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CHAPTER 2Reasoning and Proof
Wednesday, October 15, 14
SECTION 2-1Inductive Reasoning and Conjecture
Wednesday, October 15, 14
ESSENTIAL QUESTIONS
How do you make conjectures based on inductive reasoning?
How do you find counterexamples?
Wednesday, October 15, 14
VOCABULARY
1. Inductive Reasoning:
2. Conjecture:
3. Counterexample:
Wednesday, October 15, 14
VOCABULARY
1. Inductive Reasoning: Coming to a conclusion based off of specific examples and observations
2. Conjecture:
3. Counterexample:
Wednesday, October 15, 14
VOCABULARY
1. Inductive Reasoning: Coming to a conclusion based off of specific examples and observations
2. Conjecture: The conclusion reached from using inductive reasoning
3. Counterexample:
Wednesday, October 15, 14
VOCABULARY
1. Inductive Reasoning: Coming to a conclusion based off of specific examples and observations
2. Conjecture: The conclusion reached from using inductive reasoning
3. Counterexample: A false example that shows the conjecture is not true
Wednesday, October 15, 14
EXAMPLE 1Write the conjecture that describes the pattern in each
sequence. Then use your conjecture to find the next item in the sequence.
a. 7, 10, 13, 16 b. 3, 12, 48, 192
Wednesday, October 15, 14
EXAMPLE 1Write the conjecture that describes the pattern in each
sequence. Then use your conjecture to find the next item in the sequence.
a. 7, 10, 13, 16 b. 3, 12, 48, 192
Conjecture: The nth term is found by adding 3 to the previous term
Wednesday, October 15, 14
EXAMPLE 1Write the conjecture that describes the pattern in each
sequence. Then use your conjecture to find the next item in the sequence.
a. 7, 10, 13, 16 b. 3, 12, 48, 192
Conjecture: The nth term is found by adding 3 to the previous term
19
Wednesday, October 15, 14
EXAMPLE 1Write the conjecture that describes the pattern in each
sequence. Then use your conjecture to find the next item in the sequence.
a. 7, 10, 13, 16 b. 3, 12, 48, 192
Conjecture: The nth term is found by adding 3 to the previous term
19
Conjecture: The nth term is found by multiplying the
previous term by 4
Wednesday, October 15, 14
EXAMPLE 1Write the conjecture that describes the pattern in each
sequence. Then use your conjecture to find the next item in the sequence.
a. 7, 10, 13, 16 b. 3, 12, 48, 192
Conjecture: The nth term is found by adding 3 to the previous term
19
Conjecture: The nth term is found by multiplying the
previous term by 4
768
Wednesday, October 15, 14
EXAMPLE 1Write the conjecture that describes the pattern in each
sequence. Then use your conjecture to find the next item in the sequence.
c.
Wednesday, October 15, 14
EXAMPLE 1Write the conjecture that describes the pattern in each
sequence. Then use your conjecture to find the next item in the sequence.
c.
Conjecture: The nth figure is found by rotating the previous figure 90° counterclockwise
Wednesday, October 15, 14
EXAMPLE 1Write the conjecture that describes the pattern in each
sequence. Then use your conjecture to find the next item in the sequence.
c.
Conjecture: The nth figure is found by rotating the previous figure 90° counterclockwise
Wednesday, October 15, 14
EXAMPLE 2Make a conjecture about each value or geometric
relationship. List or draw some examples that support your conjecture.
a. The product of an odd and even number
Wednesday, October 15, 14
EXAMPLE 2Make a conjecture about each value or geometric
relationship. List or draw some examples that support your conjecture.
a. The product of an odd and even numberTest out some examples to see what happens
Wednesday, October 15, 14
EXAMPLE 2Make a conjecture about each value or geometric
relationship. List or draw some examples that support your conjecture.
a. The product of an odd and even numberTest out some examples to see what happens
3*2 = 6
Wednesday, October 15, 14
EXAMPLE 2Make a conjecture about each value or geometric
relationship. List or draw some examples that support your conjecture.
