Exploring Adding and Subtracting Rational Expressions

This video tutorial covers inductive reasoning, conjectures, and counterexamples. It explains how to make conclusions based on observations and patterns, and how to write conjectures. The video provides examples of number sequences to illustrate these concepts. It also introduces counterexamples, which demonstrate when a conjecture is false, and guides viewers through evaluating conjectures as true or false.

What is inductive reasoning?

• Making a conclusion based on a theory

• Making a conclusion based on a hypothesis

• Making a conclusion based on observations and patterns

• Making a conclusion based on an experiment

What is a conjecture?

• A hypothesis that needs to be tested

• An observation that needs further analysis

• A concluding statement reached using inductive reasoning

• A theory that has been proven

In the sequence 38, 31, 24, 17, what is the pattern?

• Dividing the previous term by 2

• Multiplying the previous term by 2

• Subtracting 7 from the previous term

• Adding 7 to the previous term

In the sequence where each term is double the previous term plus one, what is the next term after 47?

• 97

• 94

• 95

• 96

What is the pattern in the sequence 1, 4, 9, 16?

• Each term is a Fibonacci number

• Each term is a multiple of 3

• Each term is a perfect square

• Each term is a prime number

What is a counterexample?

• An example that is irrelevant to a conjecture

• An example that shows a conjecture is false

• An example that proves a theory

• An example that supports a conjecture

Is the conjecture 'The sum of any two consecutive integers is always odd' true or false?

• False

• True

• Cannot be determined

• Depends on the integers

Is the conjecture 'The product of two numbers is always larger than either number' true or false?

• False

• True

• Cannot be determined

• Depends on the numbers

Which of the following is a counterexample to the conjecture 'The product of two numbers is always larger than either number'?

• 5 * 1 = 5

• 3 * 4 = 12

• 2 * 0.5 = 1

• 2 * 3 = 6

Is the conjecture 'The product of two perfect squares is always a perfect square' true or false?

• Depends on the squares

• False

• True

• Cannot be determined