unit 3 shadow of a doubt-logic+proof · pdf filewith the converse, inverse, and contrapositive...

Geometry Unit 3 Beyond a Shadow of a Doubt: Logic and Proof

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QCG3.2.0

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Note QualityCore™ instructional units illustrate how the rigorous, empirically

researched course objectives can be incorporated into the classroom. For more information about how the instructional units fit into the QualityCore program, please see the Educator’s Guide included with the other QualityCore materials.

ACT recognizes that, as you determine how best to serve your students, you will take into consideration your teaching style as well as the academic needs of your students; the standards and policies set by your state, district, and school; and the curricular materials and resources that are available to you.

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C o n t e n t s

Unit 3 Beyond a Shadow of Doubt: Logic and Proof Purpose............................................................................................................ vi Overview ......................................................................................................... vi Time Frame ..................................................................................................... vi Prerequisites ..................................................................................................... 1 Selected Course Objectives .............................................................................. 1 Research-Based Strategies ............................................................................... 2 Essential Questions .......................................................................................... 2 Suggestions for Assessment ............................................................................. 2

Preassessment ............................................................................................ 2 Embedded Assessments............................................................................. 2 Unit Assessment ........................................................................................ 3

Unit Description ............................................................................................... 3 Introduction................................................................................................ 3 Suggested Teaching Strategies/Procedures................................................ 4

Enhancing Student Learning Selected Course Objectives ..................................................................... 24 Unit Extension ......................................................................................... 24 Reteaching ............................................................................................... 24

Bibliography................................................................................................... 26

Appendix A: Record Keeping ......................................................................A-1 Appendix B: Days 1–2 ................................................................................. B-1 Appendix C: Day 3....................................................................................... C-1 Appendix D: Day 4 ......................................................................................D-1 Appendix E: Day 5....................................................................................... E-1 Appendix F: Day 6 ........................................................................................F-1 Appendix G: Day 7 ......................................................................................G-1 Appendix H: Day 8 ......................................................................................H-1 Appendix I: Day 9 ......................................................................................... I-1 Appendix J: Day 10....................................................................................... J-1 Appendix K: Enhancing Student Learning...................................................K-1 Appendix L: Secondary Course Objectives ................................................. L-1 Appendix M: Course Objectives Measured by Assessments ...................... M-1

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P u r p o s e , O v e r v i e w , a n d T i m e F r a m e

Purpose This unit introduces students to logic, the science of reasoning. It begins

with an exploration of conditional and biconditional statements that leads to an introduction of inductive and deductive reasoning. Finally, students begin to learn formal proof techniques.

Overview In a rigorous Geometry course, students focus on producing and

presenting logical arguments to explain and justify conjectures about geometric figures and real-world experiences. This unit is designed to give students opportunities to investigate different proof methods as they work independently and in small groups and independently. They will begin to understand logic and proof by investigating statements made by advertisers. They will then use formal proof techniques to justify conjectures. Assessment will often be informal, although as a final activity students will work in pairs to answer culminating questions about logic and proof.

Time Frame This unit requires approximately ten 45–50 minute class periods.

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UNIT 3 BEYOND A SHADOW OF A DOUBT: LOGIC AND PROOF

Prerequisites Apply algebraic properties to simplify algebraic expressions Solve equations with one variable Identify angle pairs and use them to solve problems

Selected Course Objectives The primary objectives, which represent the central focus of this unit, are

listed below and highlight skills useful not only in Geometry, but in other disciplines as well. Secondary objectives are listed in Appendix L.

B.1. Process Objectives a. Apply problem-solving skills (e.g., identifying irrelevant or

missing information, making conjectures, extracting mathematical meaning, recognizing and performing multiple steps when needed, verifying results in the context of the problem) to the solution of real-world problems

d. Use the language of mathematics to communicate increasingly complex ideas orally and in writing, using symbols and notations correctly

f. Make mathematical connections among concepts, across disciplines, and in everyday experiences

h. Apply previously learned algebraic concepts in geometric contexts

C.1. Logic and Proof a. Use definitions, basic postulates, and theorems about points,

segments, lines, angles, and planes to write proofs and to solve problems

b. Use inductive reasoning to make conjectures and deductive reasoning to arrive at valid conclusions

Logic is the anatomy of thought. —John Locke (1632–1704)

Logic is the technique by which we add conviction to truth. —Jean de la Bruyère (1645–1696)

What’s the most difficult aspect of your life as a mathematician, Diane Maclagan, an assistant professor at Rutgers, was asked. “Trying to prove theorems,” she said. And the most fun? “Trying to prove theorems.” —Fran Schumer (2005, p. 14)

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c. Identify and write conditional and biconditional statements along with the converse, inverse, and contrapositive of a conditional statement; use these statements to form conclusions

e. Read and write different types and formats of proofs including two-column, flowchart, paragraph, and indirect proofs

Research-Based Strategies Hand Signals (pp. 5, 7) Index Cards (p. 11) Designing Exam Questions (p. 16) Visual Representation: Concept Maps (p. 19)

Essential Questions 1. How do advertisers use conditional statements? 2. What evidence can be used to validate a

mathematical conclusion? 3. What makes an argument logical? 4. What is the best format for a proof? 5. What qualities/components should a good proof

possess?

Suggestions for Assessment Except where otherwise noted, assessments can be given a point value or

they can simply be marked complete.

Preassessment Discussion—Ask students to share an experience in which they have

asked someone to “Prove it.” Determine students’ prior knowledge of logic by using questioning techniques that will reveal their understanding. (Introduction)

Embedded Assessments Rubric—By circulating through the room and listening, analyze group

discussions with the Group Work Scoring Rubric (p. B-4) to determine whether each student understands the mathematical content, uses the language of mathematics to communicate, participates in the activity, and interacts with the other group members. (Days 1–3, 5, 8, 10)

Homework—The homework is an extension of the worksheets (Conditional Statements, pp. B-7–B-10) completed in class on Days 1–2. (Day 2)

Homework—Have students record five examples of conditional statements suggested by different advertisements that they see on television, hear on the radio, or read in print (Advertising Statements, p. C-2). Then, have students create their own ads to “sell” mathematics. (Day 3)

Proof—Ask students to solve a linear equation and justify each step in a two-column proof. (Day 4)

Activity—Have students either cut apart the statements and reasons on the Jumbled Proof Steps activity, (pp. E-3–E-5) and reassemble them in the correct order to complete a two-column proof. (Day 5)

Homework—Have students answer questions based on the Triangle Angle Sum Theorem activity. (Day 6)

Tips for Teachers

The essential questions and the primary course objectives for this unit should be prominently displayed in the classroom.

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Writing—Students respond to Essential Questions 2 and 3 in their math journals. (Day 7)

Homework—Students complete a proof of the Congruent Complements Theorem in two-column, flowchart, and paragraph form. (Day 8)

Homework—Students complete an indirect proof. (Day 9)

Unit Assessment Quiz—Students work in pairs to answer the questions (Assessing Your

Understanding, pp. J-2–J-4). Use the Group Work Scoring Rubric to assess each student’s communication and conceptual understanding for each problem. For Problem 6, use the rubric to assess problem solving. (Day 10)

Unit Description

Introduction

Materials & Resources Unit Assignments and Assessments (pp. A-2–A-3) Grouping Statements (one copy, cut apart) (p. B-2) Best Pizza Deal transparency (p. B-3)

Proof is a difficult concept for many students to grasp. Throughout the unit, use several methods to reinforce the idea that a proof is like a chain that links, by a series of logical steps, the given information to what is to be proved. Many students require exposure to different types of proofs before grasping the concept. Therefore, proofs are addressed throughout the course, giving students many opportunities to make the connections.

Before the lesson begins, fill out according to the example (p. A-2) the Unit Assignments and Assessments record keeping table (p. A-3) and make copies for students. Then, cut apart the statements from the Grouping Statements activity (p. B-2) and tape each statement (Conditional, Converse, Inverse, and Contrapositive) to an index card. Arrange desks or tables so that students are working in groups of four. Have the following warm-up prompt on the board for students to respond to as they enter the room:

Think about a time when you have asked someone to “Prove it.” Write your thoughts in your math journal and be prepared to tell the class about your experiences. What did you ask the person to prove? How did the person prove what you asked?

Direct students’ attention to the warm-up prompt as you greet them at the door. When students have finished writing, ask volunteers to share their experiences. Determine students’ prior knowledge of logic by asking follow-up questions such as “Why does that reasoning seem logical?” or “What is a better way to prove that?” In the course of the discussion, reveal that this unit is an exploration of proof. Then, display the Best Pizza Deal transparency (p. B-3). Explain that it is an example of a type of problem that students encounter daily. It requires them to use logic and proof to make an informed decision. Later, students will revisit the problem write a proof to justify their choice for the best deal.

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Suggested Teaching Strategies/Procedures Days 1–2

The logical concept of a conditional statement and its converse, inverse, and contrapositive are introduced through students’ real-world experiences.

Materials & Resources Index cards* Invisible or masking tape* Scissors* Math journals* Overhead projector* Hat or box* (optional) Group Work Scoring Rubric (p. B-4) Statement Types transparency (p. B-5) Statement Types Key (p. B-6) Conditional Statements (one per group) (pp. B-7–B-10); one copy of the

homework (p. B-10) per student Conditional Statements Key (p. B-11)

*Materials or resources not included in the published unit

After completing the warm-up activity, “Prove it,” and displaying the Best Pizza Deal transparency, randomly distribute the index cards, one per student, or have each student pick a card out of a hat or box. Each of the nine different sets of statements contain a conditional and its converse, inverse, and contrapositive. Tell students to create a group of four by finding three other people who have cards with related statements. (If one group must have fewer than four students, be sure that one of those students has the conditional statement.) Students will work in these groups during Days 1–6 of this unit. Review with students how to work cooperatively in groups, if needed. Hagberg (1999) offers helpful information on cooperative learning. Introduce the Group Work Scoring Rubric (p. B-4) that you or the students will use to assess group work throughout the course.

After students have formed groups of four, ask each group to determine the relationship between the four statements on their cards. Allow several groups to share their ideas with the class; note those that support the formal definitions of the terms and use this information during the next part of the lesson. Then, using the Statement Types transparency (p. B-5), reference students’ ideas as you present the formal definitions of a conditional statement and its converse, inverse, and contrapositive. From the entire class, select a

student volunteer to write on the transparency the converse of the original statement, “If you drive a car, then you are at least 16 years old.” Select another volunteer to write the inverse of the statement and a third to write its contrapositive. The students will likely rely on the formal definitions that are on the transparency. Have class members refine the students’ work, clarifying the statements if necessary. Help students to understand that each statement has a hypothesis and a conclusion, and

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Tips for Teachers

Many teachers use a timer, a chime, or a song to help students complete a task in a reasonable time. Usually 2 minutes and 30 seconds is adequate time to group students together.

