1 INDUCTIVE AND DEDUCTIVE REASONING

2y ago
34 Views
2 Downloads
4.09 MB
62 Pages
Last View : 1m ago
Last Download : 2m ago
Upload by : Baylee Stein
Transcription

1 INDUCTIVE AND DEDUCTIVEREASONINGSpecific Outcomes Addressed in the ChapterPrerequisite Skills Neededfor the ChapterWNCP1. Analyze and prove conjectures, using inductive and deductive reasoning, to solveproblems. [C, CN, PS, R] [1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7]2. Analyze puzzles and games that involve spatial reasoning, using problem solvingstrategies. [CN, PS, R, V] [1.7]This chapter, while focusing onnew learning related to inductiveand deductive reasoning, providesan opportunity for students toreview the following skills andconcepts:Achievement Indicators Addressed in the ChapterShape and SpaceLogical Reasoning Determine parallel side lengthsin parallelograms and otherquadrilaterals. Draw diagonals in rectanglesand medians in triangles. Identify vertically oppositeangles and supplementaryangles in intersecting lines.Copyright 2011 by Nelson Education Ltd.Logical Reasoning1.1 Make conjectures by observing patterns and identifying properties, and justify thereasoning. [1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7]1.2 Explain why inductive reasoning may lead to a false conjecture. [1.1, 1.2, 1.3, 1.4,1.5, 1.6, 1.7]1.3 Compare, using examples, inductive and deductive reasoning. [1.4, 1.6, 1.7]1.4 Provide and explain a counterexample to disprove a given conjecture. [1.3, 1.4,1.5, 1.6, 1.7]1.5 Prove algebraic and number relationships, such as divisibility rules, numberproperties, mental mathematics strategies or algebraic number tricks [1.4]1.6 Prove a conjecture, using deductive reasoning (not limited to two column proofs).[1.4]1.7 Determine if a given argument is valid, and justify the reasoning. [1.2, 1.4, 1.5, 1.6,1.7]1.8 Identify errors in a given proof; e.g., a proof that ends with 2 1. [1.5]1.9 Solve a contextual problem that involves inductive or deductive reasoning. [1.4,1.6, 1.7]2.1 Determine, explain and verify a strategy to solve a puzzle or to win a game. [1.7]2.2 Identify and correct errors in a solution to a puzzle or in a strategy for winning agame. [1.7]2.3 Create a variation on a puzzle or a game, and describe a strategy for solving thepuzzle or winning the game. [1.7]Patterns and Relations Represent a situationalgebraically. Simplify, expand, and evaluatealgebraic expressions. Solve algebraic equations. Factor algebraic expressions,including a difference ofsquares. Apply and interpret algebraicreasoning and proofs. Interpret Venn diagrams.Number Identify powers of 2,consecutive perfect squares,prime numbers, and multiples. Determine square roots andsquares.Chapter 1 IntroductionMath11 TR Ch01W p001-062.indd 119/17/10 2:31 PM

Chapter 1: Planning ChartKey Question/CurriculumCharts (TR)Getting Started,pp. 4–5Planning, p. 4Assessment, p. 62 days1.1: Making Conjectures:Inductive Reasoning,pp. 6–15Planning, p. 7Assessment, p. 121 dayQ9LR1 [C, CN, PS, R]calculator,compass, protractor, andruler, or dynamic geometrysoftware,tracing paper (optional)1.2: Exploring the Validityof Conjectures, pp. 16–17Planning, p. 14Assessment, p. 161 dayLR1 [CN, PS, R]Explore the Math: OpticalIllusions,ruler,calculator1.3: Using Reasoning toFind a Counterexample toa Conjecture, pp. 18–25Planning, p. 17Assessment, p. 201 dayQ14LR1 [C, CN, R]calculator,ruler,compass1.4: Proving Conjectures:Deductive Reasoning,pp. 27–33Planning, p. 24Assessment, p. 281 dayQ10LR1 [PS, R]calculator,ruler1.5: Proofs That Are NotValid, pp. 36–44Planning, p. 30Assessment, p. 331 dayQ7LR1 [C, CN, PS, R]grid paper,ruler,scissors1.6: Reasoning to SolveProblems, pp. 45–51Planning, p. 35Assessment, p. 381 dayQ10LR1 [C, CN, PS, R]calculator1.7: Analyzing Puzzlesand Games, pp. 52–57Planning, p. 39Assessment, p. 421 dayQ7LR2 [CN, PS, R]counters in two colours orcoins of two denominations,toothpicks (optional),paper clips (optional),Solving Puzzles (Questions 10to 13)Materials/MastersReview of Terms andConnections,Diagnostic Test5 daysDeveloping a Strategy to SolveArithmagons,Solving Puzzles,Project Connection 1: Creatingan Action Plan,Chapter TestCopyright 2011 by Nelson Education Ltd.Applying Problem-Solving Strategies, p. 26Mid-Chapter Review, pp. 34–35Chapter Self-Test, p. 58Chapter Review, pp. 59–62Chapter Task, p. 63Project Connection, pp. 64–652Pacing(14 days)Lesson (SB)Foundations of Mathematics 11: Chapter 1: Inductive and Deductive ReasoningMath11 TR Ch01W p001-062.indd 29/17/10 2:31 PM

