Dominique Davis, Molly Duffy, Julie Hernandez, Kelly Holton, Brandi Zars CI 473 Pre-Reading Lesson Plan Square Roots and Pythagorean Theorem Lesson Plan Grade Level: 9th grade Algebra I 334 Demographics: 20 students (10AA, 7 Caucasian, 2 Latino/a, 1 Asian) (15 freshman, 4 sophomores, 1 junior) Estimated Time: 1 class period (about 50 min) Concept: Students will first thoroughly reexamine the concepts of perfect squares and square roots. The main focus of this lesson will demonstrate to students how to use square roots and the Pythagorean Theorem to find distance. They will also become familiar with real world applications. By the end of the lesson students will have a clear understanding of the importance of the Pythagorean Theorem and how to apply it. Objectives: Students will review the meaning of a perfect square Student will be practice how to find the square root of a number Students will understand real world applications of the Pythagorean Theorem Students will use visual representation to measure and compute distance Students will use the Pythagorean Theorem to compute distances Materials: - 11” x 17” paper - ruler - protractor - pencil - calculator with square root function - principles of square roots and Pythagorean Theorem worksheet (homework) Procedure: 1) Good Morning Class! While I am taking attendance, I want you to write all the numbers I have written on the board in your notebook and circle the ones that are perfect squares. You should all be familiar with the concept of perfect squares, but if you cannot remember what a perfect square is – ask your partner what they are, then circle them on your own! 2, 4, 9, 12, 16, 25, 36, 44, 55, 64, 81, 100 (After attendance is complete) Alright, could someone please raise their hand and tell me which of these numbers are perfect squares? (Call on student) The perfect squares are 4, 9, 16, 25, 36, 64, 81, 100 When we have a number multiplied by itself, it gives us a perfect square Let’s take a look at these numbers that were written on the board. Ask students what number multiplied by itself gives us these numbers. 2x2 = 4 3x3 = 9 4x4 = 25 6x6 = 36 8x8 = 64 9x9 = 100 When we take the square root of any perfect square, we get an integer. The symbol for a square root is called a radical. To remind you all, this is what a radical looks like So could someone raise their hand and tell me what the answer to √9 is? What about √25? Can anyone find √8? If you are having difficult that is okay. This problem is more difficult to solve because the number under the square root, 8, is not a perfect square. When we do not have perfect square under the radical it becomes more complicated. For today we are going to use the square root button on our calculators to find the approximate decimal answers of square roots such as these. (Show class where and how to use square root button on calculator. Later on in this unit we will take a look at how to rewrite √8. Alright, now that we have re-familiarized ourselves with how to do square roots, let us understand where in the world we are ever going to have to use them! Have all the students look up at the board. Draw a Rectangle on the board and label one side 3 and the other side 4. Ask them if they can see clearly the lengths of all four sides of this rectangle? Now draw a diagonal through the rectangle and erase everything above the diagonal. Now ask the class if they know how to find the length of this side (the diagonal). We’ve never had to figure out something like this before, but today we will learn using something called the Pythagorean Theorem! I bet some of you are still wondering, “What is the importance of right triangles and the Pythagorean Theorem?” Quickly explain the following to the class Electricians: use right triangle to make off-sets when bending conduit Carpenters: apply the Pythagorean Theorem to check the diagonal foundation or framer wall to determine if it is a square Plumbers: have fittings that allow them to make perfect 90° angles Surveyors: use sophisticated instruments All of these professions need this kind of math! Before we go into the details of the Pythagorean Theorem, let us just review a few terms most of us should remember from before which have to do with the Pythagorean Theorem (draw pictures to explain): Altitude: The perpendicular distance from the vertex to the base Angle: The union of two rays with a common endpoint Base: The bottom of a plane figure or three-dimensional figure Diagonal: The line segment connecting two nonadjacent vertices in a polygon Right angle: An angle whose measure in 90° Hypotenuse: The side opposite the right angle in a right triangle Square Root (√): The square root of x is the number that, when multiplied by itself, gives the number x. It is especially important that we remember what a right triangle is, because the Pythagorean Theorem will ONLY work with right triangles. Recall the following two triangles on the board: 30°-60°-90° 45°, 45°, 90° Alright, I think we are all caught up on everything we are going to need to know for the Pythagorean Theorem: Demonstration for 3-4-5 triangle: Now what I want everyone to do is take out your 11x17 paper that I handed out. (Demonstrate on board while students follow on their paper) 1. First measure the corner of your paper with your protractor to discover the degree of the angle of the corner (90°) – which is what we need to use the Pythagorean Theorem. 