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Connecting Reasoning and Proof Make conjectures Use laws of logic Solve problems by looking for a pattern Write algebraic proofs Write proofs involving segment and angle theorems 2.1 Inductive reasoning and conjecturing Inductive reasoning – observe the same thing happening again and again and form a conclusion from those observations. Conjecture – educated guess, unproven … based on observations False example – counter example Examples of using Inductive Reasoning Sally was late 6 days in a row. Conclusion???? Counterexamples Counterexamples disprove conclusions. It only takes one counterexample to disprove the conclusion. Draw 4 points A,B,C,and D. Connect the dots. Did you create a quadrilateral? Will it always work? Can you write a counterexample? 2.1 examples continued 3.Shelby was preparing toast for breakfast. After a few minutes the bread popped up but was not toasted. Make a list of conjectures that Shelby can make as to why the bread was not toasted. 4.Given that points A,B, and C are collinear and B is between A and C, Ashley made a conjecture that B is the midpoint of AC. Determine if her conjecture is true or false. Explain. EXAMPLE 1 Describe a visual pattern 1) Describe how to sketch the fourth figure in the pattern. Then sketch the fourth figure. SOLUTION Each circle is divided into twice as many equal regions as the figure number. Sketch the fourth figure by dividing a circle into eighths. Shade the section just above the horizontal segment at the left. EXAMPLE 2 Describe a number pattern 2) Describe the pattern in the numbers –7, –21, –63, –189,… and write the next three numbers in the pattern. Notice that each number in the pattern is three times the previous number. ANSWER Continue the pattern. The next three numbers are –567, –1701, and –5103. GUIDED PRACTICE 3). Describe the pattern in the numbers 5.01, 5.03, 5.05, 5.07,… Write the next three numbers in the pattern. Notice that each number in the pattern is increasing by 0.02. 5.01 5.03 +0.02 5.05 +0.02 5.07 +0.02 5.09 +0.02 5.11 +0.02 5.13 +0.02 ANSWER Continue the pattern. The next three numbers are 5.09, 5.11 and 5.13 EXAMPLE 4 Make a conjecture 4) Given five collinear points, make a conjecture about the number of ways to connect different pairs of the points. SOLUTION Make a table and look for a pattern. Notice the pattern in how the number of connections increases. You can use the pattern to make a conjecture. EXAMPLE 3 Make a conjecture ANSWER Conjecture: You can connect five collinear points 6 + 4, or 10 different ways. EXAMPLE 5 Make and test a conjecture 5) Numbers such as 3, 4, and 5 are called consecutive integers. Make and test a conjecture about the sum of any three consecutive numbers. SOLUTION STEP 1 Find a pattern using a few groups of small numbers. 3 + 4 + 5 = 12 = 4 3 7 + 8 + 9 = 24 = 8 3 10 + 11 + 12 = 33 = 11 3 16 + 17 + 18 = 51 = 17 3 ANSWER Conjecture: The sum of any three consecutive integers is three times the second number. EXAMPLE 5 Make and test a conjecture STEP 1 Test your conjecture using other numbers. For example, test that it works with the groups –1, 0, 1 and 100, 101, 102. –1 + 0 + 1 = 0 = 0 3 100 + 101 + 102 = 303 = 101 3 GUIDED PRACTICE 6) Make and test a conjecture about the sign of the product of any three negative integers. ANSWER Conjecture: The result of the product of three negative number is a negative number. Test: Test conjecture using the negative integer –2, –5 and –4 –2 –5 –4 = –40 EXAMPLE 7 Find a counterexample 7) A student makes the following conjecture about the sum of two numbers. Find a counterexample to disprove the student’s conjecture. Conjecture: The sum of two numbers is always greater than the larger number. SOLUTION To find a counterexample, you need to find a sum that is less than the larger number. EXAMPLE 7 Find a counterexample –2 + –3 = –5 –5 > –2 ANSWER Because a counterexample exists, the conjecture is false. Practice Determine if the conjecture is true or false. Explain and give a counterexample if false. 1. Given: ےA and ےB are supplementary Conjecture: ےA and ےB are not congruent Is the conjecture True or False? Give counterexample if false. m ےA > m ےB, m ےB > m ےC Conjecture: m ےA > m ےC 2. Given: 3. Given: AB, BC, AC Conjecture: A, B, and C are collinear 4. Given: ےA and ےB are vertical angles Conjecture: ےA and ےB are congruent EXAMPLE 6 X X Standardized Test Practice Daily Homework Quiz 1. Describe a pattern in the numbers. Write the next number in the pattern. 20, 22, 25, 29, 34, . . . ANSWER Start by adding 2 to 22, then add numbers that successively increase by 1; 40. 2. Find a counterexample for the following conjecture: If the sum of two numbers is positive, then the two numbers must be positive. ANSWER Sample: 20 + (– 10) = 10 Daily Homework Quiz 3. The scatter plot shows the average number of hours of homework done per week by a student during the first 10 weeks of a school term. Make a conjecture that could be true. Explain your reasoning. ANSWER Sample answer: The student will do about 11 hours of homework in week 11. The number of hours of homework per week increased steadily during the first 10 weeks.