Geometric Structure Inductive & Deductive Reasoning

IntroductionInstructionExamplesPractice In this lesson you will Perform geometry investigations and make some discoveries by observing common features or patterns inductive reasoning. Use your discoveries to solve problems through a process called inductive reasoning. Use inductive reasoning to discover patterns. deductive reasoning. Learn to use deductive reasoning. conjectures. Make conjectures. See how some conjectures are logically related to each other. EQ: What is reasoning?

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List of Instructional Pages 1.Inductive ReasoningInductive Reasoning 2.Patterns in all PlacesPatterns in all Places 3.More inductive reasoningMore inductive reasoning 4.One PossibilityOne Possibility 5.ConjectureConjecture 6.Goldbach's ConjectureGoldbach's Conjecture 7. More GoldbachMore Goldbach 8. ExamplesExamples 9. Deductive ReasoningDeductive Reasoning 10. Algebra ExampleAlgebra Example 11. Number TrickNumber Trick 12. Number Trick DeductionNumber Trick Deduction

IntroductionInstructionExamplesPractice This is page 1 of 11 Page listLast View presentation Inductive reasoning conjectures Definition: Inductive reasoning is the process of observing data, recognizing patterns, and making generalizations about those patterns. These generalizations are often called conjectures. Next view the presentation called Patterns in all Places. Pay close attention (maybe take notes?) to the use of inductive reasoning to arrive at a conjecture. You will be expected to follow a similar process on your own in subsequent examples. Patterns in all Places

Connecting Geometric and Numeric Patterns ©2005 Valerie Muller Certain materials are included under the fair use exemption of the U.S. Copyright Law and have been prepared according to the educational multi-media fair use guidelines and are restricted from further use.

T o find an algebraic rule for a pattern … E xamine the figure closely C ount the number of suns R ecord the number of suns in the table Figure 1 Figure Number Number of Suns 1 7 IntroNextEndVocabulary

T o find an algebraic rule for a pattern … Examine the new figure closely Focus on the “new” objects added to the previous figure Count the suns Record the number Figure 2 Figure Number Number of Suns =11 IntroNextEndVocabulary

T o find an algebraic rule for a pattern … E xamine the new figure F ocus on the “new” C ount R ecord Figure 3 Figure Number Number of Suns =15 11 IntroNextEndVocabulary

What is the next figure in the pattern? What is the next term in the pattern? Figure 4 – What must be added to the third figure to create the fourth figure? Figure Number Number of Suns IntroNextEndVocabulary

What is the next figure in the pattern? What is the next term in the pattern? Figure 4 – What must be added to the third figure to create the fourth figure? Figure Number Number of Suns add 4 19 IntroNextEndVocabulary add 4

What is the 10th term in the pattern? The table may be extended The function rule may be found “Shortcut” for repeatedly adding 4 is multiplying by 4 Figure Number Suns … 10? IntroNextEndVocabulary

What is the 10th term in the pattern? The table may be extended: 7, 11, 15, 19, 23, 27, 31, 35, 39, 43 The function rule may be found: 4*Figure # + 3 = suns “Shortcut” for repeatedly adding 4 is multiplying by 4 Figure Number Suns … 10? IntroNextEndVocabulary

What is the nth term in the pattern? The table may be extended: 7, 11, 15, 19, 23, 27, 31, 35, …, 4x + 3 The function rule is f(x) = 4x + 3 Multiply figure by 4 and add 3 Figure Number Suns x 4x + 3 IntroNextEndVocabulary

Associated terms … R elation : pairing between two sets of numbers to create a set of ordered pairs F unction: special relation where pairs are formed such that each element of the first set is paired with exactly one element of the second set IntroNextEndnth term

IntroductionInstructionExamplesPractice This is page 2 of 11 Page listLastNext Inductive reasoning conjectures Definition: Inductive reasoning is the process of forming conjectures that are based on observations. Example: The numbers 72, 963, 10854, and are all divisible by 9. Add the digits in each number. Do you see a pattern? Can you make a conjecture? One possibility

72: = 9 963: = 18 = = 9 10,854: = 18 = = 9 7,236,261: = 27 = = 9 INDUCTIVE observed Remember: This is INDUCTIVE logic. You observed a pattern and made a conjecture that you believe to be true for all examples. IntroductionInstructionExamplesPractice This is page 3 of 11 Page list Last Next One Possibility CONJECTURE: In order for a number to be divisible by 9, the sum of the digits must be divisible by 9.

