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Unit 2- LOGICAL REASONING & PROOFS
1-1: Patterns and Inductive Reasoning
(G.3)(C) To find counterexamples to disprove statements that are false.
(G.3)(D) To use inductive reasoning to formulate a conjecture.
1-1
Patterns and Inductive Reasoning
Objectives: 1) To use inductive reasoning to make conjectures.
2) To find counterexamples to disprove statements that are false.
Vocabulary
reasoning based on patterns you observe.
Inductive reasoning is ________________________________________________
a conclusion you reach using inductive reasoning.
A conjecture is ________________________________________________
an example for which the conjecture is incorrect.
A counterexample is ________________________________________________
Find a pattern for the sequence. Use the pattern to
find the next two terms in the sequence.
384, 192, 96, 48, . . .
half
2
24
2
12
2
2
2
1-1
Patterns and Inductive Reasoning
Objectives: 1) To use inductive reasoning to make conjectures.
2) To find counterexamples to disprove statements that are false.
Make a conjecture about the sum of the cubes of the first 25 counting numbers.
Find the first few sums. Notice that each sum is a perfect square and that the perfect squares form a pattern.
The sum of the first two cubes equals the square of the sum of the first two counting numbers.
The sum of the first three cubes equals the square of the sum of the first
three
counting
numbers. This pattern continues for the fourth and fifth rows. So a conjecture might be that
_____________________________________________________________________________
𝟏𝟑 + 𝟐𝟑 + 𝟑𝟑 + ⋯ + 𝟐𝟓𝟑 = (1 + 2 + 3 + … + 25)2
1-1
Patterns and Inductive Reasoning
Objectives: 1) To use inductive reasoning to make conjectures.
2) To find counterexamples to disprove statements that are false.
Find a counterexample for the conjecture.
Since 32 + 42 = 52, the sum of the squares of two consecutive numbers
is the square of the next consecutive number.
5
41
36
false
1-1
Patterns and Inductive Reasoning
Objectives: 1) To use inductive reasoning to make conjectures.
2) To find counterexamples to disprove statements that are false.
The price of overnight shipping was $8.00 in 2000, $9.50 in 2001,
and $11.00 in 2002. Make a conjecture about the price in 2003.
$8.00
$9.50
$11.00
$1.50
$1.50
$1.50
$12.50
1-1
Patterns and Inductive Reasoning
Objectives: 1) To use inductive reasoning to make conjectures.
2) To find counterexamples to disprove statements that are false.
1. Find the next two terms in each sequence.
29
37
Thursday
Friday
Patterns and Inductive Reasoning
1-1
Objectives: 1) To use inductive reasoning to make conjectures.
2) To find counterexamples to disprove statements that are false.
2. Make a conjecture about the sum of the first 35 odd numbers.
Use your calculator to verify your conjecture.
5
3
5
7
3
5
7
9
2
1
1
4
22
9
32
16
42
25
52
Conjecture: The sum of the first 35 odd numbers is 352 = 1225.
1-1
Patterns and Inductive Reasoning
Objectives: 1) To use inductive reasoning to make conjectures.
2) To find counterexamples to disprove statements that are false.
3. Find a counterexample for the conjecture.
Some products of 5 and other numbers are shown in the table.
Therefore, the product of 5 and any positive integer ends in 5.
Counterexample: Since 5 x 2 = 10 and 10 does not end in 5.
1-1
Patterns and Inductive Reasoning
Objectives: 1) To use inductive reasoning to make conjectures.
2) To find counterexamples to disprove statements that are false.
4. Suppose the price of two-day shipping was $6.00 in 2000, $7.00 in 2001,
and $8.00 in 2002. Make a conjecture about the price in 2003.
2000
$6.00
2001
$7.00
2002
$8.00
2003
?
Conjecture: The price increased by $1.00 each year and so the price
in 2003 would be $8.00 + 1.00 = $9.00.
1-1
Patterns and Inductive Reasoning
Objectives: 1) To use inductive reasoning to make conjectures.
2) To find counterexamples to disprove statements that are false.
Find a pattern for each sequence. Use the pattern to show the next two terms.
Draw the next figure in each sequence.
textbook
page 6
Use the table and inductive reasoning. Make a conjecture about each value.
19. the sum of the first 6 positive even #s
The sum … is 6 x 7 = 42.
20. the sum of the first 30 positive even #s
The sum … is 30 x 31 = 930.
21. the sum of the first 100 positive even #s
The sum … is 100 x 101 = 10,100.
22. Use the pattern in Example 2 to make a conjecture about the sum of the
first 100 odd numbers. The sum … is 1002 = 10,000.
1-1
Patterns and Inductive Reasoning
Objectives: 1) To use inductive reasoning to make conjectures.
2) To find counterexamples to disprove statements that are false.
Predict the next term in each sequence. Use your calculator to verify your answer.
Textbook
page 7
Find one counterexample to show that each conjecture is false.
25. The sum of two numbers is greater than either number.
26. The product of two positive numbers is greater than either number.
27. The difference of two integers is less than either integer.
28. The quotient of two proper fractions is a proper fraction.
1-1
Patterns and Inductive Reasoning
Objectives: 1) To use inductive reasoning to make conjectures.
2) To find counterexamples to disprove statements that are false.
textbook
page 9