2. With your ruler, measure 3” up on the end of the paper and mark. 3. Measure from that same corner in the other direction out 4” and mark. 4. Draw a line from Mark to Mark 5. Measure the length of the line you drew from mark to mark (5”) Ask the class if everyone got a number for the hypotenuse approximately equal to 5”. The teacher should walk around the room to see if it looks like everyone has gotten approximately 5”. Great Job Class. I think that everyone is ready to learn the formula to find the Hypotenuse without a ruler! If we take a right triangle and label the altitude a, the base b, and the hypotenuse c The Pythagorean Theorem is the following formula: a² + b² = c² So using the rectangle that we just drew on our sheet of paper, if we only knew the lengths 3 and 4, how would we figure out the hypotenuse? We would plug our 3 and 4 in as our a and b. So we would get 3² + 4² = c² 9 + 16 = c² 25 = c² 5=c The answer is 5, which matches what we measured on our paper! Now I will draw another triangle on the board with altitude 5, and base 12. To find the hypotenuse of this how would we use the equation? We would plug 5 and 12 in as our a and b, so we would get: 5² +12² = c² 25 + 144 = c² 169 = c² 13 = c² We were lucky on these last two problems. These were special right triangles that gave us perfect squares. Sometimes a² + b² will not always give us a perfect square: What if a=2 and b=5, then what does c=? It is okay to use your calculator on this one. 2² + 5² = c² 4 + 25 = c² 29 = c² 5.39 = c I want you to do this next one on your own. When you have the answer put your pencil down and I will come around the room to see if you got the right answer. Here is the problem: (write on board) Given a and c, find b A = 3, c = 7 then b = ? a² + b² = c² 3² + b² = 7² 9 + b² = 49 b² = 49-9 b² = 40 b = 6.32 Now, on the other corner of your paper, on your own pick any two different lengths for sides of your triangle and make your marks. Then draw and measure the hypotenuse with your ruler and see the length. After you have done this, use the Pythagorean Theorem to see that c is the same length as your hypotenuse. It should match! (Go around the room and see if students successfully matched their hypotenuse measure with their c calculation) Great Job today! Tonight’s homework will be principles of square roots and Pythagorean Theorem worksheet. It will be more problems like what we were doing in class today, Solving for a, b, or c. This will be due tomorrow after you ask questions at the start of class. Assessment: Assessment is done throughout class by observing students and walking around the room to check answers and see if students are understanding. At the end of class students will receive a worksheet which goes over the Pythagorean Theorem. Homework is collected the following class period and graded. Modifications: If we run out of time, students can measure and do Pythagorean Theorem calculations at home. If we have extra time, students can get started on tonight’s homework. Rationale for strategy use: The lesson is begun with perfect squares and square roots to refresh students’ memory on how to do these. This should be very elementary for most students, and is intended for students who have struggled with these concepts in the past. The section of the lesson dedicated to the actual profession is intended to target those who have limited interest in math as well as other students who enjoy it and understand its applications. This helps students realize where they may apply these concepts in the real world. The “triangle terms” are revisited and defined again to refresh students memory with an appropriate background of what is needed for the Pythagorean Theorem, before the teacher begins using terms they have forgotten. Finally the lesson slowly moves into the main idea of the lesson, the Pythagorean Theorem, using hands on materials and basic methods on the board. Standards: Illinois Learning Standards: 9.D.3 Compute distances, lengths, and measures of angles using proportions, the Pythagorean theorem and its converse NCTM Standards: Create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean Dominique Davis, Molly Duffy, Julie Hernandez, Kelly Holton, Brandi Zars CI 473 During-Reading Lesson Plan Pythagorean Theorem Lesson Plan Grade Level: 9th grade Algebra I 334 Demographics: 20 students (10 AA, 7 Caucasian, 2 Latino/a, 1 Asian (15 freshman, 4 sophomores, 1 junior) Estimated Time: 1 class period (about 50 minutes) Concept: Students will have a deeper understanding of the Pythagorean Theorem and how it is used. After this lesson is completed, students will be able to find the length of the hypotenuse, the length of a side of a right triangle, understand the concept of a Pythagorean Triple, and they will be able to figure out if a triangle is a right triangle when given its side lengths. Objectives: Students will review and fully understand the Pythagorean Theorem. Students will understand real world applications of the Pythagorean Theorem. Students will use the Pythagorean Theorem to