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each statement has a hypothesis and a conclusion, and explain how to identify each. It may be useful to have students underline the hypothesis once and the conclusion twice. When statements convey geometric ideas, encourage them to draw a sketch and determine the meaning of the mathematical terminology used in the statements.

Next, ask the group that has the statement, “If you are a teenager, then you are at least 13 years old,” to record and label on the board their four statements as the conditional, the converse, the inverse, and the contrapositive. Have the rest of the class discuss which category they think each statement belongs in.

Then, have students work in their groups to label their own statements. As students work in groups, circulate through the room observing group discussions and encouraging students to reflect on the definitions so that they can correctly categorize their statements.

When students have demonstrated familiarity with each of the four statement types, distribute Parts I, II, and III of the Conditional Statements worksheet (pp. B-7–B-9) to each group. Each group should identify one student to serve as a recorder. Part I asks students to write their statement, its converse, inverse, and contrapositive, and to identify the hypothesis and conclusion for each. Then they should determine the truth value of each of the statements, writing True if they think the statement is correct and False if they think it is incorrect. Each group must explain its reasoning on the worksheet. Introduce the term counterexample and ask students to comment on its meaning and to provide examples. Students can use counterexamples as part of their convincing arguments. In Part II, students identify the hypothesis and conclusion of new statements and rewrite them in if-then form. Problems 4 and 5 of Part II may be difficult for many students. Encourage them to try their best on these problems or make them optional. Then have students analyze one of the five statements to determine if it is true or false. They should rely on the example given and the discussion in their small groups to complete the task. Finally, in order to encourage the habit of drawing precise illustrations of geometric problems, have students draw and label geometric sketches for Statements 3–5 on the back of the Part II page. Allow students time to finish their work.

To complete Part III, groups should exchange completed worksheets and evaluate each other’s responses. Remind students to use the language of mathematics precisely when making mathematical corrections and recommendations. When they have finished, they should return the worksheets to the original group; the original group should read and discuss the corrections marked. Use the Group Work Scoring Rubric to evaluate students’ understanding for this activity.

If time allows, have students make a second exchange to get another group’s opinion about their reasoning. Finally, have several groups share their results with the entire class. Make a table on the board to record the truth values of each group’s statements. Ask students to look for a pattern and make a conjecture about which statements are always “logically equivalent.” If students are unsure about

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Tips for Teachers

Students may ask which of the four statements is the conditional. Ask probing questions that will help the students conclude that any one of the four statements can be considered the conditional and that the labels of the remaining three will depend on which one the students choose to use as the conditional. For example, “What is your group’s best guess?” “Could you explain your reasons?” “Is there another possibility?” “What would be the effect on the other statements?”

Tips for Teachers

It may also be helpful to have the class create a set of hand signals to remind them of the meaning of each statement type. For example, if a conditional statement can be symbolized by a double thumbs-up sign, then a converse can be shown by crossing the arms with thumbs-up; the inverse by thumbs-down; and the contrapositive by crossing the arms with thumbs-down.

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always “logically equivalent.” If students are unsure about the meaning of conjecture or logically equivalent, rephrase the question: Which statements always have the same truth value? Encourage them to create their own conditional statements to test their conjectures.

Next, use a diagram and a true conditional statement whose converse is false to show the logical equivalence of a conditional statement and its contrapositive. Because the hypothesis of the statement implies the conclusion, the conclusion is true when the hypothesis is true. However, the hypothesis does not have to be true when the conclusion is true. Using a statement such as, “If I am on page 6, then my book is open,” write the hypothesis in the inner circle and the conclusion in the outer circle. Since the outer circle is “my book is open,” anything outside of the outer circle must be “my book is not open.” The hypothesis, “I am on page 6,” is within the inner circle, therefore it must be within the outer circle, so the conclusion, “my book is open,” must be true. (See Figure 1.)

To test the converse, begin with the assumption, “my book is open.” According to the diagram, this statement is within the outer circle, but we can’t tell if it is within the inner circle or not. I could be on page 6, but the book could also be open to a different page; therefore, the converse is false.

Next, test the inverse with the assumption, “I am not on page 6.” This statement must be outside of the inner circle, but we can’t tell if it is within the outer circle or outside of both circles. After all, if one is not on page 6, the book could be open to a different page, or it could not be open at all.

Finally, test the contrapositive with the assumption, “my book is not open.” This statement falls outside of the outer circle, so it must fall outside of the inner circle. Because “my book is not open” implies “I am not on page 6,” the statement and its contrapositive are logically equivalent.

Use several examples to introduce the biconditional. For this discussion, use the capital letters P, Q, and R to stand for the following statements:

P: x = 2 Q: x2 = 4 R: x = ±2

Ask the students to use these three statements to form a true conditional statement whose converse is also true. There are two possible correct answers:

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Hypothesis

Conclusion

Figure 1

“I am on page 6”

“My book is open”

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If Q, then R and If R, then Q. Respond thoughtfully and ask probing questions about answers they volunteer.

Next, explain that the compound statement Q if and only if R is true precisely when the conditional If Q, then R and its converse If R, then Q are both true. It is customary to abbreviate if and only if still further by using the symbol iff. The symbol was devised by mathematician Paul Halmos, who also created many other commonly used mathematical notations and abbreviations (Halmos, 1985, p. 403).

(In general, the biconditional Q iff R will be true precisely when Q and R have the same truth value, so Q iff R is sometimes used to express the logical equivalence of the statements Q and R. Students will have difficulty understanding this point, however, because they have not yet considered the truth value of the conditional when the hypothesis is false.)

Collect the grouping statement index cards before the end of the class. As a wrap-up activity, ask students to use thumbs-up, thumbs-to-the-side, or thumbs-down to display how comfortable they are with using conditionals and their converses, inverses, and contrapositives to form conclusions. Make note of students who show thumbs-to-the-side or thumbs-down and encourage them to seek assistance with the homework assignment to clear up any confusion. Assign Part IV of the Conditional Statements worksheet (p. B-10) for homework. (Make sure everyone has a copy.)

Incorporate additional practice problems from the course textbook, ACT formative item pool, or other resources to support students’ academic learning throughout the unit.

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Tips for Teachers

Whenever possible, continue to model for students the use of problem-solving skills:

Identify irrelevant or missing information Make conjectures Draw a sketch Use previously learned knowledge Verify results Compare to a similar problem Check answers for reasonableness Guess, check, and revise Look for patterns

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Day 3 Logical pronouncements in advertising are examined, and students’

understanding of logic and its everyday uses and abuses are extended.

Materials & Resources Overhead projector* Overhead transparency of a magazine advertisement* Magazines or 15–20 print advertisements* Scissors* Poster paper* Glue* Markers* Group Work Scoring Rubric Advertising Statements (p. C-2) Index cards* Invisible or masking tape* Math journal*


Instruct students to get into the same groups formed on Day 1 to check and discuss the homework. Each group should write a list of questions it cannot answer. Allow the class time to ask questions about the homework and encourage students to respond to the questions posed by their peers.

Direct students’ attention to Essential Question 1, “How do advertisers use conditional statements?”, which should be written on the board. Briefly, invite students to respond to it. However, move on quickly. You will return to it at the end of class to discuss it more fully.

Use the overhead projector to show the class a print advertisement from a magazine. The purpose of this activity is to point out the fallacies that advertisers often rely on to sell their products. Select an advertisement that has minimal text but includes a picture that suggests a conditional statement (e.g., a picture of a beautiful woman with a bottle of perfume implies that “If you use this perfume, then you will be a beautiful woman”). Ask students to share aloud a conditional statement that is suggested by the advertisement. Record student responses on the board. Select one response and ask groups to discuss whether the statement is true or false. Have each group select one person to share their responses. Then have each group write the converse, the inverse, and the contrapositive of the conditional statement. Discuss as a class the truth value of each statement.

Give each group a print advertisement from a magazine, a large piece of poster paper, glue, and markers. (If time permits, students can be given a magazine and scissors to find their own print ad.) Each group will make a poster displaying the selected print ad, a conditional statement that the ad suggests, its inverse, its converse, and its contrapositive. This activity will reinforce the terms and their meanings and give students practice reasoning to determine the truth of conditional statements used in the world around them.

Have each group share its poster with the class. Beforehand, explain and model how they should make a presentation using the language of mathematics (e.g., “Our conditional statement is . . . and we believe the statement to be true/false because . . . . Our inverse statement is . . . .”). They

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statement to be true/false because . . . . Our inverse statement is . . . .”). They should be using reasonable justifications for their true/false conclusions. In doing so, they are beginning to practice methods of proof. Use the Group Work Scoring Rubric to assess the students’ understanding of conditional and related statements and their use of correct mathematical language.

After each group presentation, ask students to describe specific proof techniques used during the presentations. You may choose to define any new proof techniques as they arise, including deductive reasoning, inductive reasoning, or proof by the transitive property (if a = b and b = c, then a = c). During these presentations, continue to ask questions that help students refine their thinking and improve their mathematical communication. For example, ask “Could you explain how you determined that the contrapositive of your statement is false?” and “Can you find another piece of supporting evidence?” When presentations are finished and students have had time to correct any errors, the posters can be displayed around the room.

To conclude this activity, facilitate a class discussion around the first essential question, “How do advertisers use conditional statements?” Then assign the Advertising Statements homework (p. C-2).

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Day 4 Connections are made between the previous lessons on logic, including

the discussions about advertising, and the formal, more abstract method of proof in mathematics.

Materials & Resources Algebraic Proof transparency (p. D-2) Index cards* Overhead projector* Group Work Scoring Rubric Introduction to Two-Column Proof transparency (p. D-3)


Write Essential Question 2, “What evidence can be used to validate a mathematical conclusion?”, and Essential Question 3, “What makes an argument logical?”, on the board. As students walk into the room, scan their homework and record scores based on the number of examples of conditional statements they found. Have students discuss within the previous days’ groups the conditional statements from their homework. Then ask each group to write one statement in if-then form on the board. Ask, “How can we prove that this statement is true or false?”

Give students time to conjecture about how to prove (or disprove) a statement. Ask several groups to share their conclusions based on their discussions. Once again, this should provide opportunities to discuss deductive and inductive reasoning, counterexamples, and other proof techniques.

Students should notice that many of the statements they found in advertising would be difficult to prove since they lack supporting, true statements. They are easy to disprove by simply finding a counterexample.

Proving other statements are true, however, can be more complicated. Reveal to students that they will be learning how to prove conjectures using logical arguments. In general, inductive reasoning is a process of drawing conclusions by an accumulation of evidence. Inductive reasoning is the primary process of scientific inquiry; it is also frequently used in mathematics in order to make conjectures based upon many concrete examples. In contrast, deductive reasoning is a process of drawing conclusions from a sequence of true statements. In geometry, deductive reasoning is the primary method of proof. Arguments are built upon true statements such as mathematical properties, definitions, postulates, and theorems. Clarify with classroom examples, the difference between deductive reasoning and inductive reasoning.