IntroductionInstructionExamplesPractice Inductive reasoning is the process of drawing a general conclusion by observing a pattern from specific instances. conclusion hypothesis conjecture This conclusion is called a hypothesis or conjecture. This is page 4 of 11 Page list LastNext

IntroductionInstructionExamplesPractice Goldbach’s Conjecture In 1742, mathematician Christian Goldbach made the conjecture that every integer greater than 2 could be written as the sum of two prime numbers. For example: 20 = = To this day, no one has proven this conjecture but it is accepted as true because no one has found a counterexample. This is page 5 of 11 Page list LastNext Christian Goldbach

IntroductionInstructionExamplesPractice Use Goldbach’s conjecture to express the following numbers as the sum of two primes. 100 = 68 = This is page 6 of 11 Page list LastNext

IntroductionInstructionExamplesPractice Now Now click on the Examples link above to work more sample problems for inductive reasoning. This is page 7 of 11 Graphic Page list LastNext

IntroductionInstructionExamplesPractice Next This is page 8 of 11 Page listLast Definition: Deductive reasoning Deductive reasoning is the process of drawing logical, certain conclusions by using an argument. In deductive reasoning, we use accepted facts and general principles to arrive at a specific conclusion.

IntroductionInstructionExamplesPractice Algebra Example Algebra Example: You can solve an equation by adding equal amounts to each side of the equation. You can solve x – 1 = 9 by adding 1 to both sides of the equation to get x = 10. This is an example of deductive reasoning. You used the additive property of equality to solve the equation. This is page 9 of 11 Page list LastNext

IntroductionInstructionExamplesPractice You can use deductive reasoning to explain a number trick: 1.How many days a week do you eat out? 2.Multiply this number by 2 3.Add 5 to the number you got in step 2. 4.Multiply the number you obtained in step 3 by If you have already had your birthday this year add 1756; if you haven’t, add Subtract the four-digit year that you were born. 7.You should have a three-digit number. This is page 10 of 11 Page list LastNext The first digit is the number of times that you eat out each week. What are the last two digits?

IntroductionInstructionExamplesPractice You can use deductive reasoning to explain or prove a number trick: 1.Call the number of days you eat out n 2.Multiplying this number by 2 gives 2n 3.Adding 5 gives 2n Multiplying by 50 gives 100n Adding 1756 gives 100n or 100n Subtracting the four-digit year that you were born gives 100n – 1987 (for example) or 100n The hundred’s digit is n, because the 18 (or 19) will have no effect on the hundred’s digit. This is page 11 of 11 Page list LastThe end Elementary, my dear Watson!

You have now completed the instructional portion of this lesson. You may proceed to the practice assignment. practice IntroductionInstructionExamplesPractice

IntroductionInstructionExamplesPractice Example 1 Example 2 Examples Example 1: points & segments Example 2: circles & regions

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IntroductionInstructionExamplesPractice Finding Patterns GizmoFinding Patterns Gizmo What is reasoning?

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Example 1 Back to main example page Number of Points Number of Segments ? ? ? To complete the table, try looking for a pattern

Example 2 We want to divide a circle into regions by putting points on the circumference and drawing segments from each point to every other point. Make a table with the number of points and the number of regions. Back to main example page One point  One region Two points  Two regions Three points  Four regions

Example 2 Back to main example page Number of Points Number of Regions

Example 2 Back to main example page Number of Points Number of Regions

Example 2 Conjecture:Conjecture: It appears that the number of regions doubles each time a point is added. Test the conjecture:Test the conjecture: Draw a large circle & put six points on the circumference. Connect each point to every other point & count the regions. Back to main example page

Example 2 Back to main example page Number of Points Number of Regions Be careful….after you make your conjecture, try to actually count the regions.

Example 2 Test the conjecture? There are only 31 regions, so our conjecture was false. Now, how many regions do you think 7 points would make? Back to main example page Now return to the instruction.instruction