compute distances. Students will know the concept of a Pythagorean Triple. Students will use the Pythagorean Theorem to see if a triangle is a right triangle. Materials: Notes Loose Leaf Paper Pencil Transparency Overhead Homework Handout Procedure: 1) Good morning class! Who can tell me what we learned about yesterday? Hopefully at this time a student will raise their hand and tell me that they learned about perfect squares, radicals, and the Pythagorean Theorem. If not, I will remind them quickly about what we went over and what we learned. Thank you, that’s great! Yesterday we learned about the Pythagorean Theorem and how that applies to right triangles. Before we start today, I would like all of you to take out a sheet of paper and fold it into fourths. This is going to by your summary sheet. I would like you to label the first section Finding the Length of the Hypotenuse. Label the second section Finding the Length of a Leg, label the third section Pythagorean Triples, and label the fourth section Checking for Right Triangles. As we go through our lesson today, try to fill out these sections. For the first section, I want you to think about how you would find the hypotenuse for a right triangle –what do we need to know about the triangle in order to find the hypotenuse, and what do we have to do once we have that information? Do the same for the second section except we’re trying to find the length of one of the legs of a right triangle instead of the hypotenuse – so what do we need to know, and what do we need to do once we have that information? For the third section, just tell me what a Pythagorean Triple is – describe it to me. And for the fourth section, explain how you would go about checking to see if a triangle is a right triangle – how do you know when a triangle is a right triangle and when it isn’t? Any questions about what is expected of you? Great, what I need for you to do now is take out your notes and work on the problem on the board while I am taking attendance. Problem on the board: The roller coaster “Superman: Ride of Steel” in Agawam, Massachusetts, is one of the world’s tallest roller coasters at 208 feet. It also boasts one of the world’s steepest drops, measured at 78 degrees, and it reaches a maximum speed of 77 miles per hour. If at the highest point, the rollercoaster is 208 feet and the base is approximately 44 feet, what is the length of the first hill? 208 feet c 44 feet After I am done taking attendance: Was anyone able to find the answer? That’s great! Let’s take a look at how you came up with that. Can anyone first remind me of the formula for the Pythagorean Theorem? Thank you, it’s a2 + b2 = c2 where a and b are lengths of the legs of the right triangle and c is the length of the hypotenuse. Now, if we plug in for a and b we can easily solve for c. 2082 + 442 = c2 43,264 + 1936 = c2 45,200 = c2 45,200 = c 2 212.6 feet ≈ c So, not only can electricians, carpenters, plumbers, and surveyors use the Pythagorean Theorem like we learned yesterday, but it’s applicable in many other facets of life as well – such as finding the length of a drop on a rollercoaster. 2) In this last problem, what was our unknown? Yes, it was the hypotenuse. What would happen if one of our legs of the right triangle was our unknown instead of the hypotenuse? Would we be able to use the Pythagorean Theorem to solve for that side? What do you think? We looked at something like this yesterday. Here, I will let them brainstorm for a while and explain to me how we could find the length of the missing side using the Pythagorean Theorem to make sure they fully understand what it is we learned yesterday. Now let’s look at an example of this and work through it together. In a right triangle, let c = 25 and b = 10. What is the length of a? Here we will work through the problem on the board: a2 + 102 = 252 a2 + 100 = 625 a2 = 625 – 100 a2 = 525 a 2 = 525 a ≈ 22.91 That was great! Now we know how to find the length of any side of a right triangle, given that only one side is missing. 3) We had a few examples of triangles yesterday where each side of the right triangle was a whole number. When this happens, they are called Pythagorean Triples. The examples we saw yesterday included a 3-4-5 triangle and a 5-12-13 triangle. Can you think of any other examples of Pythagorean Triples? If you think you have one, use the Pythagorean Theorem to make sure it’s a right triangle. If you’re having trouble, see if you can come up with a Pythagorean Triple based off of the ones we already know. Were any of you able to find some? That’s great! I don’t expect you to memorize or remember how to do what I am about to show you. I just want you to be exposed to it in case it comes up in one of your future math classes. By seeing it now, hopefully when it does appear, you’ll be able to think back to this and it will be easier for you to retrieve. Okay, so there is a simple formula that gives you all of the Pythagorean Triples. Suppose m and n are two positive integers with m < n, then a = n2 - m2, b = 2mn, and c = n2 + m2. Look at this table: (We will make this into a transparency and put it on the overhead) m=1 2 3 4 5 6 n= 2 [3, 4, 5] 3 [8, 6, 10] [5, 12, 13] 4 [15, 8, 17] [12, 16, 20] [7, 24, 