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HSCE: L3.3.1 Know the basic structure for the proof of an “If…, then…” statement (assuming the hypothesis and ending with the conclusion) and know that proving the contrapositive is equivalent. Clarification statements: Conditional Statements: If A (hypothesis), then B (conclustion) Statement Converse Inverse Contrapositive A B B A A : B B : A If A then B If B then A Negate A and B Reverse and negate A and B A statement and its converse need not be true. A statement and its contrapositive always have the same truth value. They are either both true or both false. Thus, B : A also proves A B . This is sometimes helpful when a clear path from the original hypothesis and conclusion is not readily apparent. The inverse and converse always have the same truth value. They are either both true or both false. Given the opportunity to describe how one can go about proving an “If …, then…” statement, begin by assuming the hypothesis is true. This information is the “given” of the situation. Students then develop a chain of logical reasoning, ending with showing that the conclusion must be true. This chain of logic can take several forms: a paragraph proof, a flow proof, a two-column proof, a proof using figures on the coordinate plane, and so on. Example 1: Prove that corresponding angles formed by parallel lines cut by a transversal are congruent. Direct Proof: Rewrite the statement; “If parallel lines are cut by a transversal, then the corresponding angles formed are congruent.” Next, identify the hypothesis to determine what information we are given to use. In this case, we are working with “parallel lines cut by a transversal”. We could draw a diagram at this point and label the angles: Since the conclusion says “the corresponding angles formed 1 2 are congruent”, at some point in l the proof one would need to 4 3 identify some corresponding 6 5 7 8 t m l || m angles and state why they are identified as such. One would also need to probably work with pairs of angles and other angle relationships which are already known (through postulates, theorems, definitions). By following a step-by-step chain of logical reasoning, show that “the corresponding angles formed are congruent. Proof by using the contrapositive: First write the contrapositive: “If two lines are cut by a transversal and the corresponding angles formed are NOT congruent, then the lines are NOT parallel” We then would assume we have two lines and a transversal, but with the angles which form corresponding pairs of angles NOT being congruent. We would follow a similar chain of reasoning, which would eventually lead us to the conclusion that, in fact, the two lines cannot be parallel. VOCABULARY Proof, “If…, then…” statement, hypothesis, conclusion, assumptions, givens, contrapositive, logically equivalent. Web Resource: Finding Proof, Annenberg Video Library #9 Students are introduced to the historical context of proofs and to Greek mathematician Thales, who sought proof for mathematical c http://www.learner.org/resources/series34.html?pop=yes&vodid#onclusions. Activity 5: Different Methods of Proof. This PDF highlights proof, both in algebra and geometry. http://mdk12.org/instruction/curriculum/hsa/geometry/clg.html HSCE: L3.3.2 Construct proofs by contradiction; use counterexamples, when appropriate, to disprove a statement. Clarification statements: Proof by contradiction: The "Proof by Contradiction" is also known as reductio ad absurdum, which is probably Latin for "reduce it to something absurd". Here's the idea: 1. Assume that a given proposition is untrue. 2. Based on that assumption, reach two conclusions that contradict each other. This is based on a classical formal logic construction known as Modus Tollens: If P implies Q ( P Q ) and Q is false ( Q ), then P is false ( P ). http://www.programmingforfun.com/Programs/Math_Topics/proof_by_contra diction.htm Students will likely use direct reasoning/direct proof strategies when constructing proofs. However, there are times when a direct reasoning path is hard to see, or when using indirect reasoning is more elegant. Students should see several proofs using indirect methods (proof by contradiction) over their high school mathematics experience. They should also be able to construct fairly simple proofs by contradiction. This expectation highlights the importance of counterexamples. One good counterexample can spoil what looked like a very good generalization. Students also need to realize that just because numerous examples supporting a generalization can be provided, they themselves do not constitute a proof. They may suggest a generalization, and even point to how the generalized statement might be proved. However, if a counterexample is provided, the generalization will need to be