Lesson 1: Getting Ready for Proof



Unit Name: Unit 5: Similarity, Right Triangle Trigonometry, and ProofLesson Plan Number & Title: Lesson 1: Getting Ready for ProofGrade Level: High School Math IILesson Overview:Students will develop the foundations of logical reasoning as a prelude to validating their conjectures using rigorous methods of proof. This lesson is designed for 45 – 90 minutes depending on the sophistication of the understanding of the students. Students have considered the idea of a converse of a statement in Math 8.Focus/Driving Question:How can the basic tools of symbolic logic including the converse, inverse, and contrapositive of a conditional statement be utilized in the construction of logical arguments to validate conjectures?West Virginia College- and Career-Readiness Standards:M.2HS.42Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Implementation may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.2HS.C.3. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning.Manage the Lesson:This lesson is the first of two lessons that will provide the scaffolding for the understanding of formal proof. Up to this point, the work of geometry has been primarily one of conjecture based upon investigations. Methods of rigorous proof will be addressed later in the unit in the context of specific theorems. A variety of approaches to understanding these theorems will afford accessibility for all students, with increased demand placed upon the STEM student who will be progressing to Math III STEM.Academic Vocabulary Development:Vocabulary addressed in this lesson will include: conditionalcontrapositiveconverseinverseThese terms will be added to the glossary and/or foldable developed in Math I. A variety of opportunities will be provided in the lesson to increase the student’s understanding of these new terms.Launch/Introduction:Ask students how many of them have been involved in a sport? Remind them that every game is played by rules. Show 1.01 Rules for Sports (examples of rules for various sports), and ask students to identify the sport. [Answers: (1) football; (2) basketball; (3) soccer; (4) baseball; (5) volleyball; (6) hockey.] Ask students to determine what these statements of rules have in common? (Each statement tells what happens in the game if something else has happened. Each statement contains the word “if”.) These are called conditional statements.Show the Morton Salt advertisement 1.02 Conditional Statements in Advertisements 1. Use this example to define a conditional statement (a statement consisting of two clauses that can be written in if-then form with the “if” clause the hypothesis and the “then” clause the conclusion.) Have students identify the hypothesis and the conclusion for this statement. Then, if needed, show 1.03 Conditional Statements in Advertisements 2 and write each advertisement in if-then form and again identify the hypothesis and the conclusion. Consider a geometric example and how a declarative statement can be written in “if-then” form. For example: The diagonals of a rectangle are congruent. If a figure is a rectangle, then its diagonals are congruent. Prompt students that as we are preparing to justify our thinking more formally, our conjectures can be written as conditional statements. It will be important to be able to determine when a conditional statement is true or false. Although an in-depth study of symbolic logic is unrealistic given time constraints, students can possibly take a look at the truth values for the hypothesis and the conclusion of a given conditional. The following example may prove helpful: “If (student’s name from the class) gets all A’s, then his dad will buy him/her a car.” Refer to 1.04 When a Conditional Statement is False for additional teaching comments. Ensure that students understand that a conditional is false only when the hypothesis is true and the conclusion is false. Utilizing truth tables can be helpful in developing students’ understanding of the logical equivalence of a conditional and its contrapositive which will prove useful as the idea of more formal proof is developed. Both proof by contrapositive and the similar concept of proof by contradiction are used throughout the unit. This lesson provides a foundation for deepening the understanding of these methods of proof.Investigate/Explore:Introduce the converse. Ask students if anyone knows who is pictured on a US $100 bill. Consider the following statement: “If something is a US $100 bill, then it has a picture of Benjamin Franklin on it.” Ask if the following statement says the same thing? “If something has a picture of Benjamin Franklin on it, then it is a US $100 bill.” Show a 1.05 Picture of a $100 bill and Picture of Benjamin Franklin. The first is true while the second is false. The second statement is called the converse of the first: If q, then p. The converse is not necessarily true if the conditional is true. The converse of a conditional then switches the hypothesis and the conclusion. Have students write in conditional form Benjamin Franklin’s adage: “Early to bed and early to rise makes a man healthy, wealthy, and wise.” Then they should write the converse of this conditional. If Franklin’s adage is true, discuss if its converse is also true. Next, discuss the negation of a statement. Introduce the inverse and the contrapositive of a conditional statement. The inverse of a conditional negates the hypothesis and the conclusion. (For inclusion students, it may be helpful for them to note that to create an INverse, they will need to INsert the word NOT into both portions of the statement.) Remind students that since they are negating each part of the