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=Applet: The Pythagorean Puzzle =

====In this applet, two proofs of the Pythagorean Theorem are demonstrated visually. The Pythagorean Theorem can be defined as, “In any right-angled triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle). This applet came from the National Library of Virtual Manipulative website. This site contains countless applets that allow teachers and students from grades K5-12 to use technology in this classroom. This is a place where teachers and students from all over the world are able to use these applets to better explain and understand the concepts that are focused on in the mathematical standards. These applets range in ability and function. Some of them are simplistic and straightforward, allowing the students to perform procedural based tasks, while others are more rich in mathematics, and require that students use cognitive thought to use features involving real-world scenarios, proofs, and graphs. All of the applets on this website, however, are free of charge. It does not require any membership or payment to use these applets. ====

====The Pythagorean Puzzle is one that allows students to see various proofs of the Pythagorean Theorem. This applet comes with directions that instruct students on how to maneuver the pieces of the puzzle, and from there the students are required to complete the puzzles. This creates a "proof without words" and allows the students to see that the Pythagorean Theorem holds true. Using technology for the purpose of proving theorems and identities can be extremely advantageous to both the teacher and the students. Specifically, when dealing with pictorial proofs, like the ones in this applet, technology allows the students to visualize the proofs without having to go through the hassle of using tools like cutouts, scissors, rulers, and other items that can be time consuming for both the teacher and the student. This specific applet can be found under the Middle and High School activities section of this website. The target grade for this applet would be 8th, 9th, and 10th graders. However, this proof can be beneficial when upperclassmen are developing conceptual understanding of different proofs as well. ====

====For more information on this technology, it might be beneficial to real the parent-teacher information or the instructions for this applet. ====

The Pythagorean Theorem:
 

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=Standards = ====The standards that apply to this applet both directly and indirectly relate to this technology. The common core standards clearly express that a proof for the Pythagorean Theorem must be performed and mastered by the students. This is under 8th grade in the common core. However, in high school, the standard becomes more specific in saying that the proof of the Pythagorean Theorem must be done using triangle similarity. Although this applet does not provide a proof of the Pythagorean Theorem using triangle similarity, the proofs provided could be useful in other proofs that are required of the students. Other standards that apply to this applet include standards that are more general. These standards are under the process standards and include things like develop and evaluate proofs, recognize proofs as fundamental aspects of mathematics, and create and use representations to communicate mathematical ideas. In addition, knowing and understanding a proof of the Pythagorean Theorem is beneficial in so many other ways. It will allow students to understand the concepts of this idea rather than just the procedure. As a result this theorem can be applied to many other aspects of mathematics and can fulfill various other standards that apply to solving right triangles. ====
 * ====**Common Core: **====

MCC8.G.6 Explain a proof of the Pythagorean Theorem and its converse.
====MCC8.G.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real‐world and mathematical problems in two and three dimensions. ====

__Define trigonometric ratios and solve problems involving right triangles. __
====MCC9‐12.G.SRT.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. ====

__<span style="font-family: Tahoma,Geneva,sans-serif;">Prove theorems involving similarity. __
====<span style="font-family: Tahoma,Geneva,sans-serif;">MCC9‐12.G.SRT.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. ====


 * ====**<span style="font-family: Tahoma,Geneva,sans-serif;">Process Standards: **====

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=<span style="font-family: Tahoma,Geneva,sans-serif;">Activity =

<span style="font-family: Tahoma,Geneva,sans-serif;">Allow group and whole-class discussion when working through this proof and the questions that go along with it.

 * ====<span style="font-family: Tahoma,Geneva,sans-serif;">State the Pythagorean Theorem? ====
 * ====<span style="font-family: Tahoma,Geneva,sans-serif;">For what kind of triangles does this theorem hold true? ====
 * ====<span style="font-family: Tahoma,Geneva,sans-serif;">Does the Pythagorean Theorem hold true any right triangle? What about the one below? ====

<span style="font-family: Tahoma,Geneva,sans-serif;">[[image:Screen shot 2013-05-22 at 11.04.43 AM.png width="319" height="180"]]

 * ====<span style="font-family: Tahoma,Geneva,sans-serif;">Which side(s) of a triangle form(s) the side of the square? ====
 * ====<span style="font-family: Tahoma,Geneva,sans-serif;">What is the length of that segment? ====
 * ====<span style="font-family: Tahoma,Geneva,sans-serif;">What can you say about the areas of the two white regions? Are they the same or different? How do you know? ====
 * ====<span style="font-family: Tahoma,Geneva,sans-serif;">Without using a formula, state what the Pythagorean Theorem is. ====