25] 5 [24, 10, 26] [21, 20, 29] [16, 30, 34] [9, 40, 41] 6 [35, 12, 37] [32, 24, 40] [27, 36, 45] [20, 48, 52] [11, 60, 61] 7 [48, 14, 50] [45, 28, 53] [40, 42, 58] [33, 56, 65] [24, 70, 74] [13, 84, 85] As you can see in the 3-4-5 right triangle, when m = 1 and n = 2, a = 4 – 1 = 3, b = 2(2)(1) = 4, and c = 4 + 1 = 5. The table can go on forever, and I thought it was very interesting to see that you can find EVERY Pythagorean Triple. Anyways, this is just for your own benefit, so take it as you wish. 4) Next, let’s think about what we were doing when we were figuring out if our Pythagorean Triples really were Pythagorean Triples. We were taking our a’s, b’s, and c’s and seeing if they worked in the Pythagorean Theorem. By doing this, we were determining whether the side measures of a triangle form a right triangle. Therefore, if a2 + b2 = c2, then the triangle is a right triangle. However, if a2 + b2 ≠ c2, then the triangle is not a right triangle. Let’s see if these next triangles are actually right triangles. First let’s examine the triangle with sides 20, 21, and 29. Of these three sides, which one is the hypotenuse? Great, and why is the longest side the hypotenuse? What angle is always opposite the hypotenuse? Why? Excellent, now let’s plug in these values. 202 + 212 =(?) 292 400 + 441 =(?) 841 841 = 841 As we can see, a triangle with side lengths of 20, 21, and 29 form a right triangle. What about a triangle with the sides 8, 10, 12? Let’s try this one too. 82 + 102 =(?) 122 64 + 100 =(?) 144 164 ≠ 144 As we can see, this triangle is not a right triangle. 5) So, who can tell me what we did today? What did you fill out in your four sections of your paper? As you explain it to me, I will write it out on the board so everyone can see. I will let the students take their time in summarizing the main topics of today and then we will have a class discussion about it. Modifications: If we run out of time, we will continue our summary on the board the following day. If we have extra time, they will be able to start on their homework. Assessment: Assessment is done throughout class by observing students and walking around the room to make sure they are staying on task. At the end of class, students will receive a handout with the following questions on it: 1) Draw a right triangle and label each side and angle. Be sure to indicate the right angle. 2) Explain how you can determine which angle is the right angle of a right triangle if you are given the lengths of the three sides. 3) Write an equation you could use to find the length of the diagonal d of a square with side length s. 4) What was your favorite thing you learned today and why? Explain it to me. 5) Can you think of where else you might use the Pythagorean Theorem in your life? They will have to answers these on a separate sheet of paper and hand them in the following class period. Rational: We began our lesson by discussing what was learned the previous day in hopes of getting the students thinking and getting them engaged with the text at the very beginning of class. We also let the students brainstorm by having them tell us how to find the length of a side of a right triangle instead of just giving them the answer. We did this because it will help them synthesize and organize the information they are learning and have already learned. Furthermore, we constantly questioned them throughout the lesson which guided their understanding into what we wanted them to know. By answering these questions, not only were we guiding their learning, but they were reflecting upon the information that was being presented to them. This was also accomplished by having them fill out the summary sheet with the four sections throughout the class. We gave them explicit directions and told them exactly what we wanted from them and what we were looking for in each box. By doing this they were able to reflect up on the information we went over in class, and they were able to put it in their own words. At the end of class, we had a discussion about what we learned. We had the students describe in their own words what they wrote in their boxes. After that, we wrote this information out on the board, in sentences, so everyone could see. We did this because group summaries help the students review and remember the information that was presented to them during class. Standards: Illinois Learning Standards: 9.D.3 Compute distances, lengths, and measures of angles using proportions, the Pythagorean Theorem and its converse NCTM Standards: Create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean Theorem. Dominique Davis, Molly Duffy, Julie Hernandez, Kelly Holton, Brandi Zars CI 473 Writing in Mathematics Lesson Plan Pythagorean Theorem Lesson Plan Grade Level: 9th grade Algebra I 334 Demographics: 20 students (10 AA, 7 Caucasian, 2 Latino/a, 1 Asian (15 freshman, 4 sophomores, 1 junior) Estimated Time: 1 class period (about 50 minutes) Concept: Students will have a deeper understanding of the Pythagorean Theorem and why it actually “works”. Prior to this lesson, students applied the theorem to find lengths of missing sides of right triangles and to identify right triangles given dimensions of arbitrary triangles. After this lesson, they