Flow Chart Proofs Essential Questions, Exercises of Reasoning

Download Flow Chart Proofs Essential Questions and more Exercises Reasoning in PDF only on Docsity! Different Methods of Proof Lesson Plan – Flow Chart Proofs Essential Questions How can deductive reasoning be used to validate conjectures? What methods can be used to organize a deductive argument? Warm-Up/Opening Activity Write the directions for making a sandwich in a flow chart. Teacher Note: You might want to have the ingredients to make a sandwich available in class. Development of Ideas Arrange in order the steps to solve an algebraic equation. Teacher Note: Have the students work in groups for problems 5, 6, and 7. Each group of students will need an envelope with the statements and reasons for each problem cut into strips and a copy of the flowchart template for each problem. The master copies follow the worksheet. Justify the steps to solve the algebraic equation in a flow chart format. Worksheet: Introduction to Flow Chart Proofs Answers: 1. c. 3x-15 = 150 ! 3x = 165 ! x = 55 Given Addition Division Equation Property Property of Equality of Equality d.-e. The first box is the ‘if’ statement and the last box is the ‘then’ statement. 2. a. Given ! Multiplication Property of Equality ! Division Property of Equality b.-c. The first box is the ‘if’ statement and the last box is the ‘then’ statement. 3. 3x+28= 58 ! 3x = 30 ! x = 10 Given Subtraction Division Equation Property Property of Equality of Equality HSA Geometry Activities Activity 5 Page 90 Page 10 Different Methods of Proof Development of Ideas (Continued) Answers to Introduction to Flow Chart Proofs (Continued) 4. 5x-12=x-32 ! 4x-12=-32 ! 4x=-20 ! x =-5 Given Subtraction Addition Division Equation Property Property Property of Equality of Equality of Equality 5. a. Given: 1 ∠ and 2 ∠ are supplementary 32 ∠≅∠ Prove: 31 ∠+∠ = 180º b. ∠1 and ∠2 are supplementary m∠1 + m∠2 = 180° ∠2 ≅ ∠3 m∠2 = m∠3 m∠1 + m∠3 = 180° Given Definition of supplementary angles Given Definition of congruent angles Substitution property of equality c.-d. The first box is the ‘if’ statement and the last box is the ‘then’ statement. 6. a. Given: m 1 ∠ = m 2 ∠ Prove: m AEC ∠ = m BED ∠ b. m∠1 = m∠2 m∠1 + m∠3 = m∠2 + m∠3 m∠AEC = m∠1 + m∠3 m∠BED = m∠2 + m∠3 m∠AEC = m∠BED Given Addition property of equality Angle addition postulate Substitution property of equality 7. a. Given: PR and QS bisect each other at T Prove: PQT RST∆ ≅ ∆ PT TR≅ Definition of bisector PR and QS bisect each other at T QT TS≅ PQT RST∆ ≅ ∆ Given Definition of bisector Side-Angle-Side Congruence PTQ RTS∠ ≅ ∠ Vertical angles are congruent 8. a. Given: PR and QS bisect each other at T Prove: P R∠ ≅ ∠ c. Reasons: Definition of Bisector Given Definition of Bisector Side-angle-side Triangle Congruency Definition of Vertical Angles HSA Geometry Activities Activity 5 Page 91 Page 11 Different Methods of Proof Closure Describe the advantages and disadvantages of writing instructions for a task in a flow chart. Answer: An advantage of using flow charts is to be able to show different directions and logic pathways within the sequence of directions. A disadvantage is that the pathways can be confusing and difficult to see at first. Describe how deductive reasoning is used in flow charts. Answer: Flow charts show how deductive reasoning is developed by using the given statements, definitions, and theorems to demonstrate proofs and showing how the connections are made. HSA Geometry Activities Activity 5 Page 94 Page 14 Different Methods of Proof Introduction to Flow Chart Proofs A flow chart proof is a concept map that shows the statements and reasons needed for a proof in a structure that helps to indicate the logical order. Statements, written in the logical order, are placed in the boxes. The reason for each statement is placed under that box. 1. a. Cut out the individual boxes of statements and reasons at the bottom of the page. b. Arrange the statements and reasons to prove the following conditional: If 3x − 15 = 150 then x = 55. c. Copy the statements and reasons in the proper order on the flowchart displayed below. Place the statements in the boxes and the reasons on the lines below the boxes. d. What is the statement in the first box? How does it relate to the conditional? e. What is the statement in the last box? How does it relate to the conditional? HSA Geometry Activities Activity 5 Page 95 Page 15 Start End 3x − 15 = 150 Addition Property of Equality 3x = 165 Given equationDivision Property of Equality x = 55 Cut out: Different Methods of Proof Introduction to Flow Chart Proofs (Continued) 2. Prove the following conditional: If 4 24 7 =x , then x = 42. a. The statements are already entered into the flowchart. Write the correct reasons below each box. b. What is the c. What is the 3. Prove the follo Write the corre 4. Given the cond Write the corre HSA Geometry Activi Page 96 Start Start Start statement in the first box? How does it relate to the conditional? statement in the last box? How does it relate to the conditional? wing conditional: If 3x + 28 = 58, then x = 10. ct statements and reasons in the flowchart to prove the conditional above. itional: If 5x − 12 = x − 32, then x = -5. ct statements and reasons in the flowchart to prove the conditional above. ties Activity 5 Page 16 End4 24 7 =x x = 424x = 168 End End4x − 12 = −32 Addition Property Different Methods of Proof Introduction to Flow Chart Proofs (Continued) 9. If ∠ A and ∠ B are complementary and ∠ B and ∠ C are complimentary, then ∠ A ≅ ∠ C. a. Draw a diagram for this conditional. b. State the given and prove for this conditional in terms of the diagram. Given: Prove: c. Fill in the missing reasons in the flowchart below. HSA Geometry Activities Activity 5 Page 99 Page 19 Given Start m∠A+ m∠B = 90° Definition of complementary angles m∠B+ m∠C = 90° m∠A + m∠B = m∠B + m∠C m∠A = m∠C End Transitive Property of Equality Different Methods of Proof Introduction to Flow Chart Proofs (Continued) Statements and Reasons for problem 5 flowchart proof ∠1 and ∠2 are supplementary Given m∠1 + m∠3 = 180° Definition of congruent angles m∠1 + m∠2 = 180° Substitution property of equality ∠2 ≅ ∠3 Definition of supplementary angles m∠2 = m∠3 Given Statements and Reasons for problem 6 flowchart proof m∠1 = m∠2 Angle addition postulate m∠1 + m∠3 = m∠2 + m∠3 Substitution property of equality m∠AEC = m∠1 + m∠3 m∠BED = m∠2 + m∠3 Given m∠AEC = m∠BED Addition property of equality Statements and Reasons for problem 7 flowchart proof PR and QS bisect each other at T Given PT TR≅ Vertical angles are congruent QT TS≅ Definition of bisector PTQ RTS∠ ≅ ∠ Side-Angle-Side Congruence PQT RST∆ ≅ ∆ Definition of bisector HSA Geometry Activities Activity 5 Page 100 Page 20 Different Methods of Proof Introduction to Flow Chart Proofs (Continued) HSA Geometry Activities Activity 5 Page 101 Page 21 Fl ow ch ar t f or p ro bl em 5 Fl ow ch ar t f or p ro bl em 6