The Power Point of Proofs

1. Title Slide This project will address the overwhelming educational issues of national, state, and district guidelines. The technology will be Used to promote higher level thinking and make learning appealing, individualized and effective for all students. Title Slide This project will address the overwhelming educational issues of national, state, and district guidelines. The technology will be Used to promote higher level thinking and make learning appealing, individualized and effective for all students.

2. Proof Modules Home Instructional strategy: Scaffolding since there is sufficient support for student success from the beginning and as they learn more information, students work towards being able to create their own proofs. Home Instructional strategy: Scaffolding since there is sufficient support for student success from the beginning and as they learn more information, students work towards being able to create their own proofs.

3. Michigan Merit Curriculum Proof standards Object of this moduleObject of this module

4. Michigan Merit Curriculum continuedL4.2 Language and laws of logic L4.2.1 Know and use the terms of basic logic (e.g., proposition, negation, truth and falsity, implication, if and only if, contrapositive, and converse). L4.2.2 Use the connectives �not,� �and,� �or,� and �if..., then,� in mathematical and everyday settings. Know the truth table of each connective and how to logically negate statements involving these connectives. L4.2.3 Use the quantifiers �there exists� and �all� in mathematical and everyday settings and know how to logically negate statements involving them. L4.2.4 Write the converse, inverse, and contrapositive of an �If..., then...� statement. Use the fact, in mathematical and everyday settings, that the contrapositive is logically equivalent to the original while the inverse and converse are not. Object of this module Object of this module

5. A Proof in Geometry is� A convincing argument that uses mathematical reasoning, logic, definitions, properties and theorems to show why a statement is true. Instructional strategy: Deductive, since the audience is being told exactly what something is. Instructional strategy: Deductive, since the audience is being told exactly what something is.

6. Two-column proof A form of written proof Conclusions are numbered and written in one column Justifications are beside them in a second column. The numbered statements and reasons show the logical order of the argument. Instructional strategy: Deductive, since the audience is being told exactly what something is.Instructional strategy: Deductive, since the audience is being told exactly what something is.

7. Two-column vocabulary Given- information already known Prove- information you are trying to show true Conclusions- are statements deduced from the given information used to provide facts Justification or reason- is a definition, postulate, or theorem which enables a conclusion to be shown true Instructional strategy: Deductive, since the audience is being told exactly what something is.Instructional strategy: Deductive, since the audience is being told exactly what something is.

8. Proof Vocabulary �If�,then� statement structure (conditional) Logical Reasoning Converse Inverse Contrapositive Proof by Contradiction Using Counterexamples to disprove Conditions Necessary Sufficient Instructional strategy: Deductive, since the audience is being told exactly what something is. Instructional strategy: Deductive, since the audience is being told exactly what something is.

9. If then statements or conditionals Hypothesis � The �if� part of an if-then statement. Conclusion � The �then� part of an if-then statement. Instructional strategy: Deductive, since the audience is being told exactly what something is and scaffolding within the examples. Instructional strategy: Deductive, since the audience is being told exactly what something is and scaffolding within the examples.

10. Try it. Below are two conditionals. Click on the correct answer for each. If you are thirsty, then drink water. a. you are thirsty is the consequence b. you are thirsty is the hypothesis c. drink water is the conditional. 2. You will get your driver�s license if you pass the driving test. a. you pass the driving test is the hypothesis b. you will get your driver�s license is the hypothesis c. pass the test is the conditional. Mathetics: sufficient support has been provided for a student to be able to answer this question.Mathetics: sufficient support has been provided for a student to be able to answer this question.

11. Good Work! immediate feedback immediate feedback


13. Sorry. Try again! Immediate feedbackImmediate feedback

14. Proof Vocabulary �If�,then� statement structure (conditional) Logical Reasoning Converse Inverse Contrapositive Proof by Contradiction Using Counterexamples to disprove Conditions Necessary Sufficient Instructional strategy: Deductive, since the audience is being told exactly what something is. Instructional strategy: Deductive, since the audience is being told exactly what something is.

15. Logical Reasoning Converse: The statement formed by interchanging the hypothesis and conclusion of a conditional. Inverse: The statement that results when the hypothesis and conclusion of a conditional statement are both negated. Contrapositive: The statement that results when the hypothesis and conclusion of a conditional statement are both negated, then interchanged. Instructional strategies: Deductive, since the audience is being told exactly what something is and scaffolding within the examples. Instructional strategies: Deductive, since the audience is being told exactly what something is and scaffolding within the examples.

