Looking for Pythagoras An Investigation of the Pythagorean Theorem

Transcription

1 Looking for Pythagoras An Investigation of the Pythagorean Theorem I2t Stephen Walczyk Grade 8 7-Day Unit Plan Tools Used: Overhead Projector Overhead markers TI-83 Graphing Calculator (& class set) TI-83/84 Viewscreen for overhead Class set of Geoboards Overhead Geoboard Dot-grid paper (2) ETA The Proofs of Pythagoras Overhead kits Pythagoras proof puzzle pieces and frames Looking for Pythagoras Student Edition

2 Objectives and Standards Objectives: - Students will be able to relate the area of a square to the side length. - Students will be able to estimate the values of square roots of whole numbers. - Students will be able to locate irrational numbers on a number line. - Students will develop strategies for finding the distance between two points on a coordinate grid. - Students will understand and apply the Pythagorean Theorem. - Students will use the Pythagorean Theorem to solve everyday problems. NCTM Standards Addressed: -Number and Operations -Geometry -Problem Solving -Reasoning and Proof -Communication -Connections New York State Standards Addressed: -Number and Operations (7N15, 7N16, 7N17, 7N18) -Geometry (7G5, 7G6, 7G8, 7G9) -Problem Solving (8PS3, 8PS11) -Reasoning and Proof (8RP1, 8RP2, 8RP3, 8RP4, 8RP5) -Communication (8CM4, 8CM5) -Representation (8R6, 8R8) Resources: Pearson Prentice Hall, Looking for Pythagoras: The Pythagorean Theorem (Connected Mathematics: Teacher s Edition), by Lappen et al., Problem 2.1 Problem 3.2 pages 34-60, 2006.

3 Unit Overview: (7 38 minute classes) Day 1: Students will be expected to complete Problem 2.1 during class. They will discover the squares with 8 different areas that fit in a 5 x 5 dot grid. They will use their knowledge of finding area by dissecting the dot region as they did in a previous investigation. They will be allowed to use the equivalent section of a geoboard as a tool with their partners. For homework, the students will be expected to complete ACE questions #1, 2, and 42 starting on page 23. Day 2+3: Students will be expected to complete Problem 2.2 during class. They will be introduced to the concept of square root and also understand square root geometrically, as the side length of a square with a given area. They will be expected to estimate square roots and will have use of the TI-83 calculator as a tool. For homework, the students will be expected to complete ACE questions #4-6, 10, starting on page 23. Day 4: Students will be expected to finish Problem 2.3 in class. They will use their knowledge of square roots and drawing/constructing squares of different areas to find all possible lengths of line segments that fit in a 5 x 5 dot grid. They already have the knowledge of the first 8, now they must find the other 6. Geoboards will again be allowed for their use. For homework, they will be expected to complete ACE questions #35-37, 41 starting on page 25. Day 5+6: Students will complete Problem 3.1 in class during these 2 days. They will draw right triangles with given lengths for the legs and then use their knowledge of building squares off of line segments and finding the area of those squares, along with the side lengths. This will help them discover the length of the hypotenuse, along with examining their chart to discover the Pythagorean Theorem, where the sum of the areas of the squares off the legs will equal the area of the square off the hypotenuse. For homework, students will be expected to have ACE questions #1, 2, 5, 6, 8-11, and 12 on page 38 complete after these two class periods.

4 Day 7: Students will finish Problem 3.2 during class today. They will be working in pairs, using pre-cut puzzle pieces, to prove the Pythagorean Theorem. I will then demonstrate what they should have done on the overhead with 2 sets of ETA s The Proofs of Pythagoras overhead kit. Their homework will be ACE questions #23, 26 on page 41.