statement, they may use other words, in addition to NOT, to create the negation. Point out that the inverse, like the converse, does NOT necessarily have the same truth value as the original conditional. The contrapositive is formed by negating both the hypothesis and the conclusions AND switching the two clauses. (For inclusion students, it may be helpful to note that the contrapositive combines the strategies for the converse and the inverse.) An example 1.06 PowerPoint is provided that can be used to introduce or review the converse, inverse, and contrapositive. It also includes an activity in which students work in teams, with each member of the team writing a different form of the original conditional.Summarize/Debrief:As a culminating activity, read or use an example 1.07 PowerPoint of the Laura Joffe Numeroff book If You Give a Mouse a Cookie. Students are asked to create six frames of a 1.08 Conditional Comic following the same format of the book. This activity could be utilized as a homework assignment. The following day a follow-up activity asks students to write converses, inverses, and contrapositives based on their own comics.(Various forms of the activity can also be found on numerous websites including and . The second site also includes a rubric for evaluating the project.)Materials:1.01 Rules for Sports 1.02 Conditional Statements in Advertisements 11.03 Conditional Statements in Advertisements 21.04 When a Conditional Statement is False 1.05 Picture of a $100 bill and Picture of Benjamin Franklin1.06 PowerPoint on Converses, Inverses, and Contrapositives1.07 PowerPoint of If You Give a Mouse a Cookie1.08 Conditional Comic1.09 Teacher Reflection SheetCareer Connection:Although logical reasoning skills are advantageous in most professions, they are particularly needed in a variety of STEM careers from computer science to engineering. Additionally, they would be an asset to careers in the Health Sciences.Lesson Reflection:As an exit slip, students can create a converse-inverse-contrapositive poem by writing an “if-then” statement followed by its converse, inverse and contrapositive. Additional stanzas might include related “if-then” statements. Teachers are provided with a 1.09 Teacher Reflection Sheet to guide their thinking as they seek to improve their practice. Certainly, it may not be feasible to formally complete such a reflection after every lesson, but hopefully the questions can generate some ideas for contemplation.1.01 Rules for SportsHere are some examples of rules for various sports. Which sports can you identify?1.If an opponent touches the ball, all the passing side is eligible to receive the ball. 2.A game is forfeited if a team does not have at least two players on the court. 3.If the kicker breaks the law, the defending side is awarded an indirect free kick.4.A runner is out if he fails to beat a throw to the base to which he is running. 5.If the game is tied at 14/14, it continues until one team has a two-point lead. 6.A player is offside if he is within his opponents’ half when the ball is played by a team mate. 1.02 Conditional Statements in Advertisements 1Morton Salt’s trademark shown here in 1956 and today continues to be well-known. What is the conditional statement? Identify the hypothesis and the conclusion.1.03 Conditional Statements in Advertisements 2Write the conditional statement for each advertisement.1231.04 When a Conditional Statement is FalseIf Jason gets all A’s, then his dad will buy him a car.Hypothesis: Jason gets all A’s.Conclusion: His dad will buy him a car.The hypothesis may be true or false, and the conclusion may be true or false, so how many possibilities do we need to consider?Case 1: True Hypothesis – True ConclusionCase 2: True Hypothesis – False ConclusionCase 3: False Hypothesis – True ConclusionCase 4: False Hypothesis – False ConclusionIn case 1 Jason gets all A’s and his dad fulfills his promise and buys him a car. It certainly makes sense that if the hypothesis is true, the conclusion is true. But what happens when the hypothesis is false? In case 3, if Jason doesn’t get all A’s, his dad may say that he worked really hard and still deserves the car. In case 4, if Jason doesn’t get all A’s, his dad doesn’t buy him the car (Jason didn’t fulfill his part of the bargain, so he doesn’t receive the car.) In both instances, the promise is not put to the test, so the conditional is true. However, in case 2, Jason does get all A’s, but his dad doesn’t follow through with purchasing the car. In other words, only in case 2 is the promise broken. Hence, it is the only case in which the conditional is false. In mathematics we represent this in the form of what is called a truth table. Usually the hypothesis is represented by the letter p, and the conclusion by the letter q. Since the statement is an if-then statement, an arrow is used to show that p implies q. pqp → qTTTTFFFTTFFT1.05 Picture of a $100 bill and Picture of Benjamin Franklin1.06 PowerPoint1.07 PowerPoint1.08 Conditional Comic1.09 Teacher Reflection SheetTeacher Reflection SheetAfter teaching a lesson, use the following questions to make future instructional decisions. 1. As I reflect on the lesson, to what extent were the students productively engaged in the work? What supports that conclusion?2. Did the lesson allow for students to engage in investigations/learning situations that were consistent with the content standard(s) addressed in the lesson?3. What feedback did I receive from students that indicates their understanding of the goals for this lesson?4. If I teach this lesson again, what will I do differently? ................
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