Next, use a simple linear equation to introduce the fundamentals of a formal proof. Display the Algebraic Proof transparency (p. D-2) on the overhead projector. This activity will show, using supporting true statements, why x = 34 if 4x – 16 = 120. Students will learn that the supporting statements are algebraic properties.

Ask each student to solve the algebraic problem, showing all work and justifying each step. Circulate throughout the room while students work. If

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Tips for Teachers

On average, people look at a print ad for no more than two seconds (Franzen, 1994). People may be convinced to purchase a product based on these quick impressions unless they think logically about the advertisements they see throughout the day.

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Ask each student to solve the algebraic problem, showing all work and justifying each step. Circulate throughout the room while students work. If some students are struggling, ask questions to help them figure out the next step: “What did you do in the last step to get to this step?” or “What will you do next to get x alone on one side of the equation?” Then have students gather in their groups to discuss their answers and their reasoning. When students have finished discussing, ask a volunteer to show the algebraic steps and solution on the Algebraic Proof transparency. The rest of the class should check the students’ reasoning and work and ask clarifying questions.

Then, use the Introduction to Two-Column Proof transparency (p. D-3) to introduce a two-column proof format and the different parts of a two-column proof: the conditional statement, the given and the prove statements, and the lists of statements and reasons. As you discuss the transparency, reveal the following important guidelines for two-column proofs:

Write the statement to be proved in if-then form. The hypothesis includes the information that is given and the conclusion includes the information that is to be proven.

Include a sketch, if possible. Plan your proof before writing statements and reasons. Remember that the given information is accepted as true. Remember that all reasons must be true statements (mathematical

properties, definitions, postulates, or theorems). Remember that each step should be a logical deduction from the

previous step(s). Explain that the Reflexive Property of Equality is used to

introduce values that will be used in the next steps of the proof. Conclude with what you set out to prove.

These guidelines should be posted in the room for students to reference. Explain that this may be a laborious process to solve a simple linear

equation, but it’s important to understand it in order to learn the fundamentals of proof. To help them understand that sometimes there is more than one correct way to solve a problem or complete a proof, encourage volunteers to justify different steps that will lead to the same conclusion.

Provide additional opportunities for students to practice completing two-column algebraic proofs by first using the equation 3(x + 4) = 12 and then a more complicated equation such as

Again, ask students to first solve the algebraic equation individually, providing reasons for each step, then to discuss their proofs within groups, and finally to present their proofs to the class.

As a wrap-up, each student should complete an index card: they should summarize what they have learned in this lesson and identify one thing they do not yet fully understand. Students should bring these cards to class the following day. For homework, have students use the process introduced in the lesson to justify the steps used to solve the equation 2(9x – 1) + 1 = 20x – 5.

25 12 3 3.xx x− = + +

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Tips for Teachers

Students may need to be reminded of the algebraic properties of equality—reflexive, symmetric, transitive, additive, multiplicative, substitution—and the distributive property of multiplication over addition. This kind of review will help students use precise mathematical terminology when using these properties.

Many students may be uncomfortable with these properties. A poster, with each property listed in full (e.g., the additive property of equality), may also be helpful.

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Day 5 Students’ abilities to prove geometric problems with logic are expanded.

Materials & Resources Overhead projector* Two-Column Geometric Proof transparency (p. E-2) Jumbled Proof Steps (pp. E-3–E-5) Math journal* Group Work Scoring Rubric Jumbled Proof Steps Key (p. E-6)


Ask each student to share with a peer a question or statement from yesterday’s wrap-up activity. Then, ask them to share their questions with the class. Encourage them to address each other’s questions. Step in and out of the discussion only as needed. Ask students to share their proofs of the homework. Emphasize that there are multiple proof strategies by inviting students to reveal the full range of proofs they made.

Next, introduce a geometric proof by asking students to gather in their Day 1 groups and to prove the statement, “All right angles are congruent.” Ask them to think about the definition of right angles and to discuss with the group a plan to prove the statement. Then, ask them to prove it in a two-column format proof by following the steps presented on the Introduction to Two-Column Proof transparency and the guidelines introduced in the previous lesson. Most students will probably be unable to write a successful proof at this point. It is important that students understand that even unsuccessful attempts will help solidify their problem-solving abilities. “Rigorous instruction gets its traction when students are invited, even expected, to struggle with contexts and unclear problems and to construct their own knowledge. This knowledge construction, however, demands that teachers allow students the time and learning environment in which to do this difficult work” (Squires, 2006, p. 5).

After giving students time to attempt the proof, allow them to share their ideas and struggles with the class. To introduce a formal geometric proof, use the Two-Column Geometric Proof transparency (p. E-2) and add the following guidelines to the previous list:

Remember that the given is a specific case of the hypothesis. Remember that the prove is a specific case of the conclusion. Draw and label a sketch. Use geometric symbols and notation accurately in sketches and

in proofs. Continue to revisit all of the guidelines throughout the unit as students begin to make conjectures and plan proof strategies. Post a copy of the Introduction to Two-Column Proof and the Two-Column Geometric Proof transparencies in the classroom so that students can refer to them. (Some copiers can enlarge the masters to poster size.)

Next, have students complete Part I of the Jumbled Proof Steps activity (pp. E-3–E-5) in their groups. It scaffolds student understanding of geometric proofs by requiring them to determine the correct order of the steps and reasons for a geometric proof. The students should write the theorem in if-then

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form. The given and prove statements should be based on the information in the diagrams. Then, they should either cut apart the statements and reasons and reassemble them or rewrite them in a logical order on a separate sheet of paper. If students have trouble finding a logical order for the statements, remind them that each step must be deduced logically from the previous step, given in the diagram, or introduced using the reflexive property of equality.

When students have finished, allow them to share their proofs and reasoning with the class. Then, after distributing a copy to every student, assign Part II of the Jumbled Proof Steps activity for homework.

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Tips for Teachers

The abbreviation QED (quod erat demonstrandum, which translates to “that which was to be demonstrated”) is often found at the end of a proof. It was first used in medieval Latin translations of Euclid’s Elements, the foundation of Euclidean geometry. “No work, except the Bible,” writes Howard Eves (1990, p. 141), “has been more widely used, edited, or studied, and probably no work has exercised a greater influence on scientific thinking.” QED is used less frequently now than in past centuries, but it does provide a nice conclusion for a proof well done.

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Day 6 Students practice drawing conclusions.

Materials & Resources Drawing Conclusions (one per group) (pp. F-2–F-4) Drawing Conclusions Key (p. F-5) Group Work Scoring Rubric Scissors, straightedges, protractors, and graph paper* Justifying the Triangle Angle Sum Theorem (p. F-6) Paper triangles, approximately 5 in. × 5 in.* Math journal*


Have students pair up with someone close to them to share their jumbled proof homework. Allow time, if necessary, for them to annotate in the margins where and why their logic was in error and to revise their work. This close scrutiny requires students to revisit their thinking and to reflect on prior instruction. Then, collect students’ homework.

After students rejoin their groups, give each group a copy of the Drawing Conclusions worksheet (pp. F-2–F-4). A different student should now serve as the group recorder. Circulate through the room, listen to discussions, and pose questions that encourage them to refine their mathematical communication. After students complete Part I, create new groups by having one student from each group rotate to a different group to discuss and solve the problems in Part II, which asks students to conjecture and draw conclusions about geometric situations. Creating new groups gives students the opportunity to interact with and learn from different students in the class.

After Part II of the worksheet is completed, select students to present the conjectures written for Part I. Encourage responses from the class and make a list that includes each group’s conjectures. Use the Group Work Scoring Rubric to assess evidence of student participation

and understanding. Then create new groups again by having a different student rotate to a different group. Each group should now have two new members.

Give each student a paper triangle constructed according to the Justifying the Triangle Angle Sum Theorem Activity (p. F-6). Then, review the following terms and postulate: adjacent angles, straight angles, supplementary angles, and angle sum postulate. Read aloud the instructions for the exercise. This informal justification will help reinforce the ideas behind the theorem. Students will use this theorem before the formal proof is introduced in a later unit. If time permits, students could repeat this exploration using the exterior angles of a convex polygon to show the sum of 360°.

To conclude the lesson, ask volunteers to share with the class something they have learned during the course of the unit (i.e., an “Aha!” or “light bulb” moment). Then, assign the following questions for students to answer in their journals as homework:

D a y 6

Tips for Teachers

Students may need additional examples of conjectures, inductive reasoning, and deductive reasoning. Here is one such example:

Conjecture: In America, we never have more than five consecutive days of regular school. Inductive reasoning: In the years I have been

going to school, I’ve never had to go to school more than five consecutive days. Deductive reasoning: There are seven days in a

week and schools are not in attendance on Saturdays or Sundays.

15

Knowing that the sum of the measures of the angles of a triangle is 180°, how many obtuse angles can a triangle have? Why?

How many right angles can a triangle have? Why? If two angles of a triangle have measures that are

equal to the measures of two angles of a different triangle, what can you conclude about the measures of each triangle’s third angle? Why is this so?

D a y 6

Tips for Teachers

Many students are motivated by the infusion of technology into their mathematics learning. Technological tools are especially useful in investigations and discovery learning. The triangle angle sum theorem is a good introductory investigation for Geometry. It often works well to assign pairs of students to a computer or calculator and to give them a handout of instructions for the investigation. A Geometer’s Sketchpad investigation by the Center for Technology and Teacher Education (2002), and an investigation using Cabri, Jr., by Texas Instruments (2006), can be found online.

16

Day 7 The importance of precise definitions in geometry is emphasized, and

ways to understand what constitutes a good definition are introduced.

Materials & Resources Writing Good Definitions (one copy, cut apart) (p. G-2) Overhead projector* Math journal* Markers* Group Work Rubric Poster paper* Two-Column Geometric Proof transparency (p. E-2)


As a warm-up, in order to encourage reflection, ask students to write an exam question about any of the concepts presented so far in the unit. Encourage them to think about the wrap-up discussion from the previous class period. (These questions will be turned in on the last day of the unit.) Discuss answers to the homework questions with the class. Take note of students who seem to be having difficulty.

Introduce the lesson by writing the following statement on the board: “I have a friend who is ambidextrous and has triskaidekaphobia.” While students discuss the statement in the groups formed on Day 6, walk around the room and pose questions, helping them to realize that, in order to understand the situation, it is critical to know precise definitions of unknown terms. Define ambidextrous and triskaidekaphobia and give students time to develop new group rapport.