5 Lesson 1: Looking for Squares Time: 38 minutes Objectives: - Draw squares on a 5 x 5 dot grid or construct them on a geoboard and find their areas. Materials: - Dot paper - Overhead dot paper - Geoboards - Overhead geoboard - Overhead markers - Centimeter rulers Opening: Before getting started, the teacher will distribute Geoboards, dot paper, and centimeter rulers. The dot paper is set up in 5 x 5 dot regions so the teacher must tell the students they can only use a similar area of the Geoboard during this investigation. To prevent confusion, the students could use a rubber band to mark off the region we will be using. After that, the teacher will display the overhead Geoboard and make a square of area one square unit and tell the students to duplicate the square and draw it on their dot paper, using their centimeter ruler as a straight edge. The students should also write the area down. Next, the teacher will ask for a student volunteer to come to the overhead and make a square of a different area (Student example may vary). Students will then duplicate that example on their Geoboards and also draw it on their dot paper, marking down the area. The teacher then tells the students to do some exploration and see if they can find the last six squares that will fit into these 5 x 5 dot regions. Activity: Teacher will walk around the room as the students are making squares on their Geoboards and copying those squares down on their dot paper. This is all that Problem 2.1 entails. Teacher must expect student difficulty in finding the tilted squares. If this is the case, teacher will bring the class back and use the overhead Geoboard to display a tilted square of area 2 square units (emphasize units so the students always write them). So now they ve been shown 3 of the 8 squares and they have until ten minutes left in class to find the rest. Remind students to check the area of each square they draw to verify that the areas are all different. In regards to

6 the tilted squares, they should use the strategy they learned in a previous investigation to find area by dissecting the figure. For example, the square of area 2 square units is composed to 4 pieces of area 0.5 square units. Closing: Teacher will ask students to share the various squares they found and the teacher will draw them on the overhead dot paper. Continue this until all 8 squares are displayed. If the students do not offer all eight, teacher will suggest the missing ones. Discuss the strategies used to find the squares. - Which squares were easy to find? Why? (Students should say upright squares because their sides match up with the horizontal and vertical lines of dots on the grid. - Which squares were not easy to find? Why? (Students should say something to the effect of tilted squares because their sides must meet at right angles but they do not align with horizontal and vertical lines of dots on the grid. - How do you know the figures you drew were squares? Students should say they checked that the side lengths were equal and all angles were right angles. Students will be asked to leave their materials on the desk for the next class to use. Assessment: Students will drop off their sheet of dot paper with the eight squares drawn and it will be graded as a homework assignment. Students who may have been absent during this lesson will be excused from this grade, but not necessarily this assignment. Homework: Students will be expected to complete ACE questions #1, 2, and 42 starting on page 23. Teacher will provide students with a sheet of dot paper to complete their homework.

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8 Lesson 2: Square Roots Time: (2) 38 minute classes Objectives: - Introduce the concept of square root. - Understand square root geometrically, as the side length of a square with a known area. Materials: - TI-83 calculators (class set) - TI-83/84 Viewscreen for overhead - Overhead dot paper - Overhead markers - Centimeter rulers Opening: Before starting, the teacher will pass back to the students their sheet of dot paper they used during the previous class to find their eight squares. Then, the teacher will draw a square with area 4 square units on the overhead dot paper. The teacher will then ask, This square has an area of 4 square units. What is the length of a side? The students should say two units because you can easily count two units along any side and they already know that finding the area of a square involves multiplying one side by itself. Here the teacher asks What number, when multiplied by itself, equals four? The students should say two. We can say this another way, the square root of 4 is 2. The teacher tells them that a square root is a number that, when squared, or multiplied by itself, equals the number. 2 is the square root of 4 because 2 x 2 = 4. The teacher then writes 4 on the overhead and tells the students that this is how we write the square root of 4. The teacher then draws the square with area 2 square units on the overhead dot paper. What is the length of a side of this square with area 2? The students should say 2 units. (Here, the teacher should emphatically restate that, to find the length of one side of a square where the area is given, you must take the square root of that area.) Is this length greater than 1 unit? Is it greater than 2 units? (They should recognize that it is in between 1 and 2 units because 2 is between 1 and 4. Is 1.5 a good estimate for 2? Here, the teacher shows the students that 2 are closer to 1 than 4 so 2 would be closer to 1 than 2. So what would be a

9 better estimate for 2? Here, hopefully the students say something along the lines of 1.3 or 1.4. This is where the teacher sets up the TI-Viewscreen and has the students take out their TI-83 calculators to check if their estimate is correct. The teacher shows them where the square root button is and tells them that by pressing enter, they get an approximation of the value of 2. On their homework or any other assessment, when asked for the length of one side of a square with area 2, the exact answer would be 2 units. Activity: Students have the rest of the class period to start working on Problem 2.2 in class. It is ok if they don t finish because they will pick up where they left off when they come in the next day. Remind them that they should use their calculators only when the text tells them to do so. The teacher should pass out centimeter rulers so the students can find estimates of side lengths of the 8 squares they ve already drawn. During the activity, the teacher should just walk around the room checking to see if the students are using the square root notation properly along with affixing units to their answers. Closing: Talk about the square with an area of 2 square units. How can we prove that this area is 2? The students should tell the teacher to subdivide the square into 4 sections of area 0.5. What is the exact side length of this square? The students should say 2 units. You estimated the side of a square by measuring. What did you get? Most will say about 1.4. Is this the exact value of 2? Does 1.4 squared give you 2? The students should say no because equals 1.96 so 1.4 is too small. You could also find 2 by putting into your calculator. What value did the calculator give? The teacher writes this number down on the board or the overhead. Now put this number into your calculator and square it. What did you get? Is it exactly equal to two? They should get and say no. So the value you get from measuring and putting it into your calculator are both estimates and writing 2 is the exact side length of a square with area 2 square units. Just as clarification for the students, the teacher should put 2 in the calculator and hit enter. Now, if you square the calculator s result you will get two. But, if you enter the estimate and square it you will get

