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Worksheet
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\n\"Who\n\n
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\nCurated OER\n
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\nWho Dun It?\n

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\n\nFor Students\n\n9th - 12th\n
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\nIn this proofs by contradiction activity, high schoolers solve 1 word problem about a crime.  Students use proof by contradiction to determine the criminal in the word problem.\n
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\n\nGet Free Access\n\nSee Review\n
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Lesson Plan
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\n\"The\n\n
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\nEngageNY\n
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\nThe Converse of the Pythagorean Theorem\n

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\n\nFor Teachers\n\n8th\n\nStandards\n
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\nIs it a right triangle or not? Introduce scholars to the converse of the Pythagorean Theorem with a lesson that also provides a proof by contradiction of the converse. Pupils use the converse to determine whether triangles with given...\n
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\n\nGet Free Access\n\nSee Review\n
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Instructional Video4:40
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\n\"Making\n\n
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\nTED-Ed\n
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\nMaking Sense of Irrational Numbers\n

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\n\nFor Students\n\n7th - 12th\n
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\nNon-terminating. Non-recurring decimal. Irrational numbers. Proof by Contradiction? Root 2, 3, 5, 6, 7, 8. Don't be afraid to explore the impossible and check out this short video that offers a rational explanation of irrational...\n
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\n\nGet Free Access\n\nSee Review\n
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Instructional Video4:21
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\n\"Everything\n\n
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\nTED-Ed\n
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\nEverything Changed When the Fire Crystal Got Stolen\n

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\n\nFor Students\n\n6th - 12th\n
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\nViewers of a short video use the Proof by Contradiction technique; to determine which apprentice monk swallowed which crystal the group attempted to steal from the Sacred Crystal Temple.\n
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\n\nGet Free Access\n\nSee Review\n
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Lesson Plan
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\n\"Three\n\n
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\nCurated OER\n
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\nThree for the Money: The Degree/Diameter Problem\n

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\n\nFor Teachers\n\n9th - Higher Ed\n
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\nStudents explore the concept of vertex-edge graphs.  In this vertex-edge graphs lesson, students try to construct a graph with a given diameter, number of vertices, size, and planarity.  Students construct various vertex-edge...\n
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Activity
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\n\"Unsolvable\n\n
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\nBeauty and Joy of Computing\n
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\nUnsolvable and Undecidable Problems\n

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\n\nFor Teachers\n\n9th - 12th\n
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\nTry as you might, some functions just cannot be computed. The lab introduces the class to the possibility of unsolvable problems. The fourth lesson in a series of seven begins with a logic problem, then progresses to looking at functions...\n
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\n\nGet Free Access\n\nSee Review\n
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Instructional Video10:04
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\n\"The\n\n
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\nMinutePhysics\n
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\nThe No Cloning Theorem\n

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\n\nFor Students\n\n11th - Higher Ed\n
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\nPerfect cloning is impossible — and that's been proven mathematically. Investigate the complexity of cloning in physics. The video instructor explains why cloning is theoretically impossible. He uses several examples to illustrate...\n
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Lesson Plan
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\n\"Applying\n\n
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\nIllustrative Mathematics\n
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\nApplying the Pythagorean Theorem in a Mathematical Context\n

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\n\nFor Teachers\n\n7th - 9th\n\nStandards\n
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\nParticipants who use this resource will apply the Pythagorean Theorem to show whether or not the shaded triangle inscribed in a rectangle is a right triangle. Once all of the sides on the shaded triangle are found, it is important that...\n
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Lesson Plan
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\n\"The\n\n
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\nCurated OER\n
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\nThe Truth About Triangles And Proofs\n

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\n\nFor Teachers\n\n9th - 12th\n
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\nStudents engage in a activity that is about the classification of triangles and the mathematical proofs involved in working with them. They work on a variety of problems that are created by the teacher with the focus upon the importance...\n
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Lesson Plan
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\n\"Babylonian\n\n
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\nCurated OER\n
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\nBabylonian Mathematics 2\n

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\n\nFor Teachers\n\n9th - 12th\n
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\nYoung scholars research Babylonian mathematics. They calculate simple surd numbers. Students find the fractional form of rational numbers expressed as decimals. They work with numbers in base 60.\n
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Instructional Video
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\n\"Khan\n\n
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\nKhan Academy\n
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\nKhan Academy: Geometry: Ca Geometry: Proof by Contradiction\n

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\n\nFor Students\n\n9th - 10th\n\nStandards\n
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\nDemonstrates solutions to problems #4-6 of California Geometry released state test questions. Problems address understanding of proof by contradiction and proving triangles congruent. [11:56]\n
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Instructional Video
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\n\"Khan\n\n
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\nKhan Academy\n
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\nKhan Academy: Ca Geometry: Proof by Contradiction\n

