Reasoning and Patterns in Geometry

Which of the following is an example of deductive reasoning?

• All observed swans are white, so all swans must be white.

• All humans are mortal. Socrates is a human. Therefore, Socrates is mortal.

• The sun rises every morning, so it will rise tomorrow.

• If it rains, the ground gets wet. It rained yesterday, so the ground was wet.

Answer explanation

The correct choice demonstrates deductive reasoning by applying a general principle (all humans are mortal) to a specific case (Socrates is a human), leading to a logical conclusion (Socrates is mortal).

Which of the following is an example of inductive reasoning?

• If a number is even, then it is divisible by 2. 4 is even, so it is divisible by 2.

• All mammals have lungs. A whale is a mammal. Therefore, a whale has lungs.

• The first four coins I pulled from my pocket are pennies, so all the coins in my pocket are pennies.

• If it is raining, then the ground is wet. It is raining, so the ground is wet.

Answer explanation

The correct choice illustrates inductive reasoning by generalizing from specific observations (the first four coins being pennies) to a broader conclusion (all coins in the pocket are pennies).

Provide a counterexample to the statement: "All prime numbers are odd."

• 3

• 5

• 2

• 7

Answer explanation

The statement "All prime numbers are odd" is false because 2 is a prime number and it is even. Therefore, 2 serves as a counterexample to the statement.

Tags

CCSS.4.OA.B.4

Apply the law of detachment: If a student studies, then they will pass the exam. John studied.

• John will pass the exam.

• John might pass the exam.

• John will not pass the exam.

• John will fail the exam.

Answer explanation

According to the law of detachment, if a student studies, then they will pass the exam. Since John studied, we can conclude that John will pass the exam.

Tags

CCSS.RI.7.1

CCSS.RI.8.1

CCSS.RI.8.8

CCSS.RL.8.1

CCSS.RL.9-10.1

Apply the law of syllogism: If it rains, then the ground gets wet. If the ground gets wet, then the grass grows.

• If it rains, then the grass grows.

• If the grass grows, then it rains.

• If the ground gets wet, then it rains.

• If the grass grows, then the ground gets wet.

Answer explanation

Using the law of syllogism, we combine the two statements: If it rains (A), then the ground gets wet (B); and if the ground gets wet (B), then the grass grows (C). Therefore, if it rains (A), then the grass grows (C).

Identify the pattern in the sequence: 2, 4, 8, 16, ...

• Add 2 to the previous number.

• Multiply the previous number by 2.

• Subtract 2 from the previous number.

• Divide the previous number by 2.

Answer explanation

The sequence doubles each number: 2 x 2 = 4, 4 x 2 = 8, 8 x 2 = 16. Thus, the pattern is to multiply the previous number by 2.

Tags

CCSS.3.OA.D.9

Which of the following statements uses deductive reasoning?

• The sun has risen every day in recorded history, so it will rise tomorrow.

• All birds can fly. Penguins are birds. Therefore, penguins can fly.

• If a shape is a square, then it has four sides. This shape is a square, so it has four sides.

• Every time I eat peanuts, I sneeze. Therefore, I am allergic to peanuts.

Answer explanation

The correct choice uses deductive reasoning by applying a general rule (squares have four sides) to a specific case (this shape is a square) to reach a conclusion (it has four sides).

Which of the following statements uses inductive reasoning?

• If a number is divisible by 4, then it is even. 8 is divisible by 4, so it is even.

• All observed crows are black, so all crows must be black.

• If a figure is a triangle, then it has three sides. This figure is a triangle, so it has three sides.

• If it is snowing, then it is cold. It is snowing, so it is cold.

Answer explanation

The statement 'All observed crows are black, so all crows must be black' uses inductive reasoning, as it generalizes from specific observations to a broader conclusion, unlike the other options which use deductive reasoning.

Provide a counterexample to the statement: "All rectangles are squares."

• A square with sides of length 4

• A rectangle with sides of length 4 and 5

• A triangle with sides of length 3, 4, and 5

• A circle with a radius of 3

Answer explanation

The statement "All rectangles are squares" is false. A rectangle with sides of length 4 and 5 is not a square, as its sides are not equal, providing a clear counterexample.

Tags

CCSS.5.G.B.4

Apply the law of detachment: If a person is a teacher, then they have a degree. Sarah is a teacher.

• Sarah does not have a degree.

• Sarah might have a degree.

• Sarah has a degree.

• Sarah is not a teacher.

Answer explanation

According to the law of detachment, if a person is a teacher, then they have a degree. Since Sarah is a teacher, it follows that Sarah has a degree.