1 OPENERUsing the Chapter OpenerCopyright 2011 by Nelson Education Ltd.Discuss the photograph, and hypothesize about what happened in theprevious half hour. You could set up a role-playing situation, in which groupsof four students could take the roles of driver 1, driver 2, an eyewitness, andan investigator. Together, the four students could develop questions andresponses that would demonstrate their conjectures about what led up to theevents seen in the photograph. This could be set up as a series of successiveinterviews between the investigator and the other three people in thesituation.Tell students that, in this chapter, they will be examining situations,information, problems, puzzles, and games to develop their reasoning skills.They will form conjectures through the use of inductive reasoning and provetheir conjectures through the use of deductive reasoning.In Math in Action on page 15 of the Student Book, students will have anopportunity to revisit an investigative scenario through conjectures, witnessstatements, and a diagram. You may want to discuss the links amongreasoning, evidence, and proof at that point.Chapter 1 OpenerMath11 TR Ch01W p001-062.indd 339/17/10 2:31 PM

1 GETTING STARTEDIntroduce the activity by showing a map of the area from New York to theBay of Gibraltar. Have students work in pairs. Ask them to imagine thechallenges of travelling this distance by water in the present time. Howwould the challenges have been different in 1872? Discuss these challengesas a class, and then ask students to read the entire activity before respondingto the prompts.After students finish, ask them to share their explanations andjustifications. Discuss whether one explanation is more plausible thananother.Sample Answers to PromptsA. Answers may vary, e.g., there were four significant pieces of evidence:1. The hull was not damaged.2. No boats were on board.3. Only one pump was working.4. The navigation instruments, ship’s register, and ship’s papers weregone.B. Answers may vary, e.g., the bad weather could have scared the crew intothinking that the alcohol they were carrying was going to catch fire. Thecaptain and crew might have opened the hatches and then got into thelifeboats to be safe.C. Answers may vary, e.g., the bad weather could have been severe enoughto cause water to be washing over the bow of the ship. Since only onepump was working, perhaps the water level was rising inside the ship. Ifthe crew could not pump all the water out of the ship, they might haveopened the hatches at the front and the back to help bail out the water.If the water continued to rise, the captain and crew might have taken thenavigational equipment and the ship’s register and papers, and abandonedship into the lifeboats. If they left the ship during bad weather, theymight have lost contact with the Mary Celeste and their lifeboats mighthave sunk.D., E., F. Answers may vary, e.g., a piece of evidence that would support theexplanation would be confirmation that lifeboats had been aboard whenthe Mary Celeste left New York Harbour.4Student Book Pages 4 5Preparation and PlanningPacing50 min30 min10 minReview of Terms andConnectionsMary CelesteWhat Do You Think?Blackline Masters Review of Terms and Connections Diagnostic TestNelson Websitehttp://www.nelson.com/mathMath BackgroundThe activity provides studentswith an opportunity to reactivatepreviously introduced topics relatedto problem solving, which include justifying a response sorting information to find what isneededCopyright 2011 by Nelson Education Ltd.The Mystery of the Mary CelesteFoundations of Mathematics 11: Chapter 1: Inductive and Deductive ReasoningMath11 TR Ch01W p001-062.indd 49/17/10 2:31 PM