will understand why they were able to apply the formula a2 + b2 = c2 to solve right triangles. Students will experiment with specific online manipulatives that demonstrate visual proofs of the theorem. After experimenting with these visual proofs, students will be challenged to write in their own words an explanation of the proof. Objectives: Students will review and fully understand the Pythagorean Theorem. Students will understand visual proofs of the Pythagorean Theorem. Students will be describe the principles of the Pythagorean Theorem using their own words. Materials: Notes Loose Leaf Paper/Journal Overhead/transparencies Pencil Computer Lab Procedure: 1) Good morning class! Yesterday we worked with the Pythagorean Theorem, and we used it to discover the missing side lengths and hypotenuses of given right triangles. We also used the theorem to determine if an arbitrary triangle was indeed a right triangle, given the dimensions of that triangle. So yesterday we actually applied the formula a2 + b2 = c2 to solve right triangles. Today, however, we are going to focus on your understanding of the theorem. We’re going to discover why we can apply the Pythagorean Theorem to triangles. Today we will ask questions like, why does this theorem work? Why can we simply use the formula a2 + b2 = c2 to solve right triangles? 2). So today, I’m going to provide you with a visual proof of the theorem, then you will individually work with online manipulatives to understand why the proof works. After we return from the computer lab, you will all write in your own words, a brief explanation of the proof, and why it works. We have written proofs in this class already, so you should be familiar with the concept of a proof. But for a quick review, a proof is a logical argument that consists of a sequence of steps, statements, or demonstrations that leads to a valid conclusion. In other words, it’s a series of logical statements of facts that “prove” something to be true. Today we will discuss why the Pythagorean Theorem holds true for any given right triangle and focus specifically on a visual proof of the theorem. 3). The Pythagorean Theorem is a statement about triangles containing a right angle. The Pythagorean Theorem actually states that: "The area of the square built upon the hypotenuse of a right triangle is equal to the sum of the areas of the squares upon the remaining sides." We have simplified that into an easily memorized formula a2 + b2 = c2, but we need to discover why we can simply use this formula. Here we have a right triangle ABC. We say that the side AB=a, side BC=b and side CA= c. A a c B b C Now, if we construct squares upon the sides and hypotenuse of the triangle we get an object that looks like this: Now our theorem states that the area of the square built upon the hypotenuse is equal to the sum of the areas of the squares upon the remaining sides. Let’s look at the picture below to get an idea of what this statement means. According to the Pythagorean Theorem then, the sum of the areas of the two red squares A and B, is equal to the area of the blue square, square C. We find the areas of the squares as follows: Area of Square A = a2. Area of Square B = b2 . And finally, Area of Square C = c2 . This is how we arrive at our algebraic formula, a2 + b2 = c2 . So our formula is simply the algebraic application of the Pythagorean Theorem for a right triangle with sides of lengths a, b, and c, where c is the length of the hypotenuse. 4). So, we have simply broken down the statement of the Pythagorean Theorem, and discovered what it means visually. Now we’ll go to the computer lab and discover a visual proof of the theorem. (Once at the lab, have students go to http://www.usna.edu/MathDept/mdm/pyth.html, also have overheads used in class available to see in the lab) So class, we have a figure similar to the ones on our overheads in class. Here we have a triangle ABC, with squares built upon each side of the triangle. So similarly in this animated proof, we start with a right triangle and squares on each side. The middle size square is cut into four congruent quadrilaterals. In our overhead, this would be square A, cut into four congruent quadrilaterals. That means that each one is equal in dimension and area. The square is separated into parts such that the cuts are made through the center of the square, and run parallel to the sides of our square C. The animation then moves the quadrilaterals. They are hinged and rotated and shifted to the big square, or square C. Finally the smallest square, in our case square B, is translated to cover the remaining middle part of the biggest square C. So it just so happens that it is a perfect fit. Thus we have just seen visually, that the sum of the squares on the smaller two sides equals the square on the biggest side. In other words, the sum of the areas of squares A and B, is equal to the area of square C. So indeed a2 + b2 = c2. 