10 Since 2 is an exact value, if you enter 2 x 2 you will get 2. Now ask students to do decimal approximations for 5. As a class, square each approximation and see whether the result is 5. So, what is the exact length of a side of a square with area 5 square units? By now, they should be saying 5 units. Lastly, we discuss the last question on Problem 2.2. What is the exact length of the sides of the squares we have already found? Students should write these answers right near where they wrote the area for all 8 squares. Homework: Homework for these two days will be ACE questions #4-6, 10, starting on page 23. Whichever questions they could not finish after the first day, they should be able to complete after the second day.

11 Name: PROBLEM 2.2 In this problem, use your calculator only when the question directs you to. A. 1. Find the side lengths of squares with areas of 1, 9, 16, and 25 square units. 2. Find the values of 1, 9, 16, 25. B. 1. What is the area of a square with a side length of 12 units? What is the area of a square with a side length of 2.5 units? 2. Find the missing numbers.? = 12? = 2.5 C. Refer to the square with an area of 2 square units you drew in Problem 2.1. The exact side length of this square is 2 units. 1. Estimate 2 by measuring a side of the square with a centimeter ruler. 2. Calculate the area of the square, using your measurement from part (1). Is the result exactly equal to 2? 3. Use the square root key on your calculator to estimate How does your ruler estimate compare to your calculator estimate? D. 1. Which two whole numbers is 5 in between? Explain. 2. Which whole number is closer to 5? Explain. 3. Without using the square root key on your calculator, estimate the value of 5 to two decimal places. E. Give the exact side length of each square you drew in Problem 2.1

12 Name: Answer Key PROBLEM 2.2 In this problem, use your calculator only when the question directs you to. A. 1. Find the side lengths of squares with areas of 1, 9, 16, and 25 square units. 1 unit, 3 units, 4 units, 5 units 2. Find the values of 1, 9, 16, 25. 1, 3, 4, 5 B. 1. What is the area of a square with a side length of 12 units? 144 square units What is the area of a square with a side length of 2.5 units? 6.25 square units 2. Find the missing numbers.? = ? = C. Refer to the square with an area of 2 square units you drew in Problem 2.1. The exact side length of this square is 2 units. 1. Estimate 2 by measuring a side of the square with a centimeter ruler. Approx. 1.4 cm 2. Calculate the area of the square, using your measurement from part (1). Is the result exactly equal to 2? No, 1.4 x 1.4 = Use the square root key on your calculator to estimate How does your ruler estimate compare to your calculator estimate? The calculator is more accurate D. 1. Which two whole numbers is 5 in between? Explain. 2 and 3 because 2 is the square root of 4 and 3 is the square root of Which whole number is closer to 5? Explain. 2 because 5 is closer to 4 than Without using the square root key on your calculator, estimate the value of 5 to two decimal places. Answers may vary (Possible answer: 2.15) E. Give the exact side length of each square you drew in Problem 2.1 1, 2, 2, 5, 8, 3, 10, and 4