To emphasize the importance of precise definitions in geometry, give an example of a situation in which students use correct reasoning but get a wrong answer because they

are thinking of an inaccurate definition. For example, ask, “How many faces does a triangular pyramid have?” Most students will not know what a triangular pyramid is and will think of the classic rectangular pyramid. Their logic will be correct if they answer five, but they will not get the correct answer, four, because they have the wrong definition in mind. Explain that, in geometry, we rely on good definitions to help prove conjectures. These definitions are considered true statements. Give students time in their groups to discuss the concept of a “good definition.” When they have finished, ask each group to share and record on the board their ideas. The following list contains characteristics of a “good definition,” but students may not address all of them:

Good definitions use words that are commonly understood, were previously defined, or are purposely undefined.

Good definitions include no more information than is necessary or appropriate.

Good definitions are biconditional statements. Students will probably have questions about the last point, in particular. Most definitions, whether in the dictionary or their textbooks, are not written as

D a y 7

Tips for Teachers

Students can be directed to a good dictionary for detailed information on the definitions and origins of words, but definitions of ambidextrous and triskaidekaphobia are easy to comprehend:

Ambidextrous—using both hands with equal ease Triskaidekaphobia—fear of the number 13

17

biconditional statements. Instead, biconditionality is often assumed. Use the discussion as an opportunity to revisit the concept of biconditional statements and to deepen students’ understanding of them. Letter each group’s members A, B, C, and D. In each group, pair students A and B together and students C and D together. Have each pair write a conditional statement about anything they wish. Ask each pair to determine if the statement and its converse are true. If so, then the biconditional is true.

Next, pair the students again so that student A is paired with student C and student B is paired with student D. Have students take turns sharing the statements they created in their original pairs and having the new partner decide if the statement and its converse are true and whether the biconditional is true. Then, have each pair invent a new conditional statement such that it and its converse are both true. Students should begin to see similarities and differences between statements that are biconditionals and those that are not.

Have students return to their original pairs to share the new conditionals. If neither partner agrees that both the statement and its converse must be true, they should revise the statements until they produce true biconditionals.

Returning again to the second pairing (A with C, and B with D), ask students to share any revisions or observations. When they have finished, randomly select students to share their best statement, its converse, and to explain why the biconditional is or is not true. Ask students to share what they notice about those examples where the conditional and its converse must both be true. Often, students will choose a biconditional statement that is a definition or part of a definition (e.g., “It is Cinco de Mayo if and only if it is May 5”). This review and practice writing biconditional statements should prepare students to write good definitions.

Distribute a term and its respective examples from the Writing Good Definitions activity (p. G-2) to each group. Students should work in their groups to write a “good definition” of the mathematical word or phrase they were given. Remind students to use the list of characteristics of good definitions to guide their work. Some of the terms were defined earlier in the course. However, students should rely on their memories and their understanding of the term to write the definition, not their notes. As they work, pose questions to help each group refine their definitions:

What features are apparent in the examples of the term? Does your term belong in a class of geometric figures? Have you used words that are imprecise, unknown,

or part of what you are trying to define? Do you have too much or too little information? Is your definition a true biconditional? After students have written a good definition, have

one student from each group present the definition to the class for review. The class should attempt to find any flaws that keep the definition from being a “good definition.”

Finally, have groups think of a way to get their peers to remember the definition they were given. They should then present their descriptions and their mnemonic devices.

D a y 7

Tips for Teachers

Often, one must consider context in order to determine if a statement is meant as a biconditional. In his 1963 “Letter from Birmingham Jail,” Martin Luther King Jr. made this explicit. He wrote, “One has not only a legal but a moral responsibility to obey just laws. Conversely, one has a moral responsibility to disobey unjust laws.” King’s first claim can be written as the conditional statement, “If a law is just, then one must obey it.” His second claim is, in fact, an equivalent form (the contrapositive) of the converse: “If a law is not just, then one must disobey it.” King’s intended meaning can therefore be summarized as a single biconditional: “One must obey a law if and only if it is just.”

Tips for Teachers

Some common difficulties students might have writing definitions include not identifying the class in which a thing belongs (e.g., perpendicular lines are lines; rectangles are quadrilaterals) or providing too much information (e.g., rectangles are quadrilaterals with four right angles and opposite sides congruent).

18

Revisit Essential Question 2, “What evidence can be used to validate a mathematical conclusion?”, and Essential Question 3, “What makes an argument logical?” Have each student discuss the questions with a peer before writing responses in his or her math journal. Remind students to support their ideas with information they have learned.

D a y 7

Tips for Teachers

As a method for helping students with proofs, encourage them to create a personal set of “Geo-Cards.”

Use three different colors of index cards. Select one color for definitions and undefined

terms. Each card contains the undefined term or a formal definition and a sketch of the term. Select a different color for postulates and axioms.

Each card contains a different postulate and appropriate sketch. Select a third color for theorems. Each card

contains the theorem and an appropriate sketch. When writing proofs, students can create new cards each time theorems are proven or postulates or definitions are introduced. As they write proofs, they can refer to and select Geo-Cards from the stack that may be useful within the proof.

19

Day 8 Students learn two new proof methods: the flowchart proof and the

paragraph proof.

Materials & Resources Markers* Poster paper* Overhead projector* Flowchart Proof and Paragraph Proof transparency (p. H-2) Two-Column Geometric Proof transparency (p. E-2) Group Work Scoring Rubric Class notebook*


Write on the board Essential Question 4, “What is the best format for a proof?”, along with the following terms:

Definitions Theorems Postulates Undefined Terms Point Line Segment Angle Bisector Angle Addition Postulate Parallel Lines Triangle Angle Sum Theorem Conditional Statement As a warm-up, encourage students to share with a

peer the journal entries they wrote on Day 7. Circulate around the room, listen to conversations, and take notes. Identify students who have provided solid support for their thinking and have them read aloud their entries.

Next, turn students’ attention to the board. To address the essential question, students will explore concept maps as an introduction to flowchart proofs. Using a real-life example, begin by showing students how to

D a y 8

Tips for Teachers

Concept maps are helpful throughout Geometry. They are particularly useful when studying the properties of quadrilaterals.

Animals

Mammals

Reptiles

Birds

Lizards and snakes

Eagles and gulls

Bears and deer

Figure 2

20

create a concept map. (See Figure 2.) Explain how a concept map links one statement to the next and uses a diagram with arrows to show how to get from one statement to another. This is how a flowchart proof works.

To deepen their understanding of flowcharts, give each group poster paper and markers and have them create a concept map using some or all of the geometry terms on the board. They may want to add other mathematical words or statements. As students work in groups, observe and assess their understanding of the geometric terms and their relationships to one another. If time permits, have two or three groups explain their logic as they present their concept maps to the class. All groups should post their concept map on the walls to provide a visual representation of the terms.

Use the Flowchart Proof and Paragraph Proof transparency (p. H-2) to display a flowchart proof (do not reveal the paragraph proof at the bottom). Discuss the features of a flowchart proof:

Begin with the given statement(s). Analyze a related statement that advances the proof and then show

the source of the statement. Provide a reason that is a property, definition, postulate, or

theorem. Advance the proof until the conclusion is justified. Organize the flowchart proof to show the logical process. Next, reveal the paragraph proof and have a student read it aloud. Ask

the class to vote on which proof is easier to understand. Sometimes a flowchart proof is easier to understand than a paragraph proof, and sometimes the opposite is true. Both are acceptable forms.

Display the Two-Column Geometric Proof transparency (p. E-2) again. In groups, have students write a flowchart proof and a paragraph proof for the conditional statement shown. (Use the Group Work Scoring Rubric to assess students’ understanding.) Have one or two students read their paragraph proofs to the class and one or two other students write their flowchart proofs on the board. The more comfortable students are sharing their work, the more they will collaborate and learn from each other. To wrap up the lesson, revisit and discuss Essential Question 4, “What is the best format for a proof?”

Finally, write the theorem and information in Figure 3 on the board. As homework, have students write a two-column proof, a flowchart proof, and a paragraph proof of this theorem. If they need help, students should refer to the notes they have taken from the Flowchart Proof and Paragraph Proof transparency and the list of guidelines for writing proofs.

Complements of the same angle are congruent. Given: ∠1 and ∠2 are complementary angles.

∠3 and ∠2 are complementary angles. Prove: ∠1 ≅ ∠3

D a y 8

Figure 3

2

3 1

21

Day 9 Students are introduced to indirect proofs.

Materials & Resources Overhead projector* Proof Methods transparency (p. I-2) Group Work Scoring Rubric


Have students hand in the homework. Then, as a warm-up, write the multiple-choice question in Figure 4 on the board. Ask students to work the problem independently. Then, ask volunteers to explain how they arrived at their answer.

Some students will try to solve the problem directly, recalling (from Day 6) that the sum of the measures of the angles in a triangle is 180°. Others will say that they found the answer by process of elimination:

1. They noticed that one of the angles given is obtuse (m∠B=92°) 2. They recalled (from Day 6) that a triangle can have at most one

obtuse angle. 3. Assuming any of the options “A,” “C,” or “D,” requires the

triangle to have two obtuse angles and contradicts 2. 4. Therefore, “A,” “C,” and “D,” must all be false; “B” must be true.

Explain to students that this method illustrates an important proof strategy in geometry called indirect proofs—specifically, proof by contradiction.

This reasoning represents a proof by contradiction. Place the Proof Methods transparency (p. I-2) on the overhead projector and facilitate a discussion that compares direct and indirect reasoning. Ask students to describe a proof or situation in which an indirect proof would be more efficient. Then have students work in groups to create a real-life situation that uses indirect proof in problem solving. For example, a student shows up to his Geometry class on the first day of school and no one is present. What’s more,

D a y 9

A B

C

Figure 4

Given: Triangle ABC m∠A = 38° m∠B = 92°

What is the m∠C ? Select the best answer. A. 190° B. 50° C. 100° D. 92°

22

he arrives at a general rule: If no one else is present in the classroom on the first day, then I am in the wrong room. As on Day 2, use capital letters to represent statements:

P: No one else is present in the classroom. Q: I am in the wrong room.

The student’s general rule is a conditional: If P, then Q. He arrives at it in the following way: If he is not in the wrong room (i.e., Q is false), then he would be in the right room, and he would find other students and a teacher there. This contradicts the hypothesis, P, proving the contrapositive of the conditional. But because the conditional and its contrapositive are logically equivalent, this also proves If P, then Q.

After groups have identified a real-life situation that uses indirect reasoning, have them present the situations to the class. Encourage the rest of the students to identify the path of indirect reasoning according to the Proof methods transparency.

As a wrap-up activity, have students find a proof in two-column, flow-chart, or paragraph format. In pairs, have students create a plan for the same proof as an indirect proof. As homework, assign the indirect proof in Figure 5.

D a y 9

B

C A

Assume that _________. Then m∠B + m∠C = ______ because ________. The triangle angle sum theorem states that m∠A + m∠B + m∠C = ________. This contradicts the given that ________ because ______. Therefore, our assumption was incorrect and ________.

Figure 5

Given: In ABC, m∠C = 90°, m∠A > 0. Prove: m∠B cannot be 90°

23

Day 10 Students summarize what they have learned in the unit and apply this

understanding to geometry and the real world.