16. Check point If the CD does not cost $15 or less, then I will not buy it. If I will not buy the CD, then it does not cost $15 or less. If I buy the CD then it costs $15 or less. Converse Inverse Contrapositive Mathetics: sufficient support has been provided for a student to be able to answer this question. Mathetics: sufficient support has been provided for a student to be able to answer this question.

17. Great Job! That is correct! Immediate feedback Immediate feedback

18. Great Job! That is correct! Immediate feedback Immediate feedback

19. Incorrect. Try again! Immediate feedback Immediate feedback

20. Your Turn Directions: Click on the Microsoft Word icon to write the converse, inverse, and contrapositive of the given statements and decide whether each is true or false. Submit your document after completing the answers and turn it in through Moodle Any Problems- email me. wrightc@pennfield.net Instructional strategies: Deductive, since the audience is being told exactly what something is and Inductive because students have to make sense of the rules for conditionals then create their own examples based on specific facts. Instructional strategies: Deductive, since the audience is being told exactly what something is and Inductive because students have to make sense of the rules for conditionals then create their own examples based on specific facts.

21. Logical Reasoning Converse: The statement formed by interchanging the hypothesis and conclusion of a conditional. Inverse: The statement that results when the hypothesis and conclusion of a conditional statement are both negated. Contrapositive: The statement that results when the hypothesis and conclusion of a conditional statement are both negated, then interchanged. Instructional strategies: Deductive, since the audience is being told exactly what something is and scaffolding within the examples. Instructional strategies: Deductive, since the audience is being told exactly what something is and scaffolding within the examples.

22. Atomic learning Tutorials Go to the atomic learning web site Your user name is pennfieldstudents Your password is panthers Logon and go to the tutorials section Instructional strategy: Deductive, since the audience is being told exactly what something is.Instructional strategy: Deductive, since the audience is being told exactly what something is.

23. In Atomic Learning Once in the tutorial section, click on the letter �W� Click on the most appropriate link for word Instructional strategy: Deductive, since the audience is being told exactly what something is.Instructional strategy: Deductive, since the audience is being told exactly what something is.

24. Proof Vocabulary �If�,then� statement structure (conditional) Logical Reasoning Converse Inverse Contrapositive Proof by Contradiction Using Counterexamples to disprove Conditions Necessary Sufficient Instructional strategy: Deductive, since the audience is being told exactly what something is. Instructional strategy: Deductive, since the audience is being told exactly what something is.

25. Proof by Contradiction Contradiction- Is an indirect proof in which the statement that you want to prove is assumed to be false; the assumption is shown to lead to a contradiction, which indicates that the original statement must be true. In simple terms, when writing an indirect proof: Assume that the opposite of what you want to prove is true. Use logical reasoning to reach a contradiction of an earlier statement, such as the given information. Then state that the assumption you made was false. State that what you wanted to prove must be true. Instructional strategy: Deductive, since the audience is being told exactly what something is. Instructional strategy: Deductive, since the audience is being told exactly what something is.

26. Proof Vocabulary �If�,then� statement structure (conditional) Logical Reasoning Converse Inverse Contrapositive Proof by Contradiction Using Counterexamples to disprove Theorem Conditions Necessary Sufficient Instructional strategy: Deductive, since the audience is being told exactly what something is. Instructional strategy: Deductive, since the audience is being told exactly what something is.

27. Example of Proof by Contradiction Given: Kelly spent more than $60 on two video games at a store. Prove: At least one game cost more than $30. Step 1: Assume neither game cost more than $30. Step 2: Then Kelly could not have spent $60 at the store. Since this contradicts the given information, the assumption that neither game cost more than $30 must be false. Step 3: Therefore, at least one game cost $30 or more. Instructional strategy: Deductive, since the audience is being told exactly what something is. Instructional strategy: Deductive, since the audience is being told exactly what something is.

28. Counterexamples A counterexample is an example used to show that a given statement is false. Instructional strategy: Deductive, since the audience is being told exactly what something is and scaffolding within the examples. Instructional strategy: Deductive, since the audience is being told exactly what something is and scaffolding within the examples.