5). Now I want you to try a few online manipulatives to further discover what we have been learning today. Please go to the websites listed on the board: 1) http://www.dynamicgeometry.com/javasketchpad/pythagoras.php 2) http://nlvm.usu.edu/en/nav/frames_asid_164_g_4_t_3.html?open=instructions 3) http://cinderella.de/files/HTMLDemos/1G01_Pythagoras.html The first and second websites are visual applications of the Pythagorean theorem. I want you to play around with each manipulative and make your own discoveries about the theorem. The first website uses a similar object to the one in our overhead and the visual proof website. It is a geometer’s sketchpad interactive applet that allows you to move the pieces by dragging and dropping the red points on the polygons. It shows that the area of square BCDE = the area of square BFGA + the area of square AHIC. Our second website is a little bit different. In this you are asked to solve two puzzles that illustrate the proof of the Pythagorean theorem. In this manipulative, we have a right triangle with its sides labeled. You can move any triangle or square by clicking the piece and dragging it around. You rotate the pieces by clicking on the corner of the piece and rotate your mouse in a circular motion. At the bottom, there are ways to reset your pieces to start from scratch. You can also go onto the next puzzle. As you solve the puzzles, I want you to draw a picture of what the shape should look like. You will turn this in along with an explanation in your own words of what you learned today. Our third website is optional, and it is simply another visual proof of the Pythagorean Theorem. You may go to this site if you need further clarification on the subject matter. (Allow students 20 minutes to work in the lab) 6). Alright class, now that we have returned from the lab, I want you to take out a sheet of paper. Please redraw your sketches of your solutions to the two different puzzles from our second website. There should be a total of four sketches, since each puzzle had two different objects. When you have done that, I would then like for you to tell me IN YOUR OWN WORDS, what the Pythagorean Theorem states, and why we are able to use the formula a2 + b2 = c2 to solve right triangles. You are allowed to use any explanation that you see fit, whether or not you want to describe what we did in class, or what you learned from the online manipulatives, it is up to you. Please write at least four to five sentences about what you learned from today. You will be turning these in at the end of class as an evaluation of the effort you put into today’s activities. (Allow students at least 10 minutes to summarize their thoughts. Have a class discussion in which individual students share their writings, if time permits.) Modifications: If we run out of time, we will have a short review of the activity, and students will summarize and put into their own words their explanations of the Pythagorean theorem on the following day. If we have extra time, we will have a class discussion in which individual students are to share their writings with the class. Assessment: Assessment is done throughout class by observing students and walking around the computer lab to make sure they are staying on task. Also, the in-class writing activity includes their solutions to the online puzzles, which should keep students focused during lab time. Each individual student’s written submissions should also give the instructor feedback of student progress and comprehension. Rationale: We began the lesson by reviewing what was accomplished in previous lessons to help the students understand the material for the day, to get them thinking about previous material and to help them become engaged with the day’s lesson. We want our students to have a deeper understanding of the Pythagorean theorem and also see why they are able to apply a simple formula in order to solve right triangles. We prefaced the lesson by explaining that in previous lessons we simply applied the formula a2 + b2 = c2 , but that this alone, will not tell us why or how the Pythagorean theorem actually works. We gave them clear objectives for the day by telling them we were going to see visual proofs of the Pythagorean theorem and that by the end of the period, they were to be able to write in their own words an explanation of the proofs. In order to introduce the material, we wanted to first break down the theorem and give the students a visual representation of the theorem. This would provide students with further clarification of the theorem, allowing them to synthesize and organize the information with a visual aid. Our activity in the computer lab was intended to engage students with the material and force them to think critically about what we previously learned in class. We also gave them explicit directions about what would be expected of them at the completion of the task. The writing activity was intended to the students summarize their thoughts. It was a method for checking each student’s individual comprehension and ability to synthesize the material. The written submission and the class discussion are intended to effectively summarize and review the information from the day. Standards: Illinois Learning Standards: 9.C.4,5 Select and apply technology tools for research, information analysis, problem- solving, and decision-making in content learning. NCTM Standards: Create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean Theorem.
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