides of the rectangle do not line up. To be as accurate as the ancient Egyptions (the bases of the pyramids were accurate to within a foot), you need to use a tool of an appropriate size (MP6).
Naturally, the ancient Egyptians used the Converse of the Pythagorean Theorem to do it. So, we will have to prove that theorem. Then, I will demonstrate how they did it.
In this section, students work in pairs to understand a proof of the Converse of the Pythagorean Theorem.
I begin by demonstrating the proof. This gives students a chance to hear it delivered by someone who understands it. Since my intention is to help students understand the main arguments of the proof without getting stuck on the details, I do not give away all of the answers to the puzzle proof that students are about to complete. I do take student questions, however. The point is for them to understand the proof.
Students then work to complete a Puzzle Proof by writing justifications for each statement -- either in their own words or by referring to a theorem, definition, or property by name (MP3). The activity uses the Rally Coach format. I hand out the activity, then display a slide and a digital countdown timer as students get to work. I ask that at least one student in each pair get out their course notes.
At the end of a time limit (10 minutes, but it could be extended), I display a model solution (or nearly a model solution) using student work. As a class, we correct any errors and summarize the proof (MP3).
If students finish early, I ask them to begin on the proof of the Hypotenuse-Leg Congruence Theorem, which is printed on the back side of the handout. This proof also uses the Pythagorean Theorem. Students will complete the proof for homework, but it is good if they can work with a partner to begin it now.
How They Did It
With the help of student 'slaves', I demonstrate how the rope stretcher used a knotted rope to create a 3-4-5 right triangle, which could be any size required. I use a piece of 550-pound test cord, in which 12 knots have been tied at intervals of about a foot.
It helps to have the tallest student volunteer stand in the middle, holding the right angle high for the class to see. I may remember to display a slide as a background.
I ask students to get individual White Boards, rags, and dry erase markers. I display the problems using the slide show, and students practice using the Pythagorean Theorem Converse to show that a triangle is either right or not right. I use this opportunity to coach students in how much work is acceptible to show, so that they will receive full credit on the unit quiz (MP3).
Each slide has two problems, which are more or less parallel. Students may choose to work either one. Students who are working quickly should work both, while I help those who need it.
The lesson close follows our Individual Size-Up Routine. The prompt asks students to explain how they can determine whether a triangle is a right triangle, using the Converse of the Pythagorean Theorm. They write their answers in their Learning Journals. We will not summarize the theorem in course notes for two more lessons.
For homework, I assign problems #10-12 of Homework Set 1 for this unit. Problem #10 asks students to use the Converse of the Pythagorean Theorem to show that a triangle is right, or not right. Problem #11 is a puzzle proof of the Hypotenuse-Leg Congruence Theorem, which uses the Pythagorean Theorem. Problem #12 asks students to explain why the rope stretcher's method works.