29. You are up. Which conditional below does not have a counterexample? If you passed the test, then you studied. If a game is on my computer, then it is fun to play. If I am in Hawaii, then I am in the united states. If the car is running, then the key is in the ignition. Mathetics: sufficient support has been provided for a student to be able to answer this question. Mathetics: sufficient support has been provided for a student to be able to answer this question.

30. Good Work! Immediate feedback Immediate feedback

31. Incorrect. Try again! Immediate feedback Immediate feedback

32. Try Some more Directions: Click on the Microsoft Word icon to provide counterexamples of the given statements. Submit your document after completing the answers and turn it in through Moodle Any Problems- email me. wrightc@pennfield.net Instructional strategies: Deductive, since the audience is being told exactly what something is and Inductive because students had to make sense of the rules for counterexamples then create their own examples. Instructional strategies: Deductive, since the audience is being told exactly what something is and Inductive because students had to make sense of the rules for counterexamples then create their own examples.

33. Proof Vocabulary �If�,then� statement structure (conditional) Logical Reasoning Converse Inverse Contrapositive Proof by Contradiction Using Counterexamples to disprove Conditions (click here to learn more about what a condition is) Necessary Sufficient Instructional strategy: Deductive, since the audience is being told exactly what something is. Instructional strategy: Deductive, since the audience is being told exactly what something is.

34. Conditions Necessary condition- is one that must be satisfied for a statement to be true. Sufficient condition- is one that, if satisfied, guarantees the statement will be true. Instructional strategies: Deductive, since the audience is being told exactly what something is.Instructional strategies: Deductive, since the audience is being told exactly what something is.

35. Necessary or Sufficient Directions: Click on the Microsoft Word icon to state whether a given condition is necessary or sufficient. Submit your document after completing the answers and turn it in through Moodle Any Problems- email me. wrightc@pennfield.net Instructional strategies: Deductive, since the audience is being told exactly what something is and Inductive because students had to make sense of the rules for conditions. Instructional strategies: Deductive, since the audience is being told exactly what something is and Inductive because students had to make sense of the rules for conditions.

36. Proof Vocabulary �If�,then� statement structure (conditional) Logical Reasoning Converse Inverse Contrapositive Proof by Contradiction Using Counterexamples to disprove Conditions (click here to learn more about what a condition is) Necessary Sufficient Instructional strategy: Deductive, since the audience is being told exactly what something is. Instructional strategy: Deductive, since the audience is being told exactly what something is.

37. Test your knowledge Using Microsoft Word, your task is to demonstrate your knowledge by creating a quiz on what you have learned so far in this module for someone else to take. Make it meaningful. Your quiz must have at least ten questions and include an answer key for grading another classmate. Ask a classmate who has also completed this module to take your quiz and comment on the content. You should discuss the quiz completely after grading. After completing the steps above, provide a summary of what you and the other student discussed and how both of you did (strengths and weaknesses.) Turn in all 3 documents; the original quiz with classmates solutions, answer key, and discussion summary. Instructional strategies: Inductive because students have to demonstrate their own knowledge and create a product base on it. Instructional strategies: Inductive because students have to demonstrate their own knowledge and create a product base on it.

38. Resources for Proofs in Geometry Instructional Strategy: Scaffolding since there is sufficient support for student success from the beginning and as they learn more information, students work towards being able to create their own proofs using these resources as a guide. Instructional Strategy: Scaffolding since there is sufficient support for student success from the beginning and as they learn more information, students work towards being able to create their own proofs using these resources as a guide.

39. Toolbox flip chart Instructional Strategy: Deductive, since the audience is being told exactly what something is. Instructional Strategy: Deductive, since the audience is being told exactly what something is.

40. Flip chart Use this tool as a resource for setting up a two-column proof and supplying justifications in proofs. Based on the conclusion you are trying to show, choose a section that deals with the same figure or content you are working with. Instructional strategy: Deductive, since the audience is being told exactly what something is. Instructional strategy: Deductive, since the audience is being told exactly what something is.

41. Toolbox flip chart Instructional strategy: Deductive, since the audience is being told exactly what something is. Instructional strategy: Deductive, since the audience is being told exactly what something is.

42. How to Write a Proof List the given information first. Use the information from the diagram if one is provided Give a reason for every statement Use given information, definitions, postulates and theorems as reasons. List statements in order. If a statement relies on another statement, list it later than the statement it relies on . End the proof with the statement you are trying to prove. Instructional strategy: Deductive, since the audience is being told exactly what something is. Instructional strategy: Deductive, since the audience is being told exactly what something is.

