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 firstbounce height
for a drop height of 280 cm.
Solution
If you divide each firstbounce 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 firstbounce 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, 7, . . .
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
Firstbounce height (cm)
90
74
122
30
152
59
Inductive Reasoning
L E S S O N
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 oddnumbered shape to the next?
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 Spring '08
 Staff
 alternate interior angles, Key Curriculum Press, Geometry Condensed Lessons, Vertical Angles Conjectures

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