13 Lesson 3: Using Squares to Find Lengths Time: 38 minutes Objectives: - Use geometric understanding of square roots to find lengths of line segments on a dot grid. Materials: - Geoboards - Overhead Geoboard - Dot paper (Same as above) - Centimeter rulers - Overhead dot paper - Overhead markers Opening: As a class, list all the side lengths (in units) students have found so far in their work with 5 x 5 dot grids. These lengths are 1, 2, 2, 5, 8, 3, 10, and 4. The teacher then asks if the students can find a line segment with a different measure that fits in the 5 x 5 dot grid. On the overhead dot paper, draw a segment the class suggests or draw one of your own. This could also be done on the overhead Geoboard with the students using their own. How do we know the length of this segment is different from the ones we found already? Students may mention ways to informally measure the length or they may be comparing it to others that are a bit shorter or longer. How might we find the exact length of this segment? Here, it is hopeful that some students will suggest drawing/making a square off of this segment, finding the area of the square, and from knowing the area, finding the length of the original segment. After this example has been completed, inform the students that we now have 9 different lengths of segments and there are 5 more that will fit in the 5 x 5 dot grid. Explain to them that the squares they build will not necessarily fit into the 5 x 5 dot region but the first line segment must fit. When students understand this problem, allow them to work in pairs on Problem 2.3 in finding the last 5 segments/squares. Tell them that if they draw a square and its area ends up being the same as one of the 9 we already drew, that square is not one of the 5 remaining. Activity: Students do not need to find all 5 remaining lengths. However, the teacher must make sure all students can build a square off of a line segment and find the area of the square and the length of its side. Once, the teacher

14 sees that the class as a whole has found the 5 remaining squares, or if they re struggling to find the last few, bring their attention back to the overhead and ask for volunteers to come up and share the squares that they have found. Closing: As students come to the overhead and make their squares, determine as a class if the area of the squares are different from the ones we already found. Continue this until the last 5 squares are found, even if you have to give them the last one or two. After all is said and done, the remaining side lengths we should have found are 13, 17, 18, 20, 25 or 5, and 32. As a check for understanding, ask them What two whole numbers does 17 lie? (4 and 5) Which whole number is it closest to? It is closer to four because 17 is closer to 16 then 25. Ask the same question for 32. Homework: For homework, the students will be expected to complete ACE questions #35-37, 41 starting on page 25.

15 Lesson 4: The Pythagorean Theorem Time: (2) 38-minute classes Objectives: - Deduce the Pythagorean Theorem through exploration. - Use the Pythagorean Theorem to find unknown lengths of right triangles. Materials: - Dot paper - Overhead dot paper - Overhead markers Opening: As the students are walking into class, have the words hypotenuse, legs, and conjecture written on the board so they can copy the definitions from the back of their textbook into the vocabulary section of their notebook. While students are doing this, the teacher passes out a sheet of dot paper to all students. Once that is finished, the teacher draws a right triangle on the overhead dot paper with legs of length one unit. What kind of triangle have I drawn? The students should know that it is a right triangle. Explain that, in a right triangle, the two sides that form the right angle are called legs and the side opposite the right angle is called the hypotenuse. What are the lengths of the two legs of this right triangle? (1 unit). Then the teacher asks the students what strategy we should use to find the length of the hypotenuse. Recalling what they ve been learning, students should say that we could draw a square off of the hypotenuse, find its area, and from that we know the length of one side by taking the square root of the area. The teacher then draws squares off of all sides of the triangle while the students are mimicking this on their dot paper. What is the area of the squares on the legs? (1 square unit). What is the area of the square on the hypotenuse? (2 square units). Some students might realize that the sum of the areas of the squares on the legs equals the sum of the area of the square on the hypotenuse, but don t push for this. Tell them that in Problem 3.1, they will be looking for a relationship among the three squares that can be drawn on the sides of a right triangle. They will organize their work in a table, shown on page 32 of their textbook, so they can look for patterns. Have students work in pairs on the problem because there are 6 triangles in

16 the table and it would be much more efficient if each student only drew three. Activity: Have each student make their own copy of the table found in the book. As the teacher walks around the room, make sure that each student is correctly drawing the squares on the right triangles. Geoboards are not to be used in this problem so they can have more practice actually drawing the squares. Tell them to think of a conjecture from the table their constructing. Closing: On the overhead, the teacher quickly draws up the table the students have been working on, leaving out the values they were supposed to be figuring out. The teacher then asks for volunteers to give answers to the table, and if there s confusion or disagreement on a specific triangle, draw it on overhead dot paper and do it as a class. After the table is filled out, ask the students if they have found a rule associated with the table. Hopefully they will notice that that sum of the areas of the two squares on the legs equals the area of the square on the hypotenuse. This is called the Pythagorean Theorem. The teacher then draws a right triangle on the overhead and labels the legs a and b and the hypotenuse c. If the legs are a and b, how can I represent the area of the squares off of these legs? Here, they should say a 2 and b 2. How would I represent the area of the square on the hypotenuse? Again, the students should say c 2 because if one side is c, to find the area of a square, you multiply one side by itself. So, if a and b are the lengths of the legs of a right triangle and c is the length of the hypotenuse, then the Pythagorean Theorem says that a 2 + b 2 = c 2. The teacher should then draw a non-right triangle on dot paper and ask the students to use the same method to see if the Pythagorean Theorem works for triangles that are not right triangles. After they find that it doesn t work, do ACE questions 13 and 14 as a class to see if the theorem works for triangles with sides that are not whole numbers. Question 14 has leg lengths of 5 and 5, and a hypotenuse length of 10. The squares of these sides are 5, 5, and 10 and 5+5=10 so the Pythagorean Theorem works for right triangles with side lengths that are not whole numbers. If there is time left in the 2 nd class, tell the students to add another column to their table that reads Length of Hypotenuse and have them use the Pythagorean Theorem to find the lengths. Homework:

17 For homework, students will be expected to have ACE questions #1, 2, 5, 6, 8-11, and 12 on page 38 complete after these two class periods. Length of Leg 1 (units) Length of Leg 2 (units) Area of square on Leg 1 (units 2 ) Area of square on leg 2 (units 2 ) Area of square on Hypotenuse (units 2 ) Answer Key Length of Leg 2 Area of square Area of square (units) on Leg 1 on leg 2 (units 2 ) (units 2 ) Length of Leg 1 (units) Area of square on Hypotenuse (units 2 )

18 Lesson 5: A Proof of the Pythagorean Theorem Time: 38 minutes Objectives: - Reason through a geometric proof of the Pythagorean Theorem Materials: - (2) ETA The Proofs of Pythagoras Overhead kits - (20) CMP Pythagoras Puzzle pieces (Labsheet B - precut) - Overhead markers Opening: As the students are walking into class, the teacher has the ETA proof with the blue and green squares on the legs of a right triangle on the overhead. The teacher then asks, If what we have been learning is true, what should happen with these blue and green pieces if I move them into this frame on the hypotenuse. The students should tell you that the blue and green pieces should fit perfectly inside the frame on the hypotenuse and then the teacher will move the pieces to show them that this is true. Next, tell the students that we have seen many examples of right triangles that satisfy the Pythagorean Theorem. While these examples may be convincing, we need a mathematical proof to show that it works for all right triangles. There are many proofs and one of them is shown in the puzzle I am about to hand out. I am handing out to each pair a sheet of paper that has two puzzle frames on it. Also, to each pair goes a Ziploc bag with puzzle pieces in them. Each bag contains 8 equal right triangles and 3 different squares. Your task is to arrange the pieces into the frames with the rules being that only 4 triangles can go in each frame. Your task is to look for a relationship amongst the areas of the squares. Tell the students to have one book open amongst the pair in case they forget the rules. Activity: Encourage each pair to find more than one way to fit the puzzle pieces in the frames. If there is some difficulty with this, have them look to another pair for some help. The teacher should walk around the class while this is going on to help guide the students and to see when all pairs have completed the puzzle.

19 Closing: Once the students focus their attention back on the teacher, the teacher places two square frames from the ETA kit on the overhead with the 8 yellow right triangles beneath the frames. I have two frames here just like you did with 8 equal right triangles. However, I do not have the squares but it will be easy to see where the squares would go. Now the teacher asks for a student volunteer to come to the overhead and arrange four triangles in a frame exactly how they did it in their pair. Once the teacher deems this method correct, he/she then asks for a second student volunteer to come display the other method for displaying the triangles. What relationship do these completed puzzles suggest? Help students understand this argument. The areas of the two frames are equal. Each frame contains four identical right triangles. If the four right triangles are removed from each frame, the area remaining in both frames must be equal. This means that the sum of the areas of the squares in one frame must equal the area of the square in the opposite frame. The teacher then places the yellow triangles back into the frames and labels the sides of all the triangles with letters a, b, and c. In the first frame, the teacher points out that, noticing how the triangles are arranged, one square has two side lengths labeled a so its area is a 2. In the same frame, the other square has two sides of length b so its area must be represented as b 2. In the other frame, the triangles are arranged to show that the third square has four sides labeled c so its area must be represented as c 2. Going back to what we said a few minutes ago that the sum of the areas of the squares in one frame (a 2 + b 2 ) must equal the area of the square in the other frame. So, a 2 + b 2 = c 2. Homework: The students will be expected to complete ACE questions #23, 26 on page 41 for homework.

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