43. Inside the toolbox Instructional strategies: Deductive, since the audience is being told exactly what something is and Inductive because students have to choose which section of the toolbox to use in order to find the correct information. Instructional strategies: Deductive, since the audience is being told exactly what something is and Inductive because students have to choose which section of the toolbox to use in order to find the correct information.

44. Properties from Algebra Postulates of equality or congruence for any real number a, b, and c. Reflexive: a = a Symmetric: If a = b, then b = a Transitive: If a = b & b = c, then a = c. Addition Property of Equality; If a = b, then a + c = b + c. Multiplication Property of Equality; If a = b, then ac = bc. Substitution Property; If a = b, then a may be substituted in for b in an expression. Instructional strategies: Deductive, since the audience is being told exactly what something is.Instructional strategies: Deductive, since the audience is being told exactly what something is.

45. Equal Segments Definition of midpoint Definition of Equilateral triangle Definition of Isosceles triangle Reflexive Definition of a circle Definition of perpendicular bisector CPCF theorem Definition of congruence Transitive Segment congruence theorem Definition of median In an isosceles triangle, bisector of vertex angle is perpendicular bisector o the base Definition of regular polygon Isosceles triangle base angles converse theorem Rhombus diagonal theorem Isosceles trapezoid theorem Equiangular triangle is equilateral Instructional strategies: Deductive, since the audience is being told exactly what something is. Instructional strategies: Deductive, since the audience is being told exactly what something is.

46. Equal Angles Definition of angle bisector Vertical angles theorem Corresponding angles = Alternate interior angles = Alternate exterior angles = CPCF Definition of perpendicular Transitive Angle Congruent Theorem Definition of congruence Base angle of an isosceles triangle = Isosceles triangle vertex angle bisector is perpendicular to base Equilateral triangle is equiangular Base angles of an isosceles trapezoid Definition of regular polygon Isosceles triangle base angles theorem Rhombus diagonal theorem Definition of Isosceles trapezoid Instructional strategies: Deductive, since the audience is being told exactly what something is. Instructional strategies: Deductive, since the audience is being told exactly what something is.

47. Congruent Triangles SSS- If three sides of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent. SAS- (two sides and the angle they form) If two sides and an included angle of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent ASA- If two angles and an included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent. AAS- If, two angles and a non-included side of one are congruent respectively to two angles and the corresponding non-included side of the other, then the triangles are congruent. HL- for right triangles only: If in two right triangles, the hypotenuse and a leg of one are congruent to the hypotenuse and a leg of the other, then the two triangles are congruent. Instructional strategies: Deductive, since the audience is being told exactly what something is. Instructional strategies: Deductive, since the audience is being told exactly what something is.

48. Included vs. non-included 1. Which of the diagrams below has a congruent pair of triangles marked with an included angle? Inductive. The idea of included is used in the congruent triangles definitions, but included is never defined itself. By looking at these two questions and clicking on answers, students will be able to investigate what included means. Inductive. The idea of included is used in the congruent triangles definitions, but included is never defined itself. By looking at these two questions and clicking on answers, students will be able to investigate what included means.




52. Miscellaneous Parallel lines Corresponding angles postulate Alternate interior angles Alternate exterior angles If two lines are perpendicular to the same line, then the two original lines are parallel Instructional strategies: Deductive, since the audience is being told exactly what something is. Instructional strategies: Deductive, since the audience is being told exactly what something is.

53. Geometers Sketchpad Instructional strategies: Deductive, since the audience is being told exactly what something is or resources they can use, and Inductive because students are required to make choices and decide exactly what they need to do in order to fit their own needs. Not every student will need every resource every time they do a proof. Instructional strategies: Deductive, since the audience is being told exactly what something is or resources they can use, and Inductive because students are required to make choices and decide exactly what they need to do in order to fit their own needs. Not every student will need every resource every time they do a proof.

54. Classmates Peers can often answer questions that you may have. Compare your work with others. Ask what you missed if you are absent. Having others read your arguments can often improve your skills. Take pod quizzes together Keep each other on task Instructional strategies: Deductive, since the audience is being told exactly what something is or resources they can use, and Inductive because students are required to make choices and decide exactly what they need to do in order to fit their own needs. Not every student will need every resource every time they do a proof. Instructional strategies: Deductive, since the audience is being told exactly what something is or resources they can use, and Inductive because students are required to make choices and decide exactly what they need to do in order to fit their own needs. Not every student will need every resource every time they do a proof.