Materials & Resources Assessing Your Understanding (pp. J-2–J-4) Assessing Your Understanding Key (pp. J-5–J-6) Group Work Scoring Rubric

As a warm-up, have students share with a peer the exam question they wrote on Day 7 and revise it if necessary. Collect the questions for review. Then, discuss the indirect proof homework as a class.

Give each student a copy of the Assessing Your Understanding worksheet (pp. J-2–J-4) to complete as a unit assessment. Students should work in pairs so that they can discuss each concept covered by the worksheet. Inform students that you will be using the Group Work Scoring Rubric to assess their written responses, based on their communication of the solution and their understanding of the concepts.

After students have finished the assessment, direct their attention to the five essential questions displayed in the room, especially Essential Question 5, “What qualities/components should a good proof possess?” Give groups 15–20 minutes to compose answers to them. Then have several students share their responses with the class.

To wrap up the unit, reiterate that the unit represented an introduction to proof methods that are used in everyday life as well as in a geometry course. Tell students that they will be creating and writing their own proofs about parallel lines, triangles, quadrilaterals, and other important geometric figures and concepts such as congruency. With or without a formal proof, throughout the course students will continue to justify and support their reasoning as they solve new and more interesting problems.

D a y 1 0

24

ENHANCING STUDENT LEARNING

Selected Course Objectives

C.1. Logic and Proof a. Use definitions, basic postulates, and theorems about points,

segments, lines, angles, and planes to write proofs and to solve problems

Unit Extension

Suggested Teaching Strategies/Procedures

Materials & Resources Sample Proof of the Pythagorean Theorem (p. K-2)

The Pythagorean theorem will be discussed at length later in the course. It has been proven many different ways. With a little background knowledge, some of the proofs of the Pythagorean theorem are fairly easy for students to understand. To extend students’ learning, show them some proofs of the Pythagorean theorem that are examples of more complex proofs (p. K-2).

Reteaching

Suggested Teaching Strategies/Procedures

Materials & Resources Overhead projector* Building a Taco transparency (p. K-3) Doug’s Proof (p. K-4) Doug’s Proof Key (p. K-5)


Throughout the unit, use several different methods to reinforce the concept of proof as a series of steps that link the given information to what is to be proved. For instance, introduce a process that many students will be familiar with and list the reasons for the steps as you would in a formal proof. (See, for example, the Building a Taco transparency, p. K-3.) Ask students to think of a process that they are familiar with, to describe it as a series of steps, and to provide a reason for each step. Have them share their proof with a peer to determine if the process could be recreated with the steps given

For additional practice finding the correct order for steps in a proof, assign Doug’s Proof (p. K-4). Students should work in pairs to complete a proof of the given conditional statement by choosing statements from the list and putting them in the correct order.

E n h a n c i n g S t u d e n t L e a r n i n g

25

E n h a n c i n g S t u d e n t L e a r n i n g

Reflecting on Classroom Practice How much review of prerequisite topics did students need? What other teacher resources could you add to this unit to make the

content more challenging and accessible? What kinds of mathematical thinking did the instructional

environment model?

26

Bibliography

References

Center for Technology and Teacher Education. (2002). Exploring characteristics of triangles. Retrieved July 25, 2007, from http://www.teacherlink.org/content/math/activities/skpv4-triangles/guide.html

Eves, H. (1990). An introduction to the history of mathematics (6th ed.). Orlando, FL: Saunders.

Franzen, G. (1994). Advertising effectiveness: Findings from empirical research. Oxfordshire: NTC Publications.

Halmos, P. R. (1985). I want to be a mathematician: An automathography. New York: Springer-Verlag.

Hagberg, L. (1999). Dynamics of learning groups—meeting the needs of all students. Retrieved June 13, 2006, from http://adhd.kids.tripod.com/groups.html

King, M. L., Jr. (1963). Letter from Birmingham jail. Retrieved July 13, 2007, from the Martin Luther King, Jr. Papers Project: http://www.stanford.edu/group/King/popular_requests/frequentdocs/birmingham.pdf

Schumer, F. (2005, June 19). In Princeton, Taking on Harvard’s fuss about women. The New York Times, p. 14NJ.

Squires, K. (2006). Rigorous learning. The Learning Network, 4(6), pp. 5–6. Retrieved June 13, 2006, from http://www.smallschoolsproject.org/PDFS/TLN/TLN-Mar-Apr_06.pdf

Texas Instruments. (2006). Cabri Jr.—Getting started with triangles. Retrieved July 25, 2007, from http://www.cabrijr.com/assets/pdfs/upload_dir/cabrijr_activity_1_ _construct.pdf

B i b l i o g r a p h y

Contents Unit Assignments and Assessments ................................................................................................................A-2

Example

Unit Assignments and Assessments ................................................................................................................A-3 Record Keeping

A p p e n d i x A : R e c o r d K e e p i n g A - 1

©2007 by ACT, Inc. Permission granted to reproduce this page for QualityCore™ educational purposes only.

Unit Assignments and Assessments

Name: _______________________ Period: Unit 3: Beyond a Shadow of a Doubt: Logic and Proof

Directions: Prior to starting the unit, complete the log on the next page according to the example below and distribute it to students as an organizational tool.

Day Assigned Assignment/Assessment

In Class

Home-work Date Due

Feedback (Completed/

Points) 1–2 Grouping Statements X

Conditional Statements X X

3 Advertising Presentation X

Advertising Statements X

4 Linear Equation Proof X

5 Jumbled Proof Steps X X

6 Drawing Conclusions X

Triangle Angle Sum Theorem questions (Journal) X

7 Journal Assignment X

Flowchart and Paragraph Proof X

Congruent Complements Proof X

9 Indirect Proof X X

10 Assessing Your Understanding X

8

E x a m p l e A - 2


Unit Assignments and Assessments

Name: _______________________ Period: Unit 3: Beyond a Shadow of a Doubt: Logic and Proof

R e c o r d K e e p i n g A - 3

Day Assigned Assignment/Assessment

In Class

Home-work Date Due

Feedback (Completed/

Points)

Contents Grouping Statements........................................................................................................................................B-2

Activity

Best Pizza Deal ................................................................................................................................................B-3 Transparency

Group Work Scoring Rubric............................................................................................................................B-4 Rubric

Statement Types...............................................................................................................................................B-5 Transparency

Statement Types Key .......................................................................................................................................B-6 Key

Conditional Statements ....................................................................................................................................B-7 Worksheet

Conditional Statements Key ..........................................................................................................................B-11 Key

A p p e n d i x B : D a y s 1 – 2 B - 1


A c t i v i t y B - 2

Grouping Statements

Directions: Cut each statement apart and tape to an index card, or print this sheet on card stock and cut it apart.

Conditional Converse Inverse Contrapositive

If the measure of angle A is 28° (m∠A = 28°), then ∠A is an acute angle.

If ∠A is an acute angle, then the measure of angle A is 28°(m∠A = 28°).

If the measure of angle A is not 28° (m∠A ≠ 28°), then ∠A is not an acute angle.

If ∠A is not an acute angle, then the measure of ∠A is not 28° (m∠A ≠ 28°).

If 2x – 3 = –7, then x = –2.

If x = –2, then 2x – 3 = –7.

If 2x – 3 ≠ –7, then x ≠ –2.

If x ≠ –2, then 2x – 3 ≠ –7.

If a shape is a rectangle, then it has 4 sides.

If a shape has 4 sides, then the shape is a rectangle.

If a shape is not a rectangle, then the shape does not have 4 sides.

If a shape does not have 4 sides, then the shape is not a rectangle.

If you are a teenager, then you are at least 13 years old.

If you are at least 13 years old, then you are a teenager.

If you are not a teenager, then you must be less than 13 years old.

If you are less than 13 years old, then you are not a teenager.

If she is responsible for the money, then she is guilty if it is lost.

If she is guilty if the money is lost, then she is responsible for the money.

If she is not responsible for the money, then she is not guilty if it is lost.

If she is not guilty if the money is lost, then she is not responsible for the money.

If a number is divisible by 2, then it is divisible by 4.

If a number is divisible by 4, then it is divisible by 2.

If a number is not divisible by 2, then it is not divisible by 4.

If a number is not divisible by 4, then it is not divisible by 2.

If an animal is a mouse, then it is a rodent.

If an animal is a rodent, then it is a mouse.

If an animal is not a mouse, then it is not a rodent.

If an animal is not a rodent, then it is not a mouse.

If the sum of 2 angles is 90°, then the angles are complements of each other.

If 2 angles are complements of each other, then their sum is 90°.

If the sum of 2 angles is not 90°, then the angles are not complements of each other.

If 2 angles are not complements of each other, then their sum is not 90°.

If 2 angles are congruent, then their measures are equal.

If 2 angles have equal measures, then the angles are congruent.

If 2 angles are not congruent, then their measures are not equal.

If 2 angles have unequal measures, then the angles are not congruent.

✂


T r a n s p a r e n c y B - 3

Best Pizza Deal

You are a member of the debate team. It is time to test your ability to make convincing arguments.

At an upcoming meet, you and your debate partner will be allotted $15 to buy pizza. There are two options to consider:

Option A: One large (15-inch diameter) pizza for $14.99

Option B: Two medium (10-inch diameter) pizzas for $7.50 each

Select the most economical option. Use your reasoning skills to make a convincing argument for why your selection is the best. Write a short explanation of your case.


R u b r i c B - 4

Group Work Scoring Rubric

Directions: Rate each student’s group work in each of the four categories below.

Context of Group Work: ________________________________________________________________________

Name: _________________________________________ Period: ________ Date:

Score Group

Participation Communication Conceptual

Understanding Problem Solving

3

Participates enthusiasti-cally, shares tasks and responsibilities, and provides and accepts assistance from group when needed.

Extrapolates meaning from oral/written directions without prompts and/or uses precise, appropriate mathematical terminology and symbolic notation.

Shows complete understanding of concepts and principles and demonstrates insight and reflection.

Chooses and applies the correct concept or principle to the solution of problems, makes and communicates connections among all the different aspects of a problem.

2

Participates with minimal prompts, shares the majority of tasks and responsibilities, and usually provides and accepts assistance when needed.

Requires minimal prompts to extrapolate meaning and/or uses adequate mathematical terminology and symbolic notation.

Shows understanding of most of the concepts and principles and demon-strates some reasoning ability.

Chooses and applies an appropriate concept or principle, fails to make or communicate connections among different aspects of a problem.

1

Requires many prompts to participate, shares few tasks and responsibilities, and rarely provides or accepts assistance.

Requires many prompts to extrapolate meaning and/or fails to use correct mathematical terminology or symbolic notation.

Shows little understanding of the concepts and principles and demonstrates recall with little understanding.

Chooses and applies an incorrect concept or principle or shows insufficient information to show the concept or principle used.

