Download 1.2 Inductive Reasoning

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1.2
Goal
Use inductive reasoning
to make conjectures.
Inductive Reasoning
Scientists and mathematicians look for patterns and try to draw
conclusions from them. A conjecture is an unproven statement that
is based on a pattern or observation. Looking for patterns and making
conjectures is part of a process called inductive reasoning .
Key Words
• conjecture
Geo-Activity
• inductive reasoning
Making a Conjecture
• counterexample
Work in a group of five. You will count how many ways various numbers
of people can shake hands.
2 people can shake
hands 1 way.
Student Help
STUDY TIP
Copy this table in your
notebook and complete
it. Do not write in your
textbook.
3 people can shake
hands 3 ways.
1 Count the number of ways that 4 people can shake hands.
●
2 Count the number of ways that 5 people can shake hands.
●
3 Organize your results in a table like the one below.
●
People
2
3
4
5
Handshakes
1
3
?
?
4 Look for a pattern in the table. Write a conjecture about the number
●
of ways that 6 people can shake hands.
Much of the reasoning in geometry consists of three stages.
1 Look for a Pattern
●
Look at several examples. Use diagrams and
tables to help discover a pattern.
2 Make a Conjecture
●
Use the examples to make a general
conjecture. Modify it, if necessary.
3 Verify the Conjecture
●
Use logical reasoning to verify that the
conjecture is true in all cases. (You will do this in later chapters.)
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Chapter 1
Basics of Geometry
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EXAMPLE
Science
Make a Conjecture
1
Complete the conjecture.
Conjecture: The sum of any two odd numbers is __?__.
Solution
Begin by writing several examples.
112
3 13 16
516
21 9 30
3 7 10
101 235 336
Each sum is even. You can make the following conjecture.
ANSWER
SCIENTISTS use inductive
reasoning as part of the
scientific method. They
make observations, look for
patterns, and develop
conjectures (hypotheses) that
can be tested.
The sum of any two odd numbers is even.
EXAMPLE
Make a Conjecture
2
Complete the conjecture.
Conjecture: The sum of the first n odd positive integers is __?__.
Solution
List some examples and look for a pattern.
1
13
135
1357
1 12
4 22
9 32
16 42
ANSWER
The sum of the first n odd positive integers is n2.
Make a Conjecture
Complete the conjecture based on the pattern in the examples.
1. Conjecture: The product of any two odd numbers is __?__.
EXAMPLES
Student Help
READING TIP
Recall that the symbols
and p are two ways
of expressing
multiplication.
111
7 9 63
3 5 15
11 11 121
3 11 33
1 15 15
2. Conjecture: The product of the numbers (n 1) and (n 1) is __?__.
EXAMPLES
1 p 3 22 1
7 p 9 82 1
3 p 5 42 1
9 p 11 102 1
5 p 7 62 1
11 p 13 122 1
1.2
Inductive Reasoning
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MORE EXAMPLES
More examples at
classzone.com
Counterexamples Just because something is true for several
examples does not prove that it is true in general. To prove a
conjecture is true, you need to prove it is true in all cases.
A conjecture is considered false if it is not always true. To prove a
conjecture is false, you need to find only one counterexample.
A counterexample is an example that shows a conjecture is false.
EXAMPLE
3
Find a Counterexample
Show the conjecture is false by finding a counterexample.
Conjecture: The sum of two numbers is always greater than the larger
of the two numbers.
Solution
Here is a counterexample. Let the two numbers be 0 and 3.
The sum is 0 3 3, but 3 is not greater than 3.
ANSWER
The conjecture is false.
EXAMPLE
4
Find a Counterexample
Show the conjecture is false by finding a counterexample.
Conjecture: All shapes with four sides the same length are squares.
Solution
Here are some counterexamples.
These shapes have four sides the same length, but they are
not squares.
ANSWER
The conjecture is false.
Find a Counterexample
Show the conjecture is false by finding a counterexample.
3. If the product of two numbers is even, the numbers must be even.
4. If a shape has two sides the same length, it must be a rectangle.
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Chapter 1
Basics of Geometry
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1.2 Exercises
Guided Practice
Vocabulary Check
1. Explain what a conjecture is.
2. How can you prove that a conjecture is false?
Skill Check
Complete the conjecture with odd or even.
3. Conjecture: The difference of any two odd numbers is __?__.
4. Conjecture: The sum of an odd number and an even number
is __?__.
Show the conjecture is false by finding a counterexample.
5. Any number divisible by 2 is divisible by 4.
6. The difference of two numbers is less than the greater number.
7. A circle can always be drawn around a four-sided shape so that it
touches all four corners of the shape.
Practice and Applications
Extra Practice
8. Rectangular Numbers The dot patterns form rectangles with a
length that is one more than the width. Draw the next two figures
to find the next two “rectangular” numbers.
See p. 675.
2
6
12
20
9. Triangular Numbers The dot patterns form triangles. Draw the
next two figures to find the next two “triangular” numbers.