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=<span style="font-family: Tahoma,Geneva,sans-serif;">Critique = ====<span style="font-family: Tahoma,Geneva,sans-serif;">This applet works great. When researching and working with this applet, very few technical difficulties were encountered. When first using this applet, there was some trouble with rotating the "puzzle" pieces, however, that obstacle was quickly overcome. Other than that, this applet was very simple to use. Directions are clearly laid out which leaves little ambiguity. ==== ====<span style="font-family: Tahoma,Geneva,sans-serif;">There are written materials that go along with this applet. Although the purpose of this applet is to allow the students to discover visual proofs of the Pythagorean Theorem, this applet does give adequate instructions and gives an explanation of each of the puzzles. Puzzle 1 and 2 show each piece of the puzzle and has the sides labeled as a, b, or c. This makes it clear for the students to see how they all fit into the puzzle and how they each relate to the Pythagorean Theorem. This applet also comes with a parent-teacher guide so that the teacher can better implement this into the classroom. The only thing I would add would be to have some questions that guide the students through these puzzles. Questions that guide their thinking and allow them to explore the meaning of the puzzles visually and algebraically. ==== ====<span style="font-family: Tahoma,Geneva,sans-serif;">This applet is one that not only allows the students to work with the Pythagorean Theorem, but actually allows them to understand the conceptual meaning behind this theorem. The Pythagorean Theorem is used in math classes daily, therefore, students understanding of this theorem is imperative to their performance. In many classrooms, students do not actually understand the meaning behind this theorem. They have memorized a very procedural way of using the Pythagorean Theorem. This applet allows the students to see what it actually means that a^2+b^2=c^2. They can actually see visual representations of the theorem and how it relates to right triangles and squares. The goal of this applet is to allow students to develop a more conceptual understanding of the Pythagorean Theorem, and the applet does just that. ==== ====<span style="font-family: Tahoma,Geneva,sans-serif;">The directions on this applet are clearly outlined. General instructions are first laid out, and then more specific instructions are given for Puzzle 1 and Puzzle 2. As a result, the applet is very easy to understand and to use. ==== ====<span style="font-family: Tahoma,Geneva,sans-serif;">This technology has the ability to enhance students mathematical ability. It can do this because of the pure essence of this task. The entire goal of this task is for students to use the shapes as a puzzle to prove the Pythagorean Theorem. If implemented correctly and students fully understand this proof, they most likely have a conceptual understanding of the ideas behind the Pythagorean Theorem. As a result, their conceptual knowledge will allow them to see past the procedure to the root of this theorem. They will be able to understand how this theorem works, and their implementation of it will have more depth and meaning. ==== ====<span style="font-family: Tahoma,Geneva,sans-serif;">I would recommend this product for various reasons. First and foremost is because this applet is a huge time savor in the classroom. Rather than the teacher spending hours of personal time or the students spending hours of classroom time to cute out shapes to show this proof, everything they need is already in this easy-to-use applet. There is no need for any other materials. This is not only helpful for the teacher, but it also ensures that the students will not get distracted with the materials being used. Another reason I would recommend this applet is because of its usefulness in the classroom. Many teachers overlook the importance of presenting proofs in the classroom, however, it can be very valuable to the students learning process. Because there are so many student misconceptions the involve the Pythagorean Theorem, this is one proof in particular that can be advantageous for students to learn. Lastly, I would recommend this applet because of the actual proofs that it provides. Because there are so many different proofs of the Pythagorean Theorem, it can be difficult as a teacher to decide which proof is best to present in the classroom. Both of these proofs are very straightforward and are some of the most simplistic proofs out there, yet are still rich in mathematics. Students, especially middle school, are likely to see and understand how these proofs work and the advantages to knowing them. ====
 * ====<span style="font-family: Tahoma,Geneva,sans-serif;">**How well does it work?** ====
 * ====**<span style="font-family: Tahoma,Geneva,sans-serif;">Are the written materials well organized and useful? **====
 * ====**<span style="font-family: Tahoma,Geneva,sans-serif;">What are the purposes and goals for using this technology? Does the technology reach this goal? **====
 * ====<span style="font-family: Tahoma,Geneva,sans-serif;">**Is the technology relatively easy to learn how to use?** ====
 * ====<span style="font-family: Tahoma,Geneva,sans-serif;">**Does this technology enhance or extend the teaching and learning process for the intended mathematics concepts? How and why?** ====
 * ====**<span style="font-family: Tahoma,Geneva,sans-serif;">Would you recommend this product for purchase to a school? Why or why not? **====

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=<span style="font-family: Tahoma,Geneva,sans-serif;">References =

====<span style="font-family: Tahoma,Geneva,sans-serif;">Swan, M. (2012). Mathematics assessment project. Manuscript submitted for publication, University of Nottingham, Nottingham, UK. Retrieved from http://map.mathshell.org/materials/download.php?fileid=1231 ====