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UNIT TWO REASONING, JUSTIFICATION AND PROOF 18 HOURS Math 521B Revised Dec 20, 00 34 By the end of grade 11, students will be expected to: E5 use inductive and deductive reasoning when observing patterns, developing properties and making conjectures Elaborations - Instructional Strategies/Suggestions Introduction (6.1) According to the 2000 NCTM Standards document, reasoning and proof are “not special activities reserved for special times or topics in the curriculum, but should be a natural part of classroom discussions. Critiquing arguments and discussing conjectures are delicate matters; plausible guesses should be discussed even if they turn out to be wrong.” Examples of reasoning are: < algebraic or symbolic reasoning < statistical reasoning < probabilistic reasoning < spatial reasoning Reasoning is a process of thinking about a mathematical (or other) question. Justification is a rationale or argument for some mathematical proposition. This can be inductive reasoning. Proof is a justification that is logically valid, based on initial assumptions, definitions and previously proved results. This is a more formal argument and is usually a deductive level of reasoning. There are two types of reasoning that we will examine in this unit: < inductive < deductive Inductive Reasoning Inductive reasoning is the process of recognizing patterns and making generalizations or conjectures from the observations. Scientific formulae are usually arrived at inductively. Invite students to read and discuss the Triangle Sum Theorem and the Exterior Angle Theorem on p.339. Students could try to draw a number of triangles, measure the angles and notice the pattern that the sum is always 1800. Students could then extend one of the sides of these triangles in one direction to form an exterior angle. Measuring the exterior angle and the two remote interior angles in each triangle should support the Exterior Angle Theorem. Note to Teachers: Students will be expected to keep a Journal which includes definitions, theorems, postulates and corollaries in a separate notebook which may be used in the unit exam. This journal can continue in the next unit as well. 35 Worthwhile Tasks for Instruction and/or Assessment Inductive Reasoning (6.1) Group Activity Study the given equations and complete the conjecture started below that holds for all the equations. 3 + 7 = 10 !1 + 5 = 4 9 + 13 = 22 7 + 11 = 18 Conjecture: The sum of two odd numbers is always an... Suggested Resources Inductive Reasoning Math Power 11 p.340 #13,1-9,11,12, 15,16,21 Note: # 9 in the list above can be done using a Reflect-View Journal Write a conjecture about combining two numbers using a mathematical operation. Show a number of examples supporting your conjecture. Performance/Manipulative Draw any quadrilateral. Determine the midpoints of each side using a Reflect-View. Connect these midpoints. What kind of quadrilateral seems to result? Either repeat your experiment several times or compare your result with other members in your group. What is your group’s conjecture? What kind of reasoning has lead to your conclusion? Group Discussion Here are the first few rows of Pascal’s triangle. The sum of the numbers is shown to the right. According to the pattern, make a conjecture about the sum of the numbers in the 20th row. In any row in general. 1 1 1 1 2 1 1 3 3 1 1 4 6 41 1 5 10 10 5 1 1 6 15 20 15 6 1 1 2 4 8 16 32 64 Pencil/Paper a) Consider the number patterns to the right and verify that they are correct. b) What are the next two lines in the number pattern? c) What will be the sum of the first ten positive odd integers? d) Make a conjecture about the sum in any row. 36 1 = 12 1 + 3 = 22 1 + 3 + 5 = 32 1 + 3 + 5 + 7 = 42 1 + 3 + 5 + 7 + 9 = 52 By the end of grade 11, students will be expected to: Elaborations - Instructional Strategies/Suggestions Inductive Reasoning (cont’d) Testing the veracity of a statement with numerical or specific examples is inductive reasoning. E5 use inductive and deductive reasoning when observing patterns, developing properties and making conjectures Note to Teachers: Show a diagram like the one below to demonstrate the concept of an exterior angle. E6 use deductive reasoning, construct logical arguments and be able to determine, when given a logical argument, if it is valid E27 write proofs using various axiomatic systems E28 demonstrate an understanding that reasoning and proof are essential and powerful parts of mathematics p1, p2 and p3 are the angles of the triangle p4 and p6 are the exterior angles. p5 is not classed as an angle exterior to the triangle. We will see in section 6.3 that it is called an angle opposite to p3. In the examples on p.339-340 we see