Topic 1Inductive Reasoning, Conjectures, and Counterexamples

Unit 1 Topic 1

ExploreSherlock Holmes, the fictional detective, is famous for making simple but remarkable conjectures. In one instance, Holmes decoded a message written entirely in stick figures by making a conjecture like the following.

• Statement 1: The letter occurring most frequently in written English is the letter “e”.

• Statement 2: The figure occurring most frequently in the stick-figure message is .

• Conclusion: Therefore, the figure in the stick-figure message probably stands for the letter “e” in written English.

Explore

• To the right is an array of numbers called Pascal’s triangle.

• Describe the patterns you see in the triangle.

• Use the patterns you discovered to complete the last row of triangle.

• You used inductive reasoning to develop the pattern for the last row in Pascal’s triangle.

You should notice…• There are many patterns that

can be found.

• A number is determined by adding the two numbers that are above it (one slightly to the right and one slightly to the left).

1 1520156 6 1

Information

• Inductive reasoning is a type of reasoning in which a statement or conclusion is developed based on patterns or observations.

• The pattern, statement or conclusion that is developed is called a conjecture.

• Conjectures may or may not be true at all times.

Example 1Patterns in Pictures

The pattern above is created using triangles.

a) Draw figure 4 and complete the following table:

b) What conjecture can be developed based on the table?

c) How the number of small triangles in the 10th figure.

Figure 1 2 3 4

Number of Triangles

Try this on your own first!!!!

Example 1: Solution

a)The pattern is continued by adding a fourth row onto the bottom of the pattern.

b) The number of triangles in the figure is equal to the figure number, squared.

c) Figure 10:

Figure 1 2 3 4

Number of Triangles

1 4 9 16

210 =100

Example 2Patterns in Numbers

Consider the patterns of numbers below. Complete one additional line of the pattern. What conjecture can be made?

a) b)

Try this on your own first!!!!

2 2

2 2

2 2

2 2

2 -1 3

3 - 2 5

4 -3 7

5 - 4 9

2

2

2

2

1 1

11 121

111 12321

1111 1234321

Example 2: Solution

a) b)

2 2

2 2

2 2

2 2

2 -1 3

3 - 2 5

4 -3 7

5 - 4 9

2

2

2

2

1 1

11 121

111 12321

1111 1234321

Next Line:

2 26 -5 11 211111 123454321Next Line:

Conjecture: When two consecutive numbers are squared, and the smaller is subtracted from the larger, the answer is equal to the sum of the two numbers.

Conjectures can vary.

Example 3True Examples

Give one example that shows each of the following conjectures to be true.

a) The sum of two consecutive prime numbers is even.

b) When a number is subtracted from the reverse order of that same number, and the digits of the answer are added, the answer is a multiple of 9.

c) Any animal that has wings can fly.

Try this on your own first!!!!

Example 3: Solution

a) b) c) A sparrow or

a bat

Keep in mind that any number of examples could show a conjecture to be true. Your answer may be different.

3 5 8

5 7 12

or

21 12 9

53 35 18

1 8 9

or

Example 4Counterexamples

Conjectures do not necessarily hold true for all instances. Counterexamples may be used to prove a conjecture to be false.

Give a counterexample that shows each of the following conjectures to be false. 

a) All prime numbers are odd.   b) All shapes with four right angles are squares. c) All animals that live in the water are fish.

Try this on your own first!!!!

Example 4: Solution

a) 2 is an example of a prime number that is not odd

b) a rectangle has four right angles but is not a square

c) an octopus lives in the water but is not a fish

Keep in mind that any number of counterexamples could show a conjecture to be false. Your counterexample may be different.

2 only has the factors 1 and 2.

Need to Know:

• Inductive reasoning looks at examples. By observing patterns and identifying properties, a general conclusion (conjecture) is made.

• A conjecture is based on information that you have gathered.

• A counterexample is an example that does not support your conjecture.

You’re ready! Try the homework from this section.