1
Homework Help
Example 1:
Example 2:
Example 3:
Example 4:
Exs. 8−16
Exs. 8−16
Exs. 17–19
Exs. 17–19
3
6
10
Technology Use a calculator to explore the pattern. Write a
conjecture based on what you observe.
10. 101 25 __?__
101 34 __?__
101 49 __?__
11. 11 11 __?__
111 111 __?__
1111 1111 __?__
1.2
12. 3 4 __?__
33 34 __?__
333 334 __?__
Inductive Reasoning
11
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Making Conjectures Complete the conjecture based on the pattern
you observe.
IStudent Help
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HOMEWORK HELP
Extra help with problem
solving in Exs. 13–14 is
at classzone.com
13. Conjecture: The product of an odd number and an even number
is __?__.
3 p 6 18
5 p 12 60
14 p 9 126
22 p 13 286
5 p 2 10
11 p (4) 44
43 p 102 4386
254 p 63 16,002
14. Conjecture: The sum of three consecutive integers is always three
times __?__.
3453p4
4563p5
5673p6
6783p7
7893p8
8 9 10 3 p 9
9 10 11 3 p 10
10 11 12 3 p 11
11 12 13 3 p 12
15. Counting Diagonals In the shapes below, the diagonals are
shown in blue. Write a conjecture about the number of diagonals
of the next two shapes.
0
2
5
9
Science
16. Moon Phases A full moon occurs when the moon is on the
opposite side of Earth from the sun. During a full moon, the
moon appears as a complete circle. Suppose that one year, full
moons occur on these dates:
sun
July
21day
Earth
moon
Earth’s
orbit
Thurs
F
1
6 7 8
3 4 5 13 14 15
10 11 12 20 21 22
19
18
17
27 28 29
24 25 26
31
S
moon’s
orbit
M
T
W
T
S
2
9
16
23
30
August
r October vember December
e
b
m
te
p
e
S
19
17 No
Friday
15
18day Monday
16 ay Thursday
S M T W
T F S
1 2 3
7 8 9 10 4 5 6
14 15 16 17 11 12 13
21 22 23 24 18 19 20
28 29 30 31 25 26 27
Sun
T
1
7 8
4 5 6 14 15
11 12 13 21 22
20
19
18
28 29
25 26 27
S
M
T
W
F
2
9
16
23
30
S
3
10
17
24
S M
T W
T F
S
2 3
1
9 10 4 5 6 7
16 17 11 12 13 14 8
23 24 18 19 20 21 15
30 31 25 26 27 28 22
29
e
Wedn
sd
S
T F
5
T W 3 4 12
1 2 10 11 19
8 9 17 18 26
6 7 15 16 24 25
13 14 22 23
20 21 29 30
27 28
S
M
S M T W T F S
1 2 3
4 5 6 7 8 9 10
11 12 13 14 15 16 17
18 19 20 21 22 23 24
25 26 27 28 29 30 31
Not drawn to scale
FULL MOONS happen when
Earth is between the moon
and the sun.
Application Links
CLASSZONE.COM
Determine how many days are between these full moons and
predict when the next full moon occurs.
Error Analysis Show the conjecture is false by finding a
counterexample.
17. Conjecture: If the product of two numbers is positive, then the
two numbers must both be positive.
18. Conjecture: All rectangles with a perimeter of 20 feet have the
same area. Note: Perimeter 2(length width).
19. Conjecture: If two sides of a triangle are the same length, then the
third side must be shorter than either of those sides.
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Chapter 1
Basics of Geometry
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20. Telephone Keypad Write a conjecture about the letters you
expect on the next telephone key. Look at a telephone to see
whether your conjecture is correct.
21. Challenge Prove the conjecture below by writing a variable
statement and using algebra.
Conjecture: The sum of five consecutive integers is always
divisible by five.
x (x 1) (x 2) (x 3) (x 4) __?__
Standardized Test
Practice
22. Multiple Choice Which of the following is a counterexample of
the conjecture below?
Conjecture: The product of two positive numbers is always
greater than either number.
A
2, 2
B
1
, 2
2
C
D
3, 10
2, 1
23. Multiple Choice You fold a large piece of paper in half four times,
then unfold it. If you cut along the fold lines, how many identical
rectangles will you make?
F
Mixed Review
4
G
8
H
J
16
Describing Patterns Sketch the next figure you expect in the
pattern. (Lesson 1.1)
24.
25.
1
Y
Z
Algebra Skills
32
X
2
3
Using Integers Evaluate. (Skills Review, p. 663)
26. 8 (3)
27. 2 9
28. 9 (1)
29. 7 3
30. 3(5)
31. (2)(7)
32. 20 (5)
33. (8) (2)
Evaluating Expressions Evaluate the expression when x 3.
(Skills Review, p. 670)
34. x 7
35. 5 x
36. x 9
37. 2x 5
38. x 2 6
39. x 2 4x
40. 3x 2
41. 2x 3
1.2
Inductive Reasoning
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