0 Does not participate or is disrespectful to other students in the group.

Shows no attempt to communicate.

Shows no understanding. Shows no attempt to solve problems.


T r a n s p a r e n c y B - 5

Statement Types

Conditional Statement: A statement with two parts, a hypothesis and a conclusion.

When a statement is written in if-then form, the “if” clause contains the hypothesis, and the “then” clause contains the conclusion.

Converse: A statement formed by transposing the hypothesis and conclusion of a conditional statement.

Inverse: A statement formed by negating both the hypothesis and conclusion of a conditional statement.

Contrapositive: A statement formed by negating both the hypothesis and conclusion of the converse of a conditional statement.

If you drive a car, then you are at least 16 years old.


K e y B - 6

Statement Types Key

Conditional Statement: If you drive a car, then you are at least 16 years old.

Converse: If you are at least 16 years old, then you drive a car.

Inverse: If you do not drive a car, then you are not at least 16 years old.

Contrapositive: If you are not at least 16 years old, then you do not drive a car.


W o r k s h e e t B - 7

Conditional Statements

Part I Directions: Complete the first column by writing your group’s original statement, then its converse, inverse, and contrapositive. Write the hypothesis and conclusion of each statement in the second and third columns. Then, discuss the truth value of these statements and, based on your group’s understanding, decide if each statement is true (T) or false (F). Write a brief explanation why each statement is true or false in the last column. Later, another group will agree or disagree with your group’s decision, so be sure to make your argument convincing.

Name: _________________________________________ Period: ________ Date:

Original Statement Hypothesis Conclusion T/F Explanation

Converse Hypothesis Conclusion T/F Explanation

Inverse Hypothesis Conclusion T/F Explanation

Contrapositive Hypothesis Conclusion T/F Explanation


C o n d i t i o n a l S t a t e m e n t s


Part II Directions: The statements below are conditional statements that are not written in if-then form. Study the example and then discuss the 5 statements that follow. For each statement, identify the hypothesis and the conclusion and rewrite each statement in if-then form. For Statements 2–5, include geometric sketches on the back. Finally, select one of the statements to analyze. Determine whether it is true or false, provide a rationale, and predict who might use this information and why.

Statement: ___ True/False (circle one) Rationale:

Who might use this information and why?

Conditional Hypothesis Conclusion If-Then Form Example: Strong men use aftershave.

You are a strong man.

You use aftershave.

If you are a strong man, then you use aftershave.

1. Third grade students love mathematics.

2. Two perpendicular lines form congruent adjacent angles.

3. Vertical angles are congruent.

4. Two coplanar lines either intersect, or they are parallel.

5. The sum of the measures of the interior angles of a triangle is 180°.




Part III Directions: Exchange your worksheets with another group. Complete the following:

We agree/disagree (circle one) with the work on Part I.

We agree/disagree (circle one) with the work on Part II.

We have listed below all areas where we disagree and have explained our reasoning using the language of mathematics. Please review your work according to our comments and let us know what you think.

Reviewer Name(s): __________________________________________________________________

Period: _________________________________________ Date: _____________________________



W o r k s h e e t B - 1 0

Name: _________________________________________ Period: ________ Date:

Part IV Directions: The statements below are conditional but are not written in if-then form. For each statement, identify the hypothesis and the conclusion and rewrite each statement in if-then form. For Statements 3–5, include geometric sketches on the back. In the last row, create your own real-world conditional statement about a mathematical concept, identify the hypothesis and conclusion, and write it in if-then form. Finally, complete the assignment detailed at the bottom of the worksheet.

Conditional Hypothesis Conclusion If-Then Form 1. Bullies are not well liked.

2. A dog is a man’s best friend.

3. An angle is formed by two rays that share a common endpoint.

4. For any 3 noncollinear points, there is exactly 1 plane containing them.

5. An angle bisector is a ray that divides an angle into 2 congruent angles.

6.

Write your own explanation of the logic of conditional statements that would be clear to any sixth-grade student. Include examples the student could relate to.


K e y B - 1 1

Conditional Statements Key

Part I Answers will vary.

Part II Conditional Hypothesis Conclusion If-Then Form

1. Third grade students love mathematics.

He/she is a third grade student. He/she loves mathematics. If he/she is a third grade student, then he/she loves mathematics.

2. Two perpendicular lines form congruent adjacent angles.

Two lines are perpendicular. The 2 lines form congruent adjacent angles.

If 2 lines are perpendicular, then they form congruent adjacent angles.

3. Vertical angles are congruent.

Two angles are vertical angles. The 2 angles are congruent. If 2 angles are vertical angles, then they are congruent.

4. Two coplanar lines either intersect, or they are parallel.

Two lines are coplanar. The 2 lines either intersect or are parallel.

If 2 lines are coplanar, then they either intersect or are parallel.

5. The sum of the measures of the interior angles of a triangle is 180°.

A polygon is a triangle. The sum of the measures of the interior angles is 180°.

If a polygon is a triangle, then the sum of the measures of its interior angles is 180°.

Part III Answers will vary.

Part IV

Conditional Hypothesis Conclusion If-Then Form 1. Bullies are not well liked. He/she is a bully. He she is not well-liked. If he/she is a bully, then he/she

is not well-liked.

2. A dog is a man’s best friend.

It is a dog. It is man’s best friend. If it is a dog, then it is man’s best friend.

3. An angle is a figure formed by 2 rays that share a common endpoint.

A figure is an angle. The figure is formed by 2 rays that share a common endpoint.

If a figure is an angle, then it is formed by 2 rays that share a common endpoint.

4. For any 3 noncollinear points, there is exactly 1 plane containing them.

Three points are noncollinear. The 3 points contain exactly 1 plane.

If 3 points are noncollinear, then they contain exactly 1 plane.

5. An angle bisector is a ray that divides an angle into 2 congruent angles.

A ray is an angle bisector. The ray divides an angle into 2 congruent angles.

If a ray is an angle bisector, then the ray divides the angle into 2 congruent angles.

6. Answers will vary. Answers will vary. Answers will vary. Answers will vary.

Contents Advertising Statements ....................................................................................................................................C-2

Homework

A p p e n d i x C : D a y 3 C - 1


Advertising Statements

Name: _________________________________________ Period: ________ Date:

Directions: While listening to the radio, watching television, or reading a magazine, find 5 examples of conditional statements. They may or may not be written in if-then form. If they are not, rewrite them in if-then form. Record the name of the product the ad is promoting and the name of the company that produced the ad. Then, think of an advertisement that could be used to sell mathematics. Describe the ad and its target audience, and write a conditional statement to request the ad on the back of this page.

H o m e w o r k C - 2

1. Conditional Statement





Advertised Product

Company

Advertised Product

Company

Advertised Product

Company

Advertised Product

Company

Advertised Product

Company

Contents Algebraic Proof................................................................................................................................................D-2

Transparency

Introduction to Two-Column Proof ................................................................................................................D-3 Transparency

A p p e n d i x D : D a y 4 D - 1


Algebraic Proof

Solve the following equation. Show every algebraic step in your solution.

4x – 16 = 120

T r a n s p a r e n c y D - 2


Introduction to Two-Column Proof

Step 1: Write 4x – 16 = 120 as a conditional statement.

If 4x – 16 = 120, then x = 34.

Step 2: Determine the given (the hypothesis) and the prove (the conclusion) statements.

Given: 4x – 16 = 120

Prove: x = 34

Step 3: Set up a two-column proof. List a statement and reason for each step. Reasons for knowing something can include mathematical properties, postulates, givens, definitions, or theorems.

T r a n s p a r e n c y D - 3

Statements Reasons

1. 4x – 16 = 120 1. Given

2. 16 = 16 2. Reflexive Property of Equality

3. 4x = 136 3. Additive Property of Equality (Steps 1 & 2)

4. 1 14 4

= 4. Reflexive Property of Equality

5. x = 34 ∴

means “therefore” ∴

5. Multiplicative Property of Equality (Steps 3 & 4)

Contents Two-Column Geometric Proof ........................................................................................................................ E-2

Transparency

Jumbled Proof Steps ........................................................................................................................................ E-3 Activity

Jumbled Proof Steps Key................................................................................................................................. E-6 Key

A p p e n d i x E : D a y 5 E - 1


Two-Column Geometric Proof

Step 1: Write “All right angles are congruent” as a conditional statement: If 2 angles are right angles, then they are congruent.

If ∠A and ∠B are right angles, then ∠A ≅ ∠B. Step 2: Determine the given (the hypothesis) and the prove (the conclusion) statements.

Always draw and label a sketch if possible!

Given: ∠A is a right angle. ∠B is a right angle.

Prove: ∠A ≅ ∠B

Step 3: Set up a two-column proof. List a statement and reason for each step. Reasons for knowing something can include mathematical properties, postulates, givens, definitions, and theorems.

A B

Statements Reasons

1. ∠A is a right angle. 1. Given

2. ∠B is a right angle. 2. Given

3. m∠A = 90° 3. Definition of a right angle (Step 1)

4. m∠B = 90° 4. Definition of a right angle (Step 2)

5. m∠A = m∠B 5. Substitution (Steps 3 & 4)

6. ∠A ≅ ∠B ∴ 6. Definition of congruent angles

T r a n s p a r e n c y E - 2


Jumbled Proof Steps

Part I Directions: Problems 1 and 2 list two different theorems. Each theorem includes the jumbled statements and reasons used in proving the theorem. First, write each theorem in if-then form and use the sketch provided to write the given and prove statements. Then, cut apart the statements and reasons and reassemble them so that the statements and reasons are in a logical sequence to provide a proof.

1. Complements of the same angle are congruent.

If-then statement:

Given:

Prove:

A c t i v i t y E - 3

Name: _________________________________________ Period: ________ Date:

1 2

3

Statements Reasons

∠1 ≅ ∠3 Definition of complementary angles

m∠1 = m∠3 Given

m∠1 + m∠2 = 90° Reflexive Property of Equality

∠1 and ∠2 are complementary angles. Substitution Property

m∠1 + m∠2 = m∠2 + m∠3 Definition of complementary angles

m∠2 + m∠3 = 90° Subtraction Property of Equality

∠2 and ∠3 are complementary angles. Given

m∠2 = m∠2 Definition of congruent angles


2. If m∠1 = m∠2, then m∠GRA = m∠ERT.

Given:

Prove:

J u m b l e d P r o o f S t e p s


3 1

2 R

G

E

A

T

Statements Reasons

m∠ERT = m∠3 + m∠2 Given

m∠GRA = m∠ERT Addition Property of Equality

m∠GRA = m∠1 + m∠3 Angle Addition Postulate

m∠1 + m∠3 = m∠3 + m∠2 Substitution Property

m∠3 = m∠3 Angle Addition Postulate

m∠1 = m∠2 Reflexive Property of Equality


Part II Directions: Given below is a proof problem. Cut apart the jumbled statements and reasons and reassemble them so that the statements and reasons are in a logical sequence to provide a proof. Then, create your own algebraic or geometric proof with jumbled steps and reasons that a partner may reassemble in class.

