DG4CL_895_02

DG4CL_895_02 - DG4CL_895_02.qxd 11:54 AM Page 19 CONDENSED...

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In this lesson you will Learn how inductive reasoning is used in science and mathematics Use inductive reasoning to make conjectures about sequences of numbers and shapes Inductive reasoning is the process of observing data, recognizing patterns, and making generalizations based on those patterns. You probably use inductive reasoning all the time without realizing it. For example, suppose your history teacher likes to give “surprise” quizzes. You notice that, for the first four chapters of the book, she gave a quiz the day after she covered the third lesson. Based on the pattern in your observations, you might generalize that you will have a quiz after the third lesson of every chapter. A generalization based on inductive reasoning is called a conjecture. Example A in your book gives an example of how inductive reasoning is used in science. Here is another example. EXAMPLE A In physics class, Dante’s group dropped a ball from different heights and measured the height of the first bounce. They recorded their results in this table. Make a conjecture based on their findings. Then predict the first-bounce height for a drop height of 280 cm. © Solution If you divide each first-bounce height by the corresponding drop height, you get the following results: 0.75, 0.74, 0.7625, 0.75, 0.76, 0.7375. Based on these results, you could make this conjecture: “For this ball, the height of the first bounce will always be about 75% of the drop height.” Based on this conjecture, the first-bounce height for a drop height of 280 cm would be about 280 ? 0.75, or 210 cm. Example B in your book illustrates how inductive reasoning can be used to make a conjecture about a number sequence. Here is another example. EXAMPLE B Consider the sequence 10 ,7 ,9 ,6 ,8 ,5 ,... Make a conjecture about the rule for generating the sequence. Then find the next three terms. Drop height (cm) 120 100 160 40 200 80 First-bounce height (cm) 90 74 122 30 152 59 Inductive Reasoning LESSON 2.1 CONDENSED Discovering Geometry Condensed Lessons CHAPTER 2 19 ©2008 Key Curriculum Press (continued)
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Lesson 2.1 • Inductive Reasoning (continued) © Solution Look at how the numbers change from term to term. The 1st term in the sequence is 10. You subtract 3 to get the 2nd term. Then you add 2 to get the 3rd term. You continue alternating between subtracting 3 and adding 2 to generate the remaining terms. The next three terms are 4, 6, and 3. In the investigation you look at a pattern in a sequence of shapes. Investigation: Shape Shifters Look at the sequence of shapes in the investigation in your book. Complete each step of the investigation. Below are hints for each step if you need them. Step 1 Are the shapes the same or different? How does the shaded portion of the shape change from one odd-numbered shape to the next?
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DG4CL_895_02 - DG4CL_895_02.qxd 11:54 AM Page 19 CONDENSED...

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