2-1 Use Inductive Reasoning

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Presentation transcript:

2-1 Use Inductive Reasoning Hubarth Geometry

Ex 1 Describing Visual Patterns Describe how to sketch the fourth figure in the pattern. Then sketch the fourth figure. 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.

Ex 2 Describing a Number Pattern 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. Continue the pattern. The next three numbers are –567, –1701, and –5103.

Inductive Reasoning A conjecture is an unproven statement that is based on observations. You use inductive reasoning when you find a pattern in specific cases and then write a conjecture for the general cases. Ex 3 Make a conjecture Given five collinear points, make a conjecture about the number of ways to connect different pairs of the points. 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. Conjecture: You can connect five collinear points 6 + 4, or 10 different ways.

Ex 4 Make and Test a Conjecture Numbers such as 3, 4, and 5 are called consecutive integers. Make and test a conjecture about the sum of any three consecutive numbers. STEP 1 Find a pattern using a few groups of small numbers. 3 + 4 + 5 7 + 8 + 9 10 + 11 + 12 16 + 17 + 18 = 12 = 4 3 = 24 = 8 3 = 33 = 11 3 = 51 = 17 3 Conjecture: The sum of any three consecutive integers is three times the second number.

Ex 5 Find a Counter Example 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. To find a counterexample, you need to find a sum that is less than the larger number. –2 + –3 = –5 –5 > –2 Because a counterexample exists, the conjecture is false.

Ex 6 Standardized Test Practice Choices A and C can be eliminated because they refer to facts not presented by the graph. Choice B is a reasonable conjecture because the graph shows an increase from 1990–2001, but does not give any reasons for that increase. The correct answer is B.

Practice 1. Sketch the fifth figure in the pattern in example 1. 2. Describe the pattern in the numbers 5.01, 5.03, 5.05, 5.07,… Write the next three numbers in the pattern. 5.09, 5.11 and 5.13 3. Suppose you are given seven collinear points. Make a conjecture about the number of ways to connect different pairs of the points. Conjecture: You can connect seven collinear points 15 + 6, or 21 different ways. 4. Make and test a conjecture about the sign of the product of any three negative integers. Conjecture: The result of the product of three negative number is a negative number. 5. Find a counterexample to show that the following conjecture is false. Conjecture: The value of x2 is always greater than the value of x. ( 1 2 )2 = 1 4 1 4 > 1 2