numerical examples used to make conjectures about some observed situation. The examples come from: GCO E: spatial sense GCO B: operation sense GCO C: patterns and relationships 37 Worthwhile Tasks for Instruction and/or Assessment Inductive Reasoning (cont’d) Performance Count the number of regions resulting when 2, 3, 4 and 5 points are connected and complete the table.Make a conjecture about the pattern suggested in the figures below. Then use the conjecture about the number of regions formed when 6 points are connected. Finally, draw a circle, place 6 points on the circumference join them and count the regions formed. Suggested Resources Inductive Reasoning (cont’d) Solution Activity Complete the following calculations: 1×8+1= 12 × 8 + 2 = 123 × 8 + 3 = 1234 × 8 + 4 = Make a conjecture about the pattern observed. Use the conjecture to predict the answers to: 12345 × 8 + 5 = 123456 × 8 + 6 = 123456789 × 8 + 9 = Use a calculator to check your predictions. Solution 1×8+1=9 12 × 8 + 2 = 98 123 × 8 + 3 = 987 1234 × 8 + 4 = 9876 98765 987654 987654321 This is the end of this pattern. 38 By the end of grade 11, students will be expected to: Elaborations - Instructional Strategies/Suggestions Analysing Conjectures (6.2) In the previous section we observed patterns and made conjectures. Here we will analyse conjectures by using specific examples. If we can find even one counter-example that does not support the original conjecture then the conjecture has to be revised or discarded altogether. Invite students to read the first two paragraphs on p.343. E5 use inductive and deductive reasoning when observing patterns, developing properties and making conjectures In science, the scientific method is used to make advances in scientific knowledge. The steps involved in the method are: Observation - some event or pattern is observed causing a person to think about it. Hypothesis(Conjecture) - a conjecture is made about the observations E29 develop and evaluate mathematical arguments and proofs Experimentation - the conjecture is tested by searching for a counterexample that will disagree with the conjecture After many examples have been tested and a counter-example has not been found, then the conjecture still hasn’t been proved conclusively but it can be considered a: Theory - if it is trying to explain why some event happens. Law - if it merely describes what is taking place. Many times a formula is used to aid in the description of the law. Students should be able to see that what they are essentially doing is using the Scientific Method. 39 Worthwhile Tasks for Instruction and/or Assessment Analysing Conjectures (6.2) Pencil/Paper A number raised to a positive integral power yields a larger number. a) give two examples that support this conjecture. b) give a counter-example that shows the conjecture is false. Suggested Resources Analysing Conjectures Math Power 11 p.345 #1-3,5,7-10,15, 16 Pencil/Paper Provide a counter-example to prove that each conjecture below is false: a) if x > 0, then x < x b) if the diagonals of a quadrilateral are perpendicular, then the quadrilateral is a square. Research Conjecture: If you live in a country bordering Cameroon to the north, then you live in the Central African Republic. Give two counter-examples that show the conjecture is false. Journal What is the purpose of finding counter-examples to conjectures? Written Assignment Complete each of the following statements. 23 × 64 = 26 × 93 = 41 × 28 = 32 × 46 = 62 × 39 = 14 × 82 = 69 × 64 = 96 × 46 = 84 × 36 = 48 × 63 = Make a conjecture based on the pattern above. Try to find two counter-examples to the conjecture. Group Activity/Open-ended Question Make a conjecture on anything you have observed in the world around you. Bring the conjecture to your group and find two examples that support the conjecture and two counterexamples to the conjecture. 40 Solution It seems that one can conjecture that the product of a pair of two digit numbers is the same as the product of the numbers formed by reversing their digits. Counter-examples: 51 × 48 and 15 × 84 34 × 49 and 43 × 94 By the end of grade 11, students will be expected to: E5 use inductive and deductive reasoning when observing patterns, developing properties and making conjectures E6 use deductive reasoning, construct logical arguments and be able to determine, when given a logical argument, if it is valid E27 write proofs using various axiomatic systems E28 demonstrate an understanding that reasoning and proof are essential and powerful parts of mathematics Elaborations - Instructional Strategies/Suggestions Deductive Reasoning (6.3) So far in the unit students have been using inductive reasoning by observing patterns, making conjectures and analysing the conjectures