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\n\nFor Students\n\n9th - 10th\n\nStandards\n
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\nExamples of proof by contradiction.\n
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Instructional Video
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\n\"Khan\n\n
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\nKhan Academy\n
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\nKhan Academy: Worked Examples: Geometry: Ca Geometry: Proof by Contradiction\n

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\n\nFor Students\n\n9th - 10th\n\nStandards\n
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\nStudents explore how to do a geometric proof by contradiction.\n
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Instructional Video
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\n\"Khan\n\n
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\nKhan Academy\n
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\nKhan Academy: Geometry: Corresponding Angle Equivalence Implies Parallel Lines\n

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\n\nFor Students\n\n9th - 10th\n\nStandards\n
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\nProof by contradiction that if two lines cut by a transversal create congruent corresponding angles then the lines are parallel. [5:18]\n
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Instructional Video
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\n\"Khan\n\n
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\nKhan Academy\n
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\nKhan Academy: Linear Algebra: Introduction to Orthonormal Bases\n

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\n\nFor Students\n\n9th - 10th\n
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\nVideo defining an orthonormal set as a set of vectors that all have the length of 1 and are orthogonal to each other. Using proof by contradiction, the set is proven to be linearly independent and can be considered an orthonormal basis....\n
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Activity
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\n\"Regents\n\n
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\nOswego City School District\n
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\nRegents Exam Prep Center: Indirect Euclidean Proofs\n

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\n\nFor Students\n\n9th - 10th\n
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\nThis Oswego City School District Regents Exam Prep site shows you how to proof something by showing that it is false. This look at indirect proofs includes a practice and a teacher resource page, which features a warm-up activity.\n
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\n\nGet Free Access\n\nSee Review\n
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Website
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\n\"Making\n\n
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\nEducation Development Center\n
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\nMaking Mathematics: Proof\n

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\n\nFor Teachers\n\n9th - 10th\n\nStandards\n
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\nThis is a site which links to specific techniques of proving including conditional statements, proof by contradiction, mathematical induction, parity arguments, and the pigeonhole principle. The site contains a link to various example...\n
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eBook
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\n\"Mathigon:\n\n
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\nMathigon\n
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\nMathigon: Axioms and Proof\n

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\n\nFor Students\n\n9th - 10th\n\nStandards\n
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\nExplore this course in logic, sets, and proof including topics of axioms, set theory, and proof by contradiction and induction.\n
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\n\nGet Free Access\n\nSee Review\n
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Instructional Video
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\n\"Khan\n\n
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\nKhan Academy\n
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\nKhan Academy: Parallel Lines & Corresponding Angles Proof\n

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\n\nFor Students\n\n9th - 10th\n
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\nProof by contradiction that corresponding angle equivalence implies parallel lines.\n
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\n\nGet Free Access\n\nSee Review\n
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Handout
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\n\"Platonic\n\n
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\nOther\n
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\nPlatonic Realms: Conic Sections\n

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\n\nFor Students\n\n9th - 10th\n\nStandards\n
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\nA proof to show that the square root of two is an irrational number, followed by a shorter and even more beautiful proof.\n
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\n\nGet Free Access\n\nSee Review\n
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Primary
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\n\"Kent\n\n
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\nKent State University\n
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\nKent State University: Indirect Proofs, Michael Byron\n

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\n\nFor Students\n\n9th - 10th\n
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\nUse this site to explore an excellent discussion of indirect proofs. The site explains in detail why such a method works as a proof and gives steps to obtain such a proof.\n
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\n\nGet Free Access\n\nSee Review\n
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\n\"Ck\n\n
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\nCK-12 Foundation\n
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\nCk 12: Geometry: Indirect Proof in Algebra and Geometry\n

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\n\nFor Students\n\n9th - 10th\n\nStandards\n
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\n[Free Registration/Login may be required to access all resource tools.] This concept teaches students how to write an indirect proof and provides examples of indirect proofs in Algebra and Geometry.\n
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\n\nGet Free Access\n\nSee Review\n
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Unit Plan
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\n\"Ck\n\n
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\nCK-12 Foundation\n
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\nCk 12: Geometry: Indirect Proof in Algebra and Geometry\n

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\n\nFor Students\n\n9th - 10th\n\nStandards\n
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\n[Free Registration/Login may be required to access all resource tools.] Write algebraic and geometric indirect proofs.\n
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\n\nGet Free Access\n\nSee Review\n
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Activity
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\n\"California\n\n
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\nOther\n
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\nCalifornia State University, Fresno: Counter Examples\n

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\n\nFor Students\n\n9th - 10th\n
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\nCome see how a counter example is helpful in some proofs. The site explains what a counter example says, and then offers a proof using a counter example.\n
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\n\nGet Free Access\n\nSee Review\n
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