1. Given: bisects ∠DAB, ∠1 ≅ ∠3

Prove: ∠1 ≅ ∠2

AC

J u m b l e d P r o o f S t e p s


Name: _________________________________________ Period: ________ Date:

1 2

3

A B

C

D E

Statements Reasons

∠1 ≅ ∠3 Definition of an angle bisector

∠1 ≅ ∠2 Given

∠2 ≅ ∠3 Given

bisects ∠DAB Substitution Property

AC


Jumbled Proof Steps Key

Part I 1. If 2 angles are complements of the same angle, then they are congruent.

Given: ∠1 and ∠2 are complementary angles; ∠2 and ∠3 are complementary angles. Prove: ∠1 ≅ ∠3

2. Given: m∠1 = m∠2 Prove: m∠GRA = m∠ERT

Part II

K e y E - 6

Statements Reasons 1. ∠1 and ∠2 are complementary angles. 1. Given

2. ∠2 and ∠3 are complementary angles. 2. Given

3. m∠1 + m∠2 = 90° 3. Definition of complementary angles

4. m∠2 + m∠3 = 90° 4. Definition of complementary angles

5. m∠1 + m∠2 = m∠2 + m∠3 5. Substitution Property

6. m∠2 = m∠2 6. Reflexive Property of Equality

7. m∠1 = m∠3 7. Subtraction Property of Equality

8. ∠1 ≅ ∠3 8. Definition of congruent angles

Statements Reasons 1. m∠1 = m∠2 1. Given

2. m∠3 = m∠3 2. Reflexive Property of Equality

3. m∠1 + m∠3 = m∠3 + m∠2 3. Addition Property of Equality

4. m∠GRA = m∠1 + m∠3 4. Angle Addition Postulate

5. m∠ERT = m∠3 + m∠2 5. Angle Addition Postulate

6. m∠GRA = m∠ERT 6. Substitution Property

Statements Reasons 1. bisects ∠DAB 1. Given

2. ∠2 ≅ ∠3 2. Definition of an angle bisector

3. ∠1 ≅ ∠3 3. Given

4. ∠1 ≅ ∠2 4. Substitution Property

AC

Contents Drawing Conclusions ...................................................................................................................................... F-2

Worksheet

Drawing Conclusions Key ............................................................................................................................... F-5 Key

Justifying the Triangle Angle Sum Theorem .................................................................................................. F-6 Activity

A p p e n d i x F : D a y 6 F - 1


Drawing Conclusions

Part I Directions: Review the process below for creating proofs:

1. Begin by making conjectures (statements or opinions based on observations) that you wish to prove. For example, “What goes up must come down” is a conjecture.

2. Use inductive reasoning (a reasoning method based upon several observations) to support your conjecture. For example, “I threw 10 balls in the air and each ball returned to earth.”

3. Prove using deductive reasoning (reaching conclusions using facts, definitions, logic, postulates, and theorems). For example, “Newton’s law of gravitation says that objects exert an attraction to each other that is proportional to their masses and distances apart; therefore, what goes up, comes down.”

Now, define the terms conjecture, inductive reasoning, and deductive reasoning in your own words. Illustrate each with three real-life examples to help others understand your explanations. Be prepared to present your conjectures and reasoning to the class.

Names: ________________________________________ Period: ________ Date:

Conjecture Examples

W o r k s h e e t F - 2


D r a w i n g C o n c l u s i o n s


Inductive Reasoning Example

Deductive Reasoning Example


Part II

Directions: Each group member should choose one of the following problems to explore. Individually, make conjectures about the exploration. When you have finished, share your problem with another member of your group and revise as necessary. Finally, share the results with others in your group. Then, answer the last problem as a group.

1. Draw several different large triangles and measure each angle in the triangle. What conjecture would you make about the sum of these angles? (Note: You will need a straightedge and a protractor.)

2. Use a piece of graph paper to plot the following points: A(–3,7), B(7,7), C(7,–2). Make a conjecture about the location of point D that would create rectangle ABCD. (Note: You will need graph paper.)

3. Draw a line on a piece of paper and locate a point not on the line. What conjecture would you make about the shortest distance between this point and this line?

4. Study the sketch below. Make at least 5 conjectures that seem reasonable given all of the following information:

5. Scientists frequently make careful observations to draw conclusions. They use deductive reasoning to explain their findings. For instance, a scientist records the daily high temperatures in several different urban and rural areas over a period of several months. She finds that the average daily high temperature in the urban areas is higher than in the rural areas. This phenomenon is an example of what is called the “urban heat island.” In part because of the large amount of concrete and asphalt surfaces, which tend to retain heat, average temperatures in large cities are higher than in the country. Describe another situation that a scientist might encounter and the conjectures and conclusions that can be drawn using inductive and deductive reasoning.

D r a w i n g C o n c l u s i o n s


Names: ________________________________________ Period: ________ Date:

P

Q

R

U

S

T

PU ⊥ QS

QR ⊥ QT

QR bisects ∠ PQS

QS bisects ∠ RQT


Drawing Conclusions Key

Part I Answers will vary.

Part II 1. The sum of the measures is 180°.

2. D(–3,–2)

3. The shortest distance between the point and the line is the perpendicular segment through the point to the line.

4. The following list provides some of the possiblities: ∠PQS is a right angle. ∠UQS is a right angle. ∠RQT is a right angle. ∠PQR ≅ ∠RQS ≅ ∠SQT ≅ ∠TQU ∠PQR and ∠RQS are complementary. ∠SQT and ∠RQS are complementary. ∠SQT and ∠TQU are complementary. m∠PQR, m∠RQS, m∠SQT, m∠TQU = 45° ∠PQR and ∠RQU are supplementary.

5. Answers will vary.

K e y F - 5


Justifying the Triangle Angle Sum Theorem

Directions: Create triangles that are bounded by approximate dimensions of 5 in. × 5 in. so that they will be an adequate size to explore the Triangle Angle Sum Theorem. Give each student a different triangle cut from paper. Include acute, right, obtuse, scalene, isosceles, and equilateral triangles. Read the remaining directions aloud.

1. Estimate the angle measures of each angle of the triangle without measuring. Clearly mark the angles of the triangle with darkened edges. (See Figure 1.)

2. Tear the angles of the triangle apart from one another. (It is important to tear them rather than cut them so that the angle sides are clearly recognized.) (See Figure 2.)

3. Place the sides of the angles together to create adjacent angles that form a straight angle. (See Figure 3.)

4. Based on the results of the exploration, conjecture about the sum of the measures of the angles of a triangle.

Conjecture: The sum of the measures of the angles of a triangle is 180°.

Figure 2

A c t i v i t y F - 6

Figure 1

Figure 3

Contents Writing Good Definitions ................................................................................................................................G-2

Activity

A p p e n d i x G : D a y 7 G - 1


Writing Good Definitions

Directions: Cut apart on dotted lines and give one to each group.

Group Terms Examples Group 1 Define

Perpendicular Lines These are perpendicular lines:

These are not:

Group 2 Define Adjacent Angles

These are adjacent angles:

These are not:

Group 3 Define Exterior Angles

These are exterior angles:

These are not:

Group 4 Define Rectangle

These are rectangles:

These are not:

Group 5 Define a > b

These are accurate inequalities: 7 > 5 8 > 2

These are not: 5 = 5 7 < 5

Group 6 Define Vertical Angles

These are vertical angles:

These are not:

Group 7 Define Midpoint of a Segment

A and B are segment midpoints:

A and B are not:

Group 8 Define Interior Angle of a Triangle

These are interior angles of a triangle:

These are not:

Group 9 Define Point Between 2 Other Points

A and B are points between 2 other points:

A and B are not:

A c t i v i t y G - 2

A B

A B A

B

A

B

Contents Flowchart Proof and Paragraph Proof..............................................................................................................H-2

Transparency

A p p e n d i x H : D a y 8 H - 1


T r a n s p a r e n c y H - 2

Flowchart Proof and Paragraph Proof

If ∠1 and ∠2 are supplementary and ∠2 ≅ ∠3, then m∠1 + m∠3 = 180°.

∠1 and ∠2 are supplementary.

Given

Given

Definition of supplementary angles

Definition of congruent angles

Substitution

1 2 3

We are given that ∠1 and ∠2 are supplementary. By the definition of supplementary angles, we know that m∠1 + m∠2 = 180°. We are also given that ∠2 ≅ ∠3, so by definition of congruent angles, m∠2 = m∠3. We can substitute m∠3 in the above statement and arrive at our conclusion that m∠1 + m∠3 = 180°, which is what we were trying to prove.

m∠1 + m∠2 = 180°

∠2 ≅ ∠3 m∠2 = m∠3

m∠1 + m∠3 = 180°

Contents Proof Methods................................................................................................................................................... I-2

Transparency

A p p e n d i x I : D a y 9 I - 1


T r a n s p a r e n c y I - 2

Proof Methods

Proof Methods

Direct Reasoning Indirect Reasoning

Start with a true hypothesis.

Prove that the conclusion is true. Assume the conclusion is false

Prove that the hypothesis

is false

Prove that some other accepted

fact is false

Contents Assessing Your Understanding......................................................................................................................... J-2

Worksheet

Assessing Your Understanding......................................................................................................................... J-5 Key

A p p e n d i x J : D a y 1 0 J - 1


W o r k s h e e t J - 2

Assessing Your Understanding

Directions: Work in pairs to complete the worksheet.

1. Complete the following statement with the best word or phrase.

a. The two types of reasoning used to form conjectures are ___________________________________

b. Describe the difference between reasoning types and determine which type provides a more solid proof.

2. Explain and give an example of how counterexample is used in proof.

3. Write the converse, inverse, and contrapositive of this statement:

“If I am looking at page 84 of my Geometry book, then my Geometry book is open.”

Converse: ___________________________________________________________________________ Inverse: _____________________________________________________________________________ Contrapositive: _______________________________________________________________________ Which of the statements above, converse, inverse, or contrapositive, must have the same truth value as the

original statement? Explain why this is so, using appropriate mathematical terminology.

Names: ________________________________________ __________________________________

Period: _________________________________________ Date: _____________________________


A s s e s s i n g Y o u r U n d e r s t a n d i n g


4. The following proof is written in two-column form. In the space provided, rewrite it in both flowchart and paragraph form.

