In the exploration, we used inductive reasoning. Inductive reasoning uses patterns and observations to write a conjecture. These conjectures are often helpful in identifying and generalizing patterns.

Inductive reasoning

A method of drawing conclusions from a limited set of observations

Consider the scenario:

On Monday and Tuesday, after all the students sat down, the teacher started math class.

We can use inductive reasoning to write the conjecture:

Therefore, every day, the teacher will start class after all the students sit down.

To show that a conjecture is true, you must show that it is true for all cases. However, you can show that a conjecture is false by finding just one counterexample.

If, next week, there is a field trip and once all the students sit down, they line up for the bus, this would be a counterexample to our conjecture.

Examples

Example 1

Consider these appointment times:

\text{11:00 a.m., } \text{ 11:45 a.m., } \text{ 12:30 p.m., \ldots}

Write a conjecture for the given pattern.

Worked Solution

Create a strategy

First, find the pattern. We can use inductive reasoning to write a conjecture based on the pattern.

Apply the idea

\text{11:45 a.m. } = \text{11:00 a.m. } + \text{0:45}

\text{12:30 p.m. } = \text{11:45 a.m. } + \text{0:45}

Each time is 45 minutes later than the previous time.

A possible conjecture about this pattern is "There is a new appointment every 45 minutes."

Determine the next appointment time.

Worked Solution

Create a strategy

Use the conjecture from part (a) to predict the next time.

Apply the idea

In part (a), we predicted a new appointment every 45 minutes.

\text{1:15 p.m. } = \text{12:30 p.m. } + \text{0:45}

Based on our conjecture, the next appointment will be at \text{1:15 p.m.}

Example 2

Test each conjecture.

The sum of an odd integer and an even integer is odd.

Worked Solution

Create a strategy

First, select test cases that make the hypothesis true. Then, check to see if the conclusion is true.

Apply the idea

To make the hypothesis true, we need an odd integer and an even integer.

Test case 1:

3+4 = 7

This example confirms our hypothesis since an odd integer \left(3\right) and even integer \left(4\right) sum to an odd integer \left(7\right).

Test case 2:

1+2 = 3

This example confirms our hypothesis since an odd integer \left(1\right) and even integer \left(2\right) sum to an odd integer \left(3\right).

Test case 3:

11+20 = 31

This example confirms our hypothesis since an odd integer \left(11\right) and even integer \left(20\right) sum to an odd integer \left(31\right).

All three of these tests confirm our conjecture.

The numbered day of every Wednesday in 2024 is a multiple of 7.

Worked Solution

Create a strategy

Find the Wednesdays in 2024 which make the hypothesis true. Then, check to see if the numbered day is a multiple of 7 which would confirm the conclusion.

Apply the idea

The first Wednesday in February has the numbered day "7" which is a multiple of 7.

The second Wednesday in February has the numbered day "14" which is a multiple of 7.

The third Wednesday in February has the numbered day "21" which is a multiple of 7.

The fourth Wednesday in February has the numbered day "28" which is a multiple of 7.

These all confirm the conjecture that the numbered day of every Wednesday in 2024 is a multiple of 7.

Reflect and check

Notice that testing a conjecture does not prove it to always be true. If we were to extend the calendar into March, the first Wednesday in March has the numbered day "6" which is not a multiple of 7.

This single counterexample disproves the conjecture.

Example 3

For each conjecture, find a counterexample that shows it is false.

The square of any integer is even.

Worked Solution

Create a strategy

Find a counteraxample where the square of an integer is odd.

Apply the idea

3^{2} = 3 \cdot 3 = 9

Since 9 is odd, this is a counterexample to the conjecture because the square of 3 is odd.

If \angle A \text{ and } \angle B are complementary angles, then they share a common side.

Worked Solution

Create a strategy

Find a counterexample with two complementary angles that do not share a common side.

Apply the idea

Draw any two angles that are not connected, but still sum to 90 \degree. For example:

Two angles A and B, with measurements of 55 and 35 degrees respectively.

Since \angle A \text{ and } \angle B sum to 90 \degree, they are complementary. Since they do not share a common side, this is a contradiction to the conjecture.

Idea summary

Inductive reasoning is a method of drawing conclusions from a limited set of observations.

To show that a conjecture is true, you must show that it is true for all cases.

You can show that a conjecture is false by finding just one counterexample.

Outcomes

G.RLT.1

The student will translate logic statements, identify conditional statements, and use and interpret Venn diagrams.

G.RLT.1a

Translate propositional statements and compound statements into symbolic form, including negations (~p, read â€œnot pâ€), conjunctions (p âˆ§ q, read â€œp and qâ€), disjunctions (p âˆ¨ q, read â€œp or qâ€), conditionals (p â†’ q, read â€œif p then qâ€), and biconditionals (p â†” q, read â€œp if and only if qâ€), including statements representing geometric relationships.

G.RLT.1b

Identify and determine the validity of the converse, inverse, and contrapositive of a conditional statement, and recognize the connection between a biconditional statement and a true conditional statement with a true converse, including statements representing geometric relationships.

2.03 Inductive reasoning

Ideas

Inductive Reasoning

Exploration

Examples

Example 1

Example 2

Example 3

Outcomes

G.RLT.1

G.RLT.1a

G.RLT.1b

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