Background This mystery is true and well documented in court records. Charles Edey Fay’sbook (published in 1942 and reprinted in 1988) about the mystery is a factual studyof the case, unlike Arthur Conan Doyle’s short story (published in 1884), whichblends facts of the case with many pieces of fiction. Conan Doyle used the basicfacts in the historical records but took liberties by suggesting that the crew of theMary Celeste had departed only a very short time before the crew of the Dei Gratiaspotted the ship. Suggestions of tea still steaming in cups and items still fresh inthe galley (ship’s kitchen) could not have been true, based upon the first-hand dataentered into factual evidence. In August 2001, the wreck of the Mary Celeste was located off the coast of Haiti.The key words “Mary Celeste” and “mystery” entered into an Internet search enginewill yield more information about the mystery. As well, books have been writtenabout the mystery, but some ascribe details that are not supported by the evidencein the historical accounts.What Do You Think? page 5Use this activity to activate knowledge and understanding about inductiveand deductive reasoning. Explain to students that the statements involvemath concepts or skills they will learn in the chapter—they are not expectedto know the answers now. Ask students to read each statement, think about it,and decide whether they agree or disagree with it. Have volunteers explainthe reasons for their decisions. Students can share their reasoning in smallgroups, in groups where all agree or disagree, or in a general classdiscussion. Tell students that they will revisit their decisions at the end of thechapter.Sample Answers to What Do You Think?Copyright 2011 by Nelson Education Ltd.The correct answers are indicated with an asterisk (*). Students should beable to give the correct answers by the end of the chapter.1. Agree. Answers may vary, e.g., patterns can be represented byexpressions that show how the patterns change.PatternFigure Number (f)1234Number of Dots2468The pattern is represented by the expression 2f.*Disagree. Answers may vary, e.g., a pattern over a short time may notbe true all the time. Seeing four people exit a shop with coffee cups intheir hands does not mean that the next person leaving the shop will beholding a coffee cup.2. *Agree. Answers may vary, e.g., a pattern may be seen after examiningseveral examples. After seeing four people exiting a shop with coffeecups, a prediction can be made that the shop sells coffee. However, moreevidence is needed.Chapter 1 Getting StartedMath11 TR Ch01W p001-062.indd 559/17/10 2:31 PM

Disagree. Answers may vary, e.g., a pattern shows only what was and notwhat will be. More evidence is needed to make a reliable prediction.3. Agree. Answers may vary, e.g., the pattern shows increasing squares ofnumbers: 12, 22, 32, 42, 52, so the next three terms are 62, 72, and 82.*Disagree. Answers may vary, e.g., the pattern can be described asincreasing squares, but it can also be described as the sum of thepreceding number and the next odd number: 0 1, 1 3, 4 5, 9 7,16 9. In both descriptions of the pattern, however, the next three termswould be 36, 49, and 64.Initial Assessment for LearningWhat You Will See Students Doing.When students understand.If students misunderstand.Students decide that some pieces of evidence are moreimportant than others.Students place equal value on all pieces of evidence.Students make inferences about the patterns that the evidencepresents.Students make predictions that do not take into account theevidence available.Students justify their predictions based on the evidenceavailable.Students are unable to develop a justification that is clear andreasonable.Differentiating Instruction How You Can RespondEXTRA SUPPORT2. If students have difficulty visualizing the state of the shipwhen found by the crew of the Dei Gratia, then accessingblueprints for a ship of that type and size may be helpful.Students can do a search using key words such as “ship’splans” and “boat building” to look for these blueprints.Use Review of Terms and Connections, Teacher’sResource pages 53 to 56, to activate students’ skills.6Copyright 2011 by Nelson Education Ltd.1. If students have difficulty identifying the most importantpieces of evidence, scaffold the task by examining the piecesof evidence in sets of three. Ask: Of these three pieces ofevidence, which is the most important? Limiting the rangeof possibilities makes choices easier to make.Foundations of Mathematics 11: Chapter 1: Inductive and Deductive ReasoningMath11 TR Ch01W p001-062.indd 69/17/10 2:31 PM