by trying to find a counter-example. This is the creative, investigative openended form of reasoning that people use in their every-day lives. In this section students will be encouraged to examine and discuss conjectures that are logically consistent. Deductive Reasoning, or logical reasoning, is a process of proving a conjecture in general terms and not by using specific examples of the conjecture. There are three ways to write deductive proofs: < two column proofs < flow chart proofs < paragraph proofs These three methods will be illustrated in Unit 6. In this unit we will use only the paragraph proof method. A Greek, Thales, was one of the first mathematicians to make a number of geometric conjectures and prove them deductively. One of his students, Pythagoras, continued linking chains of logical reasoning. Euclid, in his work, Elements, created a deductive system composed of the following: Undefined terms and postulates In any deductive system there must be certain assumptions (undefined terms) and ground-rules (postulates) to go by. These are simply accepted as being true without proof. These assumptions differ from Euclidean to non-Euclidean geometries. Examples of undefined terms are: point, line and plane. Examples of postulates are: For any two points there is exactly one line that contains them. Through a given point there is at most one line parallel to a given line. Definitions A definition is a statement that clarifies the meaning of a word using undefined terms or previously defined terms. Theorems If any conjecture can be proved deductively it is classed as a Theorem. Corollaries Any statement or theorem which follows directly from a previously deduced theorem is a corollary. 41 Worthwhile Tasks for Instruction and/or Assessment Deductive Reasoning (6.3) Group Activity Conjecture: The sum of four consecutive integers is equivalent to the first and last integers added, then multiplied by two. a) defend the conjecture inductively by showing 2 examples. b) prove the conjecture deductively. Group Presentation Have each group prove one of the following inductively, showing three examples. Then they are to prove the conjecture deductively. Each group will present their proofs to the class. < the sum of any 2 odd integers is even < the product of any 2 consecutive integers is even < the product of any 2 even integers is even Pencil/Paper/Discussion Prove the following inductively, showing three examples. Then prove it deductively. Conjecture: The sum of a two digit number and the number formed by reversing its digits will always be divisible by 11. Presentation Find four consecutive integers whose sum is 44. Show inductively and deductively that it is not possible. Performance Prove inductively(with 3 examples) and deductively. Choose any number ×4 + 10 ÷2 !5 ÷2 +3 Group Activity Make a series of steps (like the previous problem) of five or more steps that always results in a final answer of 6. 42 Suggested Resources Deductive Reasoning Math Power 11 p.349 # 1-4,8-10,14, 17 By the end of grade 11, students will be expected to: Elaborations - Instructional Strategies/Suggestions Deductive Reasoning (cont’d) Initiate a discussion on the definitions for complementary and supplementary angles. Supplementary angles can be visualized thus: Theorem - If two angles are equal, then their supplements are equal. E5 use inductive and deductive reasoning when observing patterns, developing properties and making conjectures In all 3 situations the sum = 1800 E13 explore constructions for various geometric configurations and deduce proofs of their validity E27 write proofs using various axiomatic systems Opposite Angles Theorem When two lines intersect, the opposite angles are equal. Students will be asked to prove the above conjecture and thus it will become a theorem. Note to Teachers: To ensure that students have a mental picture of opposite angles, generating the opposite angle diagram as shown below might be helpful. (unlike complementary or supplementary angles, opposite angles must share a common vertex), Step 2 extend each side of p1 in the opposite direction Step 1 Step 3 The angle formed by these new sides ( p2) is the angle opposite p1. Worthwhile Tasks for Instruction and/or Assessment 43 Suggested Resources Deductive Reasoning (cont’d) Deductive Reasoning (cont’d) Journal Write a definition for complementary and supplementary angles. Draw three possible diagrams for each. Performance In the August 1973 issue of Scientific American the following problem appeared. From an unused matchbook containing 20 matches, tear out from 1 to 9 matches and discard them. Count the number of remaining matches. Add the two digits of this number and tear out this many additional matches. Finally tear out two more matches. a) Try this three times. What is the result? (inductive reasoning) b) Prove that this will always work.