55. Textbook List of theorems and postulates in back of book Selected answers for odd problems Examples Extra practice Website Instructional strategies: Deductive, since the audience is being told exactly what something is or resources they can use, and Inductive because students are required to make choices and decide exactly what they need to do in order to fit their own needs. Not every student will need every resource every time they do a proof. Instructional strategies: Deductive, since the audience is being told exactly what something is or resources they can use, and Inductive because students are required to make choices and decide exactly what they need to do in order to fit their own needs. Not every student will need every resource every time they do a proof.

56. Internet Research information Learn more about specific topics Find examples Search engines such as Google can be helpful along with informative pages such as Wikipedia. Instructional strategies: Deductive, since the audience is being told exactly what something is or resources they can use, and Inductive because students are required to make choices and decide exactly what they need to do in order to fit their own needs. Not every student will need every resource every time they do a proof. Instructional strategies: Deductive, since the audience is being told exactly what something is or resources they can use, and Inductive because students are required to make choices and decide exactly what they need to do in order to fit their own needs. Not every student will need every resource every time they do a proof.

57. Mrs. Wright IN CLASS? � go on to the next screen NOT IN CLASS?- come in during seminar or before or after school check out Mrs. Wright�s website e-mail Mrs. Wright wrightc@pennfield.net call Mrs. Wright (269) 441-9821 Instructional strategies: Deductive, since the audience is being told exactly what something is or resources they can use, and Inductive because students are required to make choices and decide exactly what they need to do in order to fit their own needs. Not every student will need every resource every time they do a proof. Instructional strategies: Deductive, since the audience is being told exactly what something is or resources they can use, and Inductive because students are required to make choices and decide exactly what they need to do in order to fit their own needs. Not every student will need every resource every time they do a proof.

58. Help from the teacher Please raise your hand and quietly wait for some assistance. Instructional strategies: Deductive, since the audience is being told exactly what to do. Instructional strategies: Deductive, since the audience is being told exactly what to do.

59. Big Picture The proofs you will write in this course are mostly of statements that appear to be true, but need justifications. As you learn more, you will be able to generate your own conjectures then prove that they are true. Instructional strategy: Deductive, since the audience is being told exactly what something is. Instructional strategy: Deductive, since the audience is being told exactly what something is.

60. Where do you use reasoning? Debate To make a point To make an informed decision To rebuttal someone else�s point of view Think about the last time you knew you were right about something and someone tried to argue with you. Did you or could you have found proof that you were in fact right? Instructional strategy: Deductive, since the audience is being told exactly what something is. Instructional strategy: Deductive, since the audience is being told exactly what something is.

61. Why Proofs? Mathematical Reasoning & structure Logic Learn how to think about the validity of someone else�s comments Method used to judge any argument What is obvious to one may not be obvious to another To verify unexpected results Instructional strategy: Abductive since reasons for analyzing things and studying or applying proofs even matters. Instructional strategy: Abductive since reasons for analyzing things and studying or applying proofs even matters.

62. Try it! Since you are in Geometry, And all students must pass Algebra to take Geometry then, You liked Algebra You passed Algebra You took Algebra Mathetics: sufficient support has been provided for a student to be able to answer this question. Mathetics: sufficient support has been provided for a student to be able to answer this question.



65. Validity Example Two friends, Mary and Karla are talking about dogs Karla thinks that all dogs are friendly but Mary disagrees due to experience. How could Mary prove that she is right? Introduce Karla to her own dog. Show Karla evidence using valid websites, articles or documented incidents. Take Karla to a dog show. Mathetics: sufficient support has been provided for a student to be able to answer this question. Mathetics: sufficient support has been provided for a student to be able to answer this question.



68. Practice

69. One Step proof Practice One-step proof: A proof with an argument that has only one justified conclusion. Mathetics: sufficient support has been provided for a student to be able to answer this question.Mathetics: sufficient support has been provided for a student to be able to answer this question.



72. Try another one Mathetics: sufficient support has been provided for a student to be able to answer this question. Mathetics: sufficient support has been provided for a student to be able to answer this question.