Given: A, B, C, and D are collinear points, as shown. AC = BD

Prove: AB = CD

Flowchart Proof:

Paragraph Proof:

AB CD≅

A B C D

Statements 1. A, B, C, and D are collinear. 1. Given

2. AC = BD 2. Given

3. AC = AB + BC 3. Segment Addition Postulate or Definition of between

4. BD = BC + CD 4. Segment Addition Postulate or Definition of between

5. AB + BC = BC + CD 5. Substitution (Statements 3 & 4 into Statement 2)

6. BC = BC 6. Reflexive Property of Equality

7. AB CD∴ = 7. Subtraction Property of Equality (Steps 5 & 6)

8. AB CD∴ ≅ 8. Definition of congruent segments (Step 7)

Reasons


A s s e s s i n g Y o u r U n d e r s t a n d i n g


5. Which is a correct indirect proof of the following choices (circle the correct answer): Given: ∠1 and ∠2 are not congruent. Prove: ∠1 and ∠2 are not vertical angles.

A. Assume ∠1 and ∠2 are not vertical angles. Then ∠1 and ∠2 are not congruent because vertical angles are congruent. This is the given information. Therefore, ∠1 and ∠2 are not vertical angles.

B. Assume ∠1 and ∠2 are vertical angles. Then ∠1 ≅ ∠2 because vertical angles are congruent. This contradicts the given information. Therefore, ∠1 and ∠2 are not vertical angles.

C. Assume ∠1 and ∠2 are congruent. Then ∠1 and ∠2 are vertical angles because vertical angles are congruent. This contradicts what is to be proven. Therefore, ∠1 and ∠2 are not vertical angles.

D. Assume ∠1 and ∠2 are not congruent. Then ∠1 and ∠2 are not vertical angles because vertical angles are congruent. This is what is to be proven. Therefore, ∠1 and ∠2 are not vertical angles.

6. You are a member of the debate team. It is time to test your ability to make convincing arguments. At an upcoming meet, you and your debate partner will be allotted $15 to buy pizza. There are two options to consider:

Option A: One large (15-inch diameter) pizza for $14.99

Option B: Two medium (10-inch diameter) pizzas for $7.50 each

Select the most economical option. Use your reasoning skills to make a convincing argument for why your selection is the best. Write a short explanation of your case.

1 2


K e y J - 5

Assessing Your Understanding Key

1.

2. A counterexample is used to prove that a statement is not true. In proof, it is most often used to disprove a statement. For example, the statement “If x² = 4, then x = 2” can be disproved with the counterexample x = –2.

3. Converse: If my Geometry book is open, then I am looking at page 84.

Inverse: If I am not looking at page 84 of my Geometry book, then my Geometry book is not open.

Contrapositive: If my Geometry book is not open, then I am not looking at page 84.

The contrapositive must be logically equivalent to the original statement: If my Geometry book is not open, then I cannot be looking at page 84. On the other hand, if my Geometry book is open, I could be looking at a page other than page 84. Also, if I am not looking at page 84 of my Geometry book, the book does not have to be closed, as it could be open to a different page.

4. Flowchart Proof:

Paragraph Proof: We are given that A, B, C, and D are collinear and AC = BD. By the segment addition postulate, AC = AB + BC and BD = BC + CD. Substituting AB + BC for AC and BC + CD for BD into AC = BD gives AB + BC = BC + CD. Subtracting BC from both sides of the equation by applying the subtraction property of equality gives AB = CD. This can be done because BC = BC according to the reflexive property of equality. Since AB = CD, because of the definition of congruence.

AB CD≅

A, B, C, and D are collinear

Given AB + BC = BC + CD

AC = AB + BC

Segment Addition Postulate

BD = BC + CD

Segment Addition Postulate

AC = BD Given

Substitution

BC = BC

Reflexive Property of Equality

AB = CD Subtraction Property of

Equality

AB CD≅

Definition of Congruence

a. Inductive and deductive reasoning

b. Inductive reasoning is making conjectures based on several observations. Deductive reasoning is drawing conclusions based on true statements. Deductive reasoning provides a more solid proof because it is based on statements that are already known to be true.


A s s e s s i n g Y o u r U n d e r s t a n d i n g K e y

K e y J - 6

5. B is the best indirect proof. To see this, consider the options in order:

A. Assumes the conclusions is true—a very common error. In addition, the fact that vertical angles are congruent is misapplied.

B. Assumes the conclusion is false and argues, correctly, that the hypothesis must be false. This is a correct indirect proof by contrapositive.

C. Assumes the hypothesis is false—an inappropriate strategy for either direct or indirect proof.

D. Assumes the hypothesis is true. It relies upon the equivalence of a key fact (vertical angles are congruent) and its contrapositive and looks like an attempt at direct proof, but there’s a correct contrapositive proof hiding within.

6. To find which is the better deal, first find which option gives the most amount of pizza. The area of the pizzas will determine which option gives the greater amount. To find the area of a circle, square the radius of the circle and multiply by π. The diameters of the pizzas are given, so divide the diameters by 2 to get the radii. The area of the pizza in Option A is approximately 177 square inches (7.5² × 3.14), and the area of each pizza in Option B is approximately 78.5 square inches (5² × 3.14). Since Option B offers two of these, multiply 78.5 by 2, and the total area of pizza in Option B is 157 square inches. Since these pizzas cost $7.50 each, it would cost $15. Option A gives more pizza for less money. Therefore, Option A is the better deal.

Contents Sample Proof of the Pythagorean Theorem.....................................................................................................K-2

Handout

Building a Taco................................................................................................................................................K-3 Transparency

Doug’s Proof....................................................................................................................................................K-4 Worksheet

Doug’s Proof Key ............................................................................................................................................K-5 Key

A p p e n d i x K : E n h a n c i n g S t u d e n t L e a r n i n g K - 1


Sample Proof of the Pythagorean Theorem

Background knowledge: 1. The Pythagorean theorem states that for a right triangle with side lengths a, b, and c with c being the

hypotenuse a2 + b2 = c2.

2. The area of a square is found by squaring the length of a side.

3. The area of a triangle is × its base × its height. In a right triangle, the 2 legs are the base and the height.

Start with 2 squares, as shown:

The 4 triangles are congruent, each with side lengths a, b, and c as indicated. Thus, each side of the big square has a length (a + b).

Thus, the area of the big square is (a + b)2.

The area of each of the four triangles is a b. Since the big square is made up of 4 triangles and the smaller square with

side length c, the area of the big square can also be written as 4 · · a · b + c2.

Therefore, (a + b)2 = 4 · · a · b + c2, which simplifies to a2 + 2ab + b2 = 2ab + c2. Subtracting 2ab from both sides of the equation leaves a2 + b2 = c2.

12

12

12

12

H a n d o u t K - 2

a

b

c


T r a n s p a r e n c y K - 3

Building a Taco

Steps

1. Start with browned and seasoned ground beef, hard taco shells, shredded cheese, shredded lettuce, sour cream, and salsa.

1. Given

2. Heat the taco shells on a cookie sheet in a 400° oven for about 5 minutes; then remove them from the oven.

2. Heating the taco shells makes them crispy, warm, and easier to fill.

3. Fill the bottom of each taco shell with ground beef.

3. Putting the ground beef in first makes the taco retain heat.

4. Sprinkle shredded cheese directly over the ground beef.

4. The hot ground beef melts the cheese.

5. Put shredded lettuce over the shredded cheese.

5. This step prevents the lettuce from wilting due to the heat of the ground beef.

6. Add sour cream and salsa. 6. Sour cream and salsa help to keep the lettuce in the shell.

Reasons


W o r k s h e e t K - 4

Doug’s Proof

Directions: Work in pairs to complete the worksheet. A conditional statement is given. Choose statements from the list and put them in the correct order to prove the conditional statement. You will not use all of the statements listed.

If Doug doesn’t wash the dog, then Miguel will go fishing Saturday. If Doug doesn’t wash the dog, then Doug’s sister Amanda will do it.

If the dog makes the living room dirty, then Doug will have to stay home Saturday to clean.

If Doug puts the dog outside, then the dog will run away.

If Doug stays home Saturday to clean, then he won’t go to the baseball game with Miguel.

If the dog makes the living room dirty, then Doug will put the dog outside.

If Doug doesn’t wash the dog, then the dog will make the living room dirty.

If Miguel gets a new fishing pole, then he will go fishing Saturday.

If Doug doesn’t go to the baseball game with Miguel, then Miguel will go fishing Saturday.

If Doug’s sister Amanda washes the dog, then Doug will do the dishes for her on Monday.

If Miguel gets paid for mowing lawns Friday, then he gets a new fishing pole.

Names: ________________________________________________________________

Period: _________________________________________________________________ Date:


K e y K - 5

Doug’s Proof Key

If Doug doesn’t wash the dog, then Miguel will go fishing Saturday. 1. If Doug doesn’t wash the dog, then the dog will make the living room dirty.

2. If the dog makes the living room dirty, then Doug will have to stay home Saturday to clean.

3. If Doug stays home Saturday to clean, then he won’t go to the baseball game with Miguel.

4. If Doug doesn’t go to the baseball game with Miguel, then Miguel will go fishing Saturday.

A p p e n d i x L : S e c o n d a r y C o u r s e O b j e c t i v e s L - 1

Secondary Course Objectives A primary course objective

is the central focus of the unit and is explicitly assessed in an embedded assessment and/or in the unit

assessment. A secondary course objective

is less important to the focus of the unit, but is one that students need to know and use when completing activities for this unit and

may or may not be explicitly assessed by the unit assessment or an embedded assessment.

Course objectives considered primary for this unit are listed on pages 1–2. Below is a list of secondary course objectives associated with this unit.

Selected Secondary Course Objectives

B.1. Process Objectives b. Use a variety of strategies to set up and solve increasingly complex

problems

c. Represent data, real-world situations, and solutions in increasingly complex contexts (e.g., expressions, formulas, tables, charts, graphs, relations, functions) and understand the relationships

English 10 D.2. Application c. Give impromptu and planned presentations (e.g., debates, formal

meetings) that stay on topic and/or adhere to prepared notes

g. Actively participate in small-group and large-group discussions, assuming various roles

Course Objectives Measured by Assessments This table presents at a glance how the course objectives are employed throughout the entire unit. It identifies

those objectives that are explicitly measured by the embedded and unit assessments. The first column lists course objectives by a three-character code (e.g., B.1.a.); columns 2–11 list the assessments.

A p p e n d i x M : C o u r s e O b j e c t i v e s M e a s u r e d b y A s s e s s m e n t s M - 1

Coded Course

Objectives

Embedded Assessments Unit

Assessments

Group Work

Scoring Rubric

Conditional Statements homework

Advertising Statements

Linear Equation

Proof

Jumbled Proof Steps

Triangle Angle Sum Theorem

Journal Assignment

Congruent Complements

Proof homework

Proof Methods

homework Assessing Your Understanding

B.1.a. X

B.1.b. X

B.1.c. X

B.1.d. X X X X X X X

B.1.f. X X X X X

B.1.h. X

C.1.a. X X X X

C.1.b. X X X X X X

C.1.c. X X X X

C.1.e. X X X X X X

English 10 D.2.c. X

English 10 D.2.g. X

unit 3 shadow of a doubt-logic+proof · pdf filewith the converse, inverse, and contrapositive...

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