(deductive reasoning) Communication Prove the following inductively(showing three examples) and then prove it deductively: The square of the sum of two positive integers is greater than the sum of the squares of the same two integers. Written Assignment A motorist drove 300 km at a speed of 100 km/h. On the return trip the motorist was able to go at only 60 km/h. What was the motorist’s average speed? a) What is your intuitive answer? b) Answer the following questions to deductively arrive at the answer. < How long did it take to drive there? < How long did it take to drive back? < What was the total time spent driving? < What was the total distance travelled? < What was the average speed? 44 By the end of grade 11, students will be expected to: Elaborations - Instructional Strategies/Suggestions Conditional Statements (6.5) A statement is a sentence that is either true or false. It is not an opinion. Ex. A quadrilateral is a polygon having exactly four sides. E25 explore “converse” and understand the use of the phrases “if...then” and “if and only if” in definitions and proofs E26 construct logical arguments, and be able to determine the validity of an argument E28 demonstrate an understanding that reasoning and proof are essential and powerful parts of mathematics When a statement is written in “If...., then....” form, it is a conditional statement. Ex. If a polygon is a quadrilateral, then it has exactly four sides. » hypothesis º » conclusion º The hypothesis is what is assumed to be true. The conclusion should follow logically from the hypothesis. Symbolically, a conditional statement can be written as: a º b ( read as: if a, then b) Converse If the hypothesis and conclusion are interchanged, the resulting statement is the converse of the original statement. Symbolically it can be written:(b ºa). Generally the converse does not have the same meaning as the original statement. Ex: Original: If an animal is a horse, then it has four legs. Converse: If an animal has four legs, then it is a horse. Bi-conditional statements The statements below can be read both ways and still have the same meaning. original: If a polygon is a quadrilateral, then it has exactly four sides converse: If a polygon has exactly four sides, then it is a quadrilateral. They could have also have been written in bi-conditional form: A polygon is a quadrilateral iff it has exactly four sides. No, there is no spelling error in the above statement. It is read; A polygon is a quadrilateral if and only if it has exactly four sides. Definitions can be written in bi-conditional form. In fact this is used as a test of a good definition in any field of study (not just mathematics). If a statement’s converse is logically equivalent to the original then the statement is a good definition. Contrapositive If the original statement (a º b) is rearranged into this form (not b º not a) it is given the name contrapositive of the original statement and is always logically equivalent to the original statement. 45 Worthwhile Tasks for Instruction and/or Assessment Conditional Statements (6.5) Pencil/Paper/Discussion Write the following statement in conditional form; “A triangle has at least two acute angles. Suggested Resources Conditional Statements Math power 11 p.361 #1,2,4,5,7,9,10, 12,14-16,18, 20,21,23,26, 28,30 Pencil/Paper/Discussion Write the converse and contrapostitive of the above conditional statement. Determine the validity of these statements. Pencil/Paper/Discussion Write each of the following definitions as a bi-conditional statement; a) A rhombus is an equilateral quadrilateral. b) Supplementary angles are two angles whose sum is 1800. c) Perpendicular lines are lines that intersect at right angles. Technology Use the TI-83 to test the truth of: If 4x!3 > 7x ! 15, then x < 4. 2nd test 3: > will put the > symbol on the screen Technology p.363 Green #1 #1-5, Green #2 #1-6 Note to Teachers: Read p.363 on Boolean Algebra With the standard window zoom 6:standard and press graph press trace and cursor along to test the above conclusion Journal Try to create a good definition of some object or concept in the real world. Check the validity of its converse to see if it is a good definition. 