Question

1. state whether each conclusion is found using inductive or deductive reasoning. clearly explain how you know your answer is correct.

a. i got up at nine oclock for the past week. i will get up at nine oclock tomorrow.
b. james camerons last three movies were successful. his next movie will be successful.
c. jim has 20 pencils. he gives half of them to dan. jim has 10 pencils left.
d. every even number is divisible by two. 1986 is an even number. it is divisible by two.
e. in the sequence 1, 2, 4, 7, 11, 16 the next most probable number is 22.
f. the garbage truck comes every other tuesday. it did not come last tuesday. it will come this tuesday.
g. any number multiplied by one equals that number.
h. gas prices have gone down every day this week. the price of gas will go down tomorrow.

1. give your own example of inductive reasoning and inductive reasoning. explain how you know your example demonstrates inductive or deductive reasoning.

inductive
deductive

1. kaitlyn recently learned in her geometry class that vertical angles are congruent. she uses her protractor and measures different vertical angles in her notebook, and notices that they are in fact congruent. she thinks that this is an example of inductive reasoning.

is she correct? explain how you know.

Explanation:

• Inductive Reasoning: It is based on specific observations or examples to make a general conclusion. For example, in option A, the person uses the specific observation of getting up at 9 for the past week (a sample of days) to conclude they will get up at 9 tomorrow. In option B, using the success of the last three movies (a sample of movies) to predict the next movie's success. In option E, looking at the pattern in the sequence (specific numbers in the sequence) to predict the next number. In option H, using the gas - price trend of the week (a sample of days) to predict tomorrow's price.
• Deductive Reasoning: It starts with a general rule or principle and applies it to a specific case. In option C, the general rule could be “if you have a certain number of items and give away half, you can calculate the remaining number”. Here, \(20\times\frac{1}{2}=10\). In option D, the general rule is “every even number is divisible by 2” (a general mathematical rule) and then applied to the specific number 1986. In option F, the general rule is “the garbage truck comes every other Tuesday” (a schedule rule) and then applied to the specific case of last Tuesday and this Tuesday. Option G is a general mathematical rule “any number multiplied by one equals that number”.

Answer:

• A: Inductive reasoning. Based on past - specific observations (getting up at 9 for a week) to predict the future.
• B: Inductive reasoning. Using past - specific movie successes (a sample of movies) to predict the next movie.
• C: Deductive reasoning. Applies a general arithmetic operation (\(n\times\frac{1}{2}\) where \(n = 20\)) to get the result.
• D: Deductive reasoning. Applies the general rule of even - number divisibility to the specific number 1986.
• E: Inductive reasoning. Infers the next number in the sequence based on the observed pattern (specific numbers in the sequence).
• F: Deductive reasoning. Applies the general schedule rule (every other Tuesday) to the specific Tuesday cases.
• G: Deductive reasoning. States a general mathematical rule.
• H: Inductive reasoning. Uses the gas - price trend of the week (a sample of days) to predict tomorrow.