1.1 MAKING CONJECTURES:INDUCTIVE REASONINGLesson at a GlanceGOALUse reasoning to make predictions.Prerequisite Skills/Concepts Identify perfect squares, prime numbers and odd and even integers. Determine parallel side lengths in parallelograms and other quadrilaterals.Preparation and Planning Draw diagonals in rectangles and medians in triangles.Pacing10 min35 45 min10 15 minWNCPSpecific OutcomeLogical Reasoning1. Analyze and prove conjectures, using inductive and deductive reasoning,to solve problems. [C, CN, PS, R]Achievement Indicators1.1 Make conjectures by observing patterns and identifying properties, andjustify the reasoning.1.2 Explain why inductive reasoning may lead to a false conjecture. IntroductionTeaching and LearningConsolidationMaterials calculator compass, protractor, and ruler, ordynamic geometry software tracing paper (optional)Recommended PracticeQuestions 3, 4, 6, 10, 14, 16Key QuestionQuestion 9New Vocabulary/Symbolsconjectureinductive reasoningMath BackgroundCopyright 2011 by Nelson Education Ltd.Student Book Pages 6 15This lesson provides an opportunity for students to develop their understandingof inductive reasoning through the mathematical processes of communication,connections, problem solving, and reasoning.Communication is promoted by sharing conjectures, while connections are madeusing the contexts presented, the evidence given, and the conjectures developed.Both communication and connections become integral parts of reasoning, asstudents justify the conjectures they have developed based on the context andevidence.Mathematical Processes Communication Connections Problem Solving ReasoningNelson Websitehttp://www.nelson.com/mathProblem solving is established through the variety of interpretations possible,based on the given context and evidence. This, in turn, promotes communicationabout the different interpretations and justifications.1.1: Making Conjectures: Inductive ReasoningMath11 TR Ch01W p001-062.indd 779/17/10 2:31 PM

1Introducing the Lesson(10 min)Explore (Pairs, Class), page 6The Explore problem can be assigned for students to discuss in pairs, or itcan be discussed as a class. It provides an opportunity for students to make aconjecture based on given evidence and to develop justification for theirconjecture. The following questions may help students: Where might you have seen this sequence? How could this sequence be part of a pattern?Have students share their explanations with the class. Encouragedifferent conjectures for the given sequence, and explore the possibility thatmore than one conjecture may be correct.Sample Solutions to Explore If the colour sequence is red, orange, and yellow, the rest of the sequencemay be green, blue, and purple. These colours are the primary andsecondary colours seen on a colour wheel. If the colour sequence is red, orange, and yellow, the rest of the sequencemay be green, blue, indigo, and violet. These colours are those of arainbow. If the colour sequence is red, orange, and yellow, the rest of the sequencemay repeat these three colours.(35 to 40 min)Investigate the Math (Class), page 6This investigation allows students to discuss patterning and the predictionabout the 10th figure. Help students understand that the pattern focuses onthe congruent unit triangles, not on the different-sized triangles.Math Background 8To make conjectures that are valid, based on a pattern of evidence, students needto have a variety of sample cases to view. Since any pattern requires multiple casesto support it, more than one or two specific cases are needed to begin to formulatea conjecture. The more cases that fit the conjecture, the stronger the validity of theconjecture becomes. The strength of a conjecture, however, does not substitute forproof. Proof comes only when all cases have been considered.Copyright 2011 by Nelson Education Ltd.2Teaching and LearningFoundations of Mathematics 11: Chapter 1: Inductive and Deductive ReasoningMath11 TR Ch01W p001-062.indd 89/17/10 2:31 PM

Sample Answers to PromptsA.Figure12345678910Number of Triangles149162536496481100B. The pattern in the table shows that the number of triangles equals thesquare of the figure number.C. li\ li\ ,D. Figure 11 has 112 or 121 triangles.Figure 12 has 122 or 144 triangles.The numeric pattern in the table shows that each figure will have aperfect square of congruent triangles. The number of congruent trianglesin each figure is the square of the figure number.Reflecting, page 6Students can work on the Reflecting questions in pairs, before or instead of aclass discussion.Copyright 2011 by Nelson Education Ltd.Sample Answers to ReflectingE. Georgia’s conjecture is reasonable because, when the table is extended tothe 10th figure, the pattern of values is the same as Georgia’s prediction.F. Georgia used inductive reasoning by gathering evidence about morecases. This evidence established a pattern. Based on this pattern, Georgiamade a prediction about what the values would be for a figure not shownin the evidence.G. A different conjecture could be made because a different pattern couldhave been seen. If the focus had been only on the congruent triangleswith their vertices at the bottom and their horizontal sides at the top, thenthe following conjecture could have been made: The 5th figure will have10 congruent triangles. li\ ( li\ ) li\ * li\ li\ ,1.1: Making Conjectures: Inductive ReasoningMath11 TR Ch01W p001-062.indd 999/17/10 2:31 PM

3Consolidation(10 to 15 min)Apply the Math (Class, Pairs), pages 7 to 11Sample Answers to Your Turn QuestionsExample 1: From the evidence given, a conjecture that August is the driestmonth of the year is reasonable. For the 5 years of data, Augusthas the least rainfall: 121.7 mm.10BackgroundWeather ConjecturesLong before weather forecastsbased on weather stationsand satellites were developed,people had to rely on patternsidentified from observation of theenvironment to make predictionsabout the weather. For example: Animal behaviour: First Nationspeoples predicted spring bywatching for migratory birds. Ifsmaller birds are spotted, it is asign that spring is right aroundthe corner. When the crow isspotted, it is a sign that winteris nearly over. Seagulls tendto stop flying and take refugeat the coast when a storm iscoming. Turtles often search forhigher ground when they expecta large amount of rain. (Turtlesare more likely to be seen onroads as much as 1 to 2 daysbefore rain.) Plant behaviour: Pine conescales remain closed if thehumidity is high, but open indry air. The leaves of oak andmaple trees tend to curl in highhumidity. Personal: Many people can feelhumidity, especially in their hair(it curls up and gets frizzy). Highhumidity tends to precede heavyrain.Copyright 2011 by Nelson Education Ltd.Using the Solved ExamplesIn Example 1, a conjecture is developed based on the evidence for annualrainfall. Students should be encouraged to explain, in their own words, howand why Lila came up with her conjecture. When discussing the example,focus on the patterns that have been identified. Encourage students toexplain whether the reasoning makes sense.In Example 2, a conjecture about the product of odd integers is developed.Students are encouraged to discuss the limited number of examples that Jayused to make his conjecture. Does the quantity of evidence make theconjecture more or less believable? What other evidence might Jay haveused? How does the evidence that Jay did use show a pattern?Example 3 presents two different methods for developing a conjectureabout the difference between consecutive perfect squares: numerically andgeometrically. Students are encouraged to discuss the strengths of bothconjectures and the evidence on which each was developed.In Example 4, two different methods are used to develop conjectures aboutthe shape created by joining the midpoints of adjacent sides in a quadrilateral:using a protractor and ruler or using dynamic geometry software. Encouragestudents to test Marc’s and Tracey’s solutions to reinforce geometricunderstanding and construction skills. Sorting quadrilaterals in a Venndiagram to look for common and unique attributes of different quadrilateralscould be a reminder activity prior to studying Example 4. Ask the followingquestions to guide students through the solutions: How did Marc decide to focus upon a parallelogram? What pattern didMarc recognize before he made his conjecture? How did Marc’s use ofthree different ways to show that the joining of midpoints created aparallelogram support his conjecture? Could he have used the same wayeach time? Would using one way strengthen the conjecture? What pattern did Tracey notice that led to her conjecture? How do theattributes of the shapes Tracey has focused upon differ from those thatMarc noticed? Is there another pattern that might have been noticed from Marc’s work?from Tracey’s work? Would Tracey’s conjecture fit Marc’s work? Would Marc’s conjecture fitTracey’s work?Foundations of Mathematics 11: Chapter 1: Inductive and Deductive ReasoningMath11 TR Ch01W p001-062.indd 109/17/10 2:31 PM

Example 2: Yes. Jay’s conjecture is convincing because all the differentcombinations with positive and negative odd integers wereused as samples. These three samples showed a pattern in theirproducts, which Jay then tested with different integers. Jay’sconjecture was supported by this last sample.No. Jay looked at only three cases before he made hisconjecture, then tested it with only one more example. This isnot a lot of evidence to base a conjecture on.Example 3: It is possible to have two different conjectures about the samesituation because different samples were used to developthe conjecture. Francesca used different values for the sizesof consecutive squares. When she examined her evidence,the common feature from her examples was different from thecommon feature that Steffan found from the evidence he haddeveloped.Example 4: a) The quadrilaterals that Marc and Tracey used weredifferent. The quadrilaterals that Marc used were morevaried than those that Tracey used.b) Based on the evidence used, both conjectures seem valid.The conjecture that Marc developed would hold true forall of Tracey’s quadrilaterals, since a rhombus is a specialtype of parallelogram. But Tracey’s conjecture would nothold true for all of Marc’s quadrilaterals, since not allparallelograms are rhombuses.Sample Solution to the Key QuestionCopyright 2011 by Nelson Education Ltd.9. Sum of an odd integer and an even integer:Odd 1 1 1 1 53Even 2 2 2 2 14Sum 3 3 1 1 39Based on the evidence gathered and the pattern in the sums, thefollowing conjecture can be made: The sum of an odd integer and aneven integer will always be an odd integer.Odd 5 5 100Even 6 6 99Sum 1 1 1Closing (Pairs, Class), page 15Question 19 gives students an opportunity to make connections among theterms conjecture, inference, and hypothesis. Arguments can be developed tosupport the two given opinions. Allow students to explore the nuances ofmeaning among these terms. Use reference resources and students’knowledge of these terms to support students’ understanding of how theseterms are similar and how both Lou’s and Sasha’s opinions are valid.1.1: Making Conjectures: Inductive ReasoningMath11 TR Ch01W p001-062.indd 11119/17/10 2:31 PM

Assessment and Differentiating InstructionWhat You Will See Students Doing.When students understand.If students misunderstand.Students make conjectures that consider the patterns in theinformation given and evidence gathered.Students are unable to develop conjectures, or they makeconjectures without seeing a pattern in the evidence, or theydo not recognize the patterns within the evidence.Students justify their conjectures by drawing upon specificevidence from the examples and by developing new examplesto support their conjectures.Students make faulty connections between the conjecturesand the evidence.Key Question 9Students correctly interpret the math language of the problem.Students are unable to interpret the math language of theproblem.Students make a conjecture about the sum of an odd integerand an even integer, based on evidence they have gathered.Students are uncertain how to gather evidence about thesum of an odd integer and an even integer. Students make aconjecture that is not based on the evidence.Students justify their conjecture based on the evidencegathered and the specific patterns recognized.Students’ justifications minimally connect to the evidence or donot make any connections to specific examples.Differentiating Instruction How You Can RespondEXTRA SUPPORT1. If students have difficulty interpreting the language of theproblem, review Example 2, its language, and the steps thatwere used to develop a conjecture.2. If students have difficulty seeing a pattern in the specificexamples they try, suggest that they use a table to show theirresults for the specific examples. The table may help studentsfocus upon the patterns in the evidence.1. Students could create their own problem to investigate bygathering data, making conjectures, and then testing theirconjectures with more specific cases.122. Students could work in pairs to develop sets of data andconjectures on separate cards. These cards could be used ina concentration game.Copyright 2011 by Nelson Education Ltd.EXTRA CHALLENGEFoundations of Mathematics 11: Chapter 1: Inductive and Deductive ReasoningMath11 TR Ch01W p001-062.indd 129/17/10 2:32 PM

Math in ActionStudents can be invited to reflect on the discussion of the chapter opener whendealing with this problem. The similarities between the situation in the chapter openerand the situation here may encourage students to consider what each person saw inlight of his or her perspective and experience during the accident. Various conjecturesmay be developed, but each needs to be linked to the evidence gathered.Sample SolutionConjectures: Witness at stop sign: Yellow car did not completely stop; blue car was speeding. Driver of blue car: I was driving 60 km/h; the yellow car did not stop. Driver of yellow car: I came to a full stop. Investigator: No brake marks were evident due to snow cover.Conjecture about the cause of the accident: Driver of blue car was not familiar withthe area, its speed limits, or its traffic patterns.Evidence that supports the conjecture: Passenger in blue car was looking at a map atthe time of the accident.Three questions to ask: Investigator: What evidence showed slippery road conditions? Witness: Which car was in the intersection first? In which direction were youcrossing the street?Copyright 2011 by Nelson Education Ltd.The cause of this accident cannot be proved, since there are conflicting pieces ofevidence. Each driver contradicts the other, and there is minimal corroboration foreither driver’s allegation.1.1: Making Conjectures: Inductive ReasoningMath11 TR Ch01W p001-062.indd 13139/17/10 2:32 PM

GOALDetermine whether a conjecture is valid.Prerequisite Skills/Concepts Gather evidence to support or refute a conjecture.Student Book Pages 16–17 Use inductive reasoning to make a conjecture.Preparation and PlanningWNCPSpecific OutcomePacing10 min35–45 min10–15 minLogical Reasoning1. Analyze and prove conjectures, using inductive and deductive reasoning,to solve problems. [C, CN, PS, R]Achievement Indicators1.1 Make conjectures by observing patterns and identifying properties, andjustify the reasoning.1.2 Explain why inductive reasoning may lead to a false conjecture.1.7 Determine if a given argument is valid, and justify the reasoning.1Introducing the Lesson(10 min)To introduce a discussion about the validity of conjectures, present thefollowing situation: We know that optical illusions trick our eyes intobelieving something that may not be valid. How do these optical illusionsmake us think that things are not as they are? What methods can be used tocheck the validity of the conjectures?Caution: A web search for optical illusions will result in many examplesof optical illusions that are different from those best suited to this lesson.Care needs to be exercised when using online resources, since some opticalillusions may not be appropriate for classroom use.2Materials Explore the Math: Optical Illusions ruler calculatorRecommended PracticeQuestions 1, 2, 3Mathematical Processes Communication Connections Problem Solving ReasoningBlackline Master Explore the Math: Optical IllusionsNelson Websitehttp://www.nelson.com/mathMath Background Examining optical illusionsand how they “trick” your eyesprovides an opportunity to raisethe issue of valid versus invalidconjectures. Optical illusions also providestudents with the opportunity toexplore data that may support orrefute a conjecture. Optical illusions provide theopportunity for students torevise a conjecture based onevidence they gather.Teaching and Learning(35 to 45 min)Explore the Math (Individual, Pairs, Class), page 16IntroductionTeaching and LearningConsolidationCopyright 2011 by Nelson Education Ltd.1.2 EXPLORING THE VALIDITYOF CONJECTURESLesson at a GlanceIntroduce the exploration by asking students to identify and record their firstreaction to each optical illusion. Then,

2.1 Determine, explain and verify a strategy to solve a puzzle or to win a game. [1.7] 2.2 Identify and correct errors in a solution to a puzzle or in a strategy for winning a game. [1.7] 2.3 Create a variation on a puzzle or a game, and describe a strategy for solving the puzzle or winning t

Related Documents:

Use inductive reasoning to make a conjecture about the sum of a number and itself. Then use deductive reasoning to show that the conjecture is true. 11. Decide whether inductive reasoning or deductive reasoning is used to reach the conclusion. Explain your reasoning. All multiples of 8 are divisible by 4. 64 is a multiple of 8. So, 64 is .

Section 2.2 Inductive and Deductive Reasoning 75 2.2 Inductive and Deductive Reasoning Writing a Conjecture Work with a partner. Write a conjecture about the pattern. Then use your conjecture to draw the 10th object in the pattern. a. 1234567 b. c. Using a Venn Diagram Work with a partner. Use the Venn diagram to determine whether the statement is

Deductive Reasoning: Deductions made based on factual information. Ex. ) Mike is older than Pete and Pete is older than Diane, therefore Mike is older than Diane. We make conjectures using inductive reasoning. We use deductive reasoning to Prove Conjectures. Ex. )Conjecture: When you add t

2.1 Use Inductive Reasoning Obj.: Describe patterns and use inductive reasoning. Key Vocabulary Conjecture - A conjecture is an unproven statement that is based on observations. Inductive reasoning - You use inductive reasoning when you find a pattern in specific cases and then write a conjecture for the general case. Counterex

2-1 Using Inductive Reasoning to Make Conjectures When several examples form a pattern and you assume the pattern will continue, you are applying inductive reasoning. Inductive reasoning is the process of reasoning that a rule or statement is true because specific cases are true. You may use inductiv

These first two steps involve inductive reasoning to form the conjecture. 3. Verify that your conjecture is true in all cases by using logical reasoning. 5 For Example: . Use inductive reasoning to come up with a conjecture about the number of points and the chords that can be made. Solution: Number of Points Number of Regions 2

Use inductive reasoning to complete this statement: If a triangular number is multiplied by eight and then one is added to the product a _ number is obtained. MGF 1106 - 1.1 Inductive and Deductive Reasoning Date: _ . When you made your conjecture in step 7 using the results from the specific number you tried what kind of reasoning .

Deductive reasoning or deduction is the type of logic used in hypothesis-based science. In deductive reason, the pattern of thinking moves in the opposite direction as compared to inductive reasoning. Deductive reasoning is a form of logical thinking that uses