46 To test the endpoint to see if it is part of the solution set, press trace and simply enter the x value of the endpoint (here x = 4) and press enter Notice that if y = 0 then the endpoint is not part of the solution while if y = 1 then the endpoint is part of the solution. SCO: By the end of grade 11, students will be expected to: E6 use deductive reasoning construct logical arguments and be able to determine, when given a logical argument, if it is valid Elaborations - Instructional Strategies/Suggestions Equality properties (p.384) Challenge students to read and discuss the properties on p.384. The addition property can be read two ways: a) If a = b, then a + c = b + c or If a = b and c = d, then a + c = b + d The second statement above is used to solve Systems of Linear Equations by elimination. Initiate a discussion on the subtle differences in the logic between substitution and transitive. Substitution If you have an equality a = b then because “a” equals “b” one can be substituted for the other in any other statement containing a or b. Ex: If a + b = c and b = d, then a + d = c Transitive Many times it is interchangeable with substitution but not always. Notice the two statements are linked(ie. the 2nd part of the first sentence is the same as the 1st part of the second sentence). In logic symbolism it is written: If a 6 b and b 6 c , then a 6 c This pattern is called a syllogism. Patterns can be composed of more than 2 statements, for example: A good chess player thinks ahead a number of moves something like this: If I move to a, then my opponent will move to b. If my opponent moves to b, then I will move to c. If I move to c, then my opponent will move to d. If my opponent moves to d, then I will move to e. Notice the syllogistic pattern. A good chess player may have 3 or 4 of these possible scenarios worked out for 5 or more steps in each scenario. 47 Worthwhile Tasks for Instruction and/or Assessment Equality properties Math Power 11 p.384 Green #1,2 Equality properties (p.384) Pencil/Paper/Discussion State the reason (using the equality properties) that will justify each statement to be true. a) If pA = 300 and pB = 300, then pA = pB. b) If AB = CD, then AB ! XY = CD ! XY c) If pA = pB and pB = pC, then pA = pC. Journal Explain the differences in the reasoning between the substitution and transitive properties. Communication Communicate to the class your step-by-step solution which includes justifying each step using the equality properties for each of the following: a) 2 x − 5 = 5 x + 4 b) 3( x − 2 ) = x + 2 c) Suggested Resources 3x + 4 = −4 5 48 By the end of grade 11, students will be expected to: Elaborations - Instructional Strategies/Suggestions Congruent Triangles (p.364) Naming an angle Challenge students to give four ways to name the angle indicated; Solution E7 explore properties of and make and test conjectures about two and three dimensional figures pABC pB (if only 1 angle at vertex B) pCBA p1 Terminology for triangles Invite students to discuss the terms for triangles: < side opposite a certain angle < included angle < included side ex. side AC is the side opposite pB pA is included between sides AB and AC side BC is the side included between pB and pC Definition of congruent triangles Allow students time to brainstorm on the idea of congruent triangles. Encourage students to come to a consensus on a good definition of congruent triangles. (See top of p.364). The order in which the vertices of congruent triangles are written in the congruency statement determines the correspondence of the angles and sides. ªABC – ª LPR 49 pA = pL pB = pP pC = pR BC = PR AC = LR AB = LP Worthwhile Tasks for Instruction and/or Assessment Congruent Triangles (p.364) Pencil/Paper/Discussion For Congruent Triangles Math Power 11 p.364 #1-4 p.365 #1-8 ªABC, state: a) the side opposite pC b) the angle included between sides BC and CA c) the side included between pA and pB Group Activity ª Suggested Resources –ª Given that PNT XAL, list all pairs of congruent sides and angles. (A sketch of these triangles may be helpful). Journal Write a good definition of congruent triangles. 50 By the end of grade 11, students will be expected to: Elaborations - Instructional Strategies/Suggestions Congruent Triangles Students have a geometry set( or you could demonstrate this on the blackboard). E15 explore constructions for various geometric configurations and apply them in problem solving Invite them to construct the three following triangles: a) a triangle with sides 8 cm, 12 cm and 14 cm long b) a triangle with sides 7 cm, 9 cm and an included angle of 500 c) a triangle with angles of 600 and 750 and an included side of 9 cm. For each of the above constructions, all students should have constructed the exact same triangle( they are congruent). Inductively, the student groups should be able to conclude that each of the above gives sufficient information to construct congruent triangles. A student does not have to get all 6 pairs of sides and angles congruent before deducing the triangles to be congruent. The congruency postulates are: a) SSS 3 pairs of congruent sides b) SAS 2 pairs of congruent sides