rethought. VOCABULARY: Proof, proof by contradiction, contradiction, counterexample, prove, disprove, statement, “If..., then…” statement, hypothesis, conclusion, paragraph proof, flow proof, two-column proof, coordinate proof, direct reasoning, indirect reasoning. Helpful Web Sites: “This lesson focuses on using Venn diagrams to explore direct, indirect, and transitive reasoning.” – from the NCTM Illuminations webpages. http://illuminations.nctm.org/LessonDetail.aspx?id=L384 Indirect Reasoning http://postpositivism.wordpress.com/2006/04/27/an-exercise-inindirect-reasoning/ Quick and easy class activity involving indirect reasoning. http://regentsprep.org/Regents/mathb/1e/indirectteacher.htm Intro to Logic http://people.hofstra.edu/faculty/stefan_Waner/RealWorld/logic/logic4 .html From NCTM – short student sheet on indirect reasoning http://cpscia.k12.ar.us/Curriculum/Math/Math%20912/Geometry/LG/LG.1.G.1/INDIRECT%20REASONING%20%20SHEET%202.htm HSCE: L3.3.3 Explain the difference between a necessary and a sufficient condition within the statement of a theorem; determine the correct conclusions based on interpreting a theorem in which necessary or sufficient conditions in the theorem or hypothesis are satisfied. Clarification statements: Given a statement of a theorem, the student will explain the difference between necessary and sufficient conditions. The student will also identify conditions from given theorems as “necessary”, “sufficient”, “necessary and sufficient”. The student will also draw correct conclusions when presented with a theorem and one or more necessary or sufficient conditions, which have been fulfilled (satisfied). Condition A is a “necessary condition” for condition B to occur as long as if A is false, B is then false. Or, put another way, you can’t have B without also having A. Condition A is a “sufficient condition” for condition B to occur as long as whenever A is true, B must also be true. Or, put another way, if A happens, then B must happen. When working with “If…, then…” statements, the conclusion (phrase following the “then”) is “condition B”, and the hypothesis (phrase following the “if”) is “condition A”. The concepts of necessary and sufficient can be effectively explored in class discussion or small group discussion using a number of everyday language examples and picking them apart. This can be a humorous activity, as students explore the variety of necessary conditions for a given situation. Example 1: If a figure is a right triangle, then the figure has exactly one right angle. A B Condition A: the figure is a right triangle Condition B: the figure has exactly one right angle The condition “the figure has exactly one right angle” is a necessary condition for a “figure [to be] a right triangle”. It’s “necessary” since if a figure did not have a right angle, it certainly would not be a right triangle. Now the question is whether “the figure has exactly one right angle” is a sufficient condition for a “figure [to be] a right triangle”. In other words, if we are told someone has drawn a figure with exactly one right angle, are we certain we can say, “Ah, the figure must be a right triangle”? Of course, the answer is no, we can’t be certain. The person could have drawn a pentagon (or any n-gon, or even a non-polygon) with one right angle. So “the figure has exactly one right angle” is a necessary, but not sufficient condition for a “figure [to be] a right triangle”. Example 2: List the necessary and sufficient conditions for a person using a cell phone to have a conversation with a friend. Sample necessary conditions: The person must be initiating the conversation must be in possession of a cell phone. The cell phone must be in good working order. The batteries must have sufficient charge to supply power for the duration of the conversation. The phone must be within a cell phone service area. The friend must be available to take the call. The person is alive to place the call and for the duration of the conversation. …etc. As far as sufficient conditions…no single ones jump to mind. If an exhaustive listing of necessary conditions could be compiled, they would then constitute a set of sufficient conditions. VOCABULARY: Necessary condition, sufficient condition, theorem, hypothesis, conclusion, to satisfy a condition; necessary and sufficient; necessary but not sufficient; sufficient, but not necessary; neither necessary nor sufficient. Web Sites: A very nice explanation of these two concepts, with examples. http://www.sfu.ca/philosophy/swartz/conditions1.htm http://en.wikipedia.org/wiki/Necessary_and_sufficient_conditions http://www.msu.edu/~susse/Necessary&Sufficient.htm http://faculty.uncfsu.edu/jyoung/necessary_and_sufficient_conditions.htm Nice site for Logic in general. http://philosophy.hku.hk/think/meaning/nsc.php