Geometric Proofs
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# Geometric Proofs

When someone says geometry, people think of polygons, circles, lines, and sometimes a person mentions proofs. High school geometry is typically the first time students are introduced to the idea of reasoning and proof. Reasoning, logic, proof, and creativity are vital in mathematics, and geometric proofs are culmination of these characteristics. On this page I have curated information for myself and my future students. It contains articles on the importance of proof, activities for in and outside of the classroom, along with other activities that improve reasoning skills.

## Introduction to Geometric Proof

Meant as an introduction to constructing geometric proofs, both in the flow proof style and the two-column, or statement-reason style. As an introductory lesson this packet only includes short proofs with some of the basic structure provided.

On this website, there is a video which uses a flow chart as a means to introduce students to proof. Instead of the typical two-column proof, this method provides student with a visual of how a proof can be constructed. The example has two branches that converge to prove the statement that students are asked to show is true. This method emphasises that while proofs are logical progressions of thought with reasoning behind them, they are not strictly linear. It is important to realize this because in life there may be multiple reasons that someone holds an opinion or thinks one option is better than another, and if you are asked to persuade others on a business level, then you would need a great deal of facts strung together in a logical progression to prove your point.

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## Hands on Math in High School: Made4Math Geometry Proofs

This page provides another way to introduce students to two-column proofs. Once students have a grasp on the general concept of a proof, you can begin incorporating geometric figures into the statements that need to be proven. If a student is struggling with what step comes next, you can provide them with either the "statement" or the "reason" and have them fill in the other column. This provides some hints, which support the development of reason. I think students might get dependent on this and not really understand the logic behind a statement, so having students share their reasoning with a partner would solidify their comprehension. Sharing also builds communication skills, which is vital for thriving in society.

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## Proof-scrambling activities : Mathematical Communication pages of MathDL

This activity provides students with all of the necessary steps to complete a proof, but they are out of order and missing some transitions. This is beneficial for students who are struggling to formulate phrases that are appropriate for proofs, as they should be concise and complete. This strategy is also beneficial for students who are improving on their reasoning skills. They have the statements, but now they need to reason through the statements and arrange them in a logical progression. This could also be used as a class or small group activity as an informal assessment of how well students can reason and explain their ideas.

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## Illuminations: Area Formulas

This unit of the formulas for basic polygons allows students to explore both virtually and hands-on in order to derive the formula for triangles, parallelograms, trapezoids, and irregular shapes. This unit helps students break shapes into smaller ones, whose area is easier to calculate. This takes some creativity and logic when explaining why the method works and how the formula is derived from the manipulation.

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## NCTM Illuminations

The Gateway to Standards-Based Mathematics Education. Internet resources to improve teaching learning of school mathematics based on the NCTM Priniples and Standards for School Mathematics.

This is an interactive applet that walks students through a proof without words. At the end of the exercise, students need to explain the proof. Since there are no words, students are required to have a firm understanding of what is occurring in the proof and why.

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## Getting Kids Excited About Mathematics, One Festival at a Time - Huffington Post

Huffington Post Getting Kids Excited About Mathematics, One Festival at a Time Huffington Post I was a facilitator, and I'd come a long way---from Atlanta---for the privilege of spending four hours interacting with smart kids who were eager for...

An article about how mathematics festivals provide students a challenging, yet enjoyable experience. Students have to reason through the activities and be creative. While this article does not strictly address geometric proofs, it addresses the skills necessary to solve a problem.

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This is an exercise on Khan Academy that provides students with a few practice proofs. Students are to supply the statement by filling in the blank and provide a reason by selecting one from a pull down menu. I think this would be helpful for students to get used to the flow of two-column proofs as well as give them an extra resource to work on the thought process necessary to complete a proof. Once again it is vital to discuss the reasoning of proofs, so if students were to utalize this resource I would want them to discuss their logic with a peer or myself.

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## First day of Geometry proofs - Refining my process

Last year, I figured out about a week or two after the first introduction to proofs in Geometry last year that I should have started with a more clear connection to the ideas we had been working on...

This geometry teacher uses the traditional two-column proof, which is not bad, but it is very basic. If I were to adapt this, I would show students some of my proofs from my undergraduate mathematics courses. These proofs are similar, but my collegiate proofs are in paragraph form. To enhance my students' understanding of proofs, we can compare the proofs they do in class to the ones I did as an undergraduate. During this discussion, we would talk about the importance of proofs and how they are applicable in the world outside of math class an school.

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## High School Geometry | www.MathEdPage.org

Overview of a geometry course at the Urban School of San Francisco, with key worksheets and an emphasis on how proof is approached.

This is a quick summary of how one teacher in San Francisco sets up his geometry class. Understanding the rules of proofs is important, and it is vital that the teacher lay out the boundaries before students dive into proofs. I like how the teacher starts with the physical compass, pencil, and paper, then works in the interactive software to complete proofs. Reasoning through proof is sometimes difficult for students, because they have never been allowed to question the teacher in a traditional classroom, and with proofs, students are allowed to question why something is, they are just expected to be able to think it through and back up their thoughts.

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## Illuminations: Perplexing Parallelograms

This is a lesson that allows students to explore the relationships amongst parallelograms. Students are asked to make conjectures about parallelograms and how they proved them. These proofs are informal, but contain a logical thought process. These proofs take a bit of creativity as well as a general understanding about triangles, diagonals, and parallelograms.

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