and 1 pair of congruent included angles c) ASA 2 pairs of congruent angles and 1 pair of congruent included sides 51 Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Congruent Triangles Congruent Triangles Activity Write the reason why the triangles below are congruent: a) Math Power 11 p.382-3 Green #1,2 b) Performance Draw a diagram that illustrates two triangles that are congruent due to: a) the SSS postulate b) the SAS postulate c) the ASA postulate d) the AAS postulate e) the HL postulate 52 By the end of grade 11, students will be expected to: Elaborations - Instructional Strategies/Suggestions Direct and Indirect Proof (6.6) Invite students to read p 366 to the beginning of example 1 on p367. Direct proof is essentially deductive reasoning. See examples on p 369370. E6 use deductive reasoning, construct logical arguments and be able to determine, when given a logical argument, if it is valid E28 demonstrate an understanding that reasoning and proof are essential and powerful parts of mathematics E29 develop and evaluate mathematical arguments and proofs Indirect proof is known as “proof by contradiction”. Whatever we are asked to prove, assume the opposite to be true. Using the given information and the negation of what we are asked to prove, reason until a contradiction is reached. We can then deduce that if the opposite of what we were asked to prove is false then the other possibility must be true. See the explore exercise on p.366 and examples 1,2 and 7 on the following pages for worked examples of indirect proofs. Invite students to discuss the above direct and indirect examples. Discussion should be encouraged on the following Theorems and postulates: < Parallel Lines Theorem ( and its converse) < Parallel Line Postulate < Triangle Sum Theorem < Third angle Theorem < Exterior Angle Theorem < Isosceles Triangle Theorem (and its converse) < AAS –Postulate Note: This might be a good time to do numerical problems on some of the above theorems: (See Suggested Resources column) 53 Worthwhile Tasks for Instruction and/or Assessment Direct and Indirect Proof (6.6) Pencil/Paper Using indirect proof, what is your first assumption in trying to prove: “If a number is odd, then its square is odd”. Group Activity Use indirect proof to solve the following puzzle; John, Harry and Mary are different heights. Who is the tallest and who is the shortest if only one of the following statements is true. John is the tallest Harry is not the tallest Mary is not the shortest Group Activity Use indirect proof to solve the following. If only one of the three people is telling the truth, how many friends does Joan have? John claims that Joan has at least 100 friends Harry remarks that Joan certainly doesn’t have that many Mary states that Joan must have at least one friend Pencil/Paper Use direct proof to prove that the difference between a two digit number and the number formed by reversing the digits is divisible by 9. 54 Suggested Resources Direct and Indirect Proof Math Power 11 p.372 #1-7,10,12, 16-18 Math Power 11 p.377 #29-31 Problem Solving Strategies p.379 #1, 5, 7, 9, 10 Numerical Problems Math Power 11 p.382 #1 Green p.383 Mental Math #1-4(angle measures) Inductive Reasoning Worksheet Pencil/Paper/Technology Complete the following table and make a conjecture about a rule that describes these figurate numbers. Predict the next two terms and draw diagrams describing them. Use regression analyse to determine the rule or formula for each sequence of numbers. a) Square numbers: Solution Quadratic sequence tn = n 2 b) Rectangular numbers Solution Quadratic sequence tn = n(n + 1) c) Triangular numbers Solution Quadratic sequence tn = n ( n + 1) 2 Journal When pool balls are racked on a pool table at the start of a game what shape do they make? What is the shape of the holder? How many pool balls are there? Which of the above types of figurate numbers does 55 the number of pool balls fit into? Journal/Research Write a few paragraphs on the history and meaning of the term “figurate number” Pencil/Paper/Technology Complete the following table and make a conjecture about a rule that describes these figurate numbers. Predict the next two terms and draw diagrams describing them. Use regression analyse to determine the rule or formula for each sequence of numbers. Journal Can you speculate on the name of the above figurate number pattern? 56 Manipulative Shown on the following pages are the nets of the five Platonic Solids. Use these nets to build 3-D models of each of the solids. Then use the models to complete the table. Express in writing the pattern you have observed. Finally generate Euler’s Rule. Platonic Solid # Faces Tetrahedron Hexahedron Octahedron Dodecahedron Icosahedron 57 # Vertices # Edges