Exploring The Pythagorean Theorem

Transcription

1 Exploring The Pythagorean Theorem I 2 T 2 Project Ken Cochran Grade Level: Ninth Grade Time Span: Five-day unit plan Tools: Geometer s Sketchpad Geoboards and Rubber bands 5-foot broom handle Tape measures Overhead materials from The Proofs of Pythagoras (I2D2)

2 Unit Objectives Students will be able to: Find the length of a segment without using the Pythagorean theorem Be able to describe the Pythagorean theorem in terms of area Prove the Pythagorean theorem Use the Pythagorean theorem to find the length of a segment Be able to prove if a triangle is an acute, right or obtuse triangle Find uses outside of the classroom for the Pythagorean theorem NCTM Content Standards: Algebra Measurement Standards and Key Ideas NCTM process Standards: Problem solving Communications Representation Connections New York State Standard Key Ideas: A.PS.1 Use a variety of problem solving strategies to understand new mathematical content A.PS.1 Use a variety of problem solving strategies to understand new mathematical content A.PS.6 Use a variety of strategies to extend solution methods to other problems A.PS.7 Work in collaboration with others to propose, critique, evaluate, and value alternative approaches to problem solving A.RP.1 Recognize that mathematical ideas can be supported by a variety of strategies A.RP.4 Develop, verify, and explain an argument, using appropriate mathematical ideas and language A.RP.9 Devise ways to verify results or use counterexamples to refute incorrect statements A.CM.4 Explain relationships among different representations of a problem

3 A.CM.8 A.R 6 A.N.5 A.A.5 A.A.45 Reflect on strategies of others in relation to one s own strategy Use mathematics to show and understand physical phenomena Solve algebraic problems arising from situations that involve fractions, decimals, percents (decrease/increase and discount), and proportionality/direct variation Write algebraic equations or inequalities that represent a situation Determine the measure of a third side of a right the Pythagorean theorem, given the lengths of any two sides triangle using Resources and Materials Resources: Geometer s Sketchpad I2D2 notes and worksheets The Proofs of Pythagoras, ETA, Vernon Hills IL (I2D2) Materials: Day One: Geoboards Geopaper Day Two: Computer lab GSP Files One and Two Worksheet 1 Day Three: Different size square papers Scissors Rulers Overhead materials from The Proofs of Pythagoras Day Four: Worksheet 2 Geoboards Geopaper Day Five: Tape measures 5-Foot broom handle Day Five Worksheet Unit Overview

4 Day One Finding the length of a segment on a geoboard by using area. Day Two Show that the area of the squares made from the sides of a triangle represent what type of triangle it is Day Three Prove the Pythagorean theorem Day Four Finding the same lengths used on Day One but this time using the Pythagorean theorem Day Five Outside the classroom examples of uses of the Pythagorean theorem

5 Day One Calculating the Length of a Segment Using Area Objective: Students will learn how to calculate the length of a segment using the area of squares and triangles. Students will learn how to use a Geoboard to calculate the lengths of a segment without using the Pythagorean Theorem. Students will also start to visualize the concept of using area of squares and triangles to prove the Pythagorean Theorem. Opening Activity: To begin class, have the students find the area of several squares and triangles that are on the blackboard. If it is necessary review the formula for the area of a square and triangle. Main Activity: Geoboards and geopaper will be handed out. Using the overhead geoboard I will demonstrate to the students the process that I want them to follow. 1) Use a rubber band to create a segment that is on an angle on the geoboard. 2) Now create a right angle triangle using this segment as the hypotenuse. 3) Create three more congruent triangles in the pattern shown.

6 4) This creates a square within a square. The students should now be able to calculate the area of the outside square and the right triangles using the formulas discussed in the opening activity. At this point it is important to give the students a few moments to try to find the area of the inner square. If the students are still struggling with this I will to lead them on to the correct procedure. Once the area of the inner square is known the students should be able to find the length of the side using the formula for the area of a square. Now I will break the students into their groups and have them repeat this process on another segment. Each group will be given a different length segment to calculate. After the groups have finished I will have one of the groups demonstrate their process to the class. Closing Activity: Using a laptop computer hooked up to the overhead I will Demonstrate using Geometer Sketchpad the process that the students just did. AB = 4.99 cm BC = 4.99 cm CD = 4.99 cm DA = 4.99 cm m $%&" = m EA = 0.94 cm m BF = 0.94 cm m CG = 0.94 cm m DH = 0.94 cm m HA = 4.05 cm m EB = 4.05 cm m FC = 4.05 cm m GD = 4.05 cm Area EAH = 1.90 cm 2 Area FBE = 1.90 cm 2 Area GCF = 1.90 cm 2 Area GDH = 1.90 cm 2 AB!"# = cm 2 Area Large Square = cm 2 ( Area Large Square)'4! (&()* +&, ) = cm 2 Area Inner Square = cm 2 ( Area Inner Square) = 4.16 cm Length of Segment j = 4.16 cm A E j B j F j H j D G C