75. Two-column proof Practice Given: angle 7 and angle 8 are right angles. Prove: angle 7 is congruent to angle 8 Mathetics: sufficient support has been provided for a student to be able to answer this question.Mathetics: sufficient support has been provided for a student to be able to answer this question.



78. Try another proof Given: B is the midpoint of AC; AB=EF Prove: BC=EF Mathetics: sufficient support has been provided for a student to be able to answer this question. Mathetics: sufficient support has been provided for a student to be able to answer this question.



81. Logic application Logic will help you sharpen your critical thinking and problem solving skills. While these skills are an essential part of mathematics, they are also relevant to many areas of real life. Try a crossword puzzle and Sudoku. They require you to logically think through the object of the activity in order to complete the task.

82. SUDOKU Each Sudoku has a unique solution that can be reached logically without guessing. Enter digits from 1 to 9 into the blank spaces. Every row must contain one of each digit. So must every column, as must every 3x3 square. Instructional strategy: Problem-solving because I am having students investigate and try Sudoku which is a meaning task with centers on overcoming the limiting conditions of the puzzles.Instructional strategy: Problem-solving because I am having students investigate and try Sudoku which is a meaning task with centers on overcoming the limiting conditions of the puzzles.

83. Crossword logic Instructional strategies: Mathetics since students have to investigate what words could actually go where, but I gave them a word bank and puzzle already created. However the student needs to build on existing knowledge in order to answer the questions. Instructional strategies: Mathetics since students have to investigate what words could actually go where, but I gave them a word bank and puzzle already created. However the student needs to build on existing knowledge in order to answer the questions.

84. Proof module graduate Choose a task from the list below. Make a pamphlet Do a presentation about what you learned and how you will apply it in your everyday life. Explain and model an activity that uses mathematical reasoning Teach a proof lesson Instructional strategy: Inductive since students have to provide a summary of what they learned in one of the given ways. Instructional strategy: Inductive since students have to provide a summary of what they learned in one of the given ways.

85. Mrs. Wright IN CLASS? � go on to the next screen NOT IN CLASS?- come in during seminar or before or after school to discuss your idea. e-mail Mrs. Wright wrightc@pennfield.net Instructional strategies: Deductive, since the audience is being told exactly what something is or resources they can use, and Inductive because students are required to make choices and decide exactly what they need to do in order to fit their own needs. Not every student will need every resource every time they do a proof. Instructional strategies: Deductive, since the audience is being told exactly what something is or resources they can use, and Inductive because students are required to make choices and decide exactly what they need to do in order to fit their own needs. Not every student will need every resource every time they do a proof.

86. Ask the teacher Please raise your hand and quietly wait for some assistance. Instructional strategies: Deductive, since the audience is being told exactly what to do. Instructional strategies: Deductive, since the audience is being told exactly what to do.

87. Make a pamphlet Use Microsoft publisher to create a pamphlet that will inform future Geometry students about proofs. Include things like vocabulary, set-up, valid justifications, diagrams, and more. Mathetic: Students will build on their existing knowledge to complete this task.Mathetic: Students will build on their existing knowledge to complete this task.

88. Presentation of application Design a presentation using Microsoft PowerPoint based on how you will use �proofs� in your everyday life. Include things that you have learned by completing these modules and how logical reasoning or proof will apply to your world outside of math class. Mathetic: Students will build on their existing knowledge to complete this task. Mathetic: Students will build on their existing knowledge to complete this task.

89. Explain and model an activity that uses mathematical reasoning Use one of the activities listed or choose one of your own. If you choose your own, seek approval. Mathetic: Students will build on their existing knowledge to complete this task. Mathetic: Students will build on their existing knowledge to complete this task.

90. Activities with mathematical reasoning CLUE SUDOKO Crossword puzzles Battleship Guess Who Mathetic: Students will build on their existing knowledge to complete this task. Mathetic: Students will build on their existing knowledge to complete this task.

91. Your Task Explain the object of a game Set up the teams/players if necessary Show how to play the game List strategies for winning How does the game apply to proofs?

92. Teach a lesson UCSMP 2-2 2-3 2-8 3-4 3-5 5-3 5-4 5-5 7-3 11-1 11-2 11-3 11-4 Mathetic: Students will build on their existing knowledge to complete this task. Mathetic: Students will build on their existing knowledge to complete this task.

The Power Point of Proofs

The Power Point of Proofs

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