a. The product of an odd and even numberTest out some examples to see what happens
3*2 = 6 17*4 = 68
Wednesday, October 15, 14
EXAMPLE 2Make a conjecture about each value or geometric
relationship. List or draw some examples that support your conjecture.
a. The product of an odd and even numberTest out some examples to see what happens
3*2 = 6 17*4 = 68 5*10 = 50
Wednesday, October 15, 14
EXAMPLE 2Make a conjecture about each value or geometric
relationship. List or draw some examples that support your conjecture.
a. The product of an odd and even numberTest out some examples to see what happens
3*2 = 6 17*4 = 68 5*10 = 50
Conjecture: The product of an odd and even number will be even
Wednesday, October 15, 14
EXAMPLE 2Make a conjecture about each value or geometric
relationship. List or draw some examples that support your conjecture.
b. The radius and diameter of a circle
Wednesday, October 15, 14
EXAMPLE 2Make a conjecture about each value or geometric
relationship. List or draw some examples that support your conjecture.
b. The radius and diameter of a circleTest out some examples to see what happens
Wednesday, October 15, 14
EXAMPLE 2Make a conjecture about each value or geometric
relationship. List or draw some examples that support your conjecture.
b. The radius and diameter of a circleTest out some examples to see what happens
Wednesday, October 15, 14
EXAMPLE 2Make a conjecture about each value or geometric
relationship. List or draw some examples that support your conjecture.
b. The radius and diameter of a circleTest out some examples to see what happens
Wednesday, October 15, 14
EXAMPLE 2Make a conjecture about each value or geometric
relationship. List or draw some examples that support your conjecture.
b. The radius and diameter of a circleTest out some examples to see what happens
Wednesday, October 15, 14
EXAMPLE 2Make a conjecture about each value or geometric
relationship. List or draw some examples that support your conjecture.
b. The radius and diameter of a circleTest out some examples to see what happens
Conjecture: The diameter is twice as long as the radius
Wednesday, October 15, 14
EXAMPLE 3The table shows the total sales for the first three months that Matt Mitarnowski’s Wonderporium is open. Matt
wants to predict the sales for the fourth month.
Month
Sales
1 2 3
$400 $800 $1600
Make a conjecture about the sales in the fourth month and justify your claim.
Wednesday, October 15, 14
EXAMPLE 3The table shows the total sales for the first three months that Matt Mitarnowski’s Wonderporium is open. Matt
wants to predict the sales for the fourth month.
Month
Sales
1 2 3
$400 $800 $1600
Make a conjecture about the sales in the fourth month and justify your claim.
Conjecture: The sales double each month, so the fourth month should have $3200 in sales
Wednesday, October 15, 14
EXAMPLE 4Based on the table showing unemployment rates for
various counties in Texas, find a counterexample for the following statement: The unemployment rate is highest in the
cities with the most people.
County
Population
Rate
Armstrong Cameron El Paso Hopkins Maverick Mitchell
2,163 371,825 713,126 33,201 50,436 9,402
3.7% 7.2% 7.0% 4.3% 11.3% 6.1%
Wednesday, October 15, 14
EXAMPLE 4Based on the table showing unemployment rates for
various counties in Texas, find a counterexample for the following statement: The unemployment rate is highest in the
cities with the most people.
County
Population
Rate
Armstrong Cameron El Paso Hopkins Maverick Mitchell
2,163 371,825 713,126 33,201 50,436 9,402
3.7% 7.2% 7.0% 4.3% 11.3% 6.1%
Wednesday, October 15, 14
EXAMPLE 4Based on the table showing unemployment rates for
various counties in Texas, find a counterexample for the following statement: The unemployment rate is highest in the
cities with the most people.
County
Population
Rate
Armstrong Cameron El Paso Hopkins Maverick Mitchell
2,163 371,825 713,126 33,201 50,436 9,402
3.7% 7.2% 7.0% 4.3% 11.3% 6.1%
Wednesday, October 15, 14
PROBLEM SET
Wednesday, October 15, 14
PROBLEM SET
p. 92 #1-45 odd, 50
“Success is the sum of small efforts, repeated day in and day out.” - Robert Collier
Wednesday, October 15, 14