7 During this activity I will stress the following points 1) The four yellow triangles are congruent 2) The inner square has the side length of the segment length we are looking for. 3) This is not a special case. I will drag on one of the corners to show this works for all size squares and triangles. Notes: Before one starts this lesson it is important to make sure the students are comfortable with using a geoboard to find the area of a square and triangle. If not spend a day before this introducing the geoboard to the students. \ Homework: Have the students do Worksheet #1.

8 Worksheet # 1 Name: Period: Find the length of the following segments: 1) 2)

9 Name: Find the length of the following segments: 1) Worksheet # 1 (Answer Key) Period: J K H Area IJK = 2.51 cm 2 I Area JHNO = cm 2 ( Area JHNO)!4" (#$%& '()) = cm 2 (( Area JHNO)!4" (#$%& '()) ) = 5.11 cm L O M N 2) Area DBEF = cm 2 Area ABC = 2.05 cm 2 ( Area DBEF)!4" (#$%& #'() = 8.03 cm 2 (( Area DBEF)!4" (#$%& #'() ) = 2.83 cm B A D C H E G F

10 Day Two How Area of Squares Helps Visualize the Pythagorean theorem Objective: Students will learn that the Pythagorean Theorem is more than the sum of lengths of the legs squared equal to the hypotenuse squared in a right triangle. The students will learn that the sum of the areas of the squares made from the legs is equal to the area of the square made from the hypotenuse in a right triangle. Students will learn that when a 2 +b 2 <c 2 the triangle is an obtuse triangle and that when a 2 +b 2 >c 2 the triangle is an acute triangle. Notes: The Students are to open up GSP files one and two. Each student should come up with a different set of measures for the angles and also for the area of the squares. Opening Activity: As the students enter the classroom they will be given Worksheet #1 which they will use in this lesson. I will start the class by going over the previous days homework, clearing up any misunderstandings. I will then ask the students what they know about acute, right, and obtuse triangles. I will make sure they know the properties of each. The students should also be able to visually recognize each type of triangle. Main Activity: Students will open up GSP file one on the computer. In this file there will be a drawing of a triangle with a right triangle. The students will then change the length of the legs of the triangle by dragging a corner of it and then write down the area of the squares made from the sides of the triangle on the worksheet. After the students come up with three examples on their own of different right triangles they should be encouraged to try to answer the questions on the worksheet. Students will then open GSP file two on the computer. The students will be shown how to create an obtuse triangle and an acute triangle by dragging point B to different locations on the screen. After observing to make sure they understand this procedure the students will then finish off the rest of the class worksheet. Closing Activity: I will leave 15 minutes at the end of the class to go over the worksheet. Some key questions to ask the students are: 1) When a 2 +b 2 =c 2 what type of triangle this? Did anyone come up with a counter example? 2) When a 2 +b 2 <c 2 what type of triangle is this? Does anyone have a counter example? 3) When a 2 +b 2 >c 2 what type of triangle is this? Does anyone have a counter example? It is important to state that this discovery is not a proof of the Pythagorean Theorem but a result of it. The Pythagorean Theorem is one of the most proved theorems in all of mathematics. The students will see one version of this proof tomorrow.

11 GSP File One: a = 4.08 cm b = 3.40 cm c = 5.31 cm a 2 = cm 2 Area HCAI = cm 2 b 2 = cm 2 Area BFGA = cm 2 c 2 = cm 2 Area CDEB = cm 2 a 2 +b 2 = cm 2 D C H E a c b I A B GSP File Two: a = 3.36 cm a 2 = cm 2 b = 2.60 cm b 2 = 6.73 cm 2 c = 4.16 cm c 2 = cm 2 a 2 +b 2 = cm 2 G Area ABDE = cm 2 Area AHIC = 6.73 cm 2 Area BCFG = cm 2 F m!#$" = m!#"$ = m!"#$ = G D B a c F b E A C H I

12 Name: Worksheet 1 Period: 1a) Using GSP File 1 write the area of the squares of three different right triangles. Triangle # area Length of a Length of b Length of c a 2 b 2 c b) Using the data above, what can you deduce from the area of the squares made from the legs and the area of the square made from the hypotenuse. 1c) Can you come up with a rule for a right triangle using a 2, b 2, and c 2? 2a) Using GSP File 2 write the area of the squares of three different acute triangles. Triangle # area Length of a Length of b Length of c a 2 b 2 c b) Using the data above, what can you deduce from the area of the squares made from the legs and the area of the square made from the hypotenuse. 2c) Can you come up with a rule for an acute triangle using a2, b2, and c2? 3a) Using GSP File 2 write the area of the squares of three different obtuse triangles. Triangle # area Length of a Length of b Length of c a 2 b 2 c b) Using the data above, what can you deduce from the area of the squares made from the legs and the area of the square made from the hypotenuse. 3c) Can you come up with a rule for an obtuse triangle using a 2, b 2, and c 2?

13 Name: Worksheet 1- Key Period: 1a) Using GSP File 1 write the area of the squares of three different right triangles. Triangle # area Length of a Length of b Length of c a 2 b 2 c 2 1 Answers will vary for each student 2 3 1b) Using the data above, what can you deduce from the area of the squares made from the legs and the area of the square made from the hypotenuse. The sum of the areas formed by the legs equals the area of the square formed by the hypotenuse. 1c) Can you come up with a rule for a right triangle using a 2, b 2, and c 2? A2 + B2 = C2 2a) Using GSP File 2 write the area of the squares of three different acute triangles. Triangle # area Length of a Length of b Length of c a 2 b 2 c 2 1 Answers will vary for each student 2 3 2b) Using the data above, what can you deduce from the area of the squares made from the legs and the area of the square made from the hypotenuse. The sum of the areas formed by the legs is greater than the area of the square formed by the hypotenuse. 2c) Can you come up with a rule for an acute triangle using a 2, b2, and c 2? A 2 + B 2 > C2 3a) Using GSP File 2 write the area of the squares of three different obtuse triangles. Triangle # area Length of a Length of b Length of c a 2 b 2 c 2 1 Answers will vary for each student 2 3 3b) Using the data above, what can you deduce from the area of the squares made from the legs and the area of the square made from the hypotenuse. The sum of the areas formed by the legs is less than the area of the square formed by the hypotenuse. 3c) Can you come up with a rule for an obtuse triangle using a 2, b 2, and c 2? A 2 + B 2 < C 2

14 Homework Day Two Name: Period: State if the following lengths make an acute, right or obtuse triangle. 1) 3, 4, 5 2) 6, 12, 14 3) 8, 10, 12 4) 20, 21, 29

15 Day Three Proving the Pythagorean Theorem by Inspection Objective: Students will learn a universal proof of the Pythagorean Theorem. This proof does not use specific numerical values for a, b and c like the previous days example. Opening Activity: As students enter the room they will be handed a ticket in the door with several problems similar to yesterdays homework. The students will finish the ticket in the door and turn in so I can see that they understand the material. Notes: In this lesson the students will stay in their normal seating arrangement until the lecture is over. After the students understand the material from the lecture they will be able to work in groups on the activity. Main Activity: The main activity is broken up into 2 parts: 1) Lecture: On the overhead I will place the yellow triangles. It is important to show that the yellow triangles are congruent in this proof. Arrange the yellow triangles in the 5x5 tray as shown below. a b c a b c c b a c a Ask the students to describe the area of the square shown on the overhead. I will urge them to represent this area as c 2. Next I will arrange the same four yellow triangles in the same 5x5 tray as shown below.

16 b a a b The key to this part of the proof is to make sure the students understand that the area of the unshaded (not yellow) portion is the same as before. Label the two squares made 1 and 2. Now ask the students to give the area of the square 1 and square 2 in terms of a and b. The students should come up area of square 1 = b 2 and the area of square 2 = a 2. Now since the areas of the unshaded portions are the same we can set them equal to one another. Therefore we have a 2 + b 2 = c 2 the Pythagorean theorem. 2) Have the students break into their groups and try this proof on their own. Each group will receive two congruent squares. The squares for each group will be different sizes to show that this proof works with all size squares. Next the students will mark a point on one of the sides of the square. Measure this distance from a corner with a ruler and place three more points the same distance from each corner. Have the students connect the dots and cut the four triangles shown. The students should label their triangles. The students should now be able to arrange the four triangles on top the other squares and show the proof that was modeled in the lecture. Closing Activity: I will select two groups from the class to present their models of the proof to the rest of the class. The students should now realize that it does not matter what size triangle or square used for this proof to work. Homework: No Homework

17 Day 4 Using the Pythagorean Theorem to solve for Lengths Objective: After seeing consequences of and proving the Pythagorean Theorem the last couple of days the students now have a chance to use it. Students will revisit problems from Day One and use the Pythagorean Theorem to solve them. Students will also learn that sometimes it takes more than one step to come up with an answer. Opening Activity: As the students enter the room they will be given Geoboards and Geopaper. While taking attendance I will ask the students to review the procedure used on Day 1 to find the length of a segment. Main Activity: Using the overhead geoboard I will model how to solve the same question that was ask on Day One. 1) Use a rubber band to create a segment that is on an angle on the geoboard. 2) Now create a right angle triangle using this segment as the hypotenuse. I will remind the students that at this point we made 3 more triangles, found the area of the square and took the square root. But now since we know that a 2 + b 2 = c 2 for all right triangles, we can solve directly for the length. Let a = 1 and b = 3 then we have c = the square root of 10, c = Now I will break the students into their groups and have them solve the same lengths that they worked on in Day One. Next I will demonstrate that they do not necessarily use the Pythagorean Theorem just to find length of the hypotenuse. They can also use it to find the length of the side of a leg if given the other leg and hypotenuse length. After the students have solved several problems like this I will present them with the problem below.

18 Closing Activity: Prove if the triangle below is a right triangle. After giving the groups several minutes to discuss how to solve the problem and hopefully solve it, I will demonstrate how to solve it. Without solving the problem for the students I will demonstrate to them how to attack the problem. Homework: The first part of the homework will be to finish the closing activity. The second part of the homework will be Worksheet #

19 Name: Worksheet #2 Period: 1) Find the length of the hypotenuse of the following triangles. 2) Prove the following triangle is a right triangle.

20 Ticket In The Door Name: Period: Determine whether the following side measures form right triangles. Justify your answer. 1) 24, 30, 36 2) 20, 21, 29 Ticket In The Door - Key Name: Period: Determine whether the following side measures form right triangles. Justify your answer. 1) 8, 10, 12 2) 20, 21, not equal = 841 no yes

21 Day 5 Using The Pythagorean theorem in The Classroom Objective: After a long hard week of seatwork and crunching numbers this lesson lets the students get out of their seats and use the Pythagorean theorem in a real life situation. Students will learn that carpenters use the Pythagorean theorem in the construction of buildings. Students will learn how to tell if a room, doorway, wall, ect. were constructed square. Notes: In this lesson it is important to have at least one more measuring station than groups in the classroom. Opening Activity: While the students are entering the room they will be given Day Five Worksheet and asked to solve the first problem. After every one is situated I will go over the homework and the first problem from the worksheet. Main Activity: Students will be broken up into their groups and I will pass out one tape measure to each group. At this point I will discuss with the students how the Pythagorean theorem is used in the carpenters trade to ensure that the project that they are working on is square. After everyone is confident that they understand the material they will start the worksheet. After each group is finished with the worksheet I will have each group present their findings from one of the stations to the rest of the class. This would be a good time to let the students discuss the idea of tolerances and how close each station has to be. Closing Activity: In today s lesson the students learned several ways that the Pythagorean theorem is used in the construction business. I will now ask the students to volunteer their thoughts on other trades that they think may use it. Another question to ask is Can they think of any other uses of the Pythagorean theorem. Homework: No Homework!! HAVE FUN OVER THE WEEKEND

22 Day Five Worksheet Name: Period Warm Up: Ken has an eight-foot ladder that he leans against a wall. If the base of the ladder is two feet from the wall how high up the wall does the ladder reach. Group Work: (show all work!!!) Station 1) Determine if the opening of the door to the classroom is square. Station 2) Determine if the classroom walls were constructed square. Station 3) Determine if the chalkboard is in the shape of a rectangle Station 4) Determine if the object of your choice in the classroom is square. Station 5) How would you determine if the wall and floor were constructed square using the 5-foot broom handle provided. Is it?

23 Day Five Worksheet - Key Name: Period Warm Up: Ken has an eight-foot ladder that he leans against a wall. If the base of the ladder is two feet from the wall how high up the wall does the ladder reach feet Group Work: (show all work!!!) Station 1) Determine if the opening of the door to the classroom is square. Answer will depend on student s measurements Station 2) Determine if the classroom walls were constructed square. Answer will depend on student s measurements Station 3) Determine if the chalkboard is in the shape of a rectangle Answer will depend on student s measurements Station 4) Determine if the object of your choice in the classroom is square. Answer will depend on student s measurements Station 5) How would you determine if the wall and floor were constructed square using the 5-foot broom handle provided. Is it? One would place the end of the broom handle against the wall exactly 4 feet up from the floor and see if the other end of the handle hit the floor 3 feet